A Comparative Analysis of Physical-Layer Rateless
Coding Architectures
AN CIVVS
iAASSXHUSETTS INSThWE
by
OF TECHNOLOGY
David Luis Romero
JUN 10 2014
B.S. Electrical Engineering
New Mexico State University, 2009
IBRARI.ES
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
A uthor.
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al Enginerin
of Elect
Certified by . .
21, 2014
...................
Gregory W. Wornell
Professor of Elect ical Engineering and Computer Science
SiMay
Signature
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and Computer Science
redacted
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Thesis Supervisor
Certified by ............. . .. . . r - - - - - - - .....................
Dr. Adam R. Margetts
Technical Staff, MIT Lincoln Laboratory
Thesis Supervisor
Accepted by...........
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U U
Leslie A. Kolodziejski
Chairman, Department Committee on Graduate Students
/
2
A Comparative Analysis of Physical-Layer Rateless Coding
Architectures
by
David Luis Romero
Submitted to the Department of Electrical Engineering and Computer Science
on May 21, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering
Abstract
An analysis of rateless codes implemented at the physical layer is developed. Our model
takes into account two aspects of practical communication system design that are abstracted
away in many existing works on the subject. In particular, our model assumes that : (1)
practical error detection methods are used to determine when to terminate decoding; and
(2) performance and reliability as observed at the transport layer are the metrics of interest.
Within the context of these assumptions, we then evaluate two recently proposed highperforming rateless codes. Using our analysis to guide an empirical study, the process of
selecting the best rateless code for a given set of system constraints is illustrated.
Thesis Supervisor: Gregory W. Wornell
Title: Sumitomo Professor of Engineering
Thesis Supervisor: Dr. Adam R. Margetts
Title: Technical Staff, MIT Lincoln Laboratory
3
4
Acknowledgments
I would like to express my utmost gratitude to my advisor, Professor Gregory W. Wornell,
whose deep insight and broad expertise guided me throughout the course of this work, and
without whom this thesis would not have been possible. I would also like to thank Dr. Adam
R. Margetts, whose generous sharing of expertise and time have benefited me tremendously
since before the beginning of this work. A special thank you is reserved for Professor Uri
Erez, whose expertise and assistance is reflected in many parts of this work. I would also
like to acknowledge the generous support provided to me by the MIT Lincoln Laboratory
Lincoln Scholars fellowship, which made my graduate study possible. Finally, I would like
to acknowledge the support and encouragement that I received from my family and friends
while completing this work, and, moreover, extend immense gratitude to my Father and
Mother, who showed me to continue moving forward when faced with difficulty.
"Once upon a time they used to represent victory as winged. But her feet were heavy and
blistered, covered with blood and dust."
The Fall of Paris
Ilya Ehrenburg
5
6
Contents
1
13
Introduction
1.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2 Preliminaries
3
4
15
2.1
N otation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.2
System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
29
System Analysis
3.1
Error Detection Model and Analysis . . . . . . . . . . . . . . . . . . . . . . .
30
3.2
Message Error Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.3
Approximate Decoding Error Probability . . . . . . . . . . . . . . . . . . . .
41
3.4
Packet Error Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3.5
Packet Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3.6
Constrained System Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
System Design and Practical Rateless Codes
4.1
63
Layered C odes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Decoding Errors Under Successive Decoding . . . . . . . . . . . . . .
65
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
. . . . . . . . . . . . . . . . . . . . . . . . .
71
Latency Constrained System Design . . . . . . . . . . . . . . . . . . .
71
4.1.1
4.2
Spinal C odes
4.3
Design of Constrained Systems
4.3.1
7
4.3.2
Reliability Constrained System Design . . . . . . . . . . . . . . . . .
83
4.3.3
Short Packet Lengths . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
5 Discussion and Conclusion
95
A Layered System Architectures
99
B Design Details
101
B.1 Layered Rateless Code Design . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B.2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B .1.1
Choosing ctgt
B.1.2
Mean Squared Error Estimate of Effective SNR . . . . . . . . . . . . 102
Complexity of Practical Rateless Codes . . . . . . . . . . . . . . . . . . . . . 103
B.2.1
Layered Rateless Codes . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.2.2
Spinal Codes
B.2.3
Simulation Aided Complexity Analysis . . . . . . . . . . . . . . . . . 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8
List of Figures
2-1
System m odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2-2
Illustration of the structure of a packet. . . . . . . . . . . . . . . . . . . . . .
26
2-3
Illustration of the structure of a rateless codeword.
. . . . . . . . . . . . . .
27
3-1
Fraction of throughput dedicated to information bits, pt/R with R = 1/3, as
a function of undetected error probability, pue, for various information block
len gth s.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
3-2
Single decoding attempt probability of error example, k = 256,
= 0 dB. .
43
3-3
Upper bound on probability of message error, k = 1024, -Ydb = 5 dB. . . . . .
44
3-4
Numerical examples of upper and lower bounds on the overall packet error
'Ydb
. . . . . . . . . . . . . . . . . . .
47
3-5
Bounds on throughput, -Ydb = 1 dB. . . . . . . . . . . . . . . . . . . . . . . .
61
3-6
Bounds on throughput, -Ydb = 10 dB.
. . . . . . . . . . . . . . . . . . . . . .
62
4-1
Packet error probability upper bounds for UDP design example, -Ydb = 1 dB.
73
4-2
Packet error probability bounds for UDP design example, -Ydb = 1 dB. .....
74
4-3
Throughput upper bounds and simulation results for layered, and spinal codes
probability as a function of k,
for UDP design example,
4-4
-Ydb =
Ydb =
1 dB.
1 dB. . . . . . . . . . . . . . . . . . . . . . .
77
Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example, -Ydb = 1 dB. . . . . . . . . . . . . . . .
9
78
4-5
Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example,
4-6
-Ydb
= 8 dBe.
. . . . . . . . . . . . . . . . . . . . .
79
Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example,
-Ydb
= 8 dBe.
. . . . . . . . . . . . . .
80
4-7 Throughput upper bounds and simulation results for layered, and spinal codes
for TCP design example,
4-8
1 dB. . . . . . . . . . . . . . . . . . . . . . .
85
Throughput upper bounds and simulation results for layered, and spinal codes
for TCP design example,
4-9
-Ydb =
-Ydb =
8 dB. . . . . . . . . . . . . . . . . . . . . . .
86
Packet error probability bounds for UDP design example with short packet
length,
/db
= 1 dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
4-10 Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example with short packet length,
Ydb
=
1 dB. . . . . . . . .
90
4-11 Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example with short packet length,
-Ydb =
1 dB.
91
4-12 Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example with short packet length, Ydb = 8 dB. . . . . . . . .
92
4-13 Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example with short packet length,
-Ydb
= 8 dB.
.
93
A-1 Three layer communication system architecture. . . . . . . . . . . . . . . . . 100
B-i Throughput efficiency for M = L = 7 layered codes vs SNR. Various choices
of Etgt are show n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B-2 MSE and measured effective SNR when attempting to decoder layer 7.
. . . 103
B-3 Average receiver operations per goodbit for spinal and layered rateless codes. 112
10
List of Tables
2.1
Deterministic Scalar Quantities
. . . . . . . . . . . . . . . . . . . . . . . . .
18
2.2
E vents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3
Random Scalars, Vectors, and Matrices . . . . . . . . . . . . . . . . . . . . .
20
2.4
Layered Code Parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.5
Spinal Code Parameters
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.1
Summary Of Basic Error Events . . . . . . . . . . . . . . . . . . . . . . . . .
31
4.1
Simulation Configuration, k
{256, 512, 1024}
. . . . . . . . . . . . . . . .
76
4.2
Spinal Code Throughput Comparison, k
. . . . . . . . .
81
4.3
Receiver Ops. Per Goodbit, k = 1024,
1 dB . . . . . . . . . . . . . . .
82
4.4
Short Packet Length Simulation Configuration, k = {64, 128, 256}
=
=
1024,
_Ydb =
_Ydb =
1 dB
. . . . . .
89
B. 1 Real Arithmetic Operations Required To Compute And Apply UMMSE Combining Weights For Layered Rateless Code. . . . . . . . . . . . . . . . . . . . 106
B.2 Real Operations Required By The Turbo Decoder To Decode The Layered
R ateless C ode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
B.3 Real Operations Required By a Spinal Code Receiver. . . . . . . . . . . . . . 110
11
12
Chapter 1
Introduction
Rateless codes are forward error correcting codes for which the number of encoded symbols
used to communicate a given message varies with the channel. This is in contrast to traditional fixed rate codes, which attempt to communicate using a fixed number of encoded
symbols and are not capable of dynamically adapting to each channel realization. The highly
adaptive nature of rateless codes make them an attractive choice for systems that operate
where the communication channel is unknown, or varies over time in an unknown manner.
A number of methods exist that can be used to enable a system to adapt to uncertainties
in a channel. In a broad sense, all of these methods fall into the category of variable rate
codes 1 . One sub-class of such codes is based on shortening a block of information symbols,
which are subsequently encoded into a fixed length codeword [1]. Another sub-class of variable rate codes are designed to encode a fixed number of information symbols into a variable
length codeword using either code extension, or puncturing and incremental transmission [2].
Still other variable rate codes use some combination of shortening, and extension or puncturing [3]. In this thesis, we consider only variable rate codes that encode a fixed number of
information symbols into a variable length codeword, and refer these as rateless codes.
Rateless codes have long been of interest to the coding community.
After practical
constructions were successfully applied to erasure channel models [4], an interest in devel'Certain classes of these codes are sometimes referred to as rate compatible codes.
13
oping rateless codes for noisy, physical layer channel models emerged. It has since been
demonstrated that practical constructions of such codes are possible, and can achieve good
performance [5, 6, 7, 8].
Typical analyses of rateless codes appearing in the literature focus on performance as
observed at the physical layer (PHY) and medium access control layer (MAC) of the system
(See Appendix A).
Additionally, many existing analyses assume a model in which the
receiver behaves as if knowledge of the transmitted message is available after decoding,
enabling the decoder to detect with probability 1 whether decoding decisions are valid.
Such perspectives are useful for distinguishing code performance from other characteristics
of the system. However, once a set of candidate rateless codes with good performance has
been identified, a system designer must evaluate their performance when coupled within the
system. Such an evaluation must take into account performance and reliability as observed
at the application layer of the system, as these metrics most closely affect user experience.
Furthermore, systems must utilize an error detection scheme that provides some imperfect
level of reliability, as a "genie" aided receiver is not possible in practice.
The above considerations must be taken into account when designing any communication
system. However, when considering rateless codes it is particularly important to take the
effects of imperfect error detection into account, as decoding and error detection are typically
executed repeatedly for each message, leading to an aggregate probability of undetected
error that is substantially higher than in the case of fixed rate codes. Additionally, because
application level performance and reliability is impacted by protocols that operate within the
transport layer of the system (See Appendix A), it is important to consider such protocols
when undertaking a detailed system analysis.
In this thesis, we analyze the performance and reliability of rateless codes with the goal of
contributing a more detailed system analysis than what has typically appeared previously in
the literature. While previous works have abstracted away certain details to produce useful
course grained analyses of rateless codes, we develop a finer grained analysis by considering
14
a model employing fewer analytical simplifications.
In particular, we analyze performance
and reliability from the perspective of the transport layer of the system, while assuming use
of a practical error detection method.
Our analysis is executed in two phases.
First, a model of rateless codes is presented
and analyzed. The model is anchored at the transport layer of the system, and is used to
capture the dominant effects of a practical system employing such codes. The analysis yields
a set of bounds on the throughput and probability of error that one can expect to observe
at the application layer of the system, given a set of design parameters. Second, simulation
of two high performing practical rateless codes is used, along with our analysis, to illustrate
how a system designer can select an appropriate rateless code given a set of constraints on
the system. The primary goal of this thesis is to provide insight and practical examples of
interest to system designers who are considering the use of rateless codes.
The remainder of the thesis is organized as follows.
In the following section a brief
summary of previous work on rateless codes is provided. In Chapter 2 notation is defined,
and a detailed description of the basic system model for rateless codes is given. In Chapter
3, a basic system model of rateless codes is analyzed.
In Chapter 4, layered and spinal
codes are described, and Monte Carlo simulation results are presented within the context of
illustrative design examples. Finally, the thesis concludes with a brief discussion in Chapter
5.
1.1
Background
Several fundamentally different approaches to rateless coding have been introduced over
time. One notable example is automatic repeat request (ARQ) systems. In an ARQ system,
the transmitter resends previous messages upon receiving feedback from the receiver that
a codeword was decoded in error.
The receiver may use some method of combining the
repeated transmissions. Other examples are hybrid-ARQ (HARQ) systems, which encode
15
messages using a forward error correction (FEC) code, then send a subset of the parity
symbols, only sending additional parity symbols upon request from the receiver. HARQ
systems have been practically deployed, for example using the current LTE standard [9].
A more sophisticated approach to rateless coding is that of fountain codes, which can,
in principal, generate an infinite stream of distinct encoded symbols to be sent across the
channel. A notable example of fountain codes that were designed for erasure channel models
are the Raptor codes of [10].
models in [5].
Raptor codes were later adapted to physical layer channel
Spinal codes [7], which are one of two practical rateless codes considered
in this thesis, fit into the fountain paradigm, and have been shown to perform well when
applied to physical layer channel models.
Another approach suited to physical layer channel models is to construct a rateless code
using layering and repetition of a set of base codebooks. This is the approach taken for the
layered codes of [6], which is the other of two practical codes that are considered in this
thesis.
In general, spinal codes have been shown to perform well for short information block
lengths, while layered codes employing turbo codes for the base codebook have been shown to
perform well at long information block lengths. Because of their demonstrated performance
in different information block length regimes, it is of interest to compare and contrast the
performance of the two codes over the span of short and long information block lengths.
16
Chapter 2
Preliminaries
2.1
Notation
Events and sets are denoted using calligraphic characters (A). The complement of event (or
set) A is denoted Ac. The cardinality of a set, A, is denoted JAI, while the probability of
event A is denoted P (A). Random quantities are denoted using lower case or upper case
sans serif characters (b, or B, respectively), whereas deterministic quantities appear as either
lower case, or upper case serifed characters (b, or B, respectively). Vectors (deterministic or
random) are denoted using lower case, bold face characters (v if deterministic, v if random),
while matrices are denoted using upper case, boldface characters (G if deterministic, G if
random). Deterministic scalar quantities, events, and random scalars, vectors and matrices
are summarized in Tables 2.1, 2.2, and 2.3, respectively. Parameters that are particular
to the layered, and spinal rateless codes, which are evaluated in this thesis, are defined in
Tables 2.4, and 2.5, respectively. Though some quantities such as vectors and events are
initially defined using indices and subscripts, if there is no risk of ambiguity or confusion,
such indices and subscripts are dropped during exposition for convenience.
17
Table 2.1: Deterministic Scalar Quantities
Symbol/Defintion
Description
k
Number of information bits encoded into a single variable
rate codeword.
b
Number of bits dedicated to error detection for each k bit
message.
t
Number k - b bit subpackets contained in one packet.
n
Number of channel uses.
ni
Cumulative number of channel uses corresponding to a
single message after i decoding attempts.
nIRU
Number of channel uses observed between consecutive
decoding attempts; i.e. the length of an
incremental redundancy unit (IRU).
m
Maximum number of IRUs (i.e. decoding attempts)
corresponding to one message.
ntrials
Number of Monte Carlo simulation trials.
Pue
A P ('P|)
Probability of false CRC pass given a decoding error.
P
Overall error probability for latency constrained cases.
Pr
Overall error probability for reliability constrained cases.
C
Shannon channel capacity.
Cf b
Capacity of feedback channel.
R
System throughput.
R,
Throughput for latency constrained cases.
Rr
Throughput for reliability constrained cases.
pA
Instantaneous throughput.
Average channel signal to noise ratio, linear units.
'/db
Average channel signal to noise ratio, decibel units.
ntx
Maximum number of packet transmissions dedicated to
one packet.
18
Table 2.2: Events
Symbol/Definition
Description
Si=
Decoding error given ni channel uses.
(ni)
Pi =P(ni)
Checksum (CRC) pass event.
Zund = Zud(ini)
A Si n P,
Undetected decoding error given ni channel
uses.
Zdet(ni)
A Ein 'Pi
Detected decoding error given ni channel
uses.
Si = St(ni) A En0 P.
Detected decoding success given ni channel
uses.
Ut
TA (Zynd n (n-Zet)) Undetected decoding error,
Dm A
E U (n
F A ULID
IZiet)
1 th
codeword.
Message error event, 1 th message.
Packet error event given a packet of t
messages.
19
Table 2.3: Random Scalars, Vectors, and Matrices
Symbol/Definition
Description
p
Packet containing t length k - b bit subpackets.
U1
1"h
subpacket of length k - b bits.
m1
11h
message of length k bits,
passed to encoder.
rM1
Decoding decision corresponding to the
in a given packet.
"
Length n vector of encoded symbols.
"
nThe
yfl
yi
1 th
message
IRU, a length ni - ni_1, ni < nm vector of
encoded symbols.
ith
Length n vector of noisy received symbols.
The ith noisy received IRU, a length ni - ni- 1 , ni < nm
vector of noisy received symbols.
wn
w?niLength
Length n vector containing samples of noise process.
ni - ni_1 vector of noise samples corresponding
to ith IRU, where ni < nm.
T
Number of packet transmission before packet decoding
success.
