A Capacity-Based Search forEnergy and Bandwidth Efficient

Bit-Interleaved Coded Noncoherent GFSK

Rohit Iyer Seshadri and Matthew C. Valenti

Lane Dept. of Computer Science and Electrical EngineeringWest Virginia Universityiyerr, mvalenti @csee.wvu.edu

4/6/2006 2/21

“ Which is the optimal combination of channel coding rate and continuous phase modulation (CPM) parameters for a given

bandwidth efficiency and decoder complexity?”

Problem

4/6/2006 3/21

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

Different combination of these parameters result in different spectral characteristics and signal bandwidths

4/6/2006 4/21

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

4/6/2006 5/21

Uncoded CPM System

Bitto

Symbol

Modulatoru a RicianChannel

x r’ SymboltoBit

rFilter Detector

a^

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^

4/6/2006 6/21

An Uncoded System withGaussian Frequency Shift Keying

Bitto

Symbol

GFSKu a RicianChannel

x r’ SymboltoBit

rFilter Detector

a^

u^

Gaussian frequency shift keying (GFSK) is a widely used class of CPMe.g. Bluetooth

Baseband GFSK signal during kT ≤ t ≤ (k+1)T

GFSK phase

4/6/2006 7/21

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

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

Smaller values of BgT result in a more compact power spectrum

– Here M =2 and h =0.5

– 2B99Tb quantifies the bandwidth efficiency

( ) [ ( ) ( ( ))]/g gg t Q cB t Q cB t T T B gT =0.5

B g T =0.25

B gT =0.2

B gT =0.5

B g T =0.25

B gT =0.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (normalized by T)

Pow

er S

pect

ral D

ensi

ty (d

B)

BgT =0.5, 2B

99T

b =1.04

BgT =0.25, 2B

99T

b =0.86

BgT =0.2, 2B

99T

b =0.79

4/6/2006 8/21

Coded GFSK System

Decoder uDetector

arFilter

^ ^Encoder GFSK

u b RicianChannel

a x

( )xS f

Channel coding improves energy efficiency at the expense of bandwidth efficiency

1. Find the power spectral density for uncoded GFSK

2. PSD for GFSK using rate Rc code is now

3. must meet the required spectral efficiency

4. This implies the GFSK parameters have to be modified for the coded signal

For our system, coding must be done without bandwidth expansion, i.e. 2B99Tb should remain unchanged

( ) ( )cx c x cS f R S R f

( )cxS f

0 2 4 6 8 10

-50

-40

-30

-20

-10

0

10

Frequency (normalized by T)

Pow

er S

pect

ral D

ensi

ty (d

B)

M =2, BgT =0.5, h =0.5

M =2, BgT =0.5, h =0.125

M =2, BgT =0.075, h=0.5

Suppose we need 2B99Tb =1.04 while using a rate ½ code ,

The value of h needs to be lowered, with BgT unchanged ORThe value of BgT needs to lowered, with h unchanged

ORBoth can be lowered

It is not immediately clear if the performance loss caused be lowering h and/or BgT will be overcome by the coding gain

4/6/2006 9/21

Proposed Coded GFSK System

Decoderu

SO-SDDPDb’

^br

FilterBit

Deintrlv.

^ ^r'Encoder GFSK

u b’ b x RicianChannel

BitIntrlv.

Bit-wise interleaving between encoder and modulator and bit-wise soft-information passed from detector to decoder (BICM)

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

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

4/6/2006 10/21

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)

Received signal after filtering r(t, a) = c(t) x(t, a) + n(t)

Received signal phase

1s dP P s

d

PK

P ( ) ( )s dc t P P t

(t, a) = (t, a) + ( )t

4/6/2006 11/21

SO-SDDPD

Detector finds the phase difference between successive symbol intervals

We assume that GFSK pulse shape causes adjacent symbol interference

The phase difference space from 0 to 2 is divided into R sub-regions

Detector selects the sub-region Dk in which lies

The sequence of phase regions (D0, DI, …) is sent to a branch metric calculator

k

(k k ( ) ( )) mod 2k kt t T

0 1 1 1 1( ) mod 2k k k ka a a

TiT

iT

i dttgh )(

4/6/2006 12/21

SO-SDDPD

Let be the phase differences corresponding to any transmitted sequence

A branch metric calculator finds the conditional probabilities

Branch metrics sent to a 4-state MAP decoder whose state transition is from

to

The SO-SDDPD estimates the LLR for Bi,k

The bit-wise LLRs in Z can are arranged in a vector z, such that

1( , ,...)i io

1( , ,...)i ioa a

0 1 1( ( | ), ( | ),...)i i

oP D P D

1 1,k k kS a a 1,k k kS a a

4/6/2006 13/21

Capacity Under Modulation, Channel And Receiver Design Constraints

Channel capacity denotes maximum allowable data rate for reliable communication over noisy channels

In any practical system, the input distribution is constrained by the choice of modulation

– Capacity is mutual information between the bit at modulator input and LLR at detector output

Constrained capacity in nats is; [Caire, 1998]

( )

2( )

max ( ; )

