A Study on Joint Source-Channel CodedModulation (JSCCM)

Chen-Chia LaiDirected by: Prof. Po-Ning Chen

Institute of Communications Engineering, National Chiao Tung University

2017.7.28

Motivation

Contributions

1. Propose an encoding and decoding scheme for JSCCM.

2. Compare the performance of JSCCM with JSCC+MOD.

Notations

1. S , {1, 2, . . . ,K} denotes a set of K source alphabets.

2. The probability of source alphabet Pr(si ) = pi , where si ∈ S,1 ≤ i ≤ K .

3. C , {c1, c2, . . . , cK} is the corresponding codeword set ofbinary bit streams for JSCC.

4. Q , {q1, q2, q3, · · · , qM} is a set of M-ary modulationsymbols.

5. M , {m1,m2, . . . ,mK} is the corresponding codeword set ofM-ary modulation symbol streams for JSCCM.

I In notations, we write mi ∈ Q∗ since it is of variable length.

System Model

JSCC+MOD system

SL · · · S2S1- JointSource-Channel

Encoder

-cSL · · · cS2cS1 Modulator

g(·)-

qi∗ · · · qi2qi1

JSCCM system

SL · · · S2S1- JointSource-Channel

Coded Modulation

-qi∗ · · · qi2qi1

Average Codeword Length

RS(JSCCM)

=1

L

K∑i1=1

K∑i2=1

· · ·K∑

iL=1

pi1pi2 · · · piL (||mi1 ||+ ||mi2 ||+ · · ·+ ||miL ||)

=K∑i=1

pi‖mi‖, where

{‖ · ‖ ≡ # of modulation symbols

| · | ≡ # of bits

and

RS(JSCC+MOD)

=1

L

K∑i1=1

K∑i2=1

· · ·K∑

iL=1

pi1pi2 · · · piL⌈|ci1 |+ |ci2 |+ · · ·+ |ciL |

log2(M)

⌉

=1

L

K∑i1=1

K∑i2=1

· · ·K∑

iL=1

pi1pi2 · · · piL‖g(ci1ci2 · · · ciL)‖.

Free Distance

With i = (i1, i2, . . . , iL) and j = (j1, j2, . . . , jL) in SL = {1, 2, . . . ,K}L,

dE,free(JSCCM) = mini ,j∈SL and i 6=j

dE(mi1mi2 · · ·miL ,mj1mj2 · · ·mjL)

and

dE,free(JSCC+MOD) = mini ,j∈SL and i 6=j

dE(g(ci1ci2 · · · ciL), g(cj1cj2 · · · cjL))

where dE(·, ·) is the Euclidean distance.

Here, we define dE(mi1mi2 · · ·miL ,mj1mj2 · · ·mjL) =∞ ifmi1mi2 · · ·miL and mj1mj2 · · ·mjL are not of equal length.

M-ary Modulation

8PSK gray mapping

16QAM gray mapping

Signal-to-Noise Ratios (SNRs) per Source Letter

The signal-to-noise ratio per channel use is

SNR =E

N0,

where E is the average signal power (and is normalized to 1subject to equal prior probabilities over M-ary modulationsymbols), and N0 is single-sided power spectrum density.

The SNR per source symbol is given by

SNRS =E

N0· Rs.

Union Bound for Block Error RateSuppose there are Mj block transmissions of length Nj , 1 ≤ j ≤ k .

The received vector (r1, r2, . . . , rNj) can be formulated as:

rm = xj :m + nm, m = 1, 2, . . . ,Nj ,

where xj :m ∈ Q and nm ≡ i.i.d. N (0,N0/2) Gaussian.

Let Em→m′ be the error event that the m′th block transmission isdeclared, while the mth block transmission was transmitted.

We obtain

BLER ≤k∑

j=1

Mj∑m=1

pj :m∑

1≤m′≤Mjm′ 6=m

Q

√d2j :m,m′

2N0

,

where dj :m,m′ = dE(x j :m, x j :m′), and pj :m is the prior probability forthe mth block transmission of length Nj .

Performance Index from Union Bound

Denote by D the set of all possible dj :m,m′ .

Then, we can re-express the union bound in terms of coefficients

Ad =∑k

j=1

∑Mj

m=1 pj :m∑

1≤m′≤Mj ,m′ 6=m 1{dj :m,m′ = d}:

BLER ≤∑d∈D

Ad · Q

√ d2

2N0

≤ A · Q

√√√√(SNRS

2E

)(d2E,free

Rs

) ,

where

A =∑d∈D

Ad =k∑

j=1

Mj∑m=1

pj :m(Mj − 1).

Construction of an Optimal JSCCM

Goal: Minimizing the average codeword length, subject to a freeEuclidean distance lower bound d∗E,free.