N*
Number of channel uses dedicated to the 1th codeword
prior to terminating decoding.
20
Table 2.4: Layered Code Parameters
M
Maximum number of redundancy blocks for layered rateless
codes.
L
Number of layers corresponding to one layered rateless codeword.
G
Complex layer combining matrix.
Etgt
Target bit error rate used to design G.
rb
Rate of base code when complex transmit symbol and FEC code
rate is taken into account.
N
Length of each turbo base code, in channel uses.
Table 2.5: Spinal Code Parameters
k'
Number of message bits (out of k) which make up one
hash key in a spinal encoder.
v
Number of bits which specify each spinal value.
c'
The alphabet of complex transmit symbols has a cardinality of 22',
i.e. c' bits are mapped to each real and imaginary part of the symbol.
B
Beamwidth of spinal decoder.
sP
Number of passes over the spine.
21
2.2
System Model
Our analysis assumes a discrete time baseband forward channel model that is corrupted
by additive white Gaussian noise (AWGN). Rateless coded communication takes place over
the forward channel, which is characterized by its Shannon capacity, C, in units of bits
per channel use, where a channel use is defined as the unit of time required to send one
complex symbol over the forward channel. Note that the forward channel can equivalently
be characterized by its average signal to noise ratio (SNR),
-Ydb,
in units of decibels.
A feedback channel is used to inform the transmitter when decoding success (or failure)
has been detected by the decoder. The feedback channel can be used as frequently as once
per forward channel use. The transmitter uses feedback information to determine whether
to begin transmitting the next message, or to send additional encoded symbols so that the
receiver can again attempt to decode. The simple acknowledgment/negative acknowledgment
(ACK/NACK) feedback scheme considered in this thesis implies that one bit of information
per use of the feedback channel is required. The feedback channel is characterized by its
capacity, CfA. Throughout this thesis, it is assumed that the feedback channel has zero delay,
and Cfb ;> 1 bit per channel use.
It is common when analyzing rateless codes (e.g. [7, 8]) to assume a genie aided system.
Perfect error detection is possible under this assumption, as it enables the decoder to compare
the estimated and true messages, thus allowing the decoder to detect with probability 1
whether a decoding decision is correct. When a genie aided system is not assumed (which
is the configuration of interest in this thesis), it is common to dedicate a number of the
information bits that make up each message to error detection. If b out of k message bits
are used for error detection, then these b bits function as a "checksum" that is used to verify
the integrity of the decoding decision (i.e. the remaining k - b bits) with some level of
reliability. While other methods of error detection that do not require a b bit overhead are
possible under certain rateless coding schemes', these alternative methods are not considered
'For example, error detection for the incremental redundancy rateless coding scheme presented in [8]
22
Packet
Queue
Packet
Buffer
p
Ip
Segment/
Queue
Reass./
Buffer
w.m
Ui
CRC
Rateless
Encoder
Encoder
Zfm
i
Rateless
CRC
Decoder
Decoder
Feedback Channel
Figure 2-1: System model
in this thesis because they tend to be coupled to the particular type of rateless code under
consideration.
In many practical systems, a b bit cyclic redundancy check (CRC) is used for error
detection. Because it is ubiquitous, and is not coupled to a particular type of rateless code,
this method is assumed for the remainder of this thesis.
In terms of functional units, the system consists of one transmitter, and one receiver.
The transmitter consists of a packet queue, a segmentation/queueing unit, a CRC encoder,
and a rateless encoder. The receiver is made up of a rateless decoder, a CRC decoder, a
reassembly/buffer unit, and a packet buffer. A functional system block diagrams is illustrated
in Figure 2-1.
It is assumed that the packet queue contains an endless stream of transport layer packets
of information bits. Each packet is denoted p, and is made up of t subpackets, each of which
is made up of k - b bits. Hence, each packet consists of t(k - b) bits. It is assumed that b'
out of the t(k - b) bits in each packet form a b' bit CRC which is used to verify the integrity
can be accomplished using the syndrome of the LDPC code. As another example, rateless codes employing
iterative decoders can leverage soft reliability metrics for reliability based error detection [11].
23
of the decoding decisions after all such decisions corresponding to a given packet have been
made. Note that transport layer packets model data units that are passed from the transport
layer to the MAC and PHY layers of a communication system (See Appendix A).
We next consider the operation of the system, which can be described by the following
sequence of events.
1. A large packet of bits, p, is segmented into t distinct subpackets, each denoted u for
1 = 1, 2, ... , t, where each u1 contains k - b bits.
2. A b bit CRC is computed and appended to each u1 , resulting in t distinct k bit messages,
each denoted ml.
3. The first message, mi,
{ X , x1,
0
--
, x m-1},
is passed to the rateless encoder, which maps mi --+ xm-
where each scalar x3 E Xnm is a complex encoded transmit symbol
that costs one channel use to send to over the channel. Note that for a rateless code,
in principal, nm could be infinite.
4. The first incremental redundancy unit (IRU), a subset of the encoded symbols
xm, x1m =
{xIx
2
, ... ,x
1
xim
c
}, is sent over the channel.
5. After observing the channel output, the decoder makes a decision, denoted
on the received sequence y" 1
= Xnm
+
W"l,
where
Wfl
i1 , based
is a length n 1 vector of complex
baseband samples of a white Gaussian process.
6. The decoder computes a checksum for iin1 to verify the integrity of the decoding decision. If the checksum passes, an ACK is sent over the feedback channel, which signals
to the transmitter that it can commence encoding and transmitting m2 . If the checksum fails, a NACK is sent over the feedback channel 2 , causing the transmitter to send
the second IRU,
2
X2m,
over the forward channel.
Note that the lack of an ACK can be considered a NACK given an error free feedback channel.
24
7. If necessary, additional IRUs are transmitted, and steps 5-6 are executed after the
receiver observes each IRU until the checksum corresponding to m- 1 passes, at which
time the system begins executing Step 3 for the codeword corresponding to m 2.
8. Once all t message decisions, mi, have been acquired by the receiver, the CRC bits are
removed and the resulting subpacket estimates, U-1 for 1 E 1, 2,.-. , t, are reassembled
into an estimate of the transmitted packet, P
[1,
l2,
--
, Ut].
9. The integrity of P is checked using the b' bit CRC checksum. If the checksum passes,
the receiver informs the transmitter via the noiseless feedback channel, and the system
starts the process over at Step 1 for the next packet in queue. Otherwise, the packet
is retransmitted, or dropped.
Note that all t of the mi E p must be correctly estimated, else the packet must be retransmitted or dropped, depending on the network protocol. Figure 2-2 illustrates the structure of a
packet, and its relation to its constituent subpackets and messages. Figure 2-3 illustrates the
structure of a rateless codeword and its relation to its constituent IRUs. Note that Figures
2-2 and 2-3 also help to elucidate some of the notation defined in Tables 2.1 - 2.3.
25
p
Packet
Subpac kcet
CRC Encodi ng
Messa ge
CRC
.2 -
U1
U
U11CC
R
M2
MI
Figure 2-2: Illustration of the structure of a packet.
26
R
Mt
mZIX
Message
Rateless Codeword
x
\
0 00
XX
1|
0P
v
*
Transmitted Rateless
Codeword After j IRUs
*
*
*
*
I
I
I
I
I
I
I
I
F~1
I
I
I
I
I
I
I
I
I
I
I
I
I
\%
IRUs
I
nm
0
00
I
I
I
I
I
I
x nj
Figure 2-3: Illustration of the structure of a rateless codeword.
27
I
I
I
I
I
I
28
Chapter 3
System Analysis
Many existing works on rateless channel codes [6, 7] are based on analysis and empirical
results corresponding to performance as observed at the MAC and/or PHY layers of a
communication system architecture (see Appendix A).
Furthermore, many works assume
that the decoder is omniscient in that there exists no uncertainty pertaining to detecting
when a correct decoding decision is made. While such analysis is useful for isolating code
performance from other effects in the system, it can be inadequate for designers who are
ultimately interested in the overall performance of such codes when they are integrated into
the overall system architecture. Motivated by this fact, this chapter develops an analysis of
rateless codes for the AWGN channel. Important features that differentiate this development
from other works on rateless codes are:
1. Imperfect error detection is used at the receiver
2. Performance and reliability as observed at the transport layer is of interest
Integrating features 1) and 2) throughout, our development yields a model that is useful
to system designers who are considering the tradeoffs involved in choosing between different
rateless codes, and guides the selection of important design parameters (e.g. information
block length, CRC length). The resulting set of equations and inequalities that represent
29
the model can help guide the beginning stages of the design process. Rather than having
to immediately resort to hours (or days) of simulation, our analytical model provides a
compact representation of the performance and reliability that one can expect to observe
given a rateless code, a set of design parameters, and system constraints.
Throughout the analysis, important equations and inequalities are placed in rectangular
boxes for emphasis.
3.1
Error Detection Model and Analysis
In this section, a CRC method of error detection is defined and analyzed. Our analysis shows
that, given a b bit error detection CRC, the probability of undetected error conditioned on
an incorrect decoding decision, denoted pue, can be approximated as:
Pue ~ 2-b
(3.1)
Under rateless coding, several kinds of errors are possible. For example, say that a particular receiver has acquired a sequence of noisy observations corresponding to a transmitted
rateless codeword made up of i IRUs (See Figure 2-3). The decoding error event, Si (equivalently denoted as S(ni)), occurs if M-i f m, where iin is the tentative decoding decision
given all information available at the receiver, and m is the message that was encoded at the
transmitter. The probability of a decoding error occurring, P (Ei), is generally dependent
upon the channel SNR, the block length ni, and on the particular method of encoding and
decoding that is used.
Another error that is possible under rateless coding (and, more generally, under any
communication scheme) occurs when a decoder fails to detect an error in a tentative decoding
decision. The event associated with error detection, Pi, occurs when the error detection check
returns a positive result1 . Under practical error detection methods, P (Pi) is dependent upon
1Note that this indicates either an
undetected error, or a detected success.
30
Table 3.1: Summary Of Basic Error Events
__S_ __
_
_
_
_
_sic
Pi
Undetected Error
Detected Success
P
Detected Error
False Detected Error
the particular method of error detection, and the bit error rate of the tentative decoding
decision [12]. To simplify our analysis, we assume an error detection model that depends
only on the method of error detection, and whether the ith decoding decision is correct.
Error events corresponding to a codeword of length ni channel uses and their interactions
and associated penalties are summarized in Table 3.1.
When employing rateless coding, error events corresponding to a given message can be
characterized by a sequence of events (and their complements) such as those listed in Table
3.1. Stated more formally in terms of our notation, error events for a given message, m,
occur over a sequence of decoding attempts that are executed on a sequence of codewords
of increasing lengths {ni, n 2 , ... , ni}, where ni is the length at which either si
n Pi or E9 n Pi
occurs. In principle, ni can go to infinity, though in practice it is typically limited to some
system dependent value nm. For notational convenience, the undetected, and detected error
events are defined, respectively, for a length ni codeword as:
Zlund =
zdet
E n P,
(3.2)
S.E n 'Pi
(3.3)
Similarly, the event corresponding to a detected decoding success given a length ni codeword
is defined as:
S,
Ec n P.
31
(3.4)
Assuming a maximum codeword length of nm, the overall undetected error event,
sponding to the
1 th
message, where 1 E 1, 2,
A Zund u Z"nd
(Z
d
n
, t,
corre-
is defined as the following disjoint union:
n Zdet U ... U (Zm"
(
i=1
...
El,
n Zdetn
... n Z
"
(3.5)
zet
j=1
Next, the event ZLnd = 6i n P is considered in order to gain basic insight into the
performance-reliability tradeoff that exists for a rateless code.
The undetected error event, Z1Id, incurs a penalty on the overall probability of error of
the system, as the receiver erroneously believes that in = m. Note that the false detected
error event in Table 3.1, e
Because the event
Znd
n Pf,
is assumed to have probability 0.
is very detrimental from a system reliability standpoint, as it
potentially allows many erroneous bits to pass through the MAC and PHY layers toward
the user application, it is useful to first understand P (ZInd), the probability of an undetected
error given a decoding error on a codeword of length nI. Using elementary laws of probability,
we rewrite the probability of undetected error as:
P (Znd) =IP (
1
=ED (F1
n P1 )
S,) 1P (E1l)
(3.6)
Referring to the right side of (3.6), note that P (Si) is a function of the channel SNR (which
partially determines the probability of a decoding error), and the particular rateless code that
is used. From the perspective of analyzing a given error detection model, we are interested
in P (P 1 S) = P (Pi ES), which depends only on the method of error detection, conditional
on whether a decoding error has occurred. Note that the decomposition in 3.6 conveniently
32
enables an analysis of this perspective2
For convenience, we define:
Pue
(3.7)
A P (Pi s
Note that in the case of the genie aided system discussed in Section 2.2, we have pue = 0.
As stated previously, we consider systems that use a b-bit CRC for error detection. This
method allows a designer to decrease pue by increasing b. Thus, as reliability increases,
throughput decreases. It is probabilistically possible for a CRC to pass when a decoding
error has occurred (i.e. for a CRC collision to occur). This can be understood by thinking
of a b-bit CRC as partitioning the set of possible binary messages, A, where 1A1a sequence of 2 ' subsets, {Aj}, for j = 0,,
2 k-b,
1. Each A contains all of the length k - b
..., 2
bit binary messages which collide with each other under the given b-bit CRC; i.e. JAjj=
for each j = 0, 1,,
into
2 k-2b
2' - 1. Under the assumption that each erroneous ii- is equiprobable
given that a decoding error has occurred, the probability that a set of decoded message bits
will erroneously pass the CRC can be expressed and upper bounded as:
JAI -1
2 k-2b _1
2
k-b -
1
2 k-b 2 k-b
1
(3.8)
< 2-b
2It has been shown [12] that, for certain classes of cyclic redundancy check polynomials that can be used
for error detection, the model used in this work for pue is accurate in the SNR regime where the bit error
rate of the CRC codeword is high. This turns out to be a reasonable model for rateless coding, because
many decoding attempts occur when the rateless codeword is much shorter what is required to successfully
decode, leading to a high bit error rate at the the output of the decoder.
33
where (3.8) follows from the fact that
< 1 whenever
2 k-b 2
b ;>
2b
>
1, or equivalently, when
i.
A lower bound can be derived by considering the error in the upper bound as follows:
1
2 k-2b -
2k-b
2
-b( 2 k-b
-
1) -
2k-b
-
2 k-2b -2
-b
-
+ 1
1
2 k-2b
-
2k-b
-
2 k-2b
+ 1
1
1 -2
2k-b -
(3.9)
1
1
< 2k-
-
(3.10)
1
where (3.9) is strictly greater than 0 for b > 0 and k > b, and (3.10) is true under the former
condition on b.
Using (3.8) and (3.10) we arrive at:
2
where 6 =
2k-b-.
Clearly, c
k = 256 and b = 24 we have E
E < Pue
-
< 2-b
(3.11)
0 quickly for reasonable values of k - b (for example, if
e
1.4489 x 10-70). Thus, under the current model for error
detection, we are justified in using the approximation pue ~
2 b,
which given in (3.1).
To make the throughput-reliability tradeoff concrete, consider the relationship between
the instantaneous throughput, pt A
k
,
and pue. Using the approximation given in (3.1), pt
can be expressed as:
Pt ,r..
k + log2 (pue)
(3.12)
Note that, for a rateless code, the number of channel uses, N*, required for the CRC to pass
is a random variable. For illustration purposes, in this section we take N*
34
=
ni.
Figure 3-1 illustrates the throughput-reliability tradeoff for several information block
lengths (as labeled in the figure legend). Note that the sequence of curves is characterized
by a slope that increases as the value of k is decreased, illustrating the performance cost
associated with various levels of reliability.
---
1
0.
0.99
0)
W 0.98
*0
0.97
.dO p
oo
0.96
0
.
0 0.95
6144
- - - 1024
512
256
U.
C
LL 0.94
.c
C
.... :..
.
0.93
-
10~
10 s
4-3-
10-4
10
10-2
10-1
100
Probability of Undetected Error
Figure 3-1: Fraction of throughput dedicated to information bits, pt/R with R = 1/3, as a
function of undetected error probability, Pue, for various information block lengths.
35
3.2
Message Error Probability
In Section 3.1, it was illustrated that CRC based error detection provides a mechanism to
control the reliability of decoding decisions. Furthermore, it was shown that each measure of
reliability has some associated performance cost. A model for this method of error detection
was presented, and the associated performance-reliability tradeoff was illustrated for the case
of a single decoding attempt. In the rateless setting, a sequence of decoding attempts, in
which error detection is performed after each attempt, may be necessary. Hence, the effect
of error detection on the overall reliability of each encoded message is more complex than
was illustrated in the single decoding attempt case. In this section, the results developed in
Section 3.1 are extended to characterize the overall reliability associated with transmitting
a single message. Before proceeding with a summary of the main results and corresponding
analysis, let us define the primary event of interest.
Let D, denote the error event corresponding to the 1"h of t messages that make up a
transport layer packet. A message error event occurs if either an undetected decoding error
occurs, or some agreed upon maximum number of detected decoding attempts, m, has been
reached 3 . Thus, D, can be defined as:
SA E
(3.13)
Zet
U
The main results developed in this section are as follows. We derive lower, and upper
bounds on the overall probability of undetected error corresponding to message 1 (as defined
in (3.5)), which can be expressed as:
P (S
> EPue (I - Pue)ii=1
3
P (E)
(3.14)
j=1
In this case, the transmitter and receiver may agree to "give up" on message 1, or perhaps on the entire
packet, depending on the application of interest. This issue will be addressed when we consider constrained
system models in Section 3.6.
