A Capacity-Based Approach for
Designing Bit-Interleaved Coded GFSK with
Noncoherent Detection
Rohit Iyer Seshadri and Matthew C. Valenti
Lane Dept. of Computer Science and Electrical Engineering
West Virginia University
iyerr, mvalenti @csee.wvu.edu
Problem
“ Which is the optimal combination of channel coding rate and
continuous phase modulation (CPM) parameters for a given
bandwidth efficiency and decoder complexity?”
7/12/2006
2/24
Continuous Phase Modulation

CPM is a nonlinear modulation scheme with memory
– Modulation induces controlled inter symbol interference (ISI)

Phase continuity results in small spectral side lobes

Well suited for bandwidth constrained systems

Constant envelope makes it suitable for systems with nonlinear amplifiers

CPM is characterized by the following modulation parameters
– Modulation order M
– Type and width of the pulse shape
– Modulation index h

Different combination of these parameters result in different spectral
characteristics and signal bandwidths
7/12/2006
3/24
Challenges

CPM includes an almost infinite variations on the modulated signal
– Full response, partial response, GFSK, REC, RC etc..

CPM is nonlinear
– Problem of finding realistic performance bounds for coded CPM systems
is non-trivial

When dealing with CPM systems with bandwidth constraints, lowering
the code rate does not necessarily improve the error rate

System complexity and hence the detector complexity must be kept
feasible
7/12/2006
4/24
Uncoded CPM System
Bit
to
Symbol
u
a
Modulator
x
Channel
r’
Filter
^
r
a
Detector
Symbol
to
Bit
^
u
u: data bits
a: message stream comprised of data symbols from the set { ±1, ± 3,…, ±(M-1)}
x: modulated CPM waveform
r’: signal at the output of the channel. The filter removes out-of band noise
^
a: symbol estimates provided by the detector
^
u: bit estimates provided by the detector
7/12/2006
5/24
An Uncoded System with
Gaussian Frequency Shift Keying
u
Bit
to
Symbol
a
GFSK
x
Channel
r’
Filter
^
r
a
Detector
Symbol
to
Bit
^
u
Gaussian frequency shift keying (GFSK) is a widely used class of CPM
e.g. Bluetooth, GSM
Baseband GFSK signal during kT ≤ t ≤ (k+1)T
GFSK phase
7/12/2006
6/24
GFSK Pulse Shape and
Uncoded Power Spectrum

The pulse shape g(t) is the response of a
Gaussian filter to rectangular pulse of
width T
g (t )  [Q(cBt )  Q(cB(t  T ))]/ T 
0
BT =0.5
-5
BT =0.5, 2B T =1.04

BT is the normalized 3 dB bandwidth of
the filer
–
–

Width of the pulse shape depends on BT
Wider the pulse, greater is the ISI
Smaller values of BT result in a more
compact power spectrum
–
–
Here M =2 and h =0.5
2B99Tb quantifies the bandwidth efficiency
Power Spectral Density (dB)
99 b
-10
BT =0.25
-15
BT =0.25, 2B T =0.86
99 b
BT =0.2
-20
BT =0.2, 2B T =0.79
99 b
-25
-30
-35
-40
0
0.2
0.4
0.6
0.8
1
1.2
Frequency (normalized by T)
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7/24
Coded GFSK System
u
Encoder
b
GFSK
a
Channel
x
^
r
Filter
Detector
2.
3.
4.
PSD for GFSK using rate Rc code is now
S ( f )  Rc S x ( Rc f )
c
x
cIt is not immediately clear if the performance loss
must
meet the required
spectral
efficiency
xcaused
be lowering
h and/or
BT will
be overcome
Decoder
0
Power Spectral Density (dB)
1.
The value of BT needs to lowered, with h unchanged
Find the OR
power spectral density for uncoded GFSK S x ( f )
Both can be lowered
u
10
Suppose
weimproves
need 2B99
Tb =1.04
while using
Channel
coding
energy
efficiency
at a rate ½
code , of bandwidth efficiency
the expense
For The
our system,
belowered,
done without
value ofcoding
h needsmust
to be
with BT
bandwidth
expansion,
i.e.
2B
T
should
99 b
unchanged
remain unchanged
OR
^
a
-10
-20
M =2, BT =0.5, h =0.125, R =1/2
c
-30
M =2, BT =0.5, h =0.5, uncoded
-40
M =2, BT =0.075, h =0.5, R =1/2
c
S (f)
-50
by the coding gain
This implies the GFSK parameters have to be
modified for the coded signal
7/12/2006
0
2
4
6
8
10
Frequency (normalized by T)
8/24
Proposed Coded GFSK System
u
Encoder
b’
Bit
Intrlv.
b
x
GFSK
Channel
r'
^
r
Filter
SO-SDDPD
b
^
Bit
Deintrlv.
b’
^
Decoder
u
Noncoherent detection used to reduce complexity
Detector: Soft-Decision differential phase detector (SDDPD), [Fonseka, 2001].
Produces hard-estimates of the modulated symbols
SO-SDDPD generates bit-wise log-likelihood ratios (LLRs) for the code bits
Bit-wise interleaving between encoder and modulator and bit-wise soft-information passed
from detector to decoder (BICM)
Shannon Capacity under modulation and detector design constraints used to drive the
search for the “optimum” combination of code rates and GFSK parameters at different
spectral efficiencies
The availability of capacity-approaching turbo and LDPC codes make the capacity
under BICM a very practical indicator of system performance
7/12/2006
9/24
System Model