Search Tree

where AP = {aP1, aP2, aP3, . . .} is an ordered set of M-ary modulationsymbol streams, and AL is updated by removing all M-ary symbolstreams in AP, whose prefix is aP1.

Metric Guiding the Search Process

Assume without loss of generality that pK ≤ pK−1 · · · ≤ p1.

Then the metric of node X with |CX| = t is defined as

f (X) ,t∑

i=1

pi · ‖mXi ‖+

K∑i=t+1

pi · ‖aXi−t‖.

Algorithmic Construction of an Optimal JSCCM

Example : p1 = 0.7, p2 = 0.3 and d∗E,free = 3, L = 2

For node 7, dE,free(C2) = dE(100, 001) =√

22 + 22 = 3.464.

MAP Decoder

1. The mTP-SMAP decoder is applied for JSCCM scheme.

2. The PFSD decoder is applied for JSCC+MOD scheme.

Trellis T mJ for C = {00, 101, 0110}

Assume the receiver knows the number of transmitted symbols J.

where Sj denotes the state that the number of decoded symbolsreceived thus far is j .

Trellis T mL,J for C = {00, 101, 0110} with known L and J

Assume that the number of transmitted codewords L is also knownto the receiver.

where Si ,j denotes the state that i codewords have been receivedand their total length is j symbols.

Sequence MAP Decoding Criterion

A sequence of J M-ary modulation symbols is transmitted overAWGN channel with output

r , (r1, r2, · · · , rJ).

Then, the MAP decision v satisfies

1

N0

J∑`=1

(dE(r`, v`)

)2 − ln Pr(v) ≤ 1

N0

J∑`=1

(dE(r`, v`)

)2 − ln Pr(v)

for all v ∈ XL,J , where Pr(v) is the prior probability to transmit v .

Here, for a given M-ary symbol-based VLECPC C,

XL,J ,

{x1x2 · · · xL : ∀x i ∈ C and

L∑i=1

‖x i‖ = J

}.

Path Metric for mTP-SMAP DecoderLet the sequence of M-ary symbol-based codewords

m(i ,j)(0,0) , m1m2 · · ·mi label the path from state Sm

0,0 to state Smi ,j

over extended trellis T mL,J .

Let the corresponding M-ary symbol sequence for m(i ,j)(0,0) be

α1α2 · · ·αj , where α` ∈ Q for 1 ≤ ` ≤ j .

The path metric of m(i ,j)(0,0) is defined as

g(m

(i ,j)(0,0)

)=

1

N0

j∑`=1

(dE(r`, α`)

)2 − ln Pr(m(i ,j)(0,0)).

Let m(j)(J) = αJαJ−1 · · ·αj that labels the (backward) path from

state SmJ back to state Sm

j over trellis T mJ . The backward path

metric of m(j)(J) is define as

h(m

(j)(J)

)=

1

N0

J∑`=j

(dE(r`, α`)

)2 − ln Pr(m(j)(J)).

Backward Viterbi on T mJ in Phase 1

If the number of codewords corresponding to p(Sm0 ) is equal to L,

output p(Sm0 ) as the MAP decision; otherwise, go to Phase 2.

Phase 2 of mTP-SMAP Decoder

Path Metric for PFSD Decoder

Let the sequence of bit-based codewords c(i ,j)(0,0) , c1c2 · · · ci label

the path from state S0,0 to state Si ,j over trellis TL,N .

Let the corresponding M-ary symbol sequence for c(i ,j)(0,0) be

α1α2 · · ·ακ, where

κ =

{bj/dlog2(M)ec, for i < L;

dN/dlog2(M)ee, for i = L

and α` ∈ Q for 1 ≤ ` ≤ κ.

The path metric of c(i ,j)(0,0) is defined as

g(c(i ,j)(0,0)

)=

1

N0

κ∑`=1

(dE(r`, α`)

)2 − ln Pr(c(i ,j)(0,0)).

PFSD Decoder

Experimental Setting

1. 8PSK/16QAM modulations.

2. AWGN channel.

3. # of source letters in one transmission block L = 5

4. Three randomly generated source distributions

Experimental Results : 8PSK

Table I: JSCCM (VLEC-8PSK) and JSCC+MOD (VLEC-BPSK)schemes for the Distribution-1 source and 8PSK modulation

d∗free(L = 5) Rs(L = 5) dE,free(L = 5) d2E,free/Rs

3 1.117229 1.325654 1.5729624 1.149939 1.530734 2.037627

VLEC 5 1.558547 1.874758 2.255125-BPSK 6 1.558547 1.874758 2.255125

7 1.999786 2.220120 2.4647301.32 1.034300 1.414214 1.9336761.53 1.324400 1.530734 1.769214

VLEC1.87 1.324400 2.000000 3.020236

-8PSK2.22 1.793100 2.274109 2.884151

BLERs (dotted lines) and union bounds (solid lines) ofVLEC-BPSKs in Table I.