36
Zpue (1 -
P (s,
(3.15)
Pue)<-P(Vi)
The upper bound given in (3.15) is then used, along with a bound on the probability of the
overall detected decoding failure event, P
(nli
Ziet),
to express the following upper bound
on P (D1 ), the overall probability of error associated with message 1 :
P(D) < Zpe
pue) m P (Sm)
(ES) + (1
(1 -pue-P
(3.16)
To begin our analysis, we express the probability of the event (3.13) as:
M
P (D) = P
)
(n Zfet
U
P (E) +P (
ziet
where (3.17) uses the fact that the undetected decoding error event and
(3.17)
fli1 Ziet are disjoint
(viz., (3.3), (3.5)).
The contribution of the second term on the right side of (3.17) is first considered. This
term can be rewritten and upper bounded as:
P
(zet
= P
P
(Zre)
i=1
Zdet)
Z det i
Z )
\
_j =m-i+1
<P (Zdet ) (1 - pue)m- 1
P ('P
Em) P (Sm) (1
(1 - Pue) m P (Sm)
37
(3.18)
(3.19)
-
pue)m-
(3.20)
where (3.18) follows from the chain rule for joint probabilities, (3.19) follows from the fact
that
L1
EP
(Zdet.
fT72 m-i+
(1
)
-
Pue)m 1 ,
and (3.20) is based on the definition
(3.7).
To bring system operational insight into the sequence of expressions (3.18) - (3.20),
consider the following: if the product due to the chain rule in (3.18) is collapsed we get:
m
m
rip (zm±2
i=1
m-1
ziet)
n
M-i+1
=
n ziet gzet
(i=1
(3.21)
which is the joint probability that the first m-I transmissions result in a sequence of detected
decoding failures given that the mth decoding attempt resulted in a detected decoding failure.
Intuitively, if a message can not be decoded using a codeword of length nm, then there is a
high probability that it could not be decoded at any length ni < nm, which suggests that
(3.19) may be reasonably tight.
Given (3.20) and the discussion above, a preliminary upper bound on the overall message
error probability can be expressed as:
P (DJ) < P (Et) + (1 - pue)m I? (Sm)
(3.22)
We next focus on the first term on the right side of (3.22), the overall probability of
undetected error corresponding to the 1th of t messages associated with a given packet. The
exact evaluation of this term is difficult, even for moderate m. However, in what follows we
show that using a small amount of operational insight enables us to derive upper and lower
bounds on P (E)
that are easily evaluated. We begin by expressing P (E) as follows:
38
P ES
U
=P
nzde
n
zu
j=1
(i=1
i-1
m
J:P
i=1
n
un-dn
idet(3.23)
j=1
where (3.23) follows from the fact the E' is equal to the disjoint union in (3.5). From Equation
(3.23), it can be seen that P (9E) can be bounded by bounding each term in the summation.
Hence, we expand each term as follows:
d e
net
und
_
p
und
(j=1
j=1
j=1
det
et
et
i-1
i-1
i-1
j=1
j=1
k=j+1
j=1
j=1
k=j+1
(3.25)
where (3.24) follows from the chain rule for
joint probabilities, and (3.25) follows from the
definitions of Ziund and Zidet, along with the fact that, under the current error detection
model, P (PtISi
n A) = P (PtIS,) for any arbitrary event A.
Now, consider the term P
(si
n-
Zjet), which is the conditional probability of a de-
coding error occurring on attempt i, given that detected decoding errors have occurred for
previous decoding attempts j = 1, 2, - - -, i - 1 for the current codeword. Using Bayes rule,
this conditional probability can be expressed as follows:
39
P 6i
n Zdet
(flilz
-
et ))(3.26)
Fqt
Z
From an operational perspective, we argue that P
(-i
beau)
et
Z et
> P (Oi-i
s)
Z5 et), because
conditioning on a decoding error on the ith decoding attempt clearly can not decrease the
probability that decoding attempts
j
= 1, 2, - - - , i--1 result in errors for a stationary channel.
Stated in other words, for a fixed message size k, knowledge of future decoding errors does
not decrease the probability of past decoding errors for a codeword which increases in length
with each decoding attempt, as it does for rateless codes. Hence, it can be concluded that:
P Ei
(3.27)
> P (8i)
zdet
n
j=1
Now consider the terms in the product of (3.25). Following the same operational argument
that was used to justify (3.27), each of these terms can be lower bounded as:
P Ej
n
P (Ej)
Zdet)
(3.28)
k=j+1
Combining the inequalities (3.27), and (3.28) into (3.25) results in the lower bound given
in (3.14).
To derive an upper bound on the terms in the summation of (3.23), consider the following
sequence of equations and inequalities:
F
(Zund ~q(~zet)
det
=FZnd)Fo
-)Znd)
-1
= d (Zu)
i-i
FPzde
=1k
pue(i) (1p~)
t
Z
fJ
i--1
j=1
nd
Z
ni
k=+
Zi~ed
l
j=1
_5 Pue7 (Ei) (I - Pue)i
40
P (Fi-1 Zund)
(y
--1
(3.29)
zdej'
(3.30)
kkj±1
k=j+1
(3.31)
where (3.29) follows from the chain rule for joint probabilities, (3.30) results from the definitions of Ziet, and Zind, and from the assumption that error detection is conditionally
independent of any additional events given the current decoding outcome. The inequality
in (3.31) is based on the fact that 1 1
E Zn 0 (
operational insight that led to (3.27), and (3.28).
+1 4et)
<land the same
Note that (3.31) can be extended by
dropping the conditional probability F (Ei-1 Zund) , as this term is likely close to 1 due to
conditioning on a future decoding error. The upper bound given in (3.15) follows.
Note that applying (3.15) to (3.22) yields the message error probability given in (3.16).
As an additional point, note that if it is assumed that the system is operating in an
SNR regime where successful decoding is possible for some system capable value of nm, then
P (E,,m) = 0, and the overall message error probability can be lower bounded by (3.14).
From a computational standpoint, the bounds in (3.14) - (3.16) are attractive as a design
tool because they are each a function of only
Pue,
and the marginal probabilities of decoding
error events over m decoding attempts. The only component of these bounds that has not
yet been developed, but is required for evaluation, is a model for the marginal probability
of decoding error, P (.E). In Section 3.3, two such models are described.
3.3
Approximate Decoding Error Probability
In this section, a method of approximating the marginal probability of decoding error is
presented. The method is based on random coding arguments, thus providing an optimistic
characterization of performance and reliability that can be used as a benchmark when considering practical rateless codes.
The model we consider is based on error exponent analysis and jointly typical sets. Conceptually, the decoder for this model can be thought of as making decisions based on whether
the the received sequence is jointly typical with a unique codeword from the codebook. This
approach is referred to as jointly typical decoding [13]. The model assumes the following:
41
1. There exists a codebook of
2k
unique codewords which each correspond to a unique k
bit message. The codebook is known to both the transmitter and receiver.
2. Each codeword, Xflm
-
{X 1 ,
X2 , ... , Xfm}
for i E {1, 2, ... , 2 k}, is a length nm sequence
of encoded symbols, where each symbol is drawn iid according to some probability
distribution p(x). Additionally, each encoded symbol has power constraint P; that is,
for any symbol xj E Xfm, E[x] = P.
3. The encoded symbols are sent incrementally over the channel, which outputs yf, a
distorted version of n encoded symbols, where n < nm.
4. Each encoded symbol sent over the channel is distorted according to the transition
distribution p(ylx). As stated in Section 2.2, the channel transition distribution of
interest in this work is AP (x, U2 ).
5. Each time the channel outputs additional distorted symbols corresponding to a given
k bit message, the receiver attempts to decode by looking in the codebook for a unique
codeword of length n that is jointly typical with the received sequence, according the
distribution p(X", yn) =
1
p(xy).
Additional details on jointly typical sequences and decoding can be found in Chapter 7 of
[13], which also provides a detailed derivation of the error exponent for such a decoder. The
important result of this derivation (for the current purpose) is that the average marginal
probability of decoding error corresponding to a length ni codeword can be bounded as:
P (,Ei) < 2 "aaN--
(3.32)
with SNR -y = P , and small constant 6 > 0. The bound in (3.32) is most accurate in cases
where the code block length, ni, is long.
Figure 3-2 illustrates (3.32) for a k = 256 bit rateless code over a complex AWGN channel
with capacity C = 1 bit per channel use
(Ydb
= 0 dB) plotted as a function of coding rate,
42
k
(equivalent to spectral efficiency in this case). Note that, though the error exponent curve
represents an upper bound, for the reminder of this thesis, we use the approximation.
P (Vi) - 2
(3.33)
*
100
10-2
a-
0
L.
0
0
1-6
I..
0
10 0
10-*0
10-12
0.6
0.65
0.7
0.75
0.8
0.85
0.95
0.9
1
1.05
1.1
Spectral Efficiency (bits/channel use)
Figure 3-2: Single decoding attempt probability of error example, k = 256,
-Ydb
= 0 dB.
Figure 3-3 illustrates the upper bound in (3.16) plotted as a function of pue for a complex
AWGN channel with capacity C ~ 2.06 bits per channel use
the information block length is k = 1024 bits, nIRU
=
(-Ydb
= 5 dB). In this example,
1, and nm = 8192 (i.e. decoding
attempts are executed after every encoded symbol is received, up to nm = 8192 symbols).
43
100
-- - - -
-
--
-
-
--
-
.- .-
-
-
--
10
-
0
L.
-O
102
(U
.......
.. . ..
......
. ... .. .........
0
cc
. . .. . . . .. . . ..-..
... . .. . . . .;
..
.......
10
.0-
M
. 10
05
10-6
10
1u
10
10~-
10-
-7
10-
10~5
10
10
Probability of Undetected Error (Pue)
110
Figure 3-3: Upper bound on probability of message error, k = 1024,
3.4
10~
100
Ydb
= 5 dB.
Packet Error Probability
In Section 3.2, a bound on the overall message error probability was developed. When
evaluated assuming a particular model for encoding, decoding, error detection, and channel impairments, the upper bound given in (3.16) can provide insight into the reliability
corresponding to the transmission of a single message encoded using rateless coding. Typically, user experience is more closely associated with the reliability and performance that
is observed at the transport layer of the system, as opposed that observed at the MAC and
PHY layers. Using this fact as motivation, this section focuses on characterizing the level
of reliability that is observed at the transport layer. We begin by defining the packet error
event, F, then give a summary of the main results of this section.
44
A packet error event occurs when the message error event, D1, occurs for at least one of
the t constituent messages corresponding to a given packet. Hence, F is defined as:
7
TA
UD
(3.34)
1=1
The main results of this section consist of the following lower and upper bounds on P (F),
the probability of error corresponding to a transport layer packet:
m
P F > 1-1-
Pue (I - Pue) i-' r
-
(Pue(1
P (E9 )
(3.35)
j=1
i-1
1- (
t
i
- pue) -IP (i)
+ (1 - Pue)mP (Sm))
(3.36)
The analysis leading to (3.35) and (3.36) proceeds as follows. Using results developed in
Section 3.2, the probability corresponding to (3.34) can be expressed as:
P(
D
= E)
(1=1)
t
=1 - P (
=I -
Di
P (Dc)
= 1 1 -P (Dj))
(3.37)
(3.38)
where (3.37) and (3.38) result from the assumption that error events are independent and
identically distributed across codewords. A lower bound on (3.38) is derived as follows:
45
P(F) 1-
P (D)
> 1-
1--
E
(3.39)
Applying (3.14) to (3.39) results in (3.35).
To derive an upper bound, simply apply (3.16) to (3.38), which yields (3.36).
Figure 3-4 illustrates the bounds in (3.35) and (3.36).
In this example, C ~ 1.2 bits
per channel use (-ydb = 1 dB), ni = 1, nriU = 4, nm = 8192, and tk = 4096 bits. The
corresponding upper and lower bounds are plotted as a function of the information block
length per codeword, k. The two cases correspond to two different choices of b, as labeled
in the figure legend. The plot shows that the probability of a packet error event is tightly
bounded above and below by a nearly constant function of k. The fact the the bounds are
nearly constant for fixed b as k varies is not surprising when one considers that, because nIRU,
nm, and tk are held fixed in the example, the number of possible decoding attempts over all
codewords in a given packet remains constant as k varies. This can be further understood
by considering the union in (3.5).
In addition to illustrating the upper and lower bounds derived in this section, the curves
in Figure 3-4 show the dramatic effect increasing b can have on the reliability of a packet.
Also, note that in the example of Figure 3-4, it is assumed that decoding is attempted every
4 complex channel uses. This affects the overall probability of error because the number
of opportunities for an undetected decoding error to occur increases as decoding attempts
become more frequent. Because of this effect, there is, once again, a tradeoff between the
performance a system can achieve, and the overall probability of error. The former would
have that the system be configured to attempt decoding as frequently as possible, resulting
in a fine tuning of the realized rate to each realization of the channel. The latter would have
46
the system attempt to decode only when there is a high degree of confidence' that decoding
will be successful, so as to minimize the number of terms in the union given in (3.5). We
return to this point in Chapter 5, as it suggests an interesting area of future work that is
beyond the scope of this thesis.
100
::::::::: :
:.. ...
.....
I...........
. . . . .. . . .I ..
....
............... ....
.......
. . . . . . .. . . . I
. . . . . . . . . . .I
:::L
....I . Ii :
.
b = 16
.........................'I' ' *........:...........
........
......
..............
...I .........
.......... ........... .............
.......
.......
b = 24
................................... ................ ........
.............
..................
......
.
.......
.....
........
........
.............................
10-1
..........
*-..*
....
.......
.......
....
..
...........
.
......
. ...
......
.....
...
.....
...
..
....
........
....
.....
...
.....
...
.......
....
*
.
...........
.......
......
..I....
...
.......
..
....*'..
-.
..............
.......
.....
........
........
... .......
...I .......
........
.......
....**
.....
........
.
.
........
...I
...I .........
.... ......
..I ......
...
..............
..............
............
........... .............
...................I .....
............
...........
.........
............
........... ........... ...........
.............
............
...........
10-2
0
L.
CL
L.
........... .............
...
....................
...........
.......
............
......
...
.............
...................
..
.
...
..........
..
.......
I
.....
....
........
....
....
...
... ........
..............
..
.......I...
...................
....
....
.
.......
.......
... ......
.....
.............
...........
........
...........
..... ........... ....... .....
............
........... ....... ........I ....................
...........
........................... ...I ..............................
.........................
0
LU
............
I ..... ......
........... .
10-3
...........
.......
......................
....I .......
......... ...........................
..........
.........
..........
...........
........
........
......
..............
............
..........
................................... ............ ....................... ....... ...
.........................
........... ........... ...........
........................
cc
M
10
-4
. . .. . . . . . . .. . . . . . . .. . . .... . . . . . .. ..............
. . . . . . . .... . . . . . .. . . .... . . . . . .. .................
..................... . . . . . . . .. . . .
... ........
*. ...
. . .. . . . .. . . . . .
. .. . . . .. . .. . .. . . . . .. . .. . .
. . ... . . . . . . . . . .
..
.. . .. . . . . . ..
. . . .. . . . . . . . . . .. . . .. . . . . .. . . . . . ..
..
. . . . . . .. . . .
.. . . . .. . . . . .... . . .. . . .. . ... . .. . . . . . .. . . . . . . . .. . .
10-5
. . . . . .. . ... . . .. . . .. . ... .
. . . . . . . . . . ... . . . . . . . .
. . ... . . . . . . . . . . ... . . . . . . . . . . . . .
. . . . . . .. . . . . . . .. . . . . . . .... . . . . . . . .. . . . . . . . . .. .
. . . . . .. . . . . . . . ..
.......................
...........
..............I ........
...
. . .. . . .
. . . . . . . . .. . . .
. . .... . . . . . . . .. . . . . . . . . .
. . . . . . . . . . . . . ... . . . . . . . . . . ... . . . . . . . . . . .. . . . . . . . . . . . . .
..................
*..............
.............
..........
.............
........... ............
.........I...........4-6*::::::::i::::::::::: :::........I............I. . . . . . . . .. .
500
1WO
1500
2000
2500
3WO
3500
4000
information Block Length (bits)
Figure 3-4: Numerical examples of uppe r and lower bounds on the overall packet error
probability as a function of k, -Ydb = 1 dB.
'A confidence metric could be computed based on the noisy observations, e.g. log likelihoods could be
used for this purpose.
47
3.5
Packet Throughput
In this section, the throughput of a transport layer packet is characterized. We begin by
defining the notion of packet throughput, then give a brief summary of the main results.
Then, prior to going into the detailed derivations of these results, a generic example showing
how the results can be used to construct bounds that can be evaluated numerically is provided. Note that for the remainder of our development, it is assumed that any undetected
decoding errors are subsequently detected using the bl bit CRC embedded in the packet 5 .
First, consider a packet consisting of t(k -b)
useful information bits'. The packet through-
put is defined, in units of bits per channel use, as:
R
tA
Tb
E
where i (f)
= p
(
i
P
N*
(3.40)
is the overall packet error probability which takes into account
the possibility of up to nrx - 1 packet retransmissions, and the expectation in the denominator is taken with respect to N* and T, which are both nonnegative integer valued random
variables. Each N* is assumed iid, and takes on value n C N>o when codeword 1 is successfully decoded using a rateless code which occupies n channel uses. T is distributed as a
geometric random variable that, for a given packet, takes on the value equal to the number
of attempted packet transmissions required for all t codewords to be successfully decoded.
