Modulation codes for the deep-space optical channel Bruce Moision, Jon Hamkins, Matt Klimesh, Robert McEliece Jet Propulsion Laboratory Pasadena, CA, USA DIMACS, March 25–26, 2004 March 25–26, 2004 DIMACS Page 1
The deep-space optical channel • Mars Telesat, scheduled to launch in 2009 • 5 W , 10–100 Mbps optical link demonstration • 100W, 1 . 1 Mbps X-band • 35W, 1 . 5 Mbps Ka-band Deep-space optical communications channel Constraints non-coherent, direct detection T s = slot duration (pulse-width) ≥ 2 ns P av = average signal photons/slot P pk = maximum signal photons/pulse Model Memoryless Poisson March 25–26, 2004 DIMACS Page 2
Poisson channel p ( y | x = 0) X Y p ( y | x = 1) Deep space optical channel modeled as binary-input, memoryless, Poisson. b e − n b p 0 ( k ) = p ( y = k | x = 0) = n k k ! p 1 ( k ) = p ( y = k | x = 1) = ( n b + n s ) k e − ( n b + n s ) k ! P ( x = 1) = 1 M = duty cycle (mean pulses per slot) Peak power n s ≤ P pk photons/pulse Average power n s /M ≤ P av photons/slot ⇒ n s ≤ min { MP av , P pk } March 25–26, 2004 DIMACS Page 3
Poisson channel Capacity parameterized by P av , optimized over M . C ( M ) = 1 M E Y | 1 log p 1 ( Y ) p ( Y ) + M − 1 E Y | 0 log p 0 ( Y ) M p ( Y ) M = 2 0 10 4 n b = 0 . 01 n b = 1 . 0 Capacity ( bits per second ) 8 8 10 Capacity ( bits per slot ) 16 32 64 −1 10 128 256 512 6 10 operating points for Mars link 1024 −2 10 2048 peak power constraint ⇒ M ≤ 128 −4 −2 0 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 P av = n s M photons/slot P av = n s M photons/slot March 25–26, 2004 DIMACS Page 4
Pulse-position-modulation We can achieve low duty cycles and high peak to average power ratios by using PPM. M -PPM maps a binary log 2 M tuple to a M -ary binary vector with a single one in the slot indicated by the input. Example: M = 8, mapping of 101001. 7 6 5 4 3 2 1 0 7 6 5 4 3 2 1 0 • PPM achieves a duty cycle of 1 /M • Straight-forward to implement and analyze • Known to be an efficient modulation for the Poisson channel [Pierce, 78], [McEliece, Welch, 79], [Butman et. al., 80], [Lipes, 80],[Wyner, 88] • PPM satisfies the property that each symbol is a coordinate permutation of another • Generalized PPM : a set of vectors S such that there is a group of coordinate permu- tations that fix ∗ the set (a transitive set), e.g., PPM, multipulse PPM. ∗ a group of permutations G such that for each g ∈ G , gS = S and for each x i , x j ∈ S there exists g ∈ G such that x i = σ g ( x j ), where σ g is the mapping imposed by g . March 25–26, 2004 DIMACS Page 5
Capacity of Generalized PPM binary DMC p 0 = p ( y | x = 0) X Y p 1 = p ( y | x = 1) Let S = { x 1 , x 2 , . . . , x s } be a set of length n vectors and p X ( · ) a probability distribu- tion on S . C = max p X I ( X ; Y ) Theorem 1 If S is a transitive set, then C S if achieved by a uniform distribution on S . Theorem 2 On a binary input channel with p 1 ( y ) /p 0 ( y ) < ∞ , C = d H D ( p 1 || p 0 ) − D ( p ( y ) || p ( y | 0 )) bits/symbol where D ( ·||· ) is the Kullback-Liebler distance, d H is the symbol Hamming weight, p ( y ) is the density of n -vector Y , p ( y | 0 ) the density of n -vector of noise slots. March 25–26, 2004 DIMACS Page 6
Capacity of PPM Corollary 1 For the binary M -ary PPM channel, C ( M ) = D ( p 1 || p 0 ) − D ( p ( y ) || p ( y | 0 )) ≤ D ( p 1 || p 0 ) Theorem 3 For fixed n s , n b , lim M →∞ C ( M ) = D ( p 1 || p 0 ) . Poisson channel: D ( p 1 || p 0 ) = ( n s + n b ) log(1 + n s /n b ) − n s . This term is also tight for small n s . −1 10 unconstrained n b = 0 . 01 D ( p 0 | p 1 ) 8 Capacity ( bits per second ) 10 Capacity (bits/slot) PPM 6 C ( M ) 10 −2 10 negligible loss in expected operating region n b = 1 . 0 , M = 64 −4 −2 0 10 10 10 −2 −1 10 10 P av = n s P av = n s M M March 25–26, 2004 DIMACS Page 7
Poisson PPM Capacity: small n s asymptotes, concavity in n s • n 2 M − 1 n b + O ( n 3 s ) , n b > 0 C ( M ) s 2 M log 2 = M n s log 2 + O ( n 2 s ) , n b = 0 M = 64 8 Capacity ( bits per second ) 10 • for fixed order M , asymptotic slope n b = 0 in log-log domain is 1 for n b = 0, 2 for n b > 0 7 10 • implies 1 dB increase in signal n b = 1 power compensates for 2 dB in- 6 crease in noise power (for small n s ) time-sharing 10 −3 −2 −1 • C is concave in n s for n b = 0 but not 10 10 10 for n b > 0 (single inflection point) n s M • time-sharing (using pairs n s, 1 , n s, 2 ) is advantageous (up to peak power constraint) March 25–26, 2004 DIMACS Page 8
