Transcript Document 7592502

A Capacity-Based Search for
Energy and Bandwidth Efficient
Bit-Interleaved Coded Noncoherent GFSK
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?”
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Continuous Phase Modulation

CPM is a nonlinear modulation scheme with memory
– Modulation induces controlled inter symbol interference (ISI)

Well suited for bandwidth constrained systems

Phase continuity results in small spectral side lobes
– Reduced adjacent channel interference

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

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Different combination of these parameters result in different spectral
characteristics and signal bandwidths
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Challenges

CPM includes an almost infinite variations on the modulated signal
– Full response, partial response, GFSK, 1-REC, 2-REC, 2-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
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Uncoded CPM System
u
Bit
to
Symbol
a
Modulator
x
Rician
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
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An Uncoded System with
Gaussian Frequency Shift Keying
Bit
to
Symbol
u
a
GFSK
x
Rician
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
Baseband GFSK signal during kT ≤ t ≤ (k+1)T
GFSK phase
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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
0
g (t )  [Q(cBg t )  Q(cBg (t  T ))]/ T 
B T =0.5, 2B T =1.04
-5
g
B g T =0.5
99 b
B T =0.25, 2B T =0.86
g

BgT is the normalized 3 dB bandwidth of
the filer
–
–
Width of the pulse shape depends on BgT
Wider the pulse, greater is the ISI
Power Spectral Density (dB)
-10
-15
B g T =0.25
-20
B g T =0.2
99 b
B T =0.2, 2B T =0.79
g
99 b
-25
-30
-35
-40

Smaller values of BgT result in a more
compact power spectrum
–
–
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Here M =2 and h =0.5
2B99Tb quantifies the bandwidth efficiency
-45
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Frequency (normalized by T)
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Coded GFSK System
Encoder
b
GFSK
a
Rician
Channel
x
Filter
Suppose we need 2B99Tb =1.04 while using a rate ½
Channel
improves energy efficiency at
codecoding
,
the expense of bandwidth efficiency
value ofcoding
h needsmust
to bebelowered,
with BgT
For The
our system,
done without
unchanged
bandwidth
expansion, i.e. 2B99Tb should
OR
remain unchanged
The value of BgT needs to lowered, with h unchanged
OR
1.
Find
the can
power
density for uncoded GFSK S x ( f )
Both
bespectral
lowered
2.
3.
4.
PSD for GFSK using rate
S ( f )  R S (R f )
c
Rc code is now x
clear if the performancec
x
c
It is not immediately
loss
be lowering
h and/or
BgT efficiency
will be overcome
S xccaused
( f ) must
meet the required
spectral
by the coding gain
This implies the GFSK parameters have to be
modified for the coded signal
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^
r
^
a
Detector
Decoder
u
10
0
Power Spectral Density (dB)
u
-10
-20
M =2, B T =0.5, h =0.125
g
-30
M =2, B T =0.5, h =0.5
g
-40
M =2, B T =0.075, h=0.5
g
-50
0
2
4
6
8
10
Frequency (normalized by T)
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Proposed Coded GFSK System
u
Encoder
b’
Bit
Intrlv.
b
x
GFSK
Rician
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
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System Model

Bit-interleaved codeword b is arranged in a
matrix B, such that

Each column of B is mapped to one of M possible symbols to produce a

The baseband GFSK x is sent through a frequency nonselective Rician channel

Received signal at the output of a frequency nonselective, Rician channel, before filtering
r’(t, a) = c(t) x(t, a) + n’(t)
c(t ) 

Ps 
Pd  (t )


Received signal after filtering
r(t, a) = c(t) x(t, a) + n(t)

Received signal phase
Ps  Pd  1
K 
Ps
Pd
 (t, a) =  (t, a) +  (t )
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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
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
k
lies
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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 Bi,k

The bit-wise LLRs in Z can are arranged in a vector z, such that
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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 )]
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Capacity Under Modulation, Channel
And Receiver Design Constraints

Constrained capacity for the proposed system is now
C
log2 M

i 1

log(2)  Ea ,c,n,ss '[log{exp(0)  exp( zi (1)bi )}]
In bits per channel use
C  log 2 M 

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
i 1
1
Ea,c,n,ss '[log{exp(0)  exp( zi (1)bi )}]
log(2)
Constrained capacity hence influenced by
–
–
–

