A Capacity-Based Approach forDesigning Bit-Interleaved Coded GFSK with

Noncoherent Detection

Rohit Iyer Seshadri and Matthew C. Valenti

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

7/12/2006 2/24

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

bandwidth efficiency and decoder complexity?”

Problem

7/12/2006 3/24

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 4/24

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 5/24

Uncoded CPM System

Bitto

Symbol

Modulatoru aChannel

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

u: bit estimates provided by the detector^

7/12/2006 6/24

An Uncoded System withGaussian Frequency Shift Keying

Bitto

Symbol

GFSKu aChannel

x r’ SymboltoBit

rFilter Detector

a^

u^

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

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

GFSK phase

7/12/2006 7/24

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

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

( ) [ ( ) ( ( ))]/g t Q cBt Q cB t T T BT =0.5

BT =0.25

BT =0.2

0 0.2 0.4 0.6 0.8 1 1.2

-40

-35

-30

-25

-20

-15

-10

-5

0

Frequency (normalized by T)

Powe

r Spe

ctra

l Den

sity

(dB)

BT =0.5, 2B99

Tb =1.04

BT =0.25, 2B99

Tb =0.86

BT =0.2, 2B99

Tb =0.79

7/12/2006 8/24

Coded GFSK System

Decoder uDetector

arFilter

^ ^Encoder GFSK

u bChannel

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

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

The value of h needs to be lowered, with BT unchanged ORThe value of BT 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 BT will be overcome by the coding gain

0 2 4 6 8 10

-50

-40

-30

-20

-10

0

10

Frequency (normalized by T)

Powe

r Spe

ctra

l Den

sity

(dB)

M =2, BT =0.5, h =0.5, uncoded

M =2, BT =0.5, h =0.125, Rc =1/2

M =2, BT =0.075, h =0.5, Rc =1/2

7/12/2006 9/24

Proposed Coded GFSK System

Decoderu

SO-SDDPDb’

^br

FilterBit

Deintrlv.

^ ^r'Encoder GFSK

u b’ b xChannel

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

The availability of capacity-approaching turbo and LDPC codes make the capacity under BICM a very practical indicator of system performance

7/12/2006 10/24

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)

Received signal after filtering

r(t, a) = c(t) x(t, a) + n(t)

Received signal phase

1,s dP P s

d

PK

P ( ) ( ) ,s dc t P P t

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

7/12/2006 11/24

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 )(

7/12/2006 12/24

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 code bits

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

7/12/2006 13/24

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

7/12/2006 14/24

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 BT)– Channel – Detector design– Computed using Monte-Carlo integration

The constrained capacity is used to find the minimum Eb/No required for reliable signaling

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

7/12/2006 15/24

Capacity Under Modulation, Channel And Receiver Design Constraints

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

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

Channel specifications:Rayleigh

GFSK specifications :M =4, h =0.21, BT =0.2, 2B99Tb =0.6 with Rc =2/3

min{Es/No} if found at C=Rclog2M

min{Eb/No} = min{Es/No} /C

-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 (b

its/c

hann

el us

e)

M =4, h =0.21, BT =0.2

7/12/2006 16/24

Optimum Combination of Code Rates And GFSK Parameters In An Ergodic Channel

The search space is – 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}

At a particular Rc

– 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 17/24

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

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

Channel specifications:Rayleigh

Search specifications:At each 2B99Tb, there are 6 combinations of M, h and BT

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

At 2B99Tb =1.2, selecting M =2, h =0.7 and BT =0.25 yields the lowest min{Eb/No}

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.310

12

14

16

18

20

22

24

2B99T

b

Info

rmat

ion

theo

retic

min

imum

Eb/

No (d

B)

0.7

0.48 0.33 0.29

0.26

0.14

M =2, BT =0.5

M=2, BT =0.25

M =2, BT =0.2

M =4, BT =0.5

M=4, BT =0.25

M =4, BT =0.2

7/12/2006 18/24

Optimum Combination of Code Rates And GFSK Parameters In An Ergodic Channel

Scenario:Best GFSK parameters for various code rates at 2B99Tb =0.9

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

Channel specifications:AWGN, Rayleigh

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

Notice the trade-off between code rate and energy efficiency

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 14

6

8

10

12

14

16

18

20

22

24

Code rate

Info

rmat

ion

theo

retic

min

imum

Eb/

No

(dB

)

Rayleigh

AWGN

M =4, BT =0.5, h =0.35

M =4, BT =0.5, h =0.33

M =4, BT =0.5, h =0.285

M =4, BT =0.5, h =0.24

M =4, BT =0.5, h =0.14

M =4, BT =0.5, h =0.07

M =4, BT =0.5, h =0.046

M =4, BT =0.25, h =0.05

7/12/2006 19/24

Combination of Code Rates And GFSK Parameters

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

Rayleigh Fading

7/12/2006 20/24

Combination of Code Rates And GFSK Parameters

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

Rician Fading (K =6 dB)

7/12/2006 21/24

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 22/24

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 23/24

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

When code combining is used at the receiver, the instantaneous capacity for the entire code word is

The information outage probability

1 21

1( , ,..., ) ( )

F

F bb

C CF

2| | sb

o

Ec

N

1 2 2[ ] [ ( , ,..., ) log ]o F cp F P C R M

7/12/2006 24/24

Optimum Combination of Code Rates And GFSK Parameters In A Block Fading Channel

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

At F =1, M =4, h =0.285, BT =0.5 with Rc =3/4 has the lowest information outage probability

At F =100, M =4, h =0.24, BT =0.5 with Rc =2/3 has the lowest information outage probability

The capacity based search also helps in identifying the combination of code rates and GFSK parameters with the lowest outage probability in block fading

-10 0 10 20 30 40 50 6010

-5

10-4

10-3

10-2

10-1

100

Eb/N

o (dB)

Info

rmat

ion

Out

age

Prob

abili

ty

F =1

M =4, BT =0.5, h =0.35, Rc =6/7

M =4, BT =0.5, h =0.33, Rc =5/6

M =4, BT =0.5, h =0.285, Rc =3/4

M =4, BT =0.5, h =0.24, Rc =2/3

M =4, BT =0.5,h =0.14, Rc =1/2,

M =4, BgT =0.5, h =0.07, Rc =1/3

M =4, BgT =0.5, h =0.046, Rc =1/4

M =4, BgT =0.25, h =0.05, Rc =1/5

-5 0 5 10 15 20 2510

-6

10-5

10-4

10-3

10-2

10-1

100

Eb/N

o (dB)

Info

rmat

ion

Out

age

Prob

abili

ty

M =4, BT =0.5, h =0.35, Rc =6/7

M =4, BT =0.5, h =0.33, Rc =5/6

M =4, BT =0.5, h =0.285, Rc =3/4

M =4, BT =0.5, h =0.24, Rc =2/3

M =4, BT =0.5, h =0.14, Rc =1/2

M =4, BT =0.5, h =0.07, Rc =1/3

M =4, BT =0.5, h =0.046, Rc =1/4

M =4, BT =0.25, h =0.05, Rc =1/5

F =100