Formally, N* and T can be defined as follows:
N*
min n E Z>O Si U Znd}
T ~ P (c)
5
P (E)3 1
(3.41)
(3.42)
This assumption is equivalent to assuming a moderately large value of b'
We continue to assume the error detection model of Section 3.1, and the packet structure of Section 3.4
but, for ease of exposition, include the b' bit packet CRC in this block of "useful" information bits.
6
48
for
j
= 1, 2, ..., nt,. Note that nt., is a protocol dependent parameter, and is not necessarily
under the control of the designer in the scenarios considered in this work. In what follows,
we consider only two distinct values of ntz, in particular, nt. = 1, and nri,
= 00.
The reasoning behind the definition in (3.40) is as follows. If nriaiS distinct packets must
be communicated to the receiver, the ratio of the total number of useful information bits
sent across the channel to the total number of channel uses can be expressed as:
ntriaist(k - b)
When
rItrials
(3.43)
is large, some fraction, P (F), of the packets will be decoded in error. Hence,
we can rewrite (3.43) as:
P (fc) t(k - b)
P (fc) nriait(k - b)
t
Taking the limit of (3.44) as
rials
ZT j N*
ntrials
-K
1/ntrials
Z
als
tT
N*(3.44)
oc results in (3.40).
Before proceeding with the analysis, we provide a summary of the important results that
are developed in this section. Note that for the remainder of this thesis, we make frequent
use of the fact that events can be parameterized by the code block length at which they
occur, rather than indexing them according to the decoding attempt at which they occur.
For example, for the decoding error event we have S(ni) = 9i (as previously described in the
second paragraph of Section 3.1), and for the decoding success event we have S(ni)
=
Si.
Bounding the expectation in the denominator of (3.40) is accomplished using the law of
iterated expectation. This entails developing bounds on certain conditional expectations, as
described next. First, upper, and lower bounds are developed for the conditional expectation
of N* given
.T.
Note that this is the expected code block length given knowledge that a
packet error event does not occur for the packet containing the codeword pertaining to N*.
The bounds are given as:
49
00
(3.45)
n=1
E [N,*CF"]
E
--
E
(1 - Pue)i 1 IP (E(i - 1) S(i))
P(c(i))
Note that when evaluating (3.46), P (S(i - 1) S(i)) will be approximated as
(3.46)
IF.F(C-1))
Next, upper and lower bounds on the conditional expectation of N* given F are derived.
This is accomplished by applying the law of iterated expectation, then deriving upper and
lower bounds on the conditional expectation of N* given
El.
The bounds are shown to be:
-00
IE [N* ~
<
(
-
Pu)nIp (9(n))
(3.47)
n=1
E[N*
>
(
-
P (9(i))pue(1 - pue)i-IP (s(i - 1) zund(i)
where P (F(i - 1) Zund(i)) can be approximated as P(.(i-1))
W
)
p(Zund(i))
(3.48)
It is then shown that the
bounds in (3.47) and (3.48), along with the appropriate
bounds on P (9E) from Section
(3.2), can be used to construct upper and lower bounds based on the expression:
[N*IF] = E N* -
P (E-)
Finally, it is shown that R can be expressed as:
50
+ E [N*
E] P (E
)
(3.49)
(k - b)P (FC)
(3.50)
-T) k\E [N*Fc]
E [N*F] IP (.F)
(Fc +
±P(yc)
N( e P (EC) + E [N*|F]
and that the results in (3.45) - (3.48), along with appropriate results from previous sections
can be used to construct upper and lower bounds on (3.50), which can then be numerically
evaluated. For example, an upper bound can be constructed as follows:
1. Select the desired values of of k, and b and substitute them into (3.50). Note that
selecting b implies the value of pue.
2. Substitute (3.33) for any and all marginal decoding error probabilities involved in
the construction of the desired bound (parameterized appropriately for the desired
configuration).
3. Substitute the product of (3.46) and the quantity 1 minus (3.36) into the denominator
of (3.50), in place of the product E [N *FC] P (fC) (There are two such instances in the
denominator of (3.46).).
4. Construct a lower bound on (3.49) by substituting the product of (3.48) and (3.14),
and the product of (3.46) and the quantity 1 minus (3.15) into (3.49) in place of
E [N*
] P (-)
and E N*
the fact that E N*
Ef
EI P (-E),
respectively (Note that we have made use of
= E [N *Fe].).
5. Substitute the lower bound constructed in step 4. into (3.50) in place of E [N*F].
6. Compute the ratio
.,
using (3.35) in place of P (F), and the quantity 1 minus (3.35)
in place of P (FC). Substitute the result into (3.50).
7. Compute P (Fe) appropriately and substitute the result into (3.50). For example, if
the transport layer protocol specifies that nt = 1, compute P (se) = 1 - P (F), using
(3.35) in place of P (F).
51
The example above illustrates a generic procedure that can be used to evaluate the bounds
developed in this thesis. In a given design scenario with a particular set of system specifications, some of the steps enumerated above may require slight modifications, but the basic
procedure will remain the same.
Next, we begin the analysis and derivations of (3.45) - (3.50). To evaluate the denominator of (3.40), first apply the law of iterated expectation:
NtT ]
E EN*
tT
~
=E E
N* T
tT
=E
~tT
N* T = I P (T = 1) +IE
IN*
T > 1 P (T > 1)
(3.51)
The conditioning in the expectations of (3.51) corresponds to the packet error event F as
follows:
FC= {T = 1}
(3.52)
F= {T > 1}
(3.53)
Each expectation in (3.51) is next considered separately.
The first expectation in (3.51) is conditioned on the event for which all t constituent
codewords of a given packet are successfully decoded. In this case, there is no need for the
packet to be retransmitted. Manipulating the expectation and expressing it in terms of F,
we arrive at the expression:
[
E
TtT
N* T =1
-
=E
t
N* T = i]
t
E [N* Fc]
1=1
52
(3.54)
Because N* is a nonnegative integer valued random variable, the expectation in (3.54) can
be expressed as:
E [N* Fc] = EnP (N* > n
(3.55)
7c)
Note that, because the decoding error event corresponding to a codeword of length ni can
be expressed as
(3.56)
Si = E(ni),
the events
Zund
and
Zdet
can be similarly be parameterized by a given number of channel
uses.
To begin evaluating (3.55) numerically, we rewrite the event of interest involving N* in
terms of S(n):
n
N(*> nF}
Zdet()) 0
i=1
(Ec(i)
n P(i))
(3.57)
i=n+1
Computing the probability of the event on the right hand side of (3.57) is difficult, but an
upper bound can be derived by considering the following disjoint union:
nZdet
0
c~i n-i)
U
n
det (i
c~
i=n+1
=1i+1i=1
ncPi
)C
(3.58)
which is equivalent to the intersection:
Zdet)
(3.59)
i=1
Rewriting the probability of the event given in (3.59) in terms of (3.58), and rearranging
terms results in the following upper bound on each of the probabilities in the summation of
53
(3.55):
P (N* > nrF) < P
Zdeti
<- (I - Pue)']? (S(n))
(3.60)
where (3.60) results from the same chain rule argument that resulted in (3.19). Substituting
(3.60) into (3.55) yields the result given in (3.45).
To derive a lower bound on the probabilities in the summation of (3.55), we write the
complement of the event of interest as:
{N*
(s(i) n
< nKFc} =
Zdet(j) ))
i=1
(3.61)
j=1
Because the events in the union on the right hand side of (3.61) are disjoint, the corresponding
Cd1
probability is:
IP (N* < n .V)
=?
Ps(i)n
Zdet( ) )
(3.62)
Applying the chain rule to each term in the summation of (3.62) results in the following
sequence of relations:
P (S(i) n
Zdet(j) ))
= P (S(i)) P
(
i-1
=P
(S(i))
Zdet (j)
(-1
Zdet (j) s(i) n
l 1p
j=1
S(i)
(
n
Zdet(k)))
k=j+l
i-1
=P
(Ec(i)) (1 - Pue) i~11
j=1
h
i-1
P S(j) S(i) n
(
Zdet (k)))
k=j+1
(3.63)
< P (E(i)) (1 - Pue)i- P (E(i - 1) S(i))
54
(3.64)
where (3.63) results from the definitions 7 of S(i), and
Zdet(j),
and the fact that, given E(j),
P(j) is conditionally independent of decoding attempts for j' f j. The inequality in (3.64)
results from taking the smallest term in the product 8 in (3.63), and noting that the product
of the remaining terms are bounded above by 1. A lower bound on the probability of the
complement of (3.61) can now be expressed using (3.64):
P (N* > n yc) > 1- ZP (SC(i)) (1 - Pue)i-IP (S(i - 1) S(i))
(3.65)
which implies the lower bound given in (3.46).
The second expectation in (3.51) is conditioned on the event corresponding to a given
packet requiring at least one retransmission. This event occurs when at least one of the
decoding decisions made during a given packet's initial decoding attempt is erroneous and
the CRC fails to detect the error. We begin to characterize this expectation by expanding
it into the following recurrence relation:
t
tT[~T
E
T
~
EN*
N*T > 1 =E IN*F +E
t
~tT
N* F +E
=E
~
[
(3.66)
N* T>1 IP(T >1)
tT
+E
P(T=1)
[N*T=1
=1
(3.67)
..
where (3.66) is a consequence of the memoryless property of the AWGN channel. Rearranging
7Recall that in the error detection model defined in Section 3.1, all decoding successes are detected; that
is, P
(p Ec)
A
1
8
From an operational perspective, P (E(i - 1) S(i)) is the smallest term in the product because decoding
errors are least likely to occur for j nearly i, and the nearest j can be to i in the given product is when
j = i - 1.
55
terms in (3.67) and rewriting the events involving T in terms of F yields:
E[
N*T
> 1]
=
E [EZ_ 1 N* F] + E [E'= N*|Fc] P (FC)
(3.68)
t (E [N* F] + E [N* 1C] P (Fc)
(3.69)
The conditional expectation E [N* F] is the only term in (3.69) which we have not previously established how to evaluate. To close this gap, we once again use the law of iterated
expectation, along with the fact that N* is conditionally independent of F given
IE [N* IF] =E [N* .F,E-1 P (E-1)
+ JE [N* F,F
= E [N* -]P (E ) + E [N*
Note that E N* Ef
=
P
E
E1:
(3.70)
(3.71)
EI P E)
E [N* F'], which we have previously established how to evaluate. The
remaining expectation in (3.49) can be expressed as:
E [Ni
=EP (N*
(3.72)
> n
n=1
To compute the probabilities in the summation in (3.72), note that:
nz
i=1
Because
(
n
Zdet
(i))
n Uo
de
(W)
n (
zund(i))
(3.73)
i=n+1
c
Zund(i)
Zdet(z)),
(nm=
an upper bound on (3.72)
is given by (3.47). For a lower bound on the terms in the summation of (3.72), consider the
56
complementary event:
{N* <
rP 1
}
zun=
(
od
Zdet(j)
(3.74)
j=1
i=1
which is another union of disjoint events. The probability of the corresponding event can be
expressed as a summation of probabilities, each of which was previously upper bounded in
(3.31). Applying this result yields the following upper bound:
P (N* < n
)
ZP'((i))
pue(1 - Pue)i-lIP (8(i - 1) zund(i))
(3.75)
Approximating P (V(i - 1) Zund(i ) as P(S(i-1)) and taking the complement of (3.75) results
in the desired lower bound given in (3.48).
Finally, note that after substituting (3.54) and (3.69) into (3.40), we arrive at the packet
throughput expression given in (3.50).
The expression in (3.50) can be numerically evaluated by applying the bounds on P (SE)
and P (T) that were derived in Sections 3.1, 3.2 and 3.4, along with the approximation given
in Section 3.3. Its main utility to system designers is as a tool that can be used to begin
modeling systems that use rateless coding. In particular, our results enable different choices
of design parameters and their resulting effects on system performance to be evaluated prior
to resorting to simulation.
3.6
Constrained System Models
In this section, the results developed in Sections 3.1-3.5 are used to analyze two distinct models of communication, each of which is assumed to incorporate rateless channel codes. Motivation for the analysis of these two models is based on observations first made in Section 3.1,
where the inherent tradeoff between system throughput and reliability was illustrated. Each
model is loosely based on one of the common network communication protocols belonging to
57
the transport layer of a communication system architecture (see Appendix A). Furthermore,
each model is meant to exemplify a different system constraint that is representative of those
which system designers are typically faced with. As an interesting comparison, we compare
the bounds provided by one of our models to the variable length feedback (VLF(e)) codes of
[14]. The VLF(e) bound provides a coarse grained view of variable rate codes that is based
on transmitting a single codeword. The goal of this analysis is to provide useful insight to
system designers who may be considering incorporating rateless codes into their designs.
The first model is referred to as the latency constrainedmodel. This model is useful for
applications in which delay is more costly than dropped packets, and can be considered an
archetype for the well known User Datagram Protocol (UDP). The second model, which will
be referred to as the reliability constrainedmodel, may be used for delay tolerant applications
that require high degree of reliability, and can be thought of as modeling the Transmission
Control Protocol (TCP). In what follows, "reliability constrained model" and "TCP model"
are used interchangeably, and likewise for "latency constrained model" and "UDP model".
The most important distinction between the two models stems from the fact that the reliability constrained model attempts to decrease the probability of error observed by the
application by retransmitting a packet when a packet error event occurs. This is in contrast
to the latency constrained model, which drops erroneous packets and moves on to the next
packet in queue'.
The overall probability of error and throughput for the latency constrained model are:
A1AP (F)
R (k - b)
[N P (.FC)
Ez
E [Nr*]
(3.76)
(3.77)
Where the expectation in the denominator of (3.77) is expressed as:
9
Thus, the UDP model subjects the application layer to a level of reliability equal to the packet error
probability.
58
E [N*] = E [N* Fc] P (Fc) + E [N* F] P (F)
(3.78)
Lower and upper bounds on (3.78) can be derived directly using our previous development
in Section 3.5, along with the appropriate bounds on P (F')from Section 3.4. These lead to
upper and lower bounds on (3.77).
The probability of error and throughput for the reliability constrained model, assuming there
is no limit on the number of packet retransmissions in the event that any packet is found to
be in error are given as:
Pr A 0
R
Rr
where P (f')
=
1
(3.79)
(3.80)
Pr = 1 (see 3.50).
To illustrate the utility of Equations (3.76) - (3.80) along with the bounds derived in
previous sections, consider the following scenario: Suppose a designer would like to design a
system where the packet length may not exceed 4096 bits, and the overall system probability
of error must be less than or equal to 1 x 10-10. Because the packet length and the overall
probability of error are specified, the designer must choose k and b so that the specifications
are met or exceeded. Choosing k and b will also determine the number of codewords which
are used to transmit a packet, t. Figure 3-5 illustrates upper bounds on (3.77), and (3.80)
for a low SNR case (-sdb = 1 dB), while Figure 3-6 illustrates the high SNR case (Ydb = 10
dB). For the marginal decoding error probabilities, the error exponent model described in
Section 3.3 was used. The bounds are plotted as a function of the information block length
of each codeword, k, for choices of b which enable the system error rate specification to be
met or exceeded.
As can be seen in Figures 3-5 and 3-6, the UDP model provides a bound for both high and
59
low SNR cases. Noteworthy is the fact that the UDP model, for both SNR regimes, offers
a more pessimistic bound on throughput than what is provided by the VLF(C) lower bound
of [14]. The VLF(c) bound, plotted in green in each figure, is included in the comparison
because it could be used as a tool to gain insight into such a design. However, because the
UDP model takes a more fine grained perspective into account, it is a more appropriate
analytical tool for the current design task.
Also note that the TCP model throughput bound is higher than that of the UDP model
for shorter values of k. If it is assumed that the bounds are tight, then this would suggest
that there is significant loss incurred by utilizing such a lengthy CRC (b = 45 bits) in order
to meet the system error rate specification in the UDP case. As can be seen in both SNR
examples, the upper bounds for both models converge to C as k increases. It is interesting to
note that these effects are a direct manifestation of the throughput-reliability tradeoff first
discussed in Section 3.1, and illustrated in Figure 3-1, which showed that, when using CRC
error detection, reliability has a higher throughput cost when shorter vales of k are used.
60
1.2
4)
C 0.8
U)
0.6
0.4
- -
0.2
---
--n
0
500
1000
1500
2000
Capacity
lower bound
- VLF(E)
TCP: b = 24
UDP: Error Rate: 1e-10 (b = 45)
2500
3000
Information Block Length (bits)
Figure 3-5: Bounds on throughput, -Ydb = 1 dB.
61
3500
4000
3. 5
. . . . . ..
-
-5
I
.
....
.-.-
. . -. -. .
-.
. . .. . .
-- ...
... . . .
35
.. . .
.
. .
.
.. .
..
. ...
. . .
a)
a>
2. 5 --
2--
0
C
C
.0
1. 5-
-
5
0.
0)
500
Capacity
VLF(E) lower bound
--TCP: b =24
--0-- UDP: Error Rate: 3e-1 1 (b = 45)
-
--
1000
1500
2000
-
2500
3000
Information Block Length (bits)
Figure 3-6: Bounds on throughput, -Ydb = 10 dB.