Poisson PPM Capacity: convexity in M ? Theorem 4 For n ≤ m , C ( km ) + C ( n ) ≤ C ( kn ) + C ( m ) C ( km ) ≤ C ( k ) + C ( m ) � This is essentially a subadditivity property. Let f ( x ) = C ( e x ). Then f ( x + y ) ≤ f ( x ) + f ( y ) subadditive ? f ( αx + (1 − α ) y ) ≤ αf ( x ) + (1 − α ) f ( y ) convex ∩ In practice, M chosen to be a power of 2. Corollary 2 For M = 2 j , (take k = 2 , m = M, n = M/ 2 in above Theorem) C (2 M ) − C ( M ) ≤ C ( M ) − C ( M/ 2) convex ∩ C ( M ) is decreasing in M M � March 25–26, 2004 DIMACS Page 9
Poisson PPM Capacity: invariance to slot width 0 10 • For M a power of two, and fixed n b = 1 . 0 C ( M ) /M ( bits per slot ) n s , C ( M ) /M is monotonically −1 10 increasing M decreasing in M . −2 10 • Suppose P pk /P av is a power of two. Then optimum order sat- −3 10 isfies M ≤ P pk /P av . −2 0 2 10 10 10 • Let T s be the slot width. Nor- n s (photons/pulse) Capacity ( bits per second ) malize photon arrival rates and n b = 0 . 01 , T s = 0 . 01 1 10 n b T s = 1 capacity by the slot width. Let n b = 0 . 1 , T s = 0 . 1 0 10 λ s = n s T s photons/second, λ b = n b T s photons/second. For n b = 1 . 0 , T s = 1 . 0 −1 10 small n s , n b = 10 , T s = 10 −2 10 � λ 2 C ( M ) ≈ M ( M − 1) � s bits/second . −3 10 −2 0 2 MT s 2 ln 2 λ b 10 10 10 n s MT s photons/second March 25–26, 2004 DIMACS Page 10
Achieving capacity: Coding and Modulation modulation channel received user data interleaver u outer inner x y � code code outer code inner code Reed-Solomon ( n, k ) = ( M α − 1 , k ), RSPPM M -PPM α = 1, [McEliece, 81] , α > 1, [Hamkins, Moi- sion, 03] SCPPM convolutional code accumulate- M -PPM (w/o accumulate) [Massey, 81] , (iterate with PPM) [Hamkins, Moision, 02] PCPPM parallel concatenated convolutional code M -PPM [Kiasaleh, 98],[Hamkins, 99] ,(DTMRF, iter- ate with PPM) [Peleg, Shamai, 00] March 25–26, 2004 DIMACS Page 11
Predicting iterative decoding performance Prob(bit error) = 1 d ( u , ˆ u ) � P (ˆ u | u ) 2 k k u , ˆ u The Bhattacharrya bound is commonly used to bound the pairwise error probability � d ( x , ˆ x ) �� � =: z d ( x , ˆ x ) P (ˆ u | u ) ≤ P 2 (ˆ x | x ) < p 0 ( k ) p 1 ( k ) k For constant Hamming weight coded sequences (such as generalized binary PPM) on any channel with a monotonic likelihood ratio p 1 ( k ) /p 0 ( k ) (Gaussian, Poisson, Webb- McIntyre-Conradi), we have � x = arg max ˆ y k x k : x k =1 Hence the ML pairwise codeword error may be bounded as x | x ) = P ( S < N ) + 1 P (ˆ u | u ) ≤ P 2 (ˆ 2 P ( S = N ) x , x )) ≤ z d (ˆ x , x ) = P 2 ( d (ˆ Where S is the sum of d/ 2 signal slots, N is the sum of d/ 2 noise slots. March 25–26, 2004 DIMACS Page 12
IOWEF PPM bounds inner code outer code w x u 1 � PPM 1+ D user data interleaver PPM is a non-linear mapping, however, we can bound the distance in terms of the codeword weights � d ( w , ˆ w ) � � n � 2 ≤ d ( x , ˆ x ) ≤ 2 min log 2 M , d ( w , ˆ w ) . log 2 M Now we have � d ( x ) k n d ( u ) � �� w � � h �� � � � P b ≤ P 2 2 = k A w,h P 2 2 k log 2 M log 2 M w =1 u � = 0 h =1 where A w,h is the input-output-weight-enumerating-function (IOWEF) March 25–26, 2004 DIMACS Page 13
BER and FER bounds repeat-9 ⇒ accumulate ⇒ M = 64 PPM. Interleaver lengths 0 . 5 Kbit, 32 Kbit. 0 0 10 10 frame error rate −2 10 −2 10 bit error rate −4 10 Bhattacharyya bound Bhattacharyya bound SNR i/o threshold pairwise bound −4 10 pairwise bound capacity t i 32 Kbit b −6 K approximation 10 approximation 5 . 0 −6 simulation simulations 10 −8 10 2 interleaver sizes −18 −17 −16 −15 −14 −13 −12 −11 −17 −16 −15 −14 −13 −12 −11 photons/slot (dB) photons/slot (dB) March 25–26, 2004 DIMACS Page 14
Performance n b = 1 photon/slot M = 64 , n b = 1 . 0 photon/slot 0 10 −1 10 −1 10 uncoded −2 10 capacity bits/slot M −3 M = 64 10 P BER P M C P S −4 10 SCPPM P capacity RSPPM uncoded S R −5 10 −6 −2 10 10 −7 10 0.05 0.1 0.15 0.2 −2 −1 10 10 photons/slot photons/slot Gaps to capacity BER=10 − 6 , Poisson channel SCPPM 0 . 75 dB RSPPM 2 . 75 dB uncoded 4 . 7 dB (SCPPM: | Π | = 16384, stopping rule, max 32 operations) March 25–26, 2004 DIMACS Page 15
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