Modulation codes for the deep-space optical channel Bruce Moision, - - PowerPoint PPT Presentation

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

• 5W, 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 Ts = slot duration (pulse-width) ≥ 2 ns Pav = average signal photons/slot Ppk = maximum signal photons/pulse Model Memoryless Poisson

March 25–26, 2004 DIMACS Page 2

Poisson channel

X p(y|x = 0) p(y|x = 1) Y

Deep space optical channel modeled as binary-input, memoryless, Poisson. p0(k) = p(y = k|x = 0) = nk

be−nb

k! 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 power ns ≤ Ppk photons/pulse Average power ns/M ≤ Pav photons/slot ⇒ ns ≤ min{MPav, Ppk}

March 25–26, 2004 DIMACS Page 3

Poisson channel

Capacity parameterized by Pav, optimized over M. C(M) = 1 M EY |1 log p1(Y ) p(Y ) + M − 1 M EY |0 log p0(Y ) p(Y )

10

−3

10

−2

10

−1

10 10

1

10

−2

10

−1

10

Capacity ( bits per slot ) nb = 1.0 Pav = ns

M photons/slot M = 2 4 8 16 64 128 256 512 1024 2048 32

10

−4

10

−2

10 10

6

10

8

Capacity ( bits per second ) nb = 0.01 Pav = ns

M photons/slot

• perating points for Mars link

peak power constraint ⇒ M ≤ 128

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 log2 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 4 3 2 1 7 6 1 5

• 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 xi, xj ∈ S there exists g ∈ G

such that xi = σg(xj), where σg is the mapping imposed by g.

March 25–26, 2004 DIMACS Page 5

Capacity of Generalized PPM X p0 = p(y|x = 0) p1 = p(y|x = 1) Y binary DMC

Let S = {x1, x2, . . . , xs} be a set of length n vectors and pX(·) a probability distribu- tion on S. C = max

pX I(X; Y)

Theorem 1 If S is a transitive set, then CS if achieved by a uniform distribution on S. Theorem 2 On a binary input channel with p1(y)/p0(y) < ∞, C = dHD (p1||p0) − D (p(y)||p(y|0)) bits/symbol where D(·||·) is the Kullback-Liebler distance, dH 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(p1||p0) − D(p(y)||p(y|0)) ≤ D(p1||p0) Theorem 3 For fixed ns, nb, limM→∞ C(M) = D(p1||p0). Poisson channel: D(p1||p0) = (ns + nb) log(1 + ns/nb) − ns. This term is also tight for small ns.

10

−4

10

−2

10 10

6

10

8

Capacity ( bits per second )

negligible loss in expected operating region

nb = 0.01

PPM unconstrained

Pav = ns

M

10

−2

10

−1

10

−2

10

−1

Pav = ns

M

D(p0|p1) C(M)

Capacity (bits/slot)

nb = 1.0, M = 64

March 25–26, 2004 DIMACS Page 7

Poisson PPM Capacity: small ns asymptotes, concavity in ns

• C(M)

M =   

M−1 2M log 2 n2

s

nb + O(n3 s)

, nb > 0 ns log2 +O(n2

s)

, 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 noise power (for small ns)

• C is concave in ns for 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)

10

−3

10

−2

10

−1

10

6

10

7

10

8

Capacity ( bits per second )

ns M

nb = 1 nb = 0 time-sharing M = 64

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(ex). 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 = 2j, (take k = 2, m = M, n = M/2 in above Theorem) C(2M) − C(M) ≤ C(M) − C(M/2) convex ∩ C(M) M is decreasing in M

• March 25–26, 2004

DIMACS Page 9

Poisson 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/Pav is a power of
• two. Then optimum order sat-

isfies M ≤ Ppk/Pav.

• Let Ts be the slot width. Nor-

malize photon arrival rates and capacity by the slot width. Let λs = nsTs photons/second, λb = nbTs photons/second. For small ns, C(M) MTs ≈ M(M − 1) 2 ln 2 λ2

s

λb

• bits/second.