62
3500
4000
Chapter 4
System Design and Practical Rateless
Codes
In this chapter, two practical rateless codes are evaluated. Each code is evaluated using
Monte Carlo simulation, and the analysis of Chapter 3 is used to guide a design process that
can be used to select the "best" practical rateless code for a given system. We begin by
describing each practical code.
4.1
Layered Codes
Layered rateless codes [6] utilize an approach in which weighted linear combinations of encoded symbols from a number of distinct messages are incrementally transmitted over the
channel. The encoded symbols corresponding to each distinct message, which we refer to
as base codewords, are generated using a fixed-rate base code. After a sufficient number of
observations by the receiver, the decoder attempts to obtain an estimate of each message
using a successive cancellation decoder. If decoding fails for any one of the messages, additional weighted linear combinations of the base codewords are transmitted, and successive
cancellation decoding is restarted using all of the available observations. Though the construction is not limited to large values of k, it has been shown [6] that choosing relatively long
63
information block length turbo codes for the base codebook results in good performance.
The adjective "layered" is an epithet referring to the fact that L base codewords (i.e.
"layers") are repetitiously transmitted over the channel, where each repetition, which is
referred to as a redundancy block, is constructed as a distinct linear combination of the base
codewords. At the encoder, a set of combining weights make up a M x L matrix of complex
coefficients, denoted G, where M is the maximum number of distinct redundancy blocks. If
the l1 h base codeword is denoted cl, and the m'h redundancy block is denoted
xm,
then the
M redundancy blocks output from the layered encoder are of the form:
Xi
Ci
(4.1)
XM
CL
where, in what follows, any subset of the xm's corresponding to a particular set of L cl's will
be referred to as a layered codeword.
Because of the repetition structure of the layered architecture, some method of combining
available redundancy blocks at the receiver is required prior to executing each successive
decoding attempt. This step of the receiver algorithm can be implemented using symbol
by symbol, unbiased minimum mean square error (UMMSE) combining of the observed
redundancy blocks 1 .
Note that decoding can be attempted when a non integer number of redundancy blocks
have been observed by the receiver.
Further, it is shown in Section IX of [6] that good
performance can be achieved in such a configuration, so long as at least 1 full redundancy
block has been observed.
In [6] it was shown analytically that the layered construction can achieve capacity approaching performance given a capacity approaching base code and appropriately designed
matrix G. It was also demonstrated, using simulation of several different configurations,
'Note that maximal ratio combining (MRC) can also be used, as it is for the implementation described
in [15], though UMMSE generally enables higher performance.
64
that layered codes can achieve good performance when using an analytically optimized G,
along with a long information block length Turbo base code.
In this thesis, we further contribute to the results on the layered construction by analyzing
their expected performance when used in the kinds of systems described in Section 3.6. We
begin by describing in the following subsection how this architecture fits into the models
developed in Chapter 3.
4.1.1
Decoding Errors Under Successive Decoding
The error detection model of Section 3.1 assumes that, given n channel uses using a particular
rateless codeword, the probability of an undetected decoding error, P
(Zund(n)),
is equal to
p,4eP (9(n)) e 2-'P (E(n)), which is a function only of the number of message bits dedicated
error detection, b, and the probability of decoding error, P (E(n)).
Additionally, in our
previous treatment it was assumed that P (8(n)) could be approximated simply by setting
it equal to the error probability given by either one of the two decoding models of Section
3.3. In this section, it is shown that under the successive cancellation decoding inherent
in the layered architecture, the decoding error probability, and undetected decoding error
probability for a given number of channel uses has a slightly different characterization. In
particular, it is shown that a different mapping from the decoding models of Section 3.3 to
the decoding error probability P (9(n)) is required. The main result of this section is that
the required mapping is given by the following relation:
L
P (,E(n))
=1-
S(n, 1))
(4.2)
where S(n, 1) is similar to the detected decoding error event defined in (3.4), though for
this case it must be parameterized by the current number of channel uses, n, and the layer
1 c [1, L] that is currently being decoded.
The mapping in (4.2) suggests that successive cancellation decoding provides additional
65
protection against undetected decoding errors when' compared to a rateless code which is
accurately modeled by the simple, direct mapping.
As noted in Section 3.1, the decoding error probability for any particular code which
occupies n channel uses is a function of the channel SNR, 'y. As such, the decoding error
event can be expressed in a manner which conveys explicit dependence on the channel SNR:
S(n)
=
E(n, -y)
(4.3)
The explicit parameterization of decoding error events shown on the right hand side
of (4.3) is useful when considering the probability of decoding error under the successive
cancellation decoder employed by the layered architecture. To understand why this is the
case, consider the effective SNR at the input to the decoder when attempting to decode
layer 1 E [1, L]. In particular, this is the effective SNR after any previously decoded layers
have been subtracted from the received signal, and UMMSE combining of the remaining
redundancy blocks has been executed. This effective SNR is determined by the SNR of the
communication channel, and the layers that have yet to be decoded. As such, this effective
SNR is a function of the number of received redundancy blocks, and the particular layer that
is being decoded. This means that the decoding error events corresponding to individual
layers must be parameterized by n, and ,, the effective SNR corresponding to layer 1. Thus,
given an estimate of 'yj for each n and each 1 E [1, L], the probability of decoding error can
be estimated using one of the decoding models of Section 3.3.
The discussion in the above paragraph implies that a method for obtaining estimates of
the effective SNR is required by our modified model of the layered architecture. In practice
(and as suggested in Section IX of [6]), a reasonable estimate of this effective SNR can be
obtained by computing the expected mean square error (MSE) corresponding to each layer
after cancellation of previously decoded layers and UMMSE combining is taken into account.
See Section B.1.2 of Appendix B for an illustration of the validity of using the MSE estimate
for this purpose.
66
Let
l be an estimate of the effective SNR corresponding to layer 1 for a particular
1
layered codeword which has occupied n complex channel uses. Assume that
obtained as described above.
l has been
Let Slayer (n, -Yl) be a new error event corresponding to a
decoding error given n channel uses when the effective SNR is -Y, where the subscript 1
denotes explicit dependence on the particular layer that is being decoded by the successive
cancelation decoder. Denote the event corresponding to a detected decoding success at layer
I given n channel uses as S(n, 1), where S(n, 1) is defined as:
S(n, l)
(4.4)
Eaye, (n,yi) n P
E
Note that we need not parameterize P because we continue to assume the error detection
scheme given in Equation (3.7), which is conditionally independent of n and yl given the
associated decoding result. Similar to (4.4), the undetected decoding error event defined in
(3.2) can be parameterized by 1 as:
Zund(n,
(4.5)
1) = Elayer (n, 7y) n P
and similar for the detected decoding error event,
Zdet(n, 1)
(see definition Equation (3.3)).
Because the expected MSE for layer 1 is computed under the assumption that layers
1+1, ..., L have been successfully decoded and cancelled, the resulting effective SNR estimate
can be used to approximate the conditional probability of decoding error.
This can be
accomplished using the error exponent approximation of Section 3.3 as:
L
I?
(Slayer
(n,
ni) 1 S(n, )
~ 2-n(c(
- )
(4.6)
j=I+1
Equation (4.6) enables the use of the models of Section 3.6 to analyze the layered architecture.
We next show, in detail, how this can be accomplished.
67
Decoding success for a layered codeword is defined as the intersection of detected decoding
successes of all L layers which make up the codeword for some n. For a given n the probability
of this event can be computed as:
L
P
TS(n,
()
L
=
B =1
(=1
,
1-
I
L
(n, -yi) n S(n, 1)
j=l+1
(layer
(4.7)
where we have used the chain rule for probabilities and the fact that:
P (S(n, l)1S(n, l + 1))
P (S aye, (n, y) n PIS(n, l + 1))
=
B (P'9fayer (n, yi) , S(n, 1 + 1)) P (Slayer (n, -z) S(n, 1 + 1))
=
(1
- P (Slayer
(n, -yi)|S(n, 1 + 1)))
(4.8)
(4.9)
Note that if we take the complement of the relation given in (4.7) we obtain the probability
of decoding error given in (4.2). This result given is key to applying the results of Chapter 3
to the analysis of layered codes. In particular, Equations (4.6), (4.9), and (4.2) show how to
map the decoding models of Section 3.3 to the probability of decoding error given n channel
uses under the successive cancellation decoder employed by the layered architecture.
Next, consider an undetected decoding error given n channel uses. Under successive
cancellation decoding, such an undetected error occurs if i out of L decoding attempts result
in undetected errors, where 1 < i < L, and the remaining L - i attempts each result in
successful decodings. Hence, the number of ways in which an undetected decoding error can
occur is combinatorial in L. In particular, there are E_1 (L)
=
-
an undetected error can occur for a given n. Because each of the 2 L
1 distinct ways in which
-
1 events are distinct,
the overall event can be defined as the following disjoint union:
uj
n
l)
uZ"d(n,
68
n
(
S(n,l')
(4.10)
is the ith set of indices such that B
where B
1, 2, ... , 2 L
Note that
Pte,
-
(L)
C [1, L], B L
B for i
# j,
1. The probability corresponding to (4.10) is a summation of
of the terms are proportional to pue, while the next
where i,j -
2L
_ 1 terms.
(L) are proportional
to
etc. Thus, despite the fact that the number of terms on the sum grows exponentially
with L, many of the terms are very small, even for moderate choices of b.
Additional operational insight reveals that for a number of the first decoding attempts
executed on a codeword given n channel uses, only 1 out of the 2L
-
1 terms is present
In particular, if the probability of decoding the Lth layer (which is
in the summation.
the first layer the decoder attempts to decode) is equal to zero, then the only way for an
undetected decoding error to occur is if all L decoding attempts for that particular n result
in undetected errors, an event whose probability is upper bounded by pU . Thus, for the
layered architecture, the probability of an undetected decoding error given n channel uses
can be upper bounded as:
P (Z und (n)) <
(Z1
L ()Pje
Ej
if P (S(n, L)) > 0
p1e
otherwise
(4.11)
Note that error propagation due to cancelation of erroneous codewords makes successfully
decoding subsequent layers after an undetected error occurs less likely. This can be thought of
as a feature of the layered architecture, due to the successive cancellation decoding. Because
of this feature, it is likely that (4.11) is somewhat loose.
4.2
Spinal Codes
Spinal codes
[7] were
recently shown to achieve good performance for relatively short infor-
mation block lengths. The basic idea underlying spinal codes is to encode each message, m,
into a series of correlated spine values by sequentially applying a hash function to distinct
k' bit segments of m. A sequence of random number generators (RNGs) seeded by the re69
sulting spine values are then used to generate symbols which are mapped to an appropriate
transmit symbol before being sent out on the channel. The spinal decoder described in [7], is
an approximate maximum likelihood decoder in which decoder complexity, and performance
can be traded off by varying the configuration of the decoder.
One of the main attributes that makes spinal codes of interest in this work (and in general)
is the good performance they have been shown to achieve at short message information block
lengths. This makes spinal codes particularly appealing for use in low latency applications,
where modern high performance codes (e.g. turbo, LDPC) typically do not achieve high
performance.
Though the construction of the spinal code itself does not appear to limit its good performance to the regime of short k, the decoder presented in [7] can inhibit performance as k
increases, when the beamwidth of the decoder, B, is held constant. The beamwidth of the
spinal decoder determines the number paths that are retained as the decoder sequentially
steps through the tree of possible decoding decisions. At each step, B paths back to the root
of the tree are retained, and all other tentative paths are pruned. Hence, as k is increased
(and subsequently, the required code block length), the tree lengthens, and the number of
opportunities for the correct path to be pruned from the tree increases. This means that
B must be increased in order to maintain good performance as k increases. However, increasing B increases the computational complexity of the decoder, making it prohibitive to
increase the beamwidth in practical systems. A more detailed discussion and analysis of the
parameters associated with spinal code can be found in Section 8.5 of [7]. For the analysis at
hand, we use this reference as a guide to aid in choosing the best performing configuration,
though we are also limited to configurations that are supported by [16].
In [7], the authors analyze and demonstrate spinal code performance in various configurations for AWGN, and Rayleigh fading channels. In [17], the authors develop a link layer
protocol for spinal codes which attempts to dynamically select values of nIRU by learning
the complementary distribution function (CDF) of the decoder stopping trials (similar to
70
Equation 3.41), then applying a dynamic program to determine the IRU length after every
feedback.
In this thesis, we contribute to the work on spinal codes by analyzing their performance
when they are used in the kinds of systems described in Section 3.6. The operation of
spinal codes is such that the models of Chapter 3 can be used directly, without any need for
modifications as those which had to be made for the layered codes.
Design of Constrained Systems
4.3
The main objective of this section is to illustrate how to choose the "best" rateless code for a
given design scenario. We also seek to illustrate how the analysis of Chapter 3 can be helpful
to the design process. The illustration is carried out by comparing results obtained via
Monte Carlo simulation of layered, and spinal codes to appropriately parameterized versions
of the models developed in Chapter 3 (and refined for the layered case in Section 4.1.1).
4.3.1
Latency Constrained System Design
Consider a system in which the transport packet length is at most 7168 bits, and the transport
layer protocol is UDP. Because packet retransmissions are not allowed, the system can be
modeled using (3.76), and (3.77). Suppose it has been determined that the application can
tolerate a maximum packet error rate of 3 x 10-2, and that the system should be designed
to operate in environments with Ydb ;> 1 dB. Given this setup, we would like to answer the
following questions:
Q1) What values of k and b can be chosen to meet a given system error probability specification?
Q2) Which rateless code under consideration performs best given the answer to Q1?
71
To begin answering Q1, the recall the packet error probability upper bound given in
(3.36).
Simply evaluating this bound for different choices of k and b, yields insight into
where to begin the design. To illustrate this, Figures 4-1, and 4-2 show (3.36) plotted as a
function of k for two distinct choices of b. Figure 4-1 illustrates the worst case SNR that
the system is specified to operate in
high SNR case
(-Ydb
=
(-Ydb =
1 dB), while Figure 4-2 illustrates a relatively
8 dB). The markers on each curve specify values of k that divide
7168 an integer number of times and are greater than b, and are, thus, candidate choices for
the codeword information block length. The bounds are clearly above the specified packet
error rate for low SNR, and slightly below it at high SNR for the b
=
14 case. For b
=
16,
the bounds are below the desired packet error rate for both low, and high SNR. These
observations lead to the conclusion that b > 16.
Note that in Figures 4-1 and 4-2, it is assumed that decoding is first attempted after
16 channel uses have been observed by the receiver, and then is attempted again every 8
complex channel uses. Recall that, from the perspective of wanting to match the rate of the
code to the particular channel realization, it is desirable to attempt decoding as frequently
as possible. However, when practical error detection is used, more frequent decoding leads
to a higher probability of undetected error because there are more terms in the union of
(3.5) (i.e. m is larger).
Additional observation of Figures 4-1 and 4-2 reveals an important, and intuitive design
guideline: given a specified operating environment and associated worst case SNR, it is
most important during the early stages of design to consider the worst case to help guide
the design process. Indeed, the worst case reliability illustrated in Figure 4-1 obviated the
need to consider the results in Figure 4-2. While this observation can help guide the design
process away from unnecessary tasks, it is generally important to characterize the system
over a wide range of conditions at some critical point in the design process.
Because the upper bounds shown in Figures 4-1 and 4-2 are relatively constant over
the range of k values, it can be concluded that throughput must be taken into account to
72
properly choose k. Though the exact value of k still remains to be chosen, the packet error
model of Section 3.4 been leveraged to help answer Q1.
. . ... . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . .
........ ......................................
.....
...........
......... .....
.......
.................... ....
.........
....
....
.............................
..........
...........
............... .........................
................... . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . ... . . . . . . . .
.I ..........................
.............................................I ..............
............... ............
..........
.........................................I .........
10-1 :7
.......................
..........
...................... ............................. ...
cc
..............
0
L.
10-2
.... .... ......
.... .... ....
....
: ....* ...: ...........
.........
..* ' * * *............. .......
...... ....* .....
*......................
.....
.......
.....
.........
...................
..........
. ......
.. ...........
...................
......
...........................
..........
......
...
..
.......
.........
.............. ...... ......
........ ........... ......... .......I ...
.....
0
............................. .........
. ....
CL 10)
...................
.......................
............ ........ .........
....................
.. .......
...........
.........I ...........
...
..................
............
..........
.........*
.................
............. .........
.....
................... ........................................
.............. ................... .............
........ ..... .........
...........
...... ...
.....
...........
... .....
..........
...........
....
...........I .............. .....................
............... ..............
........ .........
...........
.....
...........
............
10
17
Erro r S pe c .
b = 14
e b = 16
.......I.......... ..................
......................
........
............
100
200
300
400
500
600
.........
.......
700
800
900
1000
Information Block Length (bits)
Figure 4-1: Packet error probability upper bounds for UDP design example, -Ydb
=
1 dB.
Given the preliminary conclusions reached for Q1, (i.e. that b > 16, and k may be
selected as any value which divides 7168 an integer number of times), we now wish to answer
Q2; that is, to select a rateless code, along with a corresponding information block length,
k, that performs best in this scenario. To begin to answer this question, bounds on the
throughput, R 1, given in (3.77) can be evaluated. Because a candidate set of b and k values
has already been determined, simulations of the layered and spinal codes 2 targeting these
configurations can also be executed. Figures 4-3, and 4-5 show such simulation results for
2
All spinal code simulations were executed using the useful and well documented libwireless package [16].
