Wireless Information Transmission System Lab.
Introduction to Digital
Communications System
Institute of Communications Engineering
National Sun Yat-sen University
Recommended Books
Digital Communications / Fourth Edition (textbook)
-- John G. Proakis, McGraw Hill
Communication Systems / 4th Edition
-- Simon Haykin, John Wiley & Sons, Inc.
Digital Communications – Fundamentals and Applications /
2nd Edition
-- Bernard Sklar, Prentice Hall
Principles of Communications / Fifth Edition
-- Rodger E. Ziemer and William H. Tranter, John Wiley &
Sons, Inc.
Modern Digital and Analog Communication Systems
-- B.P. Lathi, Holt, Rinehart and Winston, Inc.
2
Example of Communications System
Local
Loop
Switch
Transmission
Equipment
Central Office
Local
Loop
Switch
Transmission
Equipment
Central Office
Local
Loop
Switch
Transmission
Equipment
Central Office
Mobile
Switching
Center
T1/E1 Facilities
regenerator
Base
Station
A/D Conversion
(Digitization)
T1/E1 Facilities
regenerator
SONET
SDH
M
U
X
T1/E1 Facilities
A/D Conversion
(Digitization)
T1/E1 Facilities
regenerator
Mobile
Switching
Center
A/D Conversion
(Digitization)
Public Switched Telephone Network (PSTN)
3
Base
Station
Basic Digital Communication Nomenclature
Textual Message: information comprised of a sequence of
characters.
Binary Digit (Bit): the fundamental information unit for all
digital systems.
Symbol (mi where i=1,2,…M): for transmission of the bit
stream; groups of k bits are combined to form new symbol
from a finite set of M such symbols; M=2k.
Digital Waveform: voltage or current waveform representing
a digital symbol.
Data Rate: Symbol transmission is associated with a symbol
duration T. Data rate R=k/T [bps].
Baud Rate: number of symbols transmitted per second [baud].
4
Nomenclature Examples
5
Messages, Characters, and Symbols
6
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Multiplexing
Interleaving
Modulation
Frequency
Spreading
Multiple
Access
TX
RF
PA
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Demultiplexing
Deinterleaving
Channel Bits
Optional
Essential
To Other Destinations
7
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Wireless Information Transmission System Lab.
Format
Institute of Communications Engineering
National Sun Yat-sen University
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Multiplexing
Interleaving
Modulation
Frequency
Spreading
Multiple
Access
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
TX
RF
PA
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Demultiplexing
Deinterleaving
Channel Bits
Optional
Essential
To Other Destinations
9
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Formatting and Baseband Transmission
10
Sampling Theorem
11
Sampling Theorem
Sampling Theorem: A bandlimited signal having no
spectral components above fm hertz can be determined
uniquely by values sampled at uniform intervals of Ts
seconds, where
1
TS ≤
or sampling rate f S ≥ 2 f m
2 fm
In sample-and-hold operation, a switch and storage
mechanism form a sequence of samples of the
continuous input waveform. The output of the sampling
process is called pulse amplitude modulation (PAM).
12
Sampling Theorem
1
X S ( f ) = X ( f ) ∗ Xδ ( f ) =
TS
13
∞
∑ X ( f − nf
n = −∞
S
)
Spectra for Various Sampling Rates
14
Natural Sampling
15
Pulse Code Modulation (PCM)
PCM is the name given to the class of baseband
signals obtained from the quantized PAM signals by
encoding each quantized sample into a digital word.
The source information is sampled and quantized to
one of L levels; then each quantized sample is digitally
encoded into an ℓ-bit (ℓ=log2L) codeword.
16
Example of Constructing PCM Sequence
17
Uniform and Non-uniform Quantization
18
Statistical Distribution of Single-Talker
Speech Amplitudes
50% of the time, speech voltage is less than ¼ RMS.
Only 15% of the time, voltage exceeds RMS.
Typical voice signal dynamic range is 40 dB.
19
Problems with Linear Quantization
Fact: Unacceptable S/N for small signals.
Solution:
Increasing quantization levels – price is too high.
Applying nonlinear quantization – achieved by first
distorting the original signal with a logarithmic
compression characteristic and then using a uniform
quantizer.
At the receiver, an inverse compression characteristic,
called expansion, is applied so that the overall
transmission is not distorted. The processing pair is
referred to as companding.
20
Implementation of Non-linear Quantizer
21
Companding Characteristics
In North America: μ-law compression:
loge [1 + µ ( x / xmax )]
⋅ sgn x
y = ymax
loge (1 + µ )
where
⎧+ 1 for x ≥ 0
sgn x = ⎨
⎩−1 for x < 0
In Europe: A-law compression:
⎧
A( x / x max )
⋅ sgn x
⎪ y max
1 + log e A
⎪
y=⎨
⎪ y 1 + log e [ A( x / x max )] ⋅ sgn x
⎪⎩ max
1 + log e A
22
0<
x
1
≤
A
x max
x
1
<
≤1
A x max
Compression Characteristics
Standard values of μ is 255 and A is 87.6.
23
Wireless Information Transmission System Lab.
Source Coding
Institute of Communications Engineering
National Sun Yat-sen University
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Multiplexing
Interleaving
Modulation
Frequency
Spreading
Multiple
Access
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
TX
RF
PA
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Deinterleaving
Demultiplexing
Channel Bits
Optional
Essential
To Other Destinations
25
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Source Coding
Source coding deals with the task of forming efficient
descriptions of information sources.
For discrete sources, the ability to form reduced data
rate descriptions is related to the information content
and the statistical correlation among the source
symbols.
For analog sources, the ability to form reduced data
rate descriptions, subject to a fixed fidelity criterion I
related to the amplitude distribution and the temporal
correlation of the source waveforms.
26
Huffman Coding
The Huffman code is source code whose average word
length approaches the fundamental limit set by the
entropy of a discrete memoryless source.