10

−2

10 10

2

10

−3

10

−2

10

−1

10

C(M)/M ( bits per slot ) ns (photons/pulse) nb = 1.0 increasing M

10

−2

10 10

2

10

−3

10

−2

10

−1

10 10

1

Capacity ( bits per second )

ns MTs photons/second

nb = 0.1, Ts = 0.1 nb = 1.0, Ts = 1.0 nb = 0.01, Ts = 0.01 nb = 10, Ts = 10

nb Ts = 1

March 25–26, 2004 DIMACS Page 10

Achieving capacity: Coding and Modulation

• uter

code code inner modulation interleaver

x

received

y

user data

u

channel

• uter code

inner code RSPPM Reed-Solomon (n, k) = (M α − 1, k), 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 2k

• u,ˆ

u

d(u, ˆ u) k P(ˆ u|u) The Bhattacharrya bound is commonly used to bound the pairwise error probability P(ˆ u|u) ≤ P2(ˆ x|x) <

• k
• p0(k)p1(k)

d(x,ˆ

x)

=: zd(x,ˆ

x)

For constant Hamming weight coded sequences (such as generalized binary PPM) on any channel with a monotonic likelihood ratio p1(k)/p0(k) (Gaussian, Poisson, Webb- McIntyre-Conradi), we have ˆ x = arg max

x

• k:xk=1

yk Hence the ML pairwise codeword error may be bounded as P(ˆ u|u) ≤ P2(ˆ x|x) = P(S < N) + 1 2P(S = N) = P2(d(ˆ x, x)) ≤ zd(ˆ

x,x)

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

x user data PPM u

• uter code

interleaver w

1 1+D

PPM is a non-linear mapping, however, we can bound the distance in terms of the codeword weights 2 d(w, ˆ w) log2 M

• ≤ d(x, ˆ

x) ≤ 2 min

• n

log2 M , d(w, ˆ w)

• .

Now we have Pb ≤

• u=0

d(u) k P2

• 2

d(x) log2 M

• =

k

• w=1

n

• h=1

w k Aw,hP2

• 2
• h

log2 M

• where Aw,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.

−18 −17 −16 −15 −14 −13 −12 −11 10

−8

10

−6

10

−4

10

−2

10

32 Kbit . 5 K b i t

bit error rate

capacity SNR i/o threshold simulations 2 interleaver sizes

photons/slot (dB)

approximation pairwise bound Bhattacharyya bound

−17 −16 −15 −14 −13 −12 −11 10

−6

10

−4

10

−2

10

frame error rate photons/slot (dB)

approximation Bhattacharyya bound pairwise bound simulation

March 25–26, 2004 DIMACS Page 14

Performance

0.05 0.1 0.15 0.2 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

capacity SCPPM RSPPM M = 64, nb = 1.0 photon/slot

BER photons/slot

uncoded

10

−2

10

−1

10

−2

10

−1

bits/slot M = 64

photons/slot

nb = 1 photon/slot S C P P M R S P P M uncoded capacity

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

High average power, bandwidth constraints

• Can continue to use PPM at high av-

erage powers with no loss by decreasing the slot width Ts up to the Bandwidth constraints of the system.

• Past that point, we see increasing losses

by restricting modulation to PPM.

• For example, uplink has high average

power and low Bandwidth.

• How to populate this region?

−1 1 2 3 4 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

photons/slot (dB)

maximum rate for PPM

bits/slot (dB)

maximum rate for any modulation

March 25–26, 2004 DIMACS Page 16

Variable-pulse modulation

Allow variable pulses per symbol. Now sym- bol mapping may be an issue. input Gray anti-Gray symbol 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

−1 1 2 3 4 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5

bits/slot (dB) photons/slot (dB)

maximum rate for any modulation coded VPM maximum rate for PPM

March 25–26, 2004 DIMACS Page 17

March 25–26, 2004 DIMACS Page 18