73
10
0 a.100
10-
Error Spec.
b = 14
b = 16
-*--
-9-1
. .
r .. .. .. .
100
..
.
200
300
.I .
. . .
400
.
. .
..
. .
500
..
.
600
.
..
. ..
.
.
700
.
..
800
.
.
. . . .
900
. .
.
1000
Information Block Length (bits)
Figure 4-2: Packet error probability bounds for UDP design example, -Ydb = 1 dB.
low, and high SNR cases
(-Ydb =
1 and 8 dB, respectively), along with upper bounds on R1 .
Table 4.1 summarizes the parameters used to configure each simulation shown in Figures
4-3, and 4-5. Recall that notation that is specific to the layered and spinal codes is described
in Tables 2.4 and 2.5, respectively. For the spinal code simulations, k', c', and B were chosen
based on recommendations in Section 8.5 of [7], while ni and nIRU are a consequence of
the "8-way" puncturing schedule3 which is available in the libwireless package for simulating
spinal codes [16]. For the layered code simulations,
rb
was selected to correspond to the LTE
turbo code of [18] along with the use of a complex QPSK transmit symbol, L was chosen
3 Between each decoding attempt, symbols corresponding to every
8 th spine value are transmitted. See
[7], Section 5.
74
so that throughput is not limited 4 for SNRs below approximately -ydb = 14 dB, M was
chosen to maximize the number of unique redundancy blocks (given the chosen L) without
imposing any loss due to the layered architecture (see [6], Section V), and ctgt was chosen
based on previous simulation results executed over the SNR range of interest (see Figure
B-1, Appendix B).
For both codes, simulations were executed with equal values of nIRU for a given information block length, k. Hence, the frequency with which feedback is acquired by the encoder
is the same for both codes. Note that the layered codes are configured so that decoding attempts begin after the first full redundancy block is received. This configuration was chosen
because it was shown ([6],
Figure 3) that attempting to decode when less than one redun-
dancy block has been observed by the decoder results in low performance. On the other hand,
due to the puncturing schedule of the spinal code simulation, decoding attempts begin after
nIRU channel uses are observed. From a maximum throughput perspective, this means that
the spinal codes have an advantage over the layered architecture in it's current configuration.
Indeed, the maximum throughput for the spinal codes is 8 x k x 2 = 64 bits per complex
channel use, whereas the layered architecture can achieve only up to L x
rb =
14/3 bits per
complex channel use. However, in practice systems generally do not operate in environments
where C > 64 bits per channel use. There is also a disadvantage for the spinal codes given
this configuration, which can be understood by recalling the union of disjoint events given in
Equation (3.5). In particular, the number of decoding attempts may be greater in the case of
spinal codes because such attempts commence when fewer channel uses have been observed.
This generally leads to a larger number of decoding attempts for the same expected N*, as
compared to layered codes. This means that, compared to the case of the layered codes,
there will be a larger number of disjoint events in the union of (3.5), and, hence, a larger
number terms in the summation of the corresponding probability of undetected error for the
spinal codes. This effect for the spinal codes can be mitigated by computing a reliability
4
1t is required that L x rb ;>
log 2 (1 + -y) hold to avoid limiting throughput for an SNR of y.
75
metric based only on the channel observations, and using this metric to decide when to begin
decoding. This enhancement is not implemented for the simulations presented in this work.
Table 4.1: Simulation Configuration, k = {256, 512, 1024}
(a) Spinal Codes
(b) Layered Codes
k'
4
rb
2/3
V
6
L
7
c'
6
M
7
B
256
ni
{390, 774, 1542}
ni
{4, 8,16}
nIRU
{4,8,16}
nIRU
{4, 8, 16}
EQgt
1 x 10-3
Mapping
PAM-2c'/ real dim.
Mapping
QPSK
b
16
b
16
The stand-alone markers plotted in Figures 4-3 and 4-5 illustrate the results of the simulations described above. The value corresponding to each marker was computed as:
tk
(k - b) Z
tktrials
Ej=1
where
tk
tialsl{T;
n*
i
is the number of messages (codewords) that make up a packet when the information
block length is k, nTtrgias is the number of packets simulated, 1{Fjc} is an indicator function,
and n* is the realization of N*. Note that (4.12) converges to (3.77) in the limit as ntrials -+ 00.
Figures 4-4 and 4-6 show the probability of error for each simulation, along with that
predicted by the models when configured with applicable parameters from Table 4.1.
Consider the results shown in 4-3. First, note that the upper bounds on R, provided by
our model give a less optimistic forecast of performance as compared to the VLF(E) lower
bound5 of [14]. This comparison is used simply to illustrate the importance of considering
5
Note that the VLF(E) bounds shown in each figure were computed using
k's for each code.
76
E = maxpl over all candidate
1.3
-
1.2
-
4)
-
-
x
0.9 2 0 .8 -
Layered.Simulation
La. e e.Mode
...
Mo..e
..
-..
. .. .. .. . ... . . .. . . .. -.. ...
A
1.1
x
... ...
....
.
-Spina
.
C
0.7-
-
--
.
.
--- -
- --
Shannon Capacity
VLF(es na
VLF(Elaee
0.6
9- Layered Model
-
-G - Spinal Model
0.5 .. .. .
- -..
...
x
. .....
-...
..
-.
. . ... .. .
A
0.4'
0
100
200
300
400
500
600
Layered Sim ulation
Spinal Simulation
800
700
900
-
1000
Information Block Length k(bits)
Figure 4-3: Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example, -Ydb = 1 dB.
the performance limitations that are imposed by the architecture of the entire system, as
opposed to considering only the performance that the rateless code may achieve.
Next, note that for this low SNR case, spinal codes achieve higher performance when
k = 256, but there is a "crossover" in the simulated throughput for k = 512, where the
layered codes achieve higher simulated throughput. This is also the case, albeit to a larger
extent, for k = 1024. This suggests that, given each system in the current configuration,
if the system is to operate primarily in low SNR environments, the highest performance
can be achieved by choosing the layered construction configured with a base code with
corresponding information block length k > 512. One explanation for this performance is
as follows. The packet length in this example is long enough such that the turbo base code
77
10
1
10.
-
10
. .. ..
0 10..
+0
(10
s
10-
-0-
0
10
Layre.Mde
....
300
400
Spna0Mde
500
600
700
800
900
1000
Information Block Length k(bits)
Figure 4-4: Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example, -ydb = 1 dB.
can be chosen at a relatively long block length, e.g. k
=
1024. Additionally, recall that the
beamwidth of the spinal decoder, B, constrains spinal code performance as k grows and B is
held constant. The crossover observed in this case is likely caused to a combination of these
two characteristics. Further on in this example we return to this issue by experimenting
with the beamwidth of the spinal decoder in an effort to determine how sensitive spinal code
performance is to a change in B within the context of the UDP model. Moreover, in Section
(4.3.3), we investigate a short packet length design example. However, given that the system
was specified to operate in high SNR environments, results for such environments must also
be considered when deciding which rateless code to use.
78
I
I
I
I
I
-
I
I
I
I
--
--
- ----
- VLF- sp--"-0
-
F 1.5 ......
I
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...... .............
-
--
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-
Shannon Capacity
--- VLF(s,
spinal
-VLFI.laered
0
1 ... .
.5
.
... .
. .. . .
..................
.........
. . ... . .. . .... .. ...
0
100
200
300
e- Spinal Model
Layered Model
a
Spinal Simulation
a Layered Simulation
.. . ...... .-...-
400
500
600
700
800
900
1000
Information Block Length k(bits)
Figure 4-5: Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example, ?Ydb = 8 dBe.
Turning to the results shown in Figure 4-5, it can be seen that the conclusion that layered
codes of long block length should be chosen for the design does not hold as strongly for this
SNR. The highest empirical throughput at this SNR is achieved by the spinal code at a
block length of k
=
512. For k
=
1024, the spinal code has a negligible advantage over the
layered code. Because this advantage is minuscule in comparison to that which the layered
code holds in the low SNR case, the reasonable conclusion remains that, for the current
configuration, layered codes should be selected, although it has now been shown that the
information block length should be chosen as k = 1024.
The preceding discussion focused on comparing the two codes based only on throughput.
Clearly, because the system under design is specified to operate at a packet error rate of less
79
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108
. . .
300
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500
400
-
E-EX
Spinal Model
Spinal Simulation
Layered Model
Layered Simulation
.. . . .... . . . .. . . . .. . . . . .. .
600
700
800
900
1000
Information Block Length k(bits)
Figure 4-6: Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example, Yb = 8 dBe.
than 3 x 10-2, the empirical packet error rates of the two codes must be taken into account
prior to making a final design decision. Turning attention to Figures 4-4 and 4-6, note that
both codes meet or exceed the specified packet error rate of under the current configuration.
Note that there are no empirical packet error rate measurements for the layered codes
appearing in Figure 4-6 for k = 512 and k = 1024. This is due to the fact that undetected
errors were not observed when simulating layered codes in the current configuration for
these values of k at
-Ydb
= 8 dB. That is, for this set of Monte Carlo trials, ntras = 6 x 104
layered codewords were simulated for each value of k, and not a single undetected error was
observed for k = 512, and k
=
1024. This means that, with high probability, the codeword
error rate, P (Dj) is only a small multiplicative constant greater than about 1 x 10-3, meaning
80
the packet error rate is, within a narrow confidence interval6 , around 1 x 10-4. This low
observed packet error probability suggests that our claim in Section 4.1.1 that (4.11) is a loose
upper bound is correct, and also illustrates that successive cancellation decoding provides
additional robustness against undetected errors for the layered codes.
Based on the simulated throughputs and packet error rates observed thus far, it can be
concluded that one should either choose the layered codes with k = 1024, or reevaluate spinal
code performance after modifying some of the parameters of Table 4. Ia. Efforts toward the
later option can be aided by observations based on Figures 4-4 and 4-6. In particular, note
that the empirical throughput of the layered code increases monotonically for both SNR
cases. In contrast, that of the spinal code increases from k = 256 to k = 512, then decreases
for k = 1024 in both SNR cases. This decrease is not predicted by the analytical model, and
is counterintuitive from the perspective of random coding theory. However, is is shown in
Section 4 of [7] that the performance of the spinal decoder can be increased by increasing
the beamwidth, B. The results of simulating spinal codes with B = 512 and 1024 are shown
in Table 4.2. For both experiments are configured with
Ydb =
1 dB, k = 1024, and other
parameters as given in Table 4.la.
Table 4.2: Spinal Code Throughput Comparison, k
=
1024, Ydb
B
Throughput (R 1 , bits per channel use)
256
0.87
512
0.90
1024
0.90
=
1 dB
As can be seen in Table 4.2, increasing B from 256 to 512 results in a slight increase in
throughput, but increasing B further does not help. Hence, spinal codes are still outperformed by the layered codes for k = 1024,
6
_Ydb =
1 dB.
These estimates are based on the application of Chebyshev's inequality, and a normal approximation to
the binomial indicator function,
11{D}
81
An additional consideration that must be taken into account when increasing B is the
resulting increase in computational complexity of decoding, as spinal decoding complexity is
proportional to B. A detailed treatment of the computational complexity of the two codes
that considers, for example, the degree to which each decoding algorithm can be optimized is
beyond the scope of this thesis. However, one can begin to acquire an understanding of the
relative complexity cost that must be paid to achieve the throughput increases illustrated in
Table 4.2 by using the simulation aided complexity analysis given in Appendix B, Section
B.2. This analysis is used to estimate the average number of receiver operations per goodbit 7
for this case. The results, which are given in Table 4.3, show that a 14% increase in the
number of receiver operations per goodbit must be paid for increasing B from 256 to 512 as
shown in Table 4.2.
Table 4.3: Receiver Ops. Per Goodbit, k = 1024,
Spinal (B = 256)
5.743
x
-Ydb =
Spinal (B = 512)
Layered
6.554 x 106
4.289 x 106
106
1 dB
Whether or not the additional complexity of the spinal code is prohibitive in practice will
depend on aspects such as the hardware platform on which the system will be deployed, size,
weight, and power constraints on the system, as well as the degree to which each decoding
algorithm can be paralellized. In
[19],
the authors develop an efficient architecture for decod-
ing spinal codes, and a brief comparison is made to the LTE turbo decoder. The comparison
in [19], along with our simulation aided complexity analysis suggests that, although spinal
decoding requires a higher number of receiver operations per goodbit, the turbo decoder of
the layered architecture requires a larger memory footprint. A detailed discussion of these
issues is beyond the scope of this thesis.
7
Recall that a goodbit is defined as a bit which is decoded correctly. For example, if K bits are decoded
over some interval of time, and the empirical error rate for these K bits is p,, the number of goodbits recieved
is equal to K x p,.
82
Given the above analysis of throughput, reliability, and computational complexity of the
candidate rateless codes for the specified latency constrained system, a reasonable solution
to Q2 is to choose the layered architecture with an information block length of k = 1024.
Indeed, under the scenario presented in this section, it has been illustrated that this design
decision will result in a system which will exhibit higher expected throughput, a lower packet
error probability, and also a lower computational complexity than choosing the spinal code
in any of the considered configurations. If the memory resources on the implementation
fabric of the receiver are limited, spinal codes are capable of delivering throughput similar to
that of the layered codes for k = 256, and 512, although it has been shown that this decision
will result in at least an order of magnitude increase in packet error probability.
Finally, note that to further balancing the comparison of these two rateless codes, it
would be ideal if each code were configured with a CRC length, b, such that the packet error
probabilities of each code are equal. This would enable a comparison based purely on the
performance that each rateless code is capable of within the setting of interest. However, we
defer such experiments for now, due to limitations of the software simulator.
4.3.2
Reliability Constrained System Design
Consider a system in which the transport packet length is at most 7168 bits, and the transport layer protocol is TCP. In this scenario the system can be modeled using (3.79), and
(3.80). Once again, the system should be designed to operate in environments with
-Ydb >
1
dB. Because we model TCP as providing unlimited packet retransmissions, which can, in
principle, drive the system error rate as low as desired, a particular specification on the
probability of error is not given.
Because packet retransmissions incur a large penalty on throughput, one should seek to
minimize such occurrences by minimizing the probability of packet error. However, because
of the throughput-reliability tradeoff that exists, such a minimization must take into account
the cost that a given level of reliability incurs on performance. Thus, instead of seeking
83
answers to Q1 and Q2 as in the latency constrained case, we wish to answer the following:
Q3) Which one of the two codes, and associated values of k and b, provides the highest
throughput under the given constraint on packet length?
To begin designing the MAC and PHY layers of the system such that the best possible
tradeoff between performance and reliability is achieved, we turn to the models, and our
simulation results. The parameters used for simulation of layered and spinal codes are the
same as those in Table 4.1.
Upper bounds on system throughput, R,, along with corresponding simulation results
for _Ydb
1 dB, and 8 dB, are shown in Figures 4-7, and 4-8, respectively.
The figures, once again, illustrate the choices of k that divide 7168 an integer number of
times. It is interesting to note that the model suggests that the layered code can outperform the spinal code for all values of k which are considered in the low SNR case, and the
layered codes do, in simulation, outperform the layered codes for two out of thee values of
k which were simulated. In the high SNR case, the models suggest that the spinal code can
outperform the layered code for all considered values of k, and the simulation results show
that this is the case for two out of the three k's which were simulated.
84
.. . . . . .. . . . ..
.. . . . . . . . . . . . .. .. .
. . ... . . .
.
.
1.2
1.1
1
xx
0.9 F
A.
xA
0.8 k..
(0
........
................
0.7[F
Shannon Capacity
Layered Model
-...
.---
0.6 F-
-
e - Spinal Model
X
0.5 F
0.4
A
0
100
200
300
400
500
600
700
Layered Simulation
Spinal Simulation
800
900
1000
information Block Length k(bits)
Figure 4-7: Throughput upper bounds and simulation results for layered, and spinal codes
for TCP design example, -Ydb = 1 dB.
Given the configurations and results shown in Figures 4-7, and 4-8, one can reasonably
answer Q3 by choosing the layered code with base codes for which k = 1024. However, this
conclusion does not take into account the effect of alternative choices of b. Could varying the
value of b for the spinal code result in higher throughput, in spite of the throughput-reliability
tradeoff that exists? Because the simulation resources for varying b are not available for the
spinal code, we cannot explore the parameter space for the spinal code empirically. Because
of this, we turn to the model, which suggests that the throughput of the spinal code could
(based on its upper bound) exceed that of the layered code if b were increased from 16 to
24. However, one must also recall that R, does not take into account the beamwidth related
effects on the spinal decoder.
Furthermore, if we consider the effect that was observed
85
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I
I
I
I
I
I
. - - ..- - - - - - --- - - -o
e-0
2.5 F
I
.
M
U
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. .. . . . . . . . . . . . . .
.
. ..
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.
2
(0
1.5 F
.
0.
1
-
e
Shannon Capacity
- Spinal Model
9
A
Layered Model
Spinal Simulation
x
Layered Simulation
0.5
0
100
200
300
400
I
I
I
500
600
700
I
800
900
1000
Information Block Length k(bits)
Figure 4-8: Throughput upper bounds and simulation results for layered, and spinal codes
for TCP design example, -Ydb = 8 dB.
in previous simulations, in which the empirical throughput of the spinal code reached a
maximum value at k = 512, then decreased when k was increased to 1024.
Thus, it is
likely that the throughput of the spinal code, configured with b = 24, will also decrease as k
increases, as opposed to following the trajectory suggested by R,.