The Huffman code is optimum in the sense that no other
uniquely decodable set of code-words has smaller
average code-word length for a given discrete
memoryless source.
27
Huffman Encoding Algorithm
1. The source symbols are listed in order of decreasing
probability. The two source symbols of lowest
probability are assigned a 0 and a 1.
2. These two source symbols are regarded as being
combined into a new source symbol with probability
equal to the sum of the two original probabilities. The
probability of the new symbol is placed in the list in
accordance with its value.
3. The procedure is repeated until we are left with a final
list of source statistics of only two for which a 0 and a 1
are assigned.
4. The code for each (original) source symbol is found by
working backward and tracing the sequence of 0s and 1s
assigned to that symbol as well as its successors.
28
Example of Huffman Coding
Symbol
S0
S1
S2
S3
S4
Symbol
S0
S1
S2
S3
S4
Probability
0.4
0.2
0.2
0.1
0.1
Code Word
00
10
11
010
011
Stage 1
Stage 2
Stage 3
Stage 4
0.4
0.2
0.2
0.1 0
0.1
1
0.4
0.2
0.2 0
0.2 1
0.4
0.4 0
0.2
1
0.6 0
0.4
1
29
Properties of Huffman Code
Huffman encoding process is not unique.
Code words for different Huffman encoding process
can have different lengths. However, the average
code-word length is the same.
When a combined symbol is moved as high as
possible, the resulting Huffman code has a
significantly smaller variance than when it is moved
as low as possible.
Huffman code is a prefix code.
A prefix code is defined as a code in which no code-word
is the prefix of any other code-word.
30
Bit Compression Technologies for Voice
Differential PCM (DPCM)
Adaptive DPCM
Delta Modulation (DM)
Adaptive DM (ADM)
.
.
.
Speech Encoding
31
Differential PCM (DPCM)
32
Delta Modulation (DM)
Delta modulation is a one-bit DPCM.
Advantage: bit compression.
Disadvantage: slope overload.
33
Speech Coding Objective
Reduce the number of bits needed to be transmitted,
therefore lowering the bandwidth required.
34
Speech Properties
Voiced Sound
Arises in generation of vowels and latter portion of some consonants.
Displays long-term repetitive pattern corresponding to the duration of a
pitch interval
Pulse-like waveform.
Unvoiced Sound
Arises in pronunciation of certain consonants such as “s”, “f”, “p”, “j”,
“x”, …, etc.
Noise-like waveform.
35
Categories of Speech Encoding
Waveform Encoding
Treats voice as analog signal and does not use properties of
speech:
Source Model Coding or Vocoding
Treats properties of speech to preserve word information
Hybrid or parametric methods
Combines waveform and vocoding
36
Linear Predictive Coder (LPC)
37
Multi-Pulse Linear Predictive Coder
(MP-LPC)
38
Regular Pulse Excited Long Term Prediction
Coder (RPE-LPT)
39
Code-Excited Linear Predictive (CELP)
40
Speech Coder Complexity
41
Speech Processing for GSM
Composition of the 13 kbps signal:
36 bits for filter parameters every 20 ms.
9 bits for LTP every 5 ms.
47 bits for RPE every 5 ms.
Thus, in a 20 ms (2080-bit block, or 260 sample) interval,
we need a total of
36+9*20/5+47*20/5=260 bits.
Data Rate = 260/(20 ms) = 13 kbps.
42
Speech Processing for IS-54
Composition of the 7.95 kbps signal:
43 bits for filter parameters every 20 ms.
7 bits for LTP every 5 ms.
88 bits for codebook every 20 ms.
Thus, in a 20 ms (2080-bit block, or 260 samples) interval, we
need a total of:
43+7*20/5+88=159 bits.
Data Rate = 159/(20ms) = 7.95 kbps.
43
Wireless Information Transmission System Lab.
Channel Coding
Institute of Communications Engineering
National Sun Yat-sen University
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Multiplexing
Interleaving
Modulation
Frequency
Spreading
Multiple
Access
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
TX
RF
PA
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Deinterleaving
Demultiplexing
Channel Bits
Optional
Essential
To Other Destinations
45
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Channel Coding
Error detecting coding: Capability of detecting errors so
that re-transmission or dropping can be done.
Cyclic Redundancy Code (CRC)
Error Correcting Coding: Capability of detecting and
correcting errors.
Block Codes: BCH codes, RS codes, … etc.
Convolutional codes.
Turbo codes.
46
Linear Block Codes
Encoder transforms block of k successive binary digits
into longer block of n (n>k) binary digits.
Called an (n,k) code.
Redundancy = n-k; Code Rate = k/n;
There are 2k possible messages.
There are 2k possible code words corresponding to the
messages.
Code word (or code vector) is an n-tuple from the space
Vn of all n-tuple.
Storing the 2k code vector in a dictionary is prohibitive
for large k.
47
Vector Spaces
The set of all binary n-tuples, Vn, is called a vector
space over GF (2).
GF: Galois Field.
Two operations are defined:
Addition: V + U = V1 + U 1 + V2 + U 2 + ... + Vn + U n
Scalar Multiplication: aV = aV1 + aV2 + ... + aVn
Example: Vector Space V4
0000 0001 0010 0011 0100 0101 0110 0111
1000 1001 1010 1011 1100 1101 1110 1111
(0101)+(1110)=(0+1, 1+1, 0+1, 1+0)=(1, 0, 1, 1)
1·(1010)=(1·1, 1·0, 1·1, 1·0)=(1, 0, 1, 0)
48
Subspaces
A subset S of Vn is a subspace if
The all-zero vector is in S
The sum of any two vectors in S is also in S.
Example of S: V 0 = 0000
V 1 = 0101
V 2 = 1010
V 3 = 1111
49
Reducing Encoding Complexity
Key feature of linear block codes: the 2k code vectors
form a k-dimensional subspace of all n-tuples.