Bit-interleaved codeword b is mapped to symbol sequence a, which is modulated to produce x

The baseband GFSK signal x is sent through a frequency nonselective Rician channel

Received signal at the output of the channel, before filtering
r’(t, a) = c(t) x(t, a) + n’(t)
c(t ) 


Ps 

Pd  (t ) ,
Ps  Pd  1,
K 
Ps
Pd
Received signal after filtering
r(t, a) = c(t) x(t, a) + n(t)

Received signal phase
 (t, a) =  (t, a) +  (t )
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10/24
SO-SDDPD

Detector finds the phase difference between successive symbol intervals
 k  (k   (tk )  (tk  T )) mod 2

We assume that GFSK pulse shape causes adjacent symbol interference
k  (ak0  ak 11  ak 11 ) mod 2
i  h
iT T
 g (t )dt
iT

The phase difference space from 0 to 2 is divided into R sub-regions

Detector selects the sub-region Dk in which

The sequence of phase regions (D0, DI, …) is sent to a branch metric calculator
7/12/2006

k
lies
11/24
SO-SDDPD

Let ( oi , 1i ,...) be the phase differences corresponding to any transmitted sequence
( aoi , a1i ,...)
(P(Do | 0i ), P( D1 | 1i ),...)

A branch metric calculator finds the conditional probabilities

Branch metrics sent to a 4-state MAP decoder whose state transition is from

Sk 1   ak 1 , ak 
to
Sk   ak , ak 1 
The SO-SDDPD estimates the LLR for code bits
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12/24
Capacity Under Modulation, Channel
And Receiver Design Constraints

Channel capacity denotes maximum allowable data rate for reliable
communication over noisy channels
C  max I ( X ; Y )
p( x)
p ( x, y )
dxdy

p( x)
p( x) p( y )
In any practical system, the input distribution is constrained by the choice of
modulation
C  max 

p ( x, y ) log 2
– Capacity is mutual information between the bit at modulator input and LLR at
detector output
C  I ( X ;Y )

Constrained capacity in nats is; [Caire, 1998]
C  E[log(2)  log p(bi | r )]
7/12/2006
13/24
Capacity Under Modulation, Channel
And Receiver Design Constraints

Constrained capacity for the proposed system is now
C
log 2 M
 log(2)  E
a ,c , n , s s '
i 1

In bits per channel use
C  log 2 M 

7/12/2006
log 2 M

i 1
1
Ea,c,n,s s ' [log{exp(0)  exp( zi (1)bi )}]
log(2)
Constrained capacity hence influenced by
–
–
–
–