86
4.3.3
Short Packet Lengths
In Sections 4.3.1 and 4.3.2, it was shown that the chosen layered construction provides good
performance and reliability for latency and reliability constrained scenarios, so long as the
information block length of the base code is chosen to be relatively large (k = 1024 in our
examples). Allowing such a value of k implies a relatively long transport packet length, which
must be proportional to L x k. Codes such as the turbo code, which constitute the base
codebook for the layered construction, do not to perform particularly well for short block
lengths. While the layered construction does not preclude us from using a high performing,
short block length code (such as an algebraic code) for the base codebook, we differ such an
investigation for future consideration.
Many applications (e.g. voice communications) have strict latency requirements. For
the model of interest in this thesis, a strict latency requirement implies a short transport
layer packet. Performance results presented in [7] suggest that spinal codes, due to their
capacity approaching performance at short information block lengths, are particularly well
suited for such applications. Given the above considerations, one must ask: are spinal codes
the better choice when designing a system with a strict latency requirement, when compared
to the layered construction considered in this thesis? As it turns out, the answer is yes. The
following design example illustrates how to arrive at this conclusion.
Consider the design of a system for which the transport packet length is at most 1024
bits, the transport layer protocol is UDP (as would be appropriate for an application with a
strict latency requirement), and the packet error rate must not exceed 1 x 10-2. Additionally,
assume the system is expected to operate in environments for which -Ydb > 1 dB. As in Section
4.3.1, we seek to answer Q1 and Q2.
Because of the short packet length, the values of M, and L for the layered codes must
be chosen smaller than those in the previous examples. For this example, L = M = 4 is
chosen 8 .
8
Recall that this choice will limit throughput for SNRs above y =
87
2 Lxrb
-
1, or Ydb
7.3 dB.
Figure 4-9 shows that b = 16 is sufficient for both codes to meet the system error rate
specification for k E {64, 128, 256}, thus answering Q1. This knowledge, as in Section 4.3.1,
enables us to target simulations with these parameter choices. A summary of the parameters
used to configure the simulations used in this example are given in Table 4.4.
10-2
CO
0
0
10~
q
CL
-e-b
10 -
80
100
120
140
160
180
200
=18
220
240
Information Block Length k(bits)
Figure 4-9: Packet error probability bounds for UDP design example with short packet
length, Y1db = 1 dB.
Figures 4-10 and 4-12 illustrate simulation results of the short packet UDP case, along
with throughput upper bounds on R 1 , and, again, VLF(c) lower bounds for each rateless code,
for comparison. For both SNRs, the bounds on R, suggest that the spinal code may outperform the layered code at each value of k, which agrees with the simulation results. Though
the empirical throughput of the layered construction increases quickly with k (whereas, in
contrast, the spinal codes performance approaches a peak at k = 128), they are still out88
Table 4.4: Short Packet Length Simulation Configuration, k = {64, 128, 256}
(b) Layered Codes
(a) Spinal Codes
k'
4
rb
2/3
v
6
L
4
C'
6
M
4
B
256
ni
{102, 198, 390}
ni
f{1, 2, 4}
nIRU
{1,2,4}
nIRU
{1,2,4}
Etgt
I x 10-3
Mapping
PAM-2c/ real dim.
Mapping
QPSK
b
16
b
16
performed for each of the values of k which could be chosen given the packet length and
the values of L and M. These observations confirm that the spinal code is better suited for
the short block length regime, when compared to the layered construction with a turbo base
codebook.
Figures 4-11 and 4-13 illustrate the probability of error observed in simulation and the
upper bounds provide by the UDP model. Unlike the longer packet length case of Section
4.3.1, undetected packet errors were observed for the layered codes in both SNR regimes, at
each considered value of k. Note that the confidence levels corresponding to the Monte Carlo
simulations of short packet length cases are similar to those given in Section 4.3.1 for the
long packet length cases. Also, observe that the empirical packet error probabilities in Figure
4-13 corresponding to the layered codes lie above the upper bound given by our model. This
phenomena suggests that, for short information block lengths, the layered construction with
turbo base code is not well modeled by our UDP model. Further investigation as to why
this is the case is deferred.
Given the performance results in Figures 4-10 and 4-12, the clear answer to Q2 is to select
spinal codes with k = 256. This choice will provide for the highest average throughput, and
89
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.. .. . . . . . . . . . . . . . . . . . .
1
0
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........
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-
-.
--
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---
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-
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-
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-
-
-
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x
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-
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.
.yer..
_-
VLF(sp
1 n)
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-0-- Layered Model
*
0.5
-..-
-
-
.
-A
x
Spinal Simulation
Layered Simulation
-
Information Block Length k(bits)
Figure 4-10: Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example with short packet length, ~Ydb = 1 dB.
a probability of error meeting that specified for the system. As was previously mentioned,
further investigation of the layered construction configured with alternative base codes is
beyond the scope of this thesis, though it is, perhaps, an interesting area warranting future
work.
90
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. .. . . . . . .
. . .. . . . . .
. . .. . . . . . . . . ... . . . . . . . .
. . . . . . . . ... . . . . . . . . .
.....
. . . . . : ......
. . . . .. . .
1
. . . . .. . . . . ......
. . . .. . . . .. . . . . .. . . . .. . . .. . - . .
..
. .
..
..
..
. ..
. ...
. . . . .
. . .. ... .
. . .
. . . . ..
.
..................................
. . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . .
. 1 . . . . . . . : ... . . . . . . . . ... - . . . . . . I . : . . . . . . ... . . . . . . . . .
. . . . . . . . I . . . . . . . I .... . . . . . . . : .. . . . . ......................................
..........................................
LU
.. . . . . . . . . . . . .. . . . .
. . . ... . . . .
. . . . . . . . . .. . . . . . . . . .... . . . . . . .
10-6
. . . . . . . . .... . . . . . . . ... . . . . . . . .
. . . . . . ...
. . ... . . . . . . . . . . . . . . . . . ... . . . . . . . . ... . . . . . . . .
. . ... . . . . . . . . . . . . I . . . . ... . . . . . . . . ... . . . . . . . .
. . . ... . . . . . . . . ... . . . . . . . .
. . ... . . . . . . . .
. . ... . . . . . . .
X
. . . . . . . . . ..
. . . . . ... . . . . . . . . . . . . . . . .
. . . . . . . . . .. . . . . . . . . .. . . . . . . . .
. . . . . . ... . . . . . . . . ... . . . . . . . . .. . . . . . .
. . .. . . . . . .
...........
. . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . .. . . . .. .. .. . . ..
cc
CL
10 -8
.. . . . . . . . .. . . . . .
. . .. . . . . . . . . ... . . . . . . . .
. . . . . .. . . . . . . . . . .. . . .
. . .. . . . . . . . . ... . . . . . . . .
. . . . . . . . ... . . . . . . . .
. . . . . . .. . . . . . . . . . .
. . . . . . . . ... . . . . . . . . .
.. . . . . . . . .
S pinal M odel
.........
...................... ..........
..............
... A Spinal Simulation
..... ......
I .......
...............
:::::::::
....
..............
.....
...................
...........
..
..................
...
....... ..............
..
......
..
.........
..
............... E) Layered M odel
..............................
.......
..........
X Layered S im ulation
................ I.......................... ..........
......
.
...... ............
..........
..
..
......
..
........
...
.......
........
...
..
.
...
......
...........
........
...... ...........
........
........
I ....
... ....
........
..
.......
........
.........
.........
..........
.........
.........
...
.......
.
......
..
........
...
......
.....
...
.......
........
.........
........ ...........
...... .............
...........
.. ......
........ ...........
........
.....
........
60
80
100
120
140
160
180
200
220
240
260
Information Block Length k(bits)
Figure 4-11: Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example with short packet length, 7db = 1 dB.
91
I
I
I
I
a)0
2.5
I
I
-
-
-
-
- - -
I
-
co
Cx
Shannon Capacity
VLF(Elayered
-- -
-
-
VLF(cs 1a
-
e
-
Spinal Model
9
A
-
0 .5 -
60
80
100
120
-
- - -.---.-.-
140
160
Layered Model
Spinal Simulation
Layered Simulation
- --...-.
.
--.-.- - --.
x
.-
180
200
220
240
260
Information Block Length k(bits)
Figure 4-12: Throughput upper bounds and simulation results for layered, and spinal codes
for UDP design example with short packet length, -Ydb = 8 dB.
92
10
0
. . .... . . . . . . .
......
..............
...................
...............
............
....................................................
........ ..........
.....................
10-1
.
........
........
...........
...........
..........................
.....
.........
.........
.....
................................................ ...........
10-2
....
........ ...... ..................... ............
............
................
....
Spinal Simulation
E)
X
Layered M
odel
Layered Simulation
. . . . . . ... . . . . . . . .
I
. . . . . ... . . . .
S pin a l M ode l
. . . .. . . . . . . .. . . .
.. . . . . . . . . ..
. . . . . . . . . ... . .
cc
. . . . . . . . .. . . .
. . . . . . . . . . . . . . . . ... . . . . .
.... . ..
. . ...
. . . . . . . . . .. . I . . . . . . .. . . . . . . . . .
. . .. . . . . . .
10
. . . . . . . .
-4
.. . . ..
.. . ..
. . . . . . . . . .. . .
.. . . . .
........
....
. . . . . . . . . .. . . . . . . . :
. . ... . . . . . . . . . . . . . . . .
.. . . . . . .. . . . . . .. . . .
. .
. . . . . .. . . . . . .. . . .
.
. .. . . . . . . . . ... . .
. . . . . . * . ... . .
*
. .. . . .. . .
. . . . . . . ...
. . . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . .
. . . . . . .
. . . . . . . : . . . . . . ... . . . . . . . . ... . . . . . . . . . . . . . . . . . ... . . . . . . . . .
. . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . ... . . . . . . . . ... . .
. . . . . . ..
....
..
..
. .. . . . .
...
. . ....
... . . . . .. . .
.. . . . . .. . . . . .
. .. . . . . .. . .
10-6
. . . . . . . . .. . . . . . .
. . . . . . .
X
. .. . . . . . .
. . . . . .... . . . . . . . ..... . . . . . . . . . . . . . . .... . . . . . . .
. . . . . . . . .... . .
. . .. . . . . .... . . . . . . . ..
. .. . . . . . . . . . . . . . . . . .. . . . . . ..
. . . . .... . .
. . .. . . . . . .. . . . . . . . ..... . . . . . ..
. . .. . . . . .... . . .
. . . . .. .... . . . .. . . . .. . . . .. . . . . .. . . . ..*. . . . . .. . .
. . . .. .... . . . .. . . . . .. . . . .. . . . .. . . . ..... . . . .. . .
.. . . . . . .. . . . .. . . . . .. . . . .. . . ..... . . . .. . .
. . . . .. . . . . .. . . . .. .
...
cc
CL
. . . . . . . . . .. . . . . . . . .
. . . . . . . . . .. . . . .
. . . . . . . . . .. . . . . .
. . . . . . . ... . . .
.. . . . . . .. .
. . . . . . . . . . . . . . . . . . . . . ..
..
. . . .. . . . .. . . . ..
W
. . . . . . .. .
. . . . . . . . ... . . . . .
...........
.. . . . . . . . . .
. . . . .. . . . . . . . . . .. . . . . . . .
. . . . ... . . .
. ... . . . . . . . . . . . . . . . . . . .. . . .
. . . . . . . . . . . . . . . . . . .. . . . .
.. . . . . . . . . .. I . . . . . . . .
. . . . . . . ..
. . .. . . . . . . . . .. . . . . . . . . .
. . . . . .
. .. . . . . .. . . .. . . . . .. . . . . . .
. . . .. . . . . . . . .
0
&W
CL
No
. . . . .. .
. . . . . . . . . .. . . . . . . . . ... . . . . . . . .
10-3
M
. . . . . .. . . . .
.. . . . . . . ..... .
. .... .. . . . . . . . .. .... . .
...7.
..........I.........
........ .....................
.........
........ ...........
...........
....................
...........
...........
.. . . . . . . . .... . . . .. . . . .. . . . . . .. . . . . .. . .....
...............
.....
.... . . . . . . . . .... ...... .....................
. . .. .
...............................................
. . .... . . . .. . . .... . . . . . ..
. . . . .. . . .... . .. . . . . .... . ..
. . . . . .. . .... . .. . . . . .... . . .
. . . . . .. . . . . . . . .. . . ... . . . .
. . . . . . .. .
. I . . . . . . . . . . . . . . .... . . . .. . . .... . . . . . ..
..
..
. ...
...
..
.....
. . . . . . . . . ...
10
-8
60
80
. . ..... . . . . ..
. . . . . . . . . . . . . . .
100
I
I
120
140
I
I
I
160
180
200
220
240
260
Information Block Length k(bits)
Figure 4-13: Packet error probability upper bounds and simulation results for layered, and
spinal codes for UDP design example with short packet length, -Ydb = 8 dB.
93
94
Chapter 5
Discussion and Conclusion
In this thesis, we have presented an analysis of practical rateless codes. The analysis takes
into account two important features of practical communication systems that are typically
ignored in other analyses appearing in the literature. We began by defining and characterizing a model of rateless coding that accounts for practical error detection methods and
packetized communication protocols at the transport layer. Our analysis resulted in the
derivation of a set of parametric bounds corresponding to the performance and reliability
of a system that uses rateless codes for error correction, and CRC for detecting decoding
decision errors. This set of parametric bounds was then used to guide a Monte Carlo based
analysis of layered and spinal rateless codes. This analysis illustrated a process that can be
followed when investigating rateless codes for system design
Furthermore, it was shown that under the UDP and TCP like models developed in
this thesis, the layered codes perform well when the transport packet length is sufficiently
long to support a relatively long turbo base codebook. For short transport packet lengths
that are characteristic of latency constrained applications, spinal codes perform best for the
configurations considered in this work.
During the course of our analysis, a number of additional and fascinating insights emerged,
including suggestions for future work, which we discuss next.
95
Due of the spinal decoder, in particular the beamwidth parameter, spinal codes are
inherently well suited to operate using short information block lengths, and not well suited
for moderate to large informations block lengths. On the other hand, the turbo base code
that the layered code is configured with in this work is inherently well suited for cases where
the information block length is relatively large. An interesting topic for future work would
consist of characterizing the performance of layered codes when configured with algebraic
codes for the base codebook, as they have been shown to perform well in the short information
block length regime.
While analyzing a well known performance-reliability tradeoff resulting from CRC error
detection overhead, it was observed that another closely related tradeoff exists for rateless
codes. Indeed, our definition and subsequent analysis of the codeword error detection event
illustrated that a performance-reliability tradeoff also exists as a result of choosing the frequency with which decoding occurs. In particular, we saw that increasing the frequency
of decoding enables a fine tuning of the realized rate, but incurs a penalty on the overall
probability of error. Investigating the optimal frequency of decoding under a detailed system
model such as ours could be a useful area for further work. Recent efforts in this general
area can be found in [20].
An interesting area related to the above line of work is the investigation of methods for
increasing the reliability of rateless codes that use CRC error detection. This is inspired
by our analysis of layered rateless codes, as it was observed that the successive cancellation
decoding architecture inherently provides additional reliability against undetected errors
under the CRC model. Additional gains can be obtained by leveraging the reliability metrics
computed at the receiver. For example, log-likelihoods can be used to dynamically determine
how many noisy encoded symbols should be observed before decoding is attempted. Such
information can decrease the number of decoding attempts (and error detection checks),
resulting in a substantial decrease in the overall probability of undetected error.
Finally, eliminating the need for CRC overhead has been investigated previously. How96
ever, with the proliferation of soft reliability based iterative decoders (such as the turbo
decoder), useful methods of leveraging the input and output soft information utilized by
these decoders remains to be fully exploited for rateless codes. An example of recent efforts
in this direction can be found in [11].
97
98
Appendix A
Layered System Architectures
One aspect of this work which distinguishes it from previous works is that we analyze performance and reliability from the perspective of the transport layer of a communication
system. In communication system design, it is common practice to partition the system into
a layered conceptual model. Well known examples include the Open Systems Interconnection model (OSI), and the Transmission Control Protocol/Internet Protocol suit (TCP/IP).
For the purpose of our analysis, we define a simple abstraction of these common layered
architectures which contains only the layers important to the work in this thesis. Though
not as elaborate as either of the standard layered system architectures mentioned above, our
model can easily be used as a proxy by designers who wish to study system performance and
reliability tradeoffs when designing the medium access control (MAC) and physical (PHY)
layers of a communication system. The three layer model which we employ is illustrated in
Figure A-1.
The communication link is modeled as consisting of one sender, and one receiver. At
the sender, it is assumed that information, in the form of a bit stream, is sourced from
the application layer, then passed to the transport layer where a transport layer packet, or
more succinctly, a packet, is constructed. The packet is then passed to the MAC/PHY layer,
where it is segmented into distinct messages and encoded prior to being transmitted over the
99
Information
Source/Sink
Application
Layer
Packet Construction/
Retransmission
Transport
Layer
Protocol
IRa
MAC/PHY
Layer
Scheduling
&z Transmission/
Rateless Coding
Figure A-1: Three layer communication system architecture.
channel. At the receiver, the sequence channel observations are decoded at the MAC/PHY
layer.
