The PPM Poisson Channel: Finite-Length Bounds and Code Design

August 21, 2014

The PPM Poisson Channel: Finite-LengthBounds and Code Design

Flavio ZabiniDEI - University of BolognaandInstitute for Communications and NavigationGerman Aerospace Center (DLR)

Balazs Matuz, Gianluigi Liva,Enrico Paolini, Marco Chiani

Motivation

• Poisson channel is a common channel model for deep spaceoptical links for direct detection

• Under peak and average power constraints, slotted modulationschemes, such as pulse position modulation (PPM), allowoperating close to capacity

• For finite-length blocks bounds/benchmarks on the block errorprobability (BLEP) are barely available in literature

• Binary low-density parity-check (LDPC) codes show a noticeablegap to capacity→ Non-binary protograph LDPC codes where field and PPMorder are matched

Outline

1 Preliminaries

2 Finite-Length Benchmarks

3 Code Design

4 Conclusions

/17 Flavio Zabini · The PPM Poisson Channel · Preliminaries August 21, 2014

PreliminariesCommunication System

LDPCencoder

Cq(N, K)

Modulator(q-ary PPM)

Poissonchannel

Demodulator LDPCdecoder

c x y P (c|y)

encodedsymbol

symbol probabilitymass function

• The code Cq(N,K) is used to generate code symbols c• q-ary code symbols c are mapped to q-ary PPM symbols x• For q-ary PPM one slot out of q slots is pulsed• PPM symbols x transmitted on the Poisson channel• Given y we get the probabilities P(c|y)

• The decoder provides a decision on the transmitted symbols


PreliminariesChannel Model

TX RX

PPM symbol PPM symbol

P(

y[t]∣∣∣X[t] = P

)=

(ns + nb)y[t] e−(ns+nb)

y[t]!for a pulsed slot

P(

y[t]∣∣∣X[t] = P

)=

nby[t] e−nb

y[t]!for a non-pulsed slot

ns: average number of signal photons in pulsed slotnb: average number of noise photons per slott: time slot index.


PreliminariesCapacity

• Channel capacity for Poisson with q-ary PPM:

CPPM = log2 q− EY|X[0]=P

{log2

[q−1∑t=0

(ns

nb+ 1)Y[t]−Y[0]]}

[bits/channel use]

• Capacity in absence of background noise, i.e., for nb = 0

CPPM−EC =log2 q(1− e−ns

)[bits/channel use]

⇒ Capacity of q-ary erasure channel (EC) with ε = exp(−ns)



Generic example

Pe : Block errorprobabilityR : Transmissionrate[bits/channeluse]



Channel capacity provides the limit for infinite length codes, but...

Pe : Block errorprobabilityR : Transmissionrate[bits/channeluse]

Outline

1 Preliminaries


3 Code Design

4 Conclusions

/17 Flavio Zabini · The PPM Poisson Channel · Finite-Length Benchmarks August 21, 2014

Finite-Length BenchmarksGoal

..we want to evaluate Pe(N, r, ns, nb):

R = r log2 q[bits/channeluse]

the minimum block error probability achievable by a code with FINITElength N and rate r [symbols/channel use] over a q-ary PPM Poissonchannel, given ns and nb


Finite-Length BenchmarksRandom Coding Bound for Poisson PPM

• Gallager’s random coding bound (RCB):

Pe ≤ 2−NEPPM(r)

• Error exponent:

EPPM(r) = max0≤ρ≤1

{− log2 [Υq,ns,nb(ρ)] + [ρ(1− r) + 1] log2 q}+ns + qnb

ln(2)

Υq,ns,nb(ρ) ,+∞∑kT =0

nkTb

kT∑i1=0

kT−i1∑i2=0

...

kT−∑q−2

n=1 in∑iq−1=0

[∑q−1n=1 ain

ρ + akT−∑q−1

n=1 inρ

]1+ρ

∏q−1n=1 in! (kT −

∑q−1n=1 in)!

aρ ,(

1 + nsnb

) 11+ρ

• For q > 16 computation becomes numerically challenging


Finite-Length BenchmarksErasure Channel Approximation

• Use q-ary EC as a surrogate for Poisson PPM channel• By matching the capacities of both channels we obtain:

ε = 1− CPPM

log2 q

• Gallager’s error exponent for the q-ary EC:

EqEC(r) =

− log2

(1−ε

q + ε)− r log2 q 0 < ε < εc

D(B1−r||Bε) εc ≤ ε ≤ 1− r0 ε > 1− r

εc , 1−rr(q−1)+1

B1−r and Bε are two Bernoulli distributions with parameters 1− rand ε.


Finite-Length BenchmarksSingleton Bound via EC Approximation

• Singleton bound, denoting the BLEP of idealized maximumdistance separable code on an EC

P(S)e =

N∑i=N−K+1

(Ni

)εi(1− ε)N−i

P(S)e = Pr (SN > N(1− r))

where SN ∼ B{N, ε}.• Again we require:

ε = 1− CPPM

log2 q


Finite-Length BenchmarksDispersion Approach - Definitions

Let us consider a generic discrete memoryless channel (DMC).

• Information density: i(x, y) , log2

(PY|X(y|X=x)

PY (y)

)• Mutual information: I(PX; PY|X) , EX

{EY|X=x {i(x, y)}

}• Conditional information variance:

V(PX; PY|X) , EX{EY|X=x

{[i(x, y)]2

}− [EY|X=x {i(x, y)}]2

}Capacity:

C , maxPX

{I(PX; PY|X)

} Dispersion:

V , minPX∈Π

{V(PX; PY|X)

}where Π = {PX : I(PX; PY|X) = C}.


