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

August 21, 2014 The PPM Poisson Channel: Finite-Length Bounds and Code Design Flavio Zabini DEI - University of Bologna and Institute for Communications and Navigation German Aerospace Center (DLR) Balazs Matuz, Gianluigi Liva, Enrico Paolini, Marco Chiani

Motivation Poisson channel is a common channel model for deep space optical links for direct detection Under peak and average power constraints, slotted modulation schemes, such as pulse position modulation (PPM), allow operating close to capacity For finite-length blocks bounds/benchmarks on the block error probability (BLEP) are barely available in literature Binary low-density parity-check (LDPC) codes show a noticeable gap to capacity Non-binary protograph LDPC codes where field and PPM order are matched

Outline 1 Preliminaries 2 Finite-Length Benchmarks 3 Code Design 4 Conclusions

Slide 1/17 Flavio Zabini The PPM Poisson Channel Preliminaries August 21, 2014 Preliminaries Communication System LDPC encoder C q (N,K) c x y P(c y) Modulator Poisson Demodulator (q-ary PPM) channel LDPC decoder encoded symbol symbol probability mass function The code C q (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

Slide 2/17 Flavio Zabini The PPM Poisson Channel Preliminaries August 21, 2014 Preliminaries Channel Model TX RX PPM symbol PPM symbol ( ) P y [t] X [t] = P = (n s + n b ) y[t] e (n s+n b) y [t]! ( ) P y [t] X [t] = P = n b y[t] e nb y [t]! for a pulsed slot for a non-pulsed slot n s : average number of signal photons in pulsed slot n b : average number of noise photons per slot t: time slot index.

Slide 3/17 Flavio Zabini The PPM Poisson Channel Preliminaries August 21, 2014 Preliminaries Capacity Channel capacity for Poisson with q-ary PPM: { [ q 1 ( ) ]} Y [t] Y [0] ns C PPM = log 2 q E Y X [0] =P log 2 + 1 [bits/channel use] n b Capacity in absence of background noise, i.e., for n b = 0 t=0 C PPM EC =log 2 q ( 1 e ns) [bits/channel use] Capacity of q-ary erasure channel (EC) with ɛ = exp( n s )

Slide 4/17 Flavio Zabini The PPM Poisson Channel Preliminaries August 21, 2014 Preliminaries Capacity Generic example P e : Block error probability R : Transmission rate [bits/channel use]

Slide 4/17 Flavio Zabini The PPM Poisson Channel Preliminaries August 21, 2014 Preliminaries Capacity Channel capacity provides the limit for infinite length codes, but... P e : Block error probability R : Transmission rate [bits/channel use]

Outline 1 Preliminaries 2 Finite-Length Benchmarks 3 Code Design 4 Conclusions

Slide 5/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Goal..we want to evaluate P e (N, r, n s, n b ): R = r log 2 q [bits/channel use] the minimum block error probability achievable by a code with FINITE length N and rate r [symbols/channel use] over a q-ary PPM Poisson channel, given n s and n b

Slide 6/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Random Coding Bound for Poisson PPM Gallager s random coding bound (RCB): Error exponent: P e 2 NE PPM(r) E PPM (r) = max 0 ρ 1 { log 2 [Υ q,n s,n b (ρ)] + [ρ(1 r) + 1] log 2 q}+ n s + qn b ln(2) Υ q,ns,n b (ρ) + k T =0 n kt b k T k T i 1 i 1=0 i 2=0... k T q 2 n=1 in i q 1=0 [ q 1 q 1 ] 1+ρ n=1 ain ρ + a kt n=1 in ρ q 1 n=1 i n! (k T q 1 n=1 i n)! ( ) 1 a ρ 1 + ns 1+ρ n b For q > 16 computation becomes numerically challenging

Slide 7/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Erasure Channel Approximation Use q-ary EC as a surrogate for Poisson PPM channel By matching the capacities of both channels we obtain: ɛ = 1 C PPM log 2 q Gallager s error exponent for the q-ary EC: ( ) 1 ɛ log 2 q + ɛ r log 2 q 0 < ɛ < ɛ c E qec (r) = D(B 1 r B ɛ ) ɛ c ɛ 1 r 0 ɛ > 1 r 1 r r(q 1)+1 ɛ c B 1 r and B ɛ are two Bernoulli distributions with parameters 1 r and ɛ.

Slide 8/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Singleton Bound via EC Approximation Singleton bound, denoting the BLEP of idealized maximum distance separable code on an EC where S N B{N, ɛ}. Again we require: P (S) e = P (S) e N i=n K+1 ( ) N ɛ i (1 ɛ) N i i = Pr (S N > N(1 r)) ɛ = 1 C PPM log 2 q

Slide 9/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Dispersion Approach - Definitions Let us consider a generic discrete memoryless channel (DMC). ( ) PY X (y X=x) Information density: i(x, y) log 2 P Y (y) { Mutual information: I(P X ; P Y X ) E X EY X=x {i(x, y)} } Conditional information variance: V(P X ; P Y X ) E X { EY X=x { [i(x, y)] 2 } [E Y X=x {i(x, y)}] 2} Capacity: C max PX { I(PX ; P Y X ) } Dispersion: V min PX Π { V(PX ; P Y X ) } where Π = {P X : I(P X ; P Y X ) = C}.