Example: k = 3, 2k = 8, n = 6, ( 6 , 3 ) code
Message
000
100
010
110
001
101
011
111
Code Word
000000
110100
011010
101110
101001
011101
110011
000111
50
⎫
⎪
⎪
⎪
⎪
⎪ A 3 - dimensiona l subspace of
⎬
⎪ the vector space of all 6 - tuples.
⎪
⎪
⎪
⎪
⎭
Reducing Encoding Complexity
It is possible to find a set of k linearly independent n tuples v1 , v 2 , ..., v k such that each n-tuple of the suspace
is a linear combination of v1 , v 2 , ..., v k .
Code word u = m1 v1 + m2 v 2 + ... + mk v k
where mi = 0 or 1
i = 1,..., k
51
Generator Matrix
v1n ⎤
⎡ v1 ⎤ ⎡ v11 v12
⎥
⎢ v ⎥ ⎢v
v
v
2n ⎥
G = ⎢ 2 ⎥ = ⎢ 21 22
= k × n Generator Matrix
⎥
⎢ ⎥ ⎢
⎥
⎢ ⎥ ⎢
v
v
v
v
⎢⎣ k ⎥⎦ ⎢⎣ k1 k 2
kn ⎥
⎦
The 2k code vectors can be described by a set of k linearly
independent code vectors.
Let m=[m1, m2, … , mk] be a message.
Code word corresponding to message m is obtained by:
⎡ v1 ⎤
⎢v ⎥
u = mG = [m1 m2
mk ] ⎢ 2 ⎥
⎢ ⎥
⎢ ⎥
⎣v k ⎦
52
Generator Matrix
Storage is greatly reduced.
The encoder needs to store the k rows of G instead of
the 2k code vectors of the code.
For example:
⎡ v1 ⎤ ⎡ 1 1 0 1 0 0 ⎤
Let G = ⎢⎢ v 2 ⎥⎥ = ⎢⎢ 0 1 1 0 1 0 ⎥⎥ and m = [1 1 0 ]
⎢⎣ v 3 ⎥⎦ ⎢⎣ 1 0 1 0 0 1 ⎥⎦
Then
⎡ v1 ⎤ = 1 ⋅ v1 + 1 ⋅ v 2 + 0 ⋅ v 3
u = [1 1 0 ] ⎢⎢ v 2 ⎥⎥ = 1 ⋅ [110100 ] + 1 ⋅ [ 011010 ] + 0 ⋅ [101001]
⎢⎣ v 3 ⎥⎦ = [1 0 1 1 1 0] Code Vector for m = [110 ]
53
Systematic Code
54
Parity Check Matrix
For each generator matrix G, there exists a parity check matrix H
such that the rows of G are orthogonal to the rows of H. (u·h=0)
⎡ h1 ⎤ ⎡ h11
⎢ h ⎥ ⎢ h
2 ⎥
21
⎢
⎢
H=
=
⎢
⎥ ⎢
⎢
⎥ ⎢
⎢⎣ h( n − k ) ⎥⎦ ⎢⎣ h( n − k )1
u = u1 , u2 ,… , un
uH T = u1hi1 + u2 hi 2 +
h12
h22
h( n − k )2
h1n ⎤
h2 n ⎥⎥
⎥
⎥
h( n − k ) n ⎥⎦
+ un hin = 0
where i = 1, 2,… , n − k
U is a code word generated by matrix G if and only if uHT=0
55
Parity Check Matrix and Syndrome
In a systematic code with G=[Pkxr Ikxk]
H=[Irxr PTrxk]
r
u
e
Received
Vector
Code
Vector
Error
Vector
=
+
Syndrome of r used for error detection and correction
s = rH T
⎧= 0
Syndrome s ⎨
⎩≠ 0
If r is a code vector
Otherwise
56
Example of Syndrome Test
⎡
⎢1 1 0
⎢
G = ⎢0 1 1
⎢1 0 1
⎢⎣
P
⎤
1 0 0⎥
⎥
0 1 0⎥
0 0 1⎥
⎥
Ik
⎦
H = [ I n − k PT ]
⎡1 0 0 1 0 1 ⎤
H = ⎢⎢0 1 0 1 1 0 ⎥⎥
⎢⎣0 0 1 0 1 1 ⎥⎦
The 6-tuple 1 0 1 1 1 0 is the code vector corresponding to the
⎡1 0 0 ⎤
message 1 1 0.
⎢0
⎢
⎢0
T
s = u ⋅ H = [1 0 1 1 1 0] • ⎢
⎢1
⎢0
⎢
⎢⎣1
0 ⎥⎥
1⎥
⎥ = [ 0 0 0]
0⎥
1⎥
⎥
0 1 ⎥⎦
1
0
1
1
Compute the syndrome for the non-code-vector 0 0 1 1 1 0
s = [ 0 0 1 1 1 0] ⋅ H T = [1 0 0]
57
Weight and Distance of Binary Vectors
Hamming Weight of a Vector:
w(v) = Number of non-zero bits in the vector.
Hamming Distance between 2 vectors:
d(u,v) = Number of bits in which they differ.
For example: u=10010110001
v=11001010101
d(u,v) = 5.
d(u,v) =w(u+v)
The Hamming Distance between 2 vectors is equal to the
Hamming Weight of their vector sum.
58
Minimum Distance of a Linear Code
The set of all code vectors of a linear code form a
subspace of the n-tuple space.
If u and v are 2 code vectors, then u+v must also be a
code vector.
Therefore, the distance d(u,v) between 2 code vectors
equals the weight of a third code vector.
d(u,v) =w(u+v)=w(w)
Thus, the minimum distance of a linear code equals
the minimum weight of its code vectors.
A code with minimum distance dmin can be shown to
correct (dmin-1)/2 erroneous bits and detect (dmin-1)
erroneous bits.