Finite-Length BenchmarksDispersion Approach - Applications

Capacity:

R < C⇒ limN→∞ Pe = 0Shannon (1948)

Dispersion:

Pe ≈ Q[(C− R)

√VN

]Strassen (1962), recently [1,2]

[1] Hayashi, Information Spectrum Approach to Second-order Coding Rate in ChannelCoding, IEEE Trans. Inf. Theory 2009[2] Polyanskiy et al., Channel Coding Rate in the Finite Blocklength Regime, IEEETrans. Inf. Theory 2010


Finite-Length BenchmarksConverse Theorem Gaussian Approximation

• For the Poisson PPM channel the dispersion VPPM results in

VPPM = EY|X[0]=P

{log2

2

[q−1∑t=0

(ns

nb+ 1)Y[t]−Y[0]]}

− [log2 q−CPPM]2

(*** novel result ***)• Thus:

Pe(N, r, ns, nb) ≈ Q

{[CPPM(q, ns, nb)− r log2 q]

√N

VPPM(q, ns, nb)

}


Finite-Length BenchmarksConverse Theorem EC Approximation

• For the q-ary EC we have

VqEC = ε(1− ε) log22 q; [bits2/channel use2]

P(qEC)e ≈ Q

[N(1− r)− Nε√

Nε(1− ε)

]= Pr (GN > N(1− r))

where GN ∼ N{Nε,Nε(1− ε)} and ε = 1− CPPMlog2 q

• We obtain an approximation of the Singleton bound by replacingthe binomial distribution B{N, ε} by a Gaussian distributionN{Nε,Nε(1− ε)}.


Finite-Length BenchmarksResults

−12.5 −12 −11.5 −1110

−6

10−5

10−4

10−3

10−2

10−1

100

ns/q [dB]

BLE

P

Gallager RCBRCB via q-EC approx.ConverseConverse via q-EC approx.Singleton via q-EC approx.

N = 684

N = 2736

Differentbounds/benchmarks:

• q = 16

• nb = 0.02

• N = {684, 2736}• r = 0.5

Outline

1 Preliminaries


3 Code Design

4 Conclusions

/17 Flavio Zabini · The PPM Poisson Channel · Code Design August 21, 2014

Code DesignSurrogate Erasure Channel Design

• For non-binary LDPC codes extrinsic information transfer (EXIT)analysis on the Poisson PPM channel is rather complex

• We propose therefore a surrogate design for the q-ary EC byprotograph EXIT analysis

Non-binary protograph:

• Each VN a GF(q) symbol

• Edges possess edgelabels


Code DesignSelected Protographs

• Fixing the size of base matrix to 3× 5 with one puncturedcolumn, we obtain

B1 =

2 1 1 1 01 2 1 1 02 0 0 0 1

with iterative decoding threshold of the ensemble of ε∗ = 0.478

• We consider the base matrix of a (binary) LDPC code from [3]

B2 =

3 1 1 1 01 3 1 1 02 0 0 0 1

with ε∗ = 0.442

[3] Barsoum et al., EXIT Function Aided Design of Iteratively Decodable Codes for thePoisson PPM Channel, IEEE TCOM 2010


Code DesignPerformance curves

−20 −19 −18 −17 −16 −15 −14 −13 −12 −11 −1010

−6

10−5

10−4

10−3

10−2

10−1

100

ns/q [dB]

CE

R,B

ER

ConverseConverse via q-EC approx.CER code C64

1

CER code C642

BER code C642

BER code C23

nb = 2

nb = 0.2

nb = 0.002

3 LDPC codes:

• C641 (1368, 684) over

q = 64 from B1

• C642 (1368, 684) over

q = 64 from B2

• C23(8192, 4096) over

q = 2 from B2

2 benchmarks:

• Converse boundnormal approx.

• EC approx. ofconverse bound

Outline

1 Preliminaries


3 Code Design

4 Conclusions

/17 Flavio Zabini · The PPM Poisson Channel · Conclusions August 21, 2014

Conclusions

• Several finite-length benchmarks for the PPM-modulated codeperformance on the Poisson channel have been developed

I Converse bound Gaussian approximation: less complex than RCBapproach but more accurate for the PPM Poisson channel withrealistic parameters (very close to Singleton bound when anEC-equivalent channel is considered)

I EC-approximation: valid also for high background noise• A surrogate erasure channel design for protograph non-binary

LDPC codes on the Poisson PPM channel has been proposedI The density evolution analysis turns to be extremely simpleI The code performance is consistently close to the benchmarks also

with stronger background radiation (robust design)

Thank you for your attention!

Questions?

Contact:Dr.-Ing. Flavio Zabini

Department of Electrical, Electronic and Information Engineering (DEI)University of [email protected]

Code DesignDecoding Thresholds

• Decoding thresholds on the Poisson PPM channel computed viaMonte Carlo density evolution

• Thresholds in terms of ns/q [dB] for protograph code ensemblesspecified by B1, B2, q = 64, and various nb:

Background noise nb

0.002 0.02 0.2 2

B1 −18.87 −17.54 −15.19 −11.71B2 −18.47 −17.00 −14.65 −11.27

Limit −19.13 −17.79 −15.51 −12.08

Numerical ResultsLDPC Code vs. Serially Concatenated Pulse PositionModulation (SCPPM)

−20 −19.5 −19 −18.5 −18 −17.5 −17 −16.5 −16

10−4

10−3

10−2

10−1

100

γ [dB]

CE

R

Random coding boundCode C4(2720, 1360)Serially concatenatedPPM scheme (2520,1260)

• Highly irregular codeC4(2720, 1360) over finitefield of order q = 64

• Background noisenb = 0.0025

• RCB as a reference

The PPM Poisson Channel: Finite-Length Bounds and Code Design

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