Once successful decoding has been achieved with a high level of confidence, the
resulting message estimate is passed up to the transport layer. When all of the constituent
messages corresponding to a packet have been decoded, they are reassembled, resulting in
an estimate of the transmitted packet. The integrity of the packet estimate is then checked
for any undetected errors prior to being passed up to the receiver application layer.
The MAC portion of the MAC/PHY layer controls scheduling of sets of encoded transmit
symbols, referred to as incremental redundancy units (IRUs), while the transport layer is
responsible for scheduling packet transmissions. If the system employs a protocol which
prescribes retransmissions of erroneous packets, scheduling of such retransmissions is handled
by the transport layer.
100
Appendix B
Design Details
B.1
Layered Rateless Code Design
In this section, several details regarding the design and configuration of the particular layered
rateless codes discussed in Chapter 4.
B.1.1
Choosing ctgt
When designing the G matrix, L, M, and ctgt must be chosen prior to executing the numerical
optimizationi required by the design. As was discussed in Section 4.3, L, and M can be
chosen based on considerations regarding the maximum desired throughput, and number of
unique redundancy blocks of encoded symbols. However, a method for choosing
Etgt
is not
as apparent. For the layered codes which were simulated for this theses, in order to choose
an appropriate value of ctgt, that is, a value which maximizes performance for a given L and
M, Monte Carlo simulations were executed over a wide range of SNRs for several values of
ctgt. These simulations were carried out after L and M had been chosen. A subset of the
results are shown in Figure B-1. Clearly, throughput efficiency is best for Etgt
=
1
X 10-3
when then entire SNR range is taken into account.
'For this thesis, this optimization is implemented using the Matlab script layeropt.m, by Mitchell D.
Trott
101
Layered Code Throughput Efficiency: Various
Etgt
0.9-
0.85 -
0.81-
0.75[0.7-....
.........
..........
..
......
0.65 t
Egt
0.61-
E
_Etgt
0.55
0
10
5
=Ee-5
=1
-
20
15
SNR (dB)
Figure B-1: Throughput efficiency for M = L = 7 layered codes vs SNR. Various choices of
ctgt are shown.
B.1.2
Mean Squared Error Estimate of Effective SNR
This section illustrates the validity of using the computed mean squared error as an estimate
of the effective SNR under the successive cancelation decoder. For this example, a Monte
Carlo simulation of the L = M = 7 layered codes were executed, and the effective SNR after
UMMSE combining, taking into account residual interference from all undecoded layers, was
measured. That is, a set of measurements of the SNR that the turbo decoder observes were
taken. Figure B-2 illustrates the results of these measurements for -Ydb
=
1 dB when decoding
layer 7, averaged over all trials. Also plotted are the MSE based estimates. All results are
plotted versus the decoding attempt index.
102
Estimated And True Effective SNR: Channel SNR = 1 dB
-2
-2.5|
-3
M -3.51
-.
.. ....... ........ .....
.........
....~
U,)
-4
-4.5
-
--
MSE Based Estimate
--- Avg. Measured Effective
SNR: Layer 7
-5
10
20
30
40
50
60
Decoding Attempt #
Figure B-2: MSE and measured effective SNR when attempting to decoder layer 7.
From observing Figure B-2, it is clear that the MSE based estimate provides a very good
estimate of the average observed effective SNR.
B.2
Complexity of Practical Rateless Codes
In this analysis, complexity is characterized by counting the number of real operations that
must be executed by a receiver employing each respective code. The resulting parameterized
expressions are then used, along with data obtained by Monte Carlo simulations of each
code, to estimate the expected number receiver operations per goodbit for each code over a
range of operating SNRs. In our consideration, processing required to produce channel and
noise parameter estimates is not taken into account because such estimates are required by
103
each rateless code of interest in this thesis.
B.2.1
Layered Rateless Codes
Expressions for the complexity of each unique processing step for the layered rateless coding
receiver are provided in this section. When using the layered rateless code (and, more generally, any rateless code), several decoding attempts may be necessary to successfully decode all
layers, where each attempt is carried out using an increasing number of received redundancy
blocks. In the expressions given in Tables B.1 and B.2, mrb < M is the number of received
redundancy blocks 2, L is the number of layers, and N is the length of each redundancy
block, in channel uses'. For the Turbo base code, u is the constraint length of the constituent convolutional encoders (i.e. each encoder has 2' possible states), 1 is the code rate
P
at the output of the Turbo encoder, k is the number of information bits per Turbo codeword
(i.e. k is the number of information bits per layer), and I is the number of decoder iterations.
The expressions given in Tables B. 1 and B.2 correspond to the complexity of a single decoding attempt; that is, they do not take into account the cumulative complexity of decoding
a particular message. However, the numerical examples which follow illustrate cumulative
complexity based on simulated performance of the layered codes, as that is of most practical
value in an analysis such as ours.
Once a set of redundancy blocks is received they must be combined so that the successive
cancellation decoder can begin attempting to decode each of the L layers. MMSE combining
of the available redundancy blocks is used. The vector of MMSE combining weights, v, are
derived for the AWGN channel of interest as the solution to:
2
3
Note that in our implementation of layered codes, mb can be a non-integer.
Equivalently, N is the length of each base codeword.
104
(B.1)
argminE [HvIy - ciHIl
V
where the input output relationship for the channel is given as:
(B.2)
y= Gc+w
where y is the
mb
x N received vector, G is a
mb
x L matrix of layer combining weights,
c is a L x N matrix of L length N base codewords, and w is a
mb
x N matrix of iid P1(0, a2 )
noise.
The solution to Eq. B.1 is:
(B.3)
V = R-Rc,
The received auto-covariance matrix, and received cross-covariance matrix above can be
shown to be:
Ry = GmlGil
+ Rzz
(B.5)
Rcy = G:,,
where Gmbl is the upper left mb x 1 of G,
t denotes
(B.4)
the conjugate transpose of a matrix
or vector, R,, is the noise covariance matrix, and G:,i is the 1th column of Gm'bl.
To compute the MMSE combining weights v for a given layer 1, and a given number of
received redundancy blocks mb the receiver proceeds as follows:
1. Compute (B.4):
(a) Compute m, x 1 complex matrix multiply GmrblGQb: Requires m2bl complex
multiplications, and mr(l - 1) complex additions and must be computed for each
105
Table B.1: Real Arithmetic Operations Required To Compute And Apply UMMSE Combining Weights For Layered Rateless Code.
Computation
MMSE Weights
Mults.
yy
~
R14L(Y
* Rey
-_
4mb2) + 7mrbL(L + 1) + mrb
Lb
L(2+1 (4m2b)
R4L + M
"V
Adds.
b)
0
4Lm 2rb
Apply MMSE
e Apply v+y
)
4L(!2+mb-L
4 Lmrb
= R-yy1 Rccy
M
+ M2
b~
2 Lmrb(mrb
4LmrbN
-
1)
2L(mrb - 1)N
1 - [1, L] for a total of m 2 L(L+1) complex multiplications and m 2 (L-1)L complex
additions.
(b) Add RZZ
=
.2 mrbXmrb to previous result: Requires mb real additions, and must
be computed 1 time for fixed mb.
(c) Invert (B.4): Requires af + m2
-
M
complex multiplications and same number
of complex additions and must be computed L times.
2. Compute (B.5)
(a) Requires mrb complex multiplications, and must be computed for each of the L
decoded layers.
3. Compute (B.3)
(a) This is a complex mrb X mrb matrix times mrb x 1 vector multiply which requires
m2b
complex multiplies and mrb(mrb
-
1) complex additions, and must be
computed for each layer.
The total number of real arithmetic operations required for the MMSE portion of the
receiver based on layered rateless codes is summarized in Table B.1.
106
After MMSE combining is complete, log-likelihood ratios for the even and odd indexed
bits are computed as:
LLRodd(y)
-2V_9ie(v~
LLReven(y) =-2vf1m(v~
y)
(B.6)
y)
(B.7)
For the complexity listed below, QPSK mapping of the coded bits to signals is assumed.
The Turbo decoder is assumed to utilize the log-MAP algorithm, which has complexity that
was analyzed in [21]. Note that after successful decoding of each of the first L - 1 layers, the
k information bits must be re-encoded, which involves interleaving of the k information bits
for 1 of the 2 constituent convolutional encoders which make up the Turbo encoder. Because
the complexity of the Turbo encoder for successive cancellation consists only of interleaving
and a number of xor and shift register operations which is linear in k, only table lookups for
the interleaver are counted in the estimate of complexity. It is also assumed that the factor
-2v
is cached, and therefore reading it from memory is negligible.
1. Compute (B.6) and (B.7): Requires 2 real multiplications per complex received symbol
for each of the L length-N layers.
2. Log-MAP decoding: Required arithmetic complexity is reported in [21] and listed in
Table B.2 for convenience. Decoding is carried out for each of the L layers.
3. Re-encoding: Requires k real table look-ups of the interleaving indices, and must be
carried out for each of the first L - 1 layers.
4. Mapping the re-encoded layers to QPSK requires N complex (2N real) table look-ups
(1 complex table look-up for each pair of encoded bits). Must be carried out for each
of the first L - 1 decoded layers.
107
Table B.2: Real Operations Required By The Turbo Decoder To Decode The Layered Rateless Code.
Computation
LLR
* QPSK
Turbo Decoding
* Log-MAP
Cancellation
" Re-Encoding
Mults.
Adds.
Table LUs+Max
2LN
0
0
2IdkL(2u(2p + 4))
2IdkL(2u(2p + 14) - 5)
4IdkL(2u+ 2 _ 1)
4mbN(L - 1)
0
(k + 2N)(L - 1)
* Cancellation
0
2mb(L - 1)N
0
5. Multiply G:, by the re-encoded and re-mapped length N complex codeword: This
requires mbN complex multiplications for each of the first L - 1 layers.
6. Cancellation: Requires N complex additions (subtractions) per received redundancy
block.
The total number of real arithmetic and logical operations (in this case, table look ups
and max( evaluations) for the Turbo decoder are summarized in Table B.2.
B.2.2
Spinal Codes
In this section we provide expressions for the computational complexity of each unique processing step for a receiver which employs spinal codes. To begin, we explain the operation of
the spinal decoder and define notation. The spinal decoder is best viewed as an approximate
maximum likelihood decoder; i.e. it aims to solve the optimization:
M = argmin||y -
x
(M)
112
(B.8)
MEA
where x (M) is the codeword corresponding to message M, y is the received sequence, A is
the set of all possible messages, and M is the result of decoding. In [7] the authors explain
that the spinal decoder breaks the distance in Equation (B.8) into a sum over spine values
(and decoder passes):
108
k/k'
sp
(B.9)
(M) 12
-i,j
_y~
[i'
11
11Y - X (M)|2
i=1 j=1
where xi,j (M) is the encoded symbol value for spine value i on decoder pass j for a particular
hypothesis, k is the length of M in bits, k' is the number of bits (out of the k bits in M)
that are used to generate each xi,j (M), yi,j is the noisy received version of x2 ,, (M), and sp
is the number of passes over the spine which have been transmitted at the time of decoding.
The receiver decoding complexity in the expressions to follow will also be a function of the
decoder depth d, and beamwidth B (See [7]). Equation (B.9) must be computed for each
decoding attempt.
In what follows, we let x = x (M), and xi,j = xi,j (M), for convenience. To compute
(B.9), the encoder must be replayed at each symbol. This can be accomplished as follows:
1. Compute (B.9):
(a) Encode hypotheses, x: Requires
(
-
d) 2 dk' hash evaluations, and
(k
- d) 2 dk's
random number generator evaluations to generate all possible encoded symbols
for each of B branches in the beam, assuming sp passes.
(b) Compute subtractions (yi,j - xij): Requires
(
- d) 2 dk's, complex subtractions
for each of B branches in the beam, assuming s, passes.
(c) Compute products
1.12:
Requires (
- d) 2dn+1s, real multiplications, and
(
-
d)
real additions for each of B branches in the beam, assuming sp passes.
(d) Compute summations E Z, (-): Requires
(-
- d)
2 dk'(sp -
1) real additions for
each of B branches in the beam assuming, sp passes.
2. Prune decoding tree:
(a) Prune decoding tree to beamwidth B: Requires
109
(
-
d) B2k' comparisons.
2 dk's
Table B.3: Real Operations Required By a Spinal Code Receiver.
Computation
Distance Metric
Mults.
Adds.
Hash+RNG +Compares
0x
0
0
(-d)
*(yi
- xij)
0
* 1-|2
d) 2dk'+1Bsp
-
-
d) 2dn+1Bsp
0
(-L - d) 2dk'Bsp
0
0
* Z(-L)
0
(
Decode
* Prune Tree
0
0
-d) 2dk'B(s - 1)
2k'B(sp + 1)
-
d) B2k'
The total number of real arithmetic operations, as well other significant computing operations are summarized in Table B.3.
110
B.2.3
Simulation Aided Complexity Analysis
In this section we present graphical results based on the expressions presented in Sections
B.2.1, and B.2.2. All results in this section were generated by averaging over Monte Carlo
simulation results for each of the rateless codes of interest.
The Monte Carlo data was
substituted into the expressions given in the previous sections in this appendix, and average
receiver complexity was computed.
Prior presenting the results, we disclose the values of relevant constant parameters for
each code. The layered code is configured with L=7 layers, and up to M=7 redundancy blocks
and uses the rate 1/3, k
=
6144 information bit LTE turbo code with QPSK constellation
as the base code for each layer. The receiver was configured to execute partial redundancy
block decoding [61 with nrIR
= 230. The spinal code is configured for k
bits, using k' = 4 bits to generate each spine value, beamwidth B
=
=
256 information
256, and depth d = 1.
The average number of receiver operations per goodbit for each code are plotted over a wide
range of SNRs in Figure B-3.
Note that the results in Figure B-3 are not presented so that the spinal and layered codes
in their stated configurations can be directly compared (indeed, it is not a fair comparison
in this configuration, as the information block lengths are vastly different). Instead, Figure
B-3 is presented to illustrate the method which was used to generate the results given in
Table 4.3.
111
Rateless Code Receiver Complexity
10 8
V
0
0
................:: '
: . :- - : - - : : :: : : ....I .......... .......
...........
........
..
..
..
I...
....
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....................
....................
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......................
.................. ..................
.................. . . . .. . . . . . .. . . . . . . .
. ..................................................I ............
..... ...I.......I.......
... ...........I.....
....................
I ...................
............ .. ......................
........ ................................ ............. -S
p in a l, k= 256 , 13= 25 6
7
.......
........... .......
L ay e re d , k= 6 144
........................
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............................................
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.......... ........
... ............
......................................
. . . .. . . . . . . . . . .. . . . . .. . . . .. . . . . .. . . .
4)
CL
6
0 10
CL
0
L_
4)
. . .. . . . . . . .. . . . . .. . . . . . . . . . I . . . . . ..
. . . . .. . .. . .
. . . . . . . . . . ...
. . . . . . . . . .. . .
. . . . . . . . . . . . . . . . . . . ... . .
..
. . . . . . . . . . . . . . . . 4 .. .
. .
. . . . . . . .. . . .
. .. .. . .
.
. . . . .. . . . . . .. ..
. . . . .. . . . . . .. . . . .
. . . . . . . . . . . . . . . 4 . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . .. . . . . . . . .
. .. . . . .. . . . . .. . . . . .. . .. . . . . . . . . . I
. . .. . . . . . . . . . . .
. . . . I
. .. . . . . . ..
.. .
. . . . .
. . .. . . .
10'
.. . . . .
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.. .. ... . . . . . . .
. . . .. .
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.. . .
. . . . . . . . . . . . . . . . . . . ... . . . . . . . . . .
. . .. .
. . . . ..
. . . .. . . . . . .. . . . . . ..
.. . . . .
. .. . . . . .. . . .
. . .
. . . . . . . . .
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.. . . . . . . .. .
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..
. . . .. . . .
. .
. .
. .
. .
. . . . . . . . . . . . . . . .I. . . . . . . . . . . . . . . . . .
. .. . . . . . .. . . . . . .
. . . . . . . . .. . . . . . . . . . . . . . . . . . . ... . . . . . . . . I . . . . . . . . . .. . . . . . . . .
10
. . .. . . . . . I
. .. . . . . . .. . . . . . . . ..
. . . . . . .. . . . . . . . . . * . . . . . . .
. ..
. . . . .. . . . . . .. . . .
. . . . . . . . . . . . . . . . . . ... . . 4 . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . .. .
. .. . . . . . . . . .
. . .. . . . . . . 4 . . . . . . .
. . . . . .
. . . . . . . .. . . . . . . . . . . . . . .. .
.. . . . . . ..
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. . .. . . . .
. . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . .
. . . . I .... . . . . . . . . . . . . . .
. . . . . .. . . . .. . . .
*. .. .. .. .. . .
. .. . . . . . . . . . . . .
. .. . . .
. . . . . .
. .. . . . . . .
. . . . . .. .
.
.
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:.
. .
* .
: .
' .
: .
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' . .
* .
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. .
. . . . . .. . .
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. .. . . . . .. . .. . .
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. .. . . . . . .
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. .. . . . . . . . . . . .
. . . . . . .. . . . . . . . . . . .
. . . . .
4
5
0
10
5
15
20
SNR (dB)
Figure B-3: Average receiver operations per goodbit for spinal and layered rateless codes.
112
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114
This work is sponsored in part by the Department of the Air Force under Air Force
Contract #FA8721-05-C-0002.
Opinions, interpretations, conclusions and recommendations
are those of the author and are not necessarily endorsed by the United States Government.
115