59
Example of Minimum Distance
dmin=3
60
Example of Error Correction and Detection
Capability
v
u
d min (u , v ) = 7
t max
⎢ d min − 1 ⎥
=⎢
:
Error
Correcting
Strength
⎥
2
⎣
⎦
mmax = d min − 1 : Error Detecting Strength
61
Convolutional Code Structure
1
1
2
2
k
1
K
2
k
1
2
k
k bits
+
1
+
2
+
+
n-1
Output
62
n
Convoltuional Code
Convolutional codes
k = number of bits shifted into the encoder at one time
k=1 is usually used!!
n = number of encoder output bits corresponding to the k
information bits
r = k/n = code rate
K = constraint length, encoder memory
Each encoded bit is a function of the present input bits
and their past ones.
63
Generator Sequence
u
r0
r1
v
r2
g 0(1) = 1, g1(1) = 0, g 2(1) = 1, and g 3(1) = 1 .
Generator Sequence: g(1)=(1 0 1 1)
u
r0
r1
r2
r3
v
g 0( 2 ) = 1, g1( 2 ) = 1, g 2( 2 ) = 1, g 3( 2 ) = 0, and g 4( 2 ) = 1 .
Generator Sequence: g(2)=(1 1 1 0 1)
64
Convolutional Codes
An Example – (rate=1/2 with K=2)
x1
G1(x)=1+x2
G2(x)=1+x1+x2
x2
0(00)
00
Present
Next
0
00
00
00
1
00
10
11
0
01
00
11
1
01
10
00
0
10
01
01
1
10
11
10
0
11
01
10
1
11
11
01
1(11)
0(11)
Output
0(01)
01
10
1(00)
0(10)
1(10)
11
1(01)
State Diagram
65
Trellis Diagram Representation
10
0(
11
)
01
0(0
1
)
0(0
1
0(0
1
)
)
)
0(0
1
10
01
0(00)
0)
1(0
0)
0)
1(0
1 (0
10
01
00
)
01
0(00)
00
0(
11
)
0(
11
)
0(
11
)
01
00 0(00)
)
1 (1 1
00 0(00)
)
1 (1 1
)
1 (1 1
00 0(00)
0(
11
)
0(00)
)
1 (1 1
00
0(0
1
0(00)
)
1 (1 1
10
10
)
0(10
)
0(10
)
0(10
)
10
1(
)
10
1(
)
10
1(
)
)
10
1(
0(10
00
11 1(01) 11 1(01) 11 1(01) 11
Trellis termination: K tail bits with value 0 are usually added to the end of the code.
66
00
Encoding Process
01
)
0)
0)
)
)
01
1 (0
1 (0
0)
0(0
1
0(0
1
0(00)
00
10
10
10
)
0(10
)
0(10
)
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
)
10
1(
11 1(01) 11 1(01) 11 1(01) 11
67
0
11
00
0(00)
0(
11
)
0(
11
)
0(
11
)
01
1 (0
10
00 0(00)
0(
11
)
00 0(00)
0
10
)
1 (1 1
01
10
1
01
)
1 (1 1
)
1 (1 1
00 0(00)
1
10
0(0
1
1
00
0 (0
1)
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
0
01
0(
11
)
Input: 1
Output: 11
01
00
Viterbi Decoding Algorithm
Maximum Likelihood (ML) decoding rule
received sequence r
ML
detected sequence d
min(d,r) !!
Viterbi Decoding Algorithm
An efficient search algorithm
Performing ML decoding rule.
Reducing the computational complexity.
68
Viterbi Decoding Algorithm
Basic concept
Generate the code trellis at the decoder
The decoder penetrates through the code trellis level by level in
search for the transmitted code sequence
At each level of the trellis, the decoder computes and
compares the metrics of all the partial paths entering a node
The decoder stores the partial path with the larger metric and
eliminates all the other partial paths. The stored partial path is
called the survivor.
69
Viterbi Decoding Process
1 (0
0(0
1
)
)
)
0)
0)
0)
)
01
10
10
10
)
0(10
)
0(10
)
0(10
)
0(10
)
10
1(
)
10
1(
)
10
1(
11 1(01) 11 1(01) 11 1(01) 11
70
11
11
00
0(00)
0(
11
)
01
1 (0
1 (0
0(0
1
0(
11
)
01
0(00)
00
)
10
1(
0
10
00 0(00)
0(
11
)
0(
11
)
0(
11
)
00 0(00)
10
11
)
1 (1 1
01
10
01
01
)
1 (1 1
00 0(00)
10
10
0(0
1
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(0
1
Output: 11
Receive: 11
01
00
Viterbi Decoding Process
)
)
0(0
1
0(0
1
)
0)
)
10
10
10
)
0(10
)
)
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
2
11 1(01) 11 1(01) 11 1(01) 11
1
71
11
11
00
0(00)
0(
11
)
01
1 (0
0)
0(0
1
0(00)
00
0(
11
)
01
1 (0
0)
10
)
10
1(
0
01
1 (0
10
00 0(00)
0(
11
)
0(
11
)
0(
11
)
01
1
00 0(00)
10
11
)
1 (1 1
4
01
01
)
1 (1 1
00 0(00)
10
10
0(0
1
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
01
00
Viterbi Decoding Process
0(10
)
)
10
)
10
1(
)
10
)
10
1(
)
10
1(
)
0(0
1
)
0(0
1
0(0
1
)
)
0(0
1
1
11 1(01) 11 1(01) 11 1(01) 11
1
2
72
11
11
00
0(00)
0(
11
)
01
0)
2
0(00)
00
1 (0
0)
0)
10
)
01
1 (0
1 (0
)
10
1(
10
0(10
0(
11
)
0(
11
)
0(
11
)
01
2
00 0(00)
0(
11
)
3
10
11
)
1 (1 1
4
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
0(10
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