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

SNRs(dB)

10-6

10-5

10-4

10-3

10-2

10-1B

lock

Err

or R

ate

Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)

BLERs (dotted lines) and union bounds (solid lines) ofVLEC-8PSKs in Table I.

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

SNRs(dB)

10-5

10-4

10-3

10-2

10-1

100B

lock

Err

or R

ate

Distribution=1 , L=5

VLEC-8PSK(1.32)VLEC-8PSK(1.53)VLEC-8PSK(1.87)VLEC-8PSK(2.22)VLEC-8PSK(1.32)VLEC-8PSK(1.53)VLEC-8PSK(1.87)VLEC-8PSK(2.22)

BLERs of VLEC-BPSKs and VLEC-8PSKs in Table I.

6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

SNRs(dB)

10-6

10-5

10-4

10-3

10-2

10-1

100B

lock

Err

or R

ate

BLER , Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-8PSK(1.32)VLEC-8PSK(1.53)VLEC-8PSK(1.87)VLEC-8PSK(2.22)

Union bounds of VLEC-BPSKs and VLEC-8PSKs in Table I.

12 14 16 18 20 22 24 26 28 30

SNRs(dB)

10-250

10-200

10-150

10-100

10-50

Blo

ck E

rror

Rat

eBest bound , Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-8PSK(1.32)VLEC-8PSK(1.53)VLEC-8PSK(1.87)VLEC-8PSK(2.22)

Experimental Results : 16QAM

Table II: JSCCM (VLEC-16QAM) and JSCC+MOD (VLEC-BPSK)schemes for the Distribution-1 source and 16QAM modulation

d∗free(L = 5) Rs(L = 5) dE,free(L = 5) d2E,free/Rs

3 0.863056 1.095445 1.3904084 0.887838 1.264911 1.802130

VLEC 5 1.193379 1.414214 1.675915-BPSK 6 1.193379 1.549193 2.011095

7 1.528067 1.897367 2.3559191.09 or 1.26 1.034300 1.264911 1.546940

1.41 1.034300 1.414214 1.933676VLEC

1.54 1.034300 1.549194 2.320412-16QAM

1.89 1.068600 1.897366 3.368892

BLERs (dotted lines) and union bounds (solid lines) ofVLEC-BPSKs in Table II.

SNRs(dB)6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

Blo

ck E

rror

Rat

e

10-5

10-4

10-3

10-2

10-1

100Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)

BLERs (dotted lines) and union bounds (solid lines) ofVLEC-16QAMs in Table II

SNRs(dB)6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

Blo

ck E

rror

Rat

e

10-5

10-4

10-3

10-2

10-1

100Distribution=1 , L=5

VLEC-16QAM(1.09,1.26)VLEC-16QAM(1.41)VLEC-16QAM(1.54)VLEC-16QAM(1.89)VLEC-16QAM(1.09,1.26)VLEC-16QAM(1.41)VLEC-16QAM(1.54)VLEC-16QAM(1.89)

BLERs of VLEC-BPSKs and VLEC-16QAMs in Table II.

SNRs(dB)6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11

Blo

ck E

rror

Rat

e

10-5

10-4

10-3

10-2

10-1

100BLER , Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-16QAM(1.09,1.26)VLEC-16QAM(1.41)VLEC-16QAM(1.54)VLEC-16QAM(1.89)

Union bounds of VLEC-BPSKs and VLEC-16QAMs in Table II.

SNRs(dB)12 14 16 18 20 22 24 26 28 30

Blo

ck E

rror

Rat

e

10-250

10-200

10-150

10-100

10-50

Best bound , Distribution=1 , L=5

VLEC-BPSK(3)VLEC-BPSK(4)VLEC-BPSK(5)VLEC-BPSK(6)VLEC-BPSK(7)VLEC-16QAM(1.09,1.26)VLEC-16QAM(1.41)VLEC-16QAM(1.54)VLEC-16QAM(1.89)

Conclusion

1. As a generalization from [2], a novel priority-first searchalgorithm for finding an optimal JSCCM was proposed.

2. Also as a generalization from [2], mTP-SMAP decoder for theMAP decoding of JSCCMs was proposed.

3. Simulation results show that JSCCM can result in a bettersystem than JSCC+MOD, in particular when a high errorprotection capability in the form of the free Euclidean distancelower bound is required.

[2] T.-Y. Wu, P.-N. Chen, F. Alajaji and Y. S. Han, ”On the design of variable-length error-correcting codes,”

IEEE Trans. Commun., vol. 61, no. 9, pp. 3553–3565, Sept. 2013.