( , )max ( , ) log

( ) ( )

p x

p x

C I X Y

p x yC p x y dxdy

p x p y

( ; )C I X Y

[log(2) log ( | )]iC E p b r

4/6/2006 14/21

Capacity Under Modulation, Channel And Receiver Design Constraints

Constrained capacity for the proposed system is now

In bits per channel use

Constrained capacity hence influenced by– Modulation parameters (M, h and BgT)– Channel – Detector design

Computed using Monte-Carlo integration

2log

2 , , , '1

1log [log{exp(0) exp( ( 1) )}]

log(2)i

Mb

a c n s s ii

C M E z

2log

, , , '1

log(2) [log{exp(0) exp( ( 1) )}]i

Mb

a c n s s ii

C E z

4/6/2006 15/21

Capacity Under Modulation, Channel And Receiver Design Constraints

Scenario:BICM capacity under constraint of using the SO-SDDPD

SDDPD specifications:R=40 uniform sub-regions for 2-GFSKR=26 uniform sub-regions for 4-GFSK

Channel parameters:Rayleigh fading

GFSK specifications : M =2, h =0.7, BgT =0.25 M =4, h =0.21, BgT =0.2

Information theoretic minimum Es/No (min{Es/No }) is found by reading the value of Es/No for C =Rclog2M

min{Eb/No} =min{Es/No}/Rc log2M

-10 0 10 20 30 40 500

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Es/N

o (dB)

C (bi

ts/c

hann

el u

se)

M =4, h =0.21, BgT =0.2

M =2, h =0.7, BgT =0.25

4/6/2006 16/21

Capacity-Based Search for Energy and Bandwidth Efficient GFSK Parameters

The search space is over M ={2, 4}-GFSKRc ={6/7, 5/6, 3/4, 2/3, 1/2, 1/3, 1/4, 1/5}2B99Tb ={0.4, 0.6, 0.8, 0.9, 1.0, 1.2}BgT ={0.5, 0.25, 0.2}

At each Rc, find h for each value of BgT and M, that meets a desired 2B99Tb

Find min{Eb/No} for all allowable combinations of M, h, BgT, and Rc at each 2B99Tb

At every 2B99Tb, select the GFSK parameters yielding the lowest min{Eb/No}

As an example, consider a rate-5/6 coded, {2,4}-GFSK, with SO-SPDPD based BICM in Rayleigh fading0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

10

12

14

16

18

20

22

24

2Bcoded

Tb

Info

rmat

ion

theo

retic

min

imum

Eb/

No

(dB

)

M =2, BgT =0.5

M=2, BgT =0.25

M =2, BgT =0.2

M =4, BgT =0.5

M=4, BgT =0.25

M =4, BgT =0.2

0.7

0.48 0.33 0.29

0.26

0.14

At each 2B99Tb , there are 6 combinations of M, h and BgT

The numbers denote h values corresponding to GFSK parameters with the lowest min{Eb/No} at the particular min{Eb/No}

For 2B99Tb =1.2 , M =2, h =0.7, BgT =0.25 has the lowest min{Eb/No} with Rc =5/6

4/6/2006 17/21

Capacity-Based Search for Energy and Bandwidth Efficient GFSK Parameters

A similar search was conducted for all listed values of Rc

This gives the set of M, h, BgT with the lowest min{Eb/No} at different 2B99Tb

for each of the considered code rates

The search is now narrowed to find the combination of Rc and GFSK parameters that have the lowest min{Eb/No} for a particular bandwidth efficiency

As an example, consider SO-SPDPD based BICM in Rayleigh fading, at 2B99Tb =0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

12

14

16

18

20

22

24

26

Code rate

Info

rmat

ion

theo

retic

min

imum

Eb/

No

(dB

)

M =4, BgT =0.5, h =0.35

M =4, BgT =0.5, h =0.33

M =4, BgT =0.5, h =0.285

M =4, BgT =0.5, h =0.24

M =4, BgT =0.5, h =0.14

M =4, BgT =0.5, h =0.07

M =4, BgT =0.5, h =0.046

M =4, BgT =0.25, h =0.05

For the proposed system, Rc =3/4 with M =4, h =0.25, BgT =0.5 has the best energy efficiency at 2B99Tb =0.8

4/6/2006 18/21

Combination of Code Rates and GFSK Parameters in Rayleigh Fading

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

4/6/2006 19/21

Bit Error Rate Simulations

Scenario:Bit error rate for SO-SDDPD based coded and uncoded systems

Solid curve: System without codingDotted curve: Systems with coding (BICM)

Channel parameters:Rayleigh fading

GFSK specifications :Coded: M =4, h =0.315, BgT =0.5, Rc =2/3, 2B99Tb =0.9Uncoded: M =2, h =0.5, BgT =0.3, 2B99Tb =0.9

Simulated Eb/No required for an arbitrarily low error rate = 12.93 dB

Information theoretic threshold = 11.99 dB

Coding gain =16 dB (at BER =10-5)5 10 15 20 25 30 35 40

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

BER

Eb/N

o (dB)

BICPM M =4, h =0.24 BgT =0.5

Uncoded M =2, h =0.5, BgT =0.3

4/6/2006 20/21

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

4/6/2006 21/21

Future Work

Extend the search space to include– M >4

– Different values of BgT

– SO-SDDPD designed to account for additional ISI– More CPM formats (RC, REC etc..)– Alternative noncoherent receivers