01
00
Viterbi Decoding Process
10
11
3
3
1
3
)
)
10
1(
)
10
1(
)
10
1(
0(10
10
)
2
)
10
)
0(0
1
)
0(0
1
)
0(0
1
0(0
1
10
)
0)
)
10
1(
10
0(10
01
1 (0
2
0)
0)
)
01
1 (0
1 (0
2
0(
11
)
0(
11
)
0(
11
)
0(
11
)
01
11 1(01) 11 1(01) 11 1(01) 11
1
2
73
1
0(00)
00
11
11
00
0(00)
0(
11
)
4
)
1 (1 1
00 0(00)
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
0(10
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
01
00
Viterbi Decoding Process
10
11
4
3
3
0(
11
)
)
0(0
1
)
0(0
1
0(0
1
)
3
0)
0(0
1
01
1 (0
0)
0)
1 (0
1 (0
)
01
2
1
3
3
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
2
)
10
)
10
)
10
0(10
)
10
1(
10
11 1(01) 11 1(01) 11 1(01) 11
1
2
74
1
0(00)
00
11
11
00
0(00)
3
0(
11
)
0(
11
)
0(
11
)
01
2
)
1 (1 1
00 0(00)
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
1
0(
11
)
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
01
00
Viterbi Decoding Process
10
11
4
3
3
0(
11
)
)
0(0
1
)
0(0
1
0(0
1
)
0)
0(0
1
2
3
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
3
)
1
)
2
)
10
0(10
)
10
1(
10
11 1(01) 11 1(01) 11 1(01) 11
75
1
1
0(00)
3
3
10
2
00
01
10
1
11
11
01
1 (0
0)
0)
1 (0
1 (0
)
01
2
0(00)
00
3
0(
11
)
0(
11
)
0(
11
)
01
2
)
1 (1 1
00 0(00)
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
0(
11
)
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
00
Viterbi Decoding Process
10
11
4
3
3
0(
11
)
)
0(0
1
)
0(0
1
0(0
1
)
0)
0(0
1
2
3
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
3
)
1
)
2
)
10
0(10
)
10
1(
10
11 1(01) 11 1(01) 11 1(01) 11
76
1
1
0(00)
3
3
10
2
00
01
10
1
11
11
01
1 (0
0)
0)
1 (0
1 (0
)
01
2
0(00)
00
3
0(
11
)
0(
11
)
0(
11
)
01
2
)
1 (1 1
00 0(00)
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
0(
11
)
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Output: 11
Receive: 11
00
2
Viterbi Decoding Process
10
11
4
3
3
0(
11
)
)
0(0
1
)
0(0
1
0(0
1
)
0)
0(0
1
2
3
0(10
0(10
)
10
1(
)
10
1(
)
10
1(
)
3
)
1
)
2
)
10
0(10
)
10
1(
10
11 1(01) 11 1(01) 11 1(01) 11
77
1
1
0(00)
3
3
10
2
00
01
10
1
11
11
01
1 (0
0)
0)
1 (0
1 (0
)
01
2
0(00)
00
3
0(
11
)
0(
11
)
0(
11
)
01
2
)
1 (1 1
00 0(00)
)
1 (1 1
00 0(00)
1
0
01
01
00 0(00)
01
10
10
10
0(
11
)
00
00
)
1 (1 1
2
0(00)
)
1 (1 1
00
0 (0
1)
0(00)
)
1 (1 1
00
01
11
0(10
Decision:11
Receive: 11
00
2
Channel Coding in GSM
78
Channel Coding in IS-54/136
79
Turbo Codes Basic Concepts
Turbo coding uses parallel concatenation of two
recursive systematic convolutional codes joined through
an interleaver.
Information bits are encoded block by block.
Turbo codes uses iterative decoding techniques.
Soft-output decoder is necessary for iterative decoding.
Turbo codes can approach to Shannon limit.
80
Turbo Codes Encoder - An Example
X(t)
Y(t)
X(t)
Interleaver
Y’(t)
X'(t)
When the switch is placed on the low position, the tail bits are feedback
and the trellis will be terminated.
81
Turbo Codes Encoding Example
A systematic convolutional encoder with memory 2
The dotted line is for termination code
Test sequence: 1011
X0
X1
1101
D
D
82
Turbo Codes Encoding Example
X0=1
X1=1
1101
0
0
00
01
11
10
11
83
Turbo Codes Encoding Example
X0=0
X1=1
110
1
0
00
01
10
11
01
11
84
Turbo Codes Encoding Example
X0=1
X1=0
11
1
1
00
01
10
11
11
01
10
85
Turbo Codes Encoding Example
X0=1
X1=0
1
1
1
00
01
10
11
11
01
10
10
86
Turbo Codes Encoding Example
X0=0
X1=1
1
1
00
01
10
11
11
01
01
10
10
87
Turbo Codes Encoding Example
X0=1
X1=1
0
1
00
01
10
11
11
11
01
01
10
10
88
Turbo Codes Encoding Example
X0=0
X1=0
0
0
00
01
10
11
11
11
01
01
10
10
89
00
Turbo Codes Encoding Example
X0
1101
X1
D
D
Interleaver
1011
(X0)
X2
D
D
Output sequence: X0, X1, X2, X0, X1, X2, X0, X1, X2,...
90
Turbo Codes Encoding Example
The second encoder input is the interleaved
data
1 0
1101
1011
1 1
00
00
01
11
11
10
00
10
10
11
91
00
CRC in WCDMA
gCRC24(D) = D 24 + D 23 + D 6 + D 5 + D + 1;
gCRC16(D) = D 16 + D 12 + D 5 + 1;
gCRC12(D) = D 12 + D 11 + D 3 + D 2 + D + 1;
gCRC8(D) = D 8 + D 7 + D 4 + D 3 + D + 1.
