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1、Modulation codes for the deep-space optical channelBruce Moision, Jon Hamkins, Matt Klimesh, Robert McEliece Jet Propulsion Laboratory Pasadena, CA, USADIMACS, March 2526, 2004March 2526, 2004DIMACSPage 1The deep-space optical channel MarsTelesat,scheduledto launch in 2009 5W, 10100 Mbps optical lin
2、k demonstration 100W, 1.1 Mbps X-band 35W, 1.5 Mbps Ka-bandDeep-space optical communications channelConstraintsnon-coherent, direct detectionTs= slot duration (pulse-width) 2 nsPav= average signal photons/slotPpk= maximum signal photons/pulseModelMemoryless PoissonMarch 2526, 2004DIMACSPage 2Poisson
3、 channelXp(y|x = 0)p(y|x = 1)YDeep space optical channel modeled as binary-input, memoryless, Poisson.p0(k) = p(y = k|x = 0) =nkbenbk!p1(k) = p(y = k|x = 1) =(nb+ ns)ke(nb+ns) k!P(x = 1) =1 M= duty cycle (mean pulses per slot)Peak powerns Ppkphotons/pulseAverage powerns/M Pavphotons/slot ns minMPav,
4、PpkMarch 2526, 2004DIMACSPage 3Poisson channelCapacity parameterized by Pav, optimized over M.C(M) =1 MEY |1logp1(Y ) p(Y )+M 1 MEY |0logp0(Y ) p(Y )103102101100101102101100Capacity ( bits per slot )nb= 1.0Pav=ns Mphotons/slotM = 24 8 1664 128 256 512 1024 204832104102100106108Capacity ( bits per se
5、cond )nb= 0.01Pav=ns Mphotons/slotoperating points for Mars linkpeak power constraint M 128March 2526, 2004DIMACSPage 4Pulse-position-modulation We can achieve low duty cycles and high peak to average power ratios by using PPM. M-PPM maps a binary log2M tuple to a M-ary binary vector with a single o
6、ne in the slot indicated by the input. Example: M = 8, mapping of 101001.7654320432107615 PPM achieves a duty cycle of 1/M Straight-forward to implement and analyze Known to be an efficient modulation for the Poisson channelPierce, 78, McEliece,Welch, 79, Butman et. al., 80, Lipes, 80,Wyner, 88 PPM
7、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 fixthe set (a transitive set), e.g., PPM, multipulse PPM.a group of permutations G such that for each g G, gS = S and for eac
8、h xi,xj S there exists g Gsuch that xi= g(xj), where gis the mapping imposed by g.March 2526, 2004DIMACSPage 5Capacity of Generalized PPMXp0= p(y|x = 0)p1= p(y|x = 1)Ybinary DMCLet S = x1,x2,.,xs be a set of length n vectors and pX() a probability distribu- tion on S.C = maxpXI(X;Y)Theorem 1 If S is
9、 a transitive set, then CSif achieved by a uniform distribution on S.Theorem 2 On a binary input channel with p1(y)/p0(y) 0nslog2+O(n2s),nb= 0 for fixed order M, asymptotic slope in log-log domain is 1 for nb= 0, 2 for nb 0 implies 1 dB increase in signal power compensates for 2 dB in- crease in noi
10、se power (for small ns) C is concave in nsfor nb= 0 but not for nb 0 (single inflection point) time-sharing (using pairs ns,1,ns,2) is advantageous (up to peak power constraint)103102101106107108Capacity ( bits per second )ns Mnb= 1nb= 0time-sharingM = 64March 2526, 2004DIMACSPage 8Poisson PPM Capac
11、ity: 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(ex). Thenf(x + y) f(x) + f(y)subadditivef(x + (1 )y)? f(x) + (1 )f(y)convex In practice, M chosen to be a power of 2.Corollary 2 For M = 2j, (take k = 2,m = M,n
12、= M/2 in above Theorem)C(2M) C(M) C(M) C(M/2)convex C(M) Mis decreasing in MMarch 2526, 2004DIMACSPage 9Poisson PPM Capacity: invariance to slot width For M a power of two, and fixed ns, C(M)/M is monotonically decreasing in M. Suppose Ppk/Pavis a power of two. Then optimum order sat-isfies M Ppk/Pa
13、v. Let Tsbe the slot width. Nor- malize photon arrival rates and capacity by the slot width. Let s=nsTsphotons/second, b= nbTsphotons/second. For small ns,C(M) MTsM(M 1) 2ln2?2s b bits/second.102100102103102101100C(M)/M( bits per slot )ns(photons/pulse)nb= 1.0increasingM102100102103102101100101Capac
14、ity ( bits per second )ns MTsphotons/secondnb= 0.1,Ts= 0.1nb= 1.0,Ts= 1.0nb= 0.01,Ts= 0.01nb= 10,Ts= 10nb Ts= 1March 2526, 2004DIMACSPage 10Achieving capacity: Coding and ModulationQouter codecodeinnermodulation interleaver xreceivedyuser data uchannelouter codeinner codeRSPPMReed-Solomon (n,k) = (M
15、 1,k),M-PPM = 1,McEliece, 81, 1,Hamkins, Moi-sion, 03SCPPMconvolutional codeaccumulate-M-PPM(w/o accumulate)Massey, 81, (iterate with PPM)Hamkins, Moision, 02PCPPMparallel concatenated convolutional codeM-PPMKiasaleh, 98,Hamkins, 99,(DTMRF, iter- ate with PPM)Peleg, Shamai, 00March 2526, 2004DIMACSP
16、age 11Predicting iterative decoding performanceProb(bit error) =1 2kXu, ud(u, u) kP( u|u)The Bhattacharrya bound is commonly used to bound the pairwise error probabilityP( u|u) P2( x|x) Xkp p0(k)p1(k)!d(x, x)=: zd(x, x)For constant Hamming weight coded sequences (such as generalized binary PPM) on any channel with a monotonic likelihood rati