Slide 10/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Dispersion Approach - Applications Capacity: R < C lim N P e = 0 Shannon (1948) Dispersion: [ P e Q (C R) V N Strassen (1962), recently [1,2] [1] Hayashi, Information Spectrum Approach to Second-order Coding Rate in Channel Coding, IEEE Trans. Inf. Theory 2009 [2] Polyanskiy et al., Channel Coding Rate in the Finite Blocklength Regime, IEEE Trans. Inf. Theory 2010 ]

Slide 11/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Converse Theorem Gaussian Approximation For the Poisson PPM channel the dispersion V PPM results in V PPM = E Y X [0] =P { (*** novel result ***) log 2 2 [ q 1 t=0 ( ) ]} Y [t] Y [0] ns + 1 [log n 2 q C PPM ] 2 b Thus: { } N P e (N, r, n s, n b ) Q [C PPM (q, n s, n b ) r log 2 q] V PPM (q, n s, n b )

Slide 12/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Converse Theorem EC Approximation For the q-ary EC we have V qec = ɛ(1 ɛ) log 2 2 q; [bits2 /channel use 2 ] [ ] P (qec) N(1 r) Nɛ e Q Nɛ(1 ɛ) = Pr (G N > N(1 r)) where G N N {Nɛ, Nɛ(1 ɛ)} and ɛ = 1 C PPM log 2 q We obtain an approximation of the Singleton bound by replacing the binomial distribution B{N, ɛ} by a Gaussian distribution N {Nɛ, Nɛ(1 ɛ)}.

Slide 13/17 Flavio Zabini The PPM Poisson Channel Finite-Length Benchmarks August 21, 2014 Finite-Length Benchmarks Results BLEP 10 0 10 1 10 2 10 3 10 4 10 5 N = 2736 N = 684 Gallager RCB RCB via q-ec approx. Converse Converse via q-ec approx. Singleton via q-ec approx. Different bounds/benchmarks: q = 16 n b = 0.02 N = {684, 2736} r = 0.5 10 6 12.5 12 11.5 11 ns/q [db]

Outline 1 Preliminaries 2 Finite-Length Benchmarks 3 Code Design 4 Conclusions

Slide 14/17 Flavio Zabini The PPM Poisson Channel Code Design August 21, 2014 Code Design Surrogate 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 by protograph EXIT analysis Non-binary protograph: Each VN a GF(q) symbol Edges possess edge labels

Slide 15/17 Flavio Zabini The PPM Poisson Channel Code Design August 21, 2014 Code Design Selected Protographs Fixing the size of base matrix to 3 5 with one punctured column, we obtain 2 1 1 1 0 B 1 = 1 2 1 1 0 2 0 0 0 1 with iterative decoding threshold of the ensemble of ɛ = 0.478 We consider the base matrix of a (binary) LDPC code from [3] 3 1 1 1 0 B 2 = 1 3 1 1 0 2 0 0 0 1 with ɛ = 0.442 [3] Barsoum et al., EXIT Function Aided Design of Iteratively Decodable Codes for the Poisson PPM Channel, IEEE TCOM 2010

Slide 16/17 Flavio Zabini The PPM Poisson Channel Code Design August 21, 2014 Code Design Performance curves CER, BER 10 0 10 1 10 2 10 3 10 4 nb =0.002 nb =0.2 nb =2 Converse Converse via q-ec approx. 10 5 CER code C1 64 CER code C2 64 BER code C2 64 BER code C3 2 10 6 20 19 18 17 16 15 14 13 12 11 10 ns/q [db] 3 LDPC codes: C 64 1 (1368, 684) over q = 64 from B 1 C 64 2 (1368, 684) over q = 64 from B 2 C 2 3(8192, 4096) over q = 2 from B 2 2 benchmarks: Converse bound normal approx. EC approx. of converse bound

Outline 1 Preliminaries 2 Finite-Length Benchmarks 3 Code Design 4 Conclusions

Slide 17/17 Flavio Zabini The PPM Poisson Channel Conclusions August 21, 2014 Conclusions Several finite-length benchmarks for the PPM-modulated code performance on the Poisson channel have been developed Converse bound Gaussian approximation: less complex than RCB approach but more accurate for the PPM Poisson channel with realistic parameters (very close to Singleton bound when an EC-equivalent channel is considered) 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 proposed The density evolution analysis turns to be extremely simple 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 Bologna flavio.zabini2@unibo.it

Code Design Decoding Thresholds Decoding thresholds on the Poisson PPM channel computed via Monte Carlo density evolution Thresholds in terms of n s /q [db] for protograph code ensembles specified by B 1, B 2, q = 64, and various n b : Background noise n b 0.002 0.02 0.2 2 B 1 18.87 17.54 15.19 11.71 B 2 18.47 17.00 14.65 11.27 Limit 19.13 17.79 15.51 12.08

Numerical Results LDPC Code vs. Serially Concatenated Pulse Position Modulation (SCPPM) CER 10 0 10 1 10 2 10 3 10 4 Random coding bound Code C4(2720, 1360) Serially concatenated PPM scheme (2520,1260) Highly irregular code C 4(2720, 1360) over finite field of order q = 64 Background noise n b = 0.0025 RCB as a reference 20 19.5 19 18.5 18 17.5 17 16.5 16 γ [db]