92
Channel Coding Adopted in WCDMA
Type of TrCH
Coding scheme
Coding rate
Convolutional
coding
1/2
BCH
PCH
RACH
1/3, 1/2
CPCH, DCH, DSCH,
FACH
Turbo coding
No coding
93
1/3
Convolutional Coding in WCDMA
Input
D
D
D
D
D
D
D
D
Output 0
G0 = 561 (octal)
Output 1
G1 = 753 (octal)
(a) Rate 1/2 convolutional coder
Input
D
D
D
D
D
D
D
D
Output 0
G0 = 557 (octal)
Output 1
G1 = 663 (octal)
Output 2
G2 = 711 (octal)
(b) Rate 1/3 convolutional coder
94
Turbo Coder in WCDMA
xk
1st constituent encoder
zk
xk
Input
D
D
D
Output
Input
Turbo code
internal interleaver
Output
x’k
2nd constituent encoder
D
D
z’k
D
x’k
95
Wireless Information Transmission System Lab.
Interleaving
Institute of Communications Engineering
National Sun Yat-sen University
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Multiplexing
Interleaving
Modulation
Frequency
Spreading
Multiple
Access
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
TX
RF
PA
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Deinterleaving
Demultiplexing
Channel Bits
Optional
Essential
To Other Destinations
97
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Bursty Error in Fading Channel
98
Interleaving Mechanism (1/2)
x
Bit
Interleaver
y
y
x
j x n-bit
Shift registers
Write Clock
Read Clock
Bit Stream before entering bit interleaver:
x=(a11 a12 … a1n a21 a22 … a2n … aj1 aj2 … ajn)
99
Interleaving Mechanism (2/2)
Conceptually, the WRITE clock places the bit stream
x by the row while the REA clock takes the bit stream
y by the column:
⎡ a11
⎢a
⎢ 21
⎢ .
⎢
⎢ .
⎢ .
⎢
⎢⎣ a j1
a12
a 22
.
.
.
a j2
.
.
.
.
.
.
.
.
.
.
.
.
. a1n ⎤
. a 2 n ⎥⎥
.
. ⎥
⎥
.
. ⎥
.
. ⎥
⎥
. a jn ⎥⎦
Bit stream at the output of the bit interleaver:
y = (a11
a21 ... a j1
a12
a22 ... a j 2 ... a1n
100
a2 n ... a jn )
Burst Error Protection with Interleaver
101
Wireless Information Transmission System Lab.
Modulation
Institute of Communications Engineering
National Sun Yat-sen University
Typical Digital Communications System
From Other Sources
Information Bits
Source Bits
Source
Encoding
Format
Encryption
Channel Bits
Channel
Encoding
Interleaving
Multiplexing
Modulation
Frequency
Spreading
Multiple
Access
si (t )
Digital
Input
C
H
A
N
N
E
L
mi
Bit
Stream
Synchronization
Digital
Waveform
Digital
Output
m̂ i
Format
TX
RF
PA
sˆi (t )
Source
Decoding
Information Sink
Decryption
Source Bits
Channel
Decoding
Deinterleaving
Demultiplexing
Channel Bits
Optional
Essential
To Other Destinations
103
Demodulation
Frequency
Despreading
Multiple
Access
RX
RF
IF
Modulation
Digital Modulation: digital symbols are transformed into
waveforms that are compatible with the characteristics of the
channel.
In baseband modulation, these waveforms are pulses.
In bandpass modulation, the desired information signal
modulates a sinusoid called a carrier. For radio transmission,
the carrier is converted in an electromagnetic (EM) wave.
Why modulation?
Antenna size should be comparable with wave length –
baseband transmission is not possible.
Modulation may be used to separate the different signals
using a single channel.
104
PCM Waveform Representations
105
PCM Waveform Representations
PCM waveform is also called line codes.
Digital baseband signals often use line codes to provide
particular spectral characteristics of a pulse train.
NRZ-L.
NRZ-M.
NRZ-S.
Unipolar-RZ.
Polar-RZ.
Bi-φ-L.
Bi-φ-M.
Bi-φ-S.
Dicode-NRZ.
Dicode-RZ.
Delay Mode.
4B3T.
Multi-level.
… etc.
106
PCM Waveform : NRZ-L
1
0
1
1
0
0
0
1
1
0
+E
0
-E
NRZ Level (or NRZ Change)
“One” is represented by one level.
“Zero” is represented by the other level.
107
1
PCM Waveform : NRZ-M
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
NRZ Mark (Differential Encoding)
“One” is represented by a change in level.
“Zero” is represented by a no change in level.
108
PCM Waveform : NRZ-S
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
NRZ Space (Differential Encoding)
“One” is represented by a no change in level.
“Zero” is represented by a change in level.
109
PCM Waveform : Unipolar-RZ
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Unipolar - RZ
“One” is represented by a half-bit width pulse.
“Zero” is represented by a no pulse condition.
110
PCM Waveform : Polar-RZ
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Polar - RZ
“One” and “Zero” are represented by opposite
level polar pulses that are one half-bit in width.
111
PCM Waveform : Bi-φ-L
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Bi-φ-L (Biphase Level or Split Phase Manchester
11 + 180o)
“One” is represented by a 10.
“Zero” is represented by a 01.
112
PCM Waveform : Bi-φ-M
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Bi-φ-M ( Biphase Mark or Manchester 1)
A transition occurs at the beginning of every bit period.
“One” is represented by a second transition one half bit
period later.
“Zero” is represented by no second transition.
113
PCM Waveform : Bi-φ-S
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Bi-φ-S ( Biphase Space)
A transition occurs at the beginning of every bit period.
“One” is represented by no second transition.
“Zero” is represented by a second transition one-half bit
period later.
114
PCM Waveform : Dicode - NRZ
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Dicode Non-Return-to-Zero
A “One” to “Zero” or “Zero” to “One” changes polarity.
Otherwise, a “Zero” is sent.
115
PCM Waveform : Dicode - RZ
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Dicode Return-to-Zero
A “One” to “Zero” or “Zero” to “One” transition produces
a half duration polarity change.
Otherwise, a “Zero” is sent.
116
PCM Waveform : Delay Mode
1
0
1
1
0
0
0
1
1
0
1
+E
0
-E
Dicode Non-Return-to-Zero
A “One” is represented by a transition at the midpoint of
the bit interval.