[log{exp(0)  exp( zi (1)bi )}]
Modulation parameters (M, h and BT)
Channel
Detector design
Computed using Monte-Carlo integration
The constrained capacity is used to find the minimum Eb/No required for reliable
signaling
14/24
Capacity Under Modulation, Channel
And Receiver Design Constraints
Scenario:
BICM capacity under constraint of using the SOSDDPD
2
M = 4, h = 0.21, BT = 0.2
1.8
SDDPD specifications:
R=26 uniform sub-regions for 4-GFSK
1.6
C (bit s/ channel use)
1.4
Channel specifications:
Rayleigh
1.2
GFSK specifications :
M =4, h =0.21, BT =0.2, 2B99Tb =0.6 with Rc =2/3
1
0.8
min{Es/No} if found at C=Rclog2M
0.6
min{Eb/No} = min{Es/No} /C
0.4
0.2
0
-10
0
10
20
E / N (dB)
s
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30
40
50
o
15/24
Optimum Combination of Code Rates And
GFSK Parameters In An Ergodic Channel

The search space is
–
–
–
–

At a particular Rc
–
–
–

M ={2, 4}- GFSK
Rc ={6/7, 5/6, 3/4, 2/3, 1/2, 1/3, 1/4, 1/5}
BT ={0.5, 0.25, 0.25}
2B99Tb ={0.4, 0.6, 0.8, 0.9, 1.0, 1.2}
Find h for each value of BT and M that meets a desired 2B99Tb
Find min{Eb/No} for all allowable combinations of M, h, BT at every 2B99Tb
At each 2B99Tb, select GFSK parameters yielding the lowest min{Eb/No}
Select the combination of Rc and GFSK parameters that have the lowest
min{Eb/No} at the desired 99% bandwidth
7/12/2006
16/24
Optimum Combination of Code Rates And
GFSK Parameters In An Ergodic Channel
Scenario:
Information theoretic minimum Eb/No at different
2B99Tb with Rc =5/6
24
Inform a t ion t heoret ic m inim um Eb/ No (dB)
M = 2, BT = 0.5
M= 2, BT = 0.25
22
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
M = 2, BT = 0.2
20
18
M =4, BT =0.5
Channel specifications:
Rayleigh
M= 4, BT = 0.25
0.14
M = 4, BT = 0.2
Search specifications:
At each 2B99Tb, there are 6 combinations of M, h
and BT
16
0.26
14
0.33
0.29
12
The numbers denote h values corresponding to
GFSK parameters with the lowest min{Eb/No} at
the particular bandwidth efficiency
0.48
0.7
10
0.4
0.5
0.6
0.7
0.8
2B T
0.9
1
1.1
1.2
1.3
At 2B99Tb =1.2, selecting M =2, h =0.7 and
BT =0.25 yields the lowest min{Eb/No}
99 b
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Optimum Combination of Code Rates And
GFSK Parameters In An Ergodic Channel
Scenario:
Best GFSK parameters for various code rates at
2B99Tb =0.9
24
M = 4, BT = 0.5, h = 0.35
M = 4, BT = 0.5, h = 0.33
Informat ion t heoret ic minimum Eb/ No (dB)
22
20
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
M = 4, BT = 0.5, h = 0.285
Rayleigh
M = 4, BT = 0.5, h = 0.24
M = 4, BT = 0.5, h = 0.14
18
M = 4, BT = 0.5, h = 0.07
Channel specifications:
AWGN, Rayleigh
M = 4, BT = 0.5, h = 0.046
16
M = 4, BT = 0.25, h = 0.05
14
12
Search specifications:
The combination of code rates and GFSK
parameters with lowest min{Eb/No} can be
identified at the particular 2B99Tb
At 2B99Tb =0.9:
M =4, h =0.24, BT =0.5 with Rc =2/3 (Rayleigh)
M =4, h =0.285, BT =0.5 with Rc =3/4 (AWGN)
yield the best energy efficiency
AWGN
10
8
6
4
0.2
7/12/2006
0.3
0.4
0.5
0.6
Code rat e
0.7
0.8
0.9
1
Notice the trade-off between code rate and energy
efficiency
18/24
Combination of Code Rates And
GFSK Parameters
Rayleigh Fading
7/12/2006
2B99Tb
Rate
M
BT
h
min{Eb/No} dB
0.4
3/4
4
0.2
0.195
18.15
0.6
2/3
4
0.2
0.21
18.08
0.8
3/4
4
0.5
0.25
12.38
0.9
2/3
4
0.5
0.24
11.99
1.0
2/3
4
0.5
0.3
11.44
1.2
5/6
2
0.25
0.7
11.34
19/24
Combination of Code Rates And
GFSK Parameters
Rician Fading (K =6 dB)
7/12/2006
2B99Tb
Rate
M
BT
h
min{Eb/No} dB
0.4
3/4
4
0.2
0.195
15.38
0.6
5/6
4
0.5
0.18
11.67
0.8
5/6
4
0.5
0.29
9.09
0.9
3/4
4
0.5
0.285
8.87
1.0
2/3
4
0.5
0.3
8.83
1.2
6/7
2
0.25
0.76
8.39
20/24
Conclusions

