1. 1 Coded modulation So far: Binary coding Binary modulation Will send R bits/symbol (spectral efficiency = R) Constant transmission rate: Requires bandwidth expansion by factor 1/R Until 1976: ”Coding not useful for spectral effiencies  1” ”Coding gain achieved at the expense of bandwidth expansion” Quantum leap: Coded modulation Trellis coded modulation (TCM) Block coded modulation Turbo coded modulation

2. 2 Coded modulation : What is it? Concatenation of an error correcting code (convolutional code, block code, turbo code) and a signal constellation Groups of coded bits are mapped into points in the signal constellation in a way that enhances the distance properties of the code Thus a codeword can be seen as a vector of signal points Decode, ideally, to the codeword which is closest to the received vector in terms of Euclidean distance

3. 3 Coded modulation : Fair comparisons Gain of coding? Reference system: Uncoded information mapped, k bits at a time, into signal constellation with 2 k different signal points, and with average signal power E s Reference spectral efficiency: k Typical scheme with coded modulation Rate k/(k+1) error correcting code. Coded bits, k+1 bits at a time, into signal constellation with 2 k+1 different signal points, and scaled down so that the average signal power is E s Spectral efficiency: k

4. 4 Coded modulation : Constellations

5. 5 Coded modulation : Energy per symbol Assume in all cases that the Euclidean distance between points is 2 2-AM: E s = 2  (+1) 2 /2 = 1 4-AM: E s = 2  ((+1) 2 + (+3) 2 )/4 = 5 8-AM: E s = 2  ((+1) 2 + (+3) 2 +(+5) 2 + (+7) 2 )/8 = 21

6. 6 QAM : Energy per symbol 4-QAM: E s = 4  ((+1) 2 + (+1) 2 )/4 = 2 8-CROSS: E s = 4  ( ((+1) 2 + (+1) 2 )+((0) 2 + 8) )/8 = 5.5

7. 7 Coded modulation : Energy per symbol

8. 8 Symbol error probability Uncoded modulation, QPSK: P s  2Q((E s /N 0 ) 1/2 )+ Q((2E s /N 0 ) 1/2 )  2Q((E s /N 0 ) 1/2 ) Uncoded modulation, general constellation: P s  A min Q((d min 2 /2N 0 ) 1/2 )  A min /2 e -d min 2 /4N 0 A min = Number of nearest neighbour points d min 2 = minimum squared Euclidean distance (MSE) between signal points (= 2E s in QPSK) Coded modulation in general P e (y)   A dfree /2 e -d free 2 /4N 0

9. 9 Coding gain Now the uncoded reference system and the coded system have the same spectral efficiency Asymptotic coding gain  = (d free:coded / E coded )/(d free:uncoded / E unoded )  = (E unoded / E coded ) (d free:coded /d free:uncoded ) =  c -1  d = constellation expansion factor  distance gain factor. But what is d free:coded ? Binary modulation: d free:coded is proportional to the free Hamming distance of the code. Hence, design for Hamming distance Nonbinary modulation: d free:coded depends on the code as well as on the mapping from code bits to point in the signal constellation

10. 10 Average Euclidean WEF Constellation with 2 k+1 points Let: e  {0,1} k+1, v  s, (v is label of s) v’ = (v  e)  s’. For each e, there are 2 k+1 pairs (v,v’) of this type. The distance  v 2 (e) between s and s’ varies over the set of pairs {(v,v’)} For a specified constellation and for each error vector e, the average Euclidean weight enumerating function is  e 2 (X) = 2 -k-1  v X d(s,s’) For a specified constellation and for each error vector e, the minimum Euclidean ”weight enumerating function” is  e 2 (X) = X min v d(s,s’)

11. 11 Computing the WEFs The error trellis A modified Viterbi algorithm on the error trellis with the branch labels X w H (e) can compute the Hamming WEF The same algorithm applied to the error trellis with branch labels  e 2 (X) can compute the minimum free Euclidean distance of the system The same algorithm applied to the error trellis with branch labels  e 2 (X) can compute the AEWEF of the system, provided the coded-bit-to-signal mapping is uniform

12. 12 Uniform mappings Split the signal constellation in two subsets, Q(0) and Q(1) such that Q(i) consists of points with a label v with v (0) =i Let  e,i 2 (X) = be the AEWE for e with respect to Q(i) A 1-1 mapping f: v  s is uniform iff  e,0 2 (X) =  e,1 2 (X) =,  e

13. 13 Uniform mappings: Example a 2 = (1/2 0.5 ) 2 +(1- 1/2 0.5 ) 2  0.586 b 2 = 2 c 2  3.414 d 2 = 4 Q(0) and Q(1) isomorphic. One can be obtained from the other by isometric mapping. Necessary for the existence of a uniform mapping

14. 14 Nonuniform mappings: Example No isometry between Q(0) and Q(1)

15. 15 Nonuniform mappings: Example Q(0) and Q(1) isomorphic. One can be obtained from the other by isometric mapping. Necessary but not sufficient for the existence of a uniform mapping

16. 16 Uniform mappings Lemma: Consider a (k+1,k) binary convolutional code whose output is blockwisely and uniformly mapped to a 2 k+1 -ary signal constellation. Then the SE distance between two code sequences y(D) and y’(D) is  l  v l 2 (e l )   l  2 (e l ), where the summation is over the blocks where y(D) and y’(D) differ. Proof: Follows because the mapping is uniform. Thus the MFSE can be computed by a modified Viterbi algorithm on the error trellis, with MEWEs as edge labels By similar reasoning, the average WEF can be computed by using an error trellis with the AEWEs as edge labels

17. 17 Two commonly used regular mappings Gray mapping The labels of two adjacent signal points will differ in only one position Used for uncoded modulation. Also in coded modulation, as it is distance preserving in some cases, for example QPSK Natural mapping Signal points are labelled in ascending order (integerwise) Used for applications which needs to be robust against carrier phase errors

18. 18 Examples: QPSK (  min 2 =2)

19. 19 Using the error trellis to compute distance 52.47.26 3 2.410

20. 20 Example: R=2/3 on 8-PSK

21. 21 Example: R=2/3 on 8-PSK 1.76 But uncoded QPSK has  min 2 =2...

22. 22 Example: R=1/2 on 8-PSK 1.76 Parallel branches 4.0

23. 23 Example: R=1/2 on 8-PSK 4.59 Best Possible

24. 24 Initial rules for design of coded modulation MSE distance between parallel branches should be maximized Branches in the modified error trellis leaving and entering the same state should have the largest possible MSE distance

25. 25 On uniform and non-uniform mappings For nonuniform mappings: Calculations on the error trellis will provide only a lower bound on the minimum distance (Example 18.5) More difficult to analyze (but the system as such may be as good as or better than one using a uniform mapping) Stricter condition: Geometric uniform mappings Even easier to analyze Most systems are not GU

26. 26 Suggested exercises 18.1-18.10