A “Zero” is represented by a no transition unless it is
followed by another zero. In this case, a transition is
placed at the end of bit period of the first zero.
117
PCM Waveform : 4B3T
O --
118
PCM Waveform : 4B3T
Ternary words in the middle column are balanced in
their DC content.
Code words from the first and third columns are selected
alternately to maintain DC balance.
If more positive pulses than negative pulses have been
transmitted, column 1 is selected.
Notice that the all-zeros code word is not used.
119
PCM Waveform : Multilevel Transmission
Multilevel transmission with 3 bits per signal interval.
120
Criteria for Selecting PCM Waveform
DC component: eliminating the dc energy from the
signal’s power spectrum.
Self-Clocking: Symbol or bit synchronization is
required for any digital communication system.
Error detection: some schemes provide error detection
without introducing additional error-detection bits.
Bandwidth compression: some schemes increase
bandwidth utilization by allowing a reduction in
required bandwidth for a given data rate.
Noise immunity.
Cost and complexity.
121
Spectral Densities of Various PCM Waveforms
122
Linear Modulation Techniques
Digital modulation techniques may be broadly classified as linear
and nonlinear.
In linear modulation techniques, the amplitude of the transmitted
signal, s(t), varies linearly with the modulating digital signal, m(t).
Linear modulation techniques are bandwidth efficient, though
they must be transmitted using linear RF amplifiers which have
poor power efficiency.
Using power efficient nonlinear amplifiers leads to the
regeneration of filtered sidelobes which can cause severe adjacent
channel interference, and results in the loss of all the spectral
efficiency gained by linear modulation.
Clever ways have been developed to get around these difficulties:
QPSK, OQPSK, π/4-QPSK.
123
Digital Modulations
Basic digital modulated signal:
v(t) = A(t) cos (ωt + θ)
Where A(t) = Amplitude; ω = Frequency; θ = Phase;
124
Basic Digital Modulations
125
Extended Modulated Signals – M-FSK
Example: 16-FSK
Every 4 bits is encoded as: A ⋅ cos(ω j t )
Gray Coding.
126
j = 1,2,…,16
Extended Modulated Signals – M-PSK
Example: 16-PSK
Every 4 bits is encoded as: A ⋅ sin(ω t + θ j )
Gray Coding.
Dotted lines are decision boundaries.
127
j = 1, 2,… ,16
Extended Modulated Signals – 16-QAM
Every 4 bits is represented by one point in the signal constellation.
Every point has its unique “amplitude” and “phase”.
128
Binary Phase Shift Keying (BPSK)
In BPSK, the phase of a constant amplitude carrier signal is
switched between two values according to the two possible
signals m1 and m2 corresponding to binary 1 and 0. Normally,
the two phases are separated by 180o.
2 Eb
sBPSK ( t ) = m ( t )
cos ( 2π f c t + θ c )
0 ≤ t ≤ Tb
Tb
= Re { g BPSK ( t ) exp ( j 2π f c t )}
⎛ sin π fTb ⎞
2 Eb
jθc
g BPSK ( t ) =
m ( t ) e ⇒ Pg BPSK (t ) ( f ) = 2 Eb ⎜
⎟
Tb
fT
π
b
⎝
⎠
2
2
2
⎡
Eb ⎛ sin π ( f − f c ) Tb ⎞ ⎛ sin π ( − f − f c ) Tb ⎞ ⎤
⎢⎜
⎥
+
PBPSK ( f ) =
⎟
⎜
⎟
2 ⎢⎝⎜ π ( f − f c ) Tb ⎠⎟ ⎝⎜ π ( − f − f c ) Tb ⎠⎟ ⎥
⎣
⎦
129
Power Spectral Density (PSD) of a BPSK
Signal.
130
BPSK Receiver
BPSK uses coherent or synchronous demodulation,
which requires that information about the phase and
frequency of the carrier be available at the receiver.
If a low level pilot carrier signal is transmitted along
with the BPSK signal, then the carrier phase and
frequency may be recovered at the receiver using a
phase locked loop (PLL).
If no pilot carrier is transmitted, a Costas loop or
squaring loop may be used to synthesize the carrier
phase and frequency from the received BPSK signal.
131
BPSK Receiver with Carrier Recovery
Circuits
132
Operations of BPSK Receiver with Carrier
Recovery Circuits
The received signal is squared to generate a DC signal and an
amplitude varying sinusoid at twice the carrier frequency.
The DC signal is filtered out using a bandpass filter with center
frequency tuned to 2fc.
A frequency divider is used to recreate the waveform
cos(2πfct+θ).
The output of the multiplier is applied to an integrate and dump
circuit which forms the low pass filter segment of a BPSK
detector.
If the transmitter and receiver pulse shapes are matched, then the
detection will be optimum.
A bit synchronizer is used to facilitate sampling of the integrator
output precisely at the end of each bit period.
133
Differential Phase Shift Keying (DPSK)
Differential PSK is a noncoherent form of phase shift keying
which avoids the need for a coherent reference signal at the
receiver.
d k = mk ⊕ d k −1
134
Block Diagram of DPSK Receiver
135
Quadrature Phase Shift Keying (QPSK)
136
Spectrum of QPSK Signals
⎡⎛ sin π ( f − f ) T
c
s
PQPSK ( f ) = Eb ⎢⎜⎜
⎢⎝ π ( f − f c ) T
⎣
137
2
⎞ ⎛ sin π ( − f − f c ) Ts
⎟⎟ + ⎜⎜
⎠ ⎝ π ( − f − fc ) T
⎞
⎟⎟
⎠
2
⎤
⎥
⎥
⎦
Block Diagram of a QPSK Transmitter
138
Block Diagram of a QPSK Receiver
139
Offset QPSK (OQPSK)
For QPSK, the occasional phase shift of πradians can cause the
signal envelope to pass through zero for just an instant.