log2 M
Modulation parameters (M, h and BgT)
Channel
Detector design
Computed using Monte-Carlo integration
14/21
Capacity Under Modulation, Channel
And Receiver Design Constraints
2
M = 4, h = 0.21, B T = 0.2
1.8
g
SDDPD specifications:
R=40 uniform sub-regions for 2-GFSK
R=26 uniform sub-regions for 4-GFSK
1.6
1.4
C (bits/ channel use)
Scenario:
BICM capacity under constraint of using the SOSDDPD
1.2
Channel parameters:
Rayleigh fading
1
M = 2, h = 0.7, B T = 0.25
g
0.8
GFSK specifications :
M =2, h =0.7, BgT =0.25
M =4, h =0.21, BgT =0.2
0.6
0.4
Information theoretic minimum Es/No
(min{Es/No }) is found by reading the value
of Es/No for C =Rclog2M
0.2
0
-10
0
10
20
E / N (dB)
s
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o
30
40
50
min{Eb/No} =min{Es/No}/Rc log2M
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Capacity-Based Search for Energy and
Bandwidth Efficient GFSK Parameters
Inform a t ion t heoret ic m inim um Eb/ No (dB)
24
22
20
The search space is over
M = 2, B T = 0.5
At each 2B99Tb , there are 6 combinations of M,
g
M ={2, 4}-GFSK
M= 2, B T = 0.25
Rc ={6/7, 5/6, 3/4, 2/3, 1/2, 1/3, 1/4, 1/5} g
M = 2, B T = 0.2
g
2B99Tb ={0.4, 0.6, 0.8, 0.9, 1.0, 1.2}
M = 4, B T = 0.5
BgT ={0.5, 0.25, 0.2}
g
h and BgT
The numbers denote h values corresponding to
GFSK parameters with the lowest min{Eb/No} at
the particular min{Eb/No}
M= 4, B T = 0.25
g
18
16
14
12
At each Rc, find h for each valueM of
and M, that meets a desired 2B99Tb
= 4, B
B TgT
= 0.2
g
0.14
=2, h =0.7, BgT =0.25
99Tb
Find min{Eb/No} for all allowable combinationsFor
of2BM,
h, =1.2
BgT,, M
and
Rc at each 2B99Tb
has the lowest min{Eb/No} with Rc =5/6
At every0.262B99Tb, select the GFSK parameters yielding the lowest min{Eb/No}
0.33
As an example, consider
a rate-5/6 coded, {2,4}-GFSK, with SO-SPDPD based BICM
0.29
0.48
in Rayleigh fading
0.7
10
0.4
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0.5
0.6
0.7
0.8
2B
0.9
coded
T
1
1.1
1.2
1.3
b
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Capacity-Based Search for Energy and
Bandwidth Efficient GFSK Parameters
26
A similar search was conducted
for all listed values For
of Rthe
c proposed system,
M = 4, B T = 0.5, h = 0.35
Informat ion t heoret ic minimum Eb/ No (dB)
g
24
22
Rc =3/4 with M =4, h =0.25, BgT =0.5 has
the best energy efficiency
= 4, B Tthe
= 0.5,
h = 0.285min{E /N } at different 2B T
This gives the set of M, h, BgTMwith
lowest
g
o Tb =0.8
99 b
atb2B99
M = 4, B T = 0.5, h = 0.33
g
M =rates
4, B T = 0.5, h = 0.24
for each of the considered code
g
M = 4, B T = 0.5, h = 0.14
g
20
M = 4, B T = 0.5, h = 0.07
The search is now narrowed to findgthe combination of Rc and GFSK parameters that
M = 4, B T = 0.5, h = 0.046
g
18 have the lowest min{E /N } for a particular
bandwidth efficiency
b
o
M = 4, B T = 0.25, h = 0.05
g
16
As an example, consider SO-SPDPD based BICM in Rayleigh fading, at 2B99Tb =0.8
14
12
10
0.1
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0.2
0.3
0.4
0.5
0.6
Code rat e
0.7
0.8
0.9
1
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Combination of Code Rates and GFSK
Parameters in Rayleigh Fading
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2B99Tb
Rate
M
BgT
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
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Bit Error Rate Simulations
Scenario:
Bit error rate for SO-SDDPD based coded and uncoded
systems
0
10
BICPM M =4, h =0.24 B T =0.5
g
Uncoded M =2, h =0.5, B T =0.3
g
-1
10
Solid curve: System without coding
Dotted curve: Systems with coding (BICM)
-2
10
Channel parameters:
Rayleigh fading
-3
BER
10
GFSK specifications :
Coded: M =4, h =0.315, BgT =0.5, Rc =2/3, 2B99Tb =0.9
Uncoded: M =2, h =0.5, BgT =0.3, 2B99Tb =0.9
-4
10
-5
10
Simulated Eb/No required for an arbitrarily low
error rate = 12.93 dB
-6
10
Information theoretic threshold = 11.99 dB
-7
10
5
10
15
20
b
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25
E / N (dB)
30
35
40
Coding gain =16 dB (at BER =10-5)
o
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Conclusions

The Shannon capacity of BICM under modulation, channel and detector
constraints is a very practical indicator of system performance

Most CPM systems are too complex to admit closed-form solution
– The constrained capacity is evaluated using Monte-Carlo integration

A Soft-output, SDDPD is used for noncoherent detection of GFSK signals

For a select range of code rates, spectral efficiencies and GFSK parameters,
the GFSK constrained capacities have been calculated

The constrained capacity is used to identify combination of code rates and
GFSK parameters with the best energy efficiency for a desired spectral
efficiency
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Future Work

Extend the search space to include
–
–
–
–
–
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M >4
Different values of BgT
SO-SDDPD designed to account for additional ISI
More CPM formats (RC, REC etc..)
Alternative noncoherent receivers
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