BICM with a soft-output SDDPD is used for noncoherent detection of
GFSK signals

The Shannon capacity of BICM under modulation, channel and
detector constraints is evaluated using Monte-Carlo integration

The constrained capacity is used to identify combination of code rates
and GFSK parameters with the best energy efficiency and outage
probability at a desired spectral efficiency
7/12/2006
21/24
Future Work

Extend the search space to include
– M >4
– Different pulse shapes and signal bandwidths
– Alternative receivers

A smarter method to comb the search space
– Evolutionary algorithm
7/12/2006
22/24
Performance In Block Fading

In block-fading a is broken into F blocks, which are transmitted over independent
channels

Channel coefficient c(t) =c, remains constant for the entire duration of a block

Instantaneous SNR of the bth block is
b | c |2

Es
No
When code combining is used at the receiver, the instantaneous capacity for the
entire code word is
1
C (1 , 2 ,..., F ) 
F

F
 C ( )
b 1
b
The information outage probability
po [ F ]  P[C (1 , 2 ,..., F )  Rc log 2 M ]
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Optimum Combination of Code Rates And
GFSK Parameters In A Block Fading Channel
0
10 0
Scenario:
Information outage probability with code combining in
block fading at F =1 and F = 100 for SO-SDDPD based
BICM at 2B99Tb =0.9
10
-1
10-1
10
F =100
Inform
Outaage
gePProba
roba
bilit
Informaattion
ion Out
bilit
y y
F =1
10
10
At F =1, M =4, h =0.285, BT =0.5 with Rc =3/4 has the lowest
information outage probability
-2
-2
At F =100, M =4, h =0.24, BT =0.5 with Rc =2/3 has the lowest
information outage probability
M = 4, BT = 0.5, h = 0.35, R = 6/ 7
-3
c
-3
M = 4, BT = 0.5, h = 0.35, R = 6/ 7
10
M = 4, BT = 0.5, h = 0.33, R = 5/ 6
c
10
The capacity based search also helps in identifying the
combination of code rates and GFSK parameters with the lowest
outage probability in block fading
c
M = 4, BT = 0.5, h = 0.285, R = 3/ 4
M = 4, BT = 0.5, h = 0.33, R = 5/ 6c
10
c
-4
4, BT
= 0.5,
h = 0.24,
MM
= 4,= BT
= 0.5,
h = 0.285,
R =R3/ 4= 2/ 3
c
c
M = 4, BT = 0.5, h = 0.24, R = 2/ 3
M = 4, BT = 0.5,h = 0.14,
c R = 1/ 2,
c
10
-4
10
-5
M = 4, BT = 0.5, h = 0.14, R = 1/ 2
c
M = 4, B T = 0.5, h = 0.07, R = 1/ 3
M = 4, BT =g0.5, h = 0.07, R = 1/ 3c
c
4, B= 0.5,
T = h0.5,
h = 0.046,
MM
= 4,= BT
= 0.046,
R = 1/R4 = 1/ 4
g
c
c
MM
= 4,= BT
h = 0.05,
= 1/R5 = 1/ 5
4, B= 0.25,
T = 0.25,
h = R0.05,
c
g
-6
-5
10
10
-5
-10
00
c
10 5
20 10
30
EE b// N
No (dB)
(dB)
b
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20 50
25
60
o
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