The amplification of the zero-crossings brings back the filtered
sidelobes since the fidelity of the signal at small voltage levels is
lost in transmission.
To prevent the regeneration of sidelobes and spectral widening, it
is imperative that QPSK signals that use pulse shaping be
amplified only using linear amplifiers, which are less efficient.
A modified form of QPSK, called offset QPSK (OQPSK) or
staggered QPSK is less susceptible to these deleterious effects
and supports more efficient amplification.
OQPSK ensures there are fewer baseband signal transitions.
Spectrum of an OQPSK signal is identical to that of QPSK.
140
Offset QPSK (OQPSK)
The time offset waveforms that are applied to the in-phase and
quadrature arms of an OQPSK modulator. Notice that a halfsymbol offset is used.
141
π/4-DQPSK
142
Generic π/4-DQPSK Transmitter
143
π/4-DQPSK Baseband Differential
Detector
144
Detection of Binary Signals in Gaussian
Noise
145
Digital Demodulation Techniques
Coherent detection: Exact replicas of the possible arriving
signals are available at the receiver. This means that the
receiver has exact knowledge of the carrier wave’s phase
reference, in which case we say the receiver is phase-locked to
the transmitter. Coherent detection is performed by crosscorrelating the received signal with each one of the replicas,
and then making a decision based on comparisons with preselected thresholds.
Non-coherent detection: Knowledge of the carrier wave’s
phase is not required. The complexity of the receiver is
thereby reduced but at the expense of an inferior error
performance, compared to a coherent system.
146
Correlation Demodulator
147
Matched Filter Demodulator
148
Inter-Symbol Interference (ISI)
149
Inter Symbol Interference (ISI)
Inter-Symbol Interference (ISI) arises because of
imperfections in the overall frequency response of the
system. When a short pulse of duration Tb seconds is
transmitted through a band-limited system, the
frequency components constituting the input pulse
are differentially attenuated and differentially delayed
by the system. Consequently, the pulse appearing at
the output of the system is dispersed over an interval
longer than Tb seconds, thereby resulting in intersymbol interference.
Even in the absence of noise, imperfect filtering and
system bandwidth constraints lead to ISI.
150
Nyquist Channels for Zero ISI
The Nyquist channel is not physically realizable since it
dictates a rectangular bandwidth characteristic and an infinite
time delay.
Detection process would be very sensitive to small timing
errors.
Solution: Raised Cosine Filter.
151
Raised Cosine Filter
⎧1
⎪⎪ 2 π f + W − 2W0
H ( f ) = ⎨cos (
)
4 W − W0
⎪
⎪⎩ 0
1
W0 =
2T
Excess Bandwidth : W − W0
W − W0
Roll - Off Factor : r =
W0
152
for f < 2W0 − W
for 2W0 − W < f < W
for f > W
Raised Cosine Filter Characteristics
153
Raised Cosine Filter Characteristics
154
Equalization
In practical systems, the frequency response of the
channel is not known to allow for a receiver design that
will compensate for the ISI.
The filter for handling ISI at the receiver contains
various parameters that are adjusted with the channel
characteristics.
The process of correcting the channel-induced distortion
is called equalization.
155
Equalization
156
Introduction to RAKE Receiver
Multiple versions of the transmitted signal are seen at
the receiver through the propagation channels.
Very low correlation between successive chips is in
CDMA spreading codes.
If these multi-path components are delayed in time
by more than a chip duration, they appear like
uncorrelated noise at a CDMA receiver.
Combine
Coherently
Equalization is
NOT necessary
157
Introduction to RAKE Receiver
To utilize the advantages of diversity techniques,
channel parameters are necessary to be estimated.
Arrival time of each path, Amplitude, and Phase.
Maximal Ratio Combiner (MRC): The combiner that
achieves the best performance is one in which each
output is multiplied by the corresponding complexvalued (conjugate) channel gain. The effect of this
multiplication is to compensate for the phase shift in the
channel and to weight the signal by a factor that is
proportional to the signal strength.
158
Maximum Ratio Combining (MRC)
MRC: Gi=Aie-jqi
G1
G2
Coherent Combining
GL
Channel Estimation
Best Performance
Receiver
159
Maximum Ratio Combining (MRC)
L
Received Envelope:rL = ∑ Gl ⋅ rl
l =1
L
Total Noise Power: σ = ∑ Gl σ n2,l
2
n
2
l =1
L
2
L
r
=
SNR: SNRL =
2
2 ⋅σ n
∑G ⋅r
l =1
L
l
2
l
2 ⋅ ∑ Gl ⋅ σ n2,l
2
l =1
L
Since
∑G ⋅r
l =1
l
l
2
⎛ rl
= ∑ Glσ n ,l ⎜
⎜σ
l =1
⎝ n ,l
L
160
⎞
⎟⎟
⎠
2
Maximum Ratio Combining (MRC)
2
L
L
L
Chebychev's Inequality : ∑ Gl ⋅ rl ≤ ∑ Glσ n ,l ⋅ ∑
l =1
L
SNRL ≤
Gσ
∑
1
l =1
2
l
L
2
n ,l
L
⋅∑
l =1
2
l =1
l =1
rl
2
σ n ,l
2
rl
σ n ,l
2
L
rl
1
= ∑ 2 = ∑ SNRl
2 l =1 σ n ,l l =1
L
2
G
σ
∑ l n ,l
2
l =1
With equality hold : Glσ n ,l = k
rl*
σ n ,l
⇒ Output SNR = Sum of SNRs from all branches @ Gl ∝ rl*
161
Example of RAKE Receiver Structure
162
Advantages of RAKE Receiver
Consider a receiver with only one finger:
Once the output of a single correlator is corrupted by
fading, large bit error is expected.
Consider a RAKE receiver
If the output of a single correlator is corrupted by fading,
the others may NOT be.
Diversity is provided by combining the outputs
Overcome fading
Improve CDMA reception
163