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Coded modulation So far:

Binary coding Binary modulation Will send R information bits/symbol (spectral efficiency = R) Constant transmission rate: Requires bandwidth expansion by a

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 (BCM) Turbo coded modulation

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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 squared Euclidean distance (assuming an AWGN channel)

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Coded modulation: Fair comparisons Gain of coding? Reference system:

Uncoded information mapped, k bits at a time, into signal constellation with 2k different signal points, and with average signal energy Es

Reference spectral efficiency is k Typical scheme with coded modulation:

Rate k/(k+1) error correcting code. Coded bits are mapped, k+1 bits at a time, into signal constellation with 2k+1 different signal points, and scaled down so that the average signal energy is Es

Spectral efficiency is k

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Coded modulation: Constellations

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Coded modulation: Energy per symbol

Assume in all cases that the minimum Euclidean distance between two points is 2

2-AM: Es = 2(+1)2 /2 = 1

4-AM: Es = 2((+1)2 + (+3)2)/4 = 5

8-AM: Es = 2((+1)2 + (+3)2 + (+5)2 + (+7)2)/8 = 21

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QAM: Energy per symbol

Assume in all cases that the minimum Euclidean distance between two points is 2

4-QAM: Es = 4((+1)2 + (+1)2)/4 = 2

8-CROSS: Es = 4(((+1)2 + (+1)2) + ((0)2 + (+3)2))/8 = 5.5

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Coded modulation: Energy per symbol

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Symbol error probability Uncoded modulation (QPSK):

Ps 2Q((Es/N0)1/2) + Q((2Es/N0)1/2) 2Q((Es/N0)1/2)

Uncoded modulation (general constellation): Ps AminQ((dmin2/2N0)1/2) (Amin/ 2) e

-dmin2/4N0

Amin = number of nearest neighbour points dmin2 = minimum squared Euclidean distance

(MSE) between signal points (= 2Es for QPSK) Coded modulation:

Pe (Adfree/ 2) e-dfree2/4N0

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Asymptotic coding gain Now, the uncoded reference system and the coded system have the

same spectral efficiency Asymptotic coding gain = (dfree:coded2 / Ecoded)/(dmin:uncoded2 / Euncoded) = (Euncoded / Ecoded ) (dfree:coded2 / dmin:uncoded2)

= c-1 d = (constellation expansion factor)-1 distance gain factor

But what is dfree:coded?

Binary modulation: dfree:coded is proportional to the free Hamming distance of the code. Hence, design for Hamming distance

Nonbinary modulation: dfree:coded depends on the code as well as on the mapping from code bits to points in the signal constellation

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Average and minimum Euclidean WE Constellation with 2k+1 points Let e {0,1}k+1, v s (v is label of s), and v = (v e) s For each e, there are 2k+1 pairs (v,v) of this type. The

distance v2(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 WE (AEWE) ise2(X) = 2-k-1 v X

v2(e)

For a specified constellation and for each error vector e, the minimum Euclidean WE (MEWE) ise2(X) = X

minv v2(e)

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Computing the WEFs Use the error trellis Mason's gain formula on the modified error state diagram

with the branch labels X w(e) can be used to compute the Hamming WEF

The same algorithm applied to the modified error state diagram with branch labels e2(X) can compute the minimum free squared Euclidean (MFSE) distance of the system, provided the code-bit-to-signal mapping is uniform

The same algorithm applied to the modified error state diagram with branch labels e2(X) can compute the average weight enumerating function of the system, provided the code-bit-to-signal mapping is uniform

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Uniform mapping 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,i2(X) be the AEWE for e with respect to Q(i)

A 1-1 mapping f: v s is uniform iff. e,02(X) = e,12(X) e

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Uniform mapping: Example

Q(0) and Q(1) isomorphic. One can be obtained from the other by isometric mapping. Necessary for the existence of a uniform mapping

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Nonuniform mapping: ExampleNo isometry between Q(0) and Q(1)

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Nonuniform mapping: ExampleQ(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

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Uniform mapping Lemma: Consider a (k+1,k) binary convolutional code

whose output is blockwisely and uniformly mapped to a 2k+1-ary signal constellation. Then, for each binary error sequence e(D) in the error trellis, there exists a pair of sequences y(D) and y(D) such that

l vl

2(el) =

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 distance can be computed by a modified

Viterbi algorithm on the error trellis, with the MEWEs as edge labels

By similar reasoning, the average WEF can be computed by using a modified error state diagram with the AEWEs as edge labels

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Two commonly used mappingsGray mapping: The labels of two adjacent signal points will differ in only

one position Used for uncoded modulation. Also used in coded

modulation, as it is distance preserving in some cases, for example for QPSK

Natural mapping: Signal points are labeled in ascending order (integerwise) Used for applications which need to be robust against carrier

phase errors

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Examples: QPSK (min2 = 2)

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Using the error trellis to compute distance

5 2.4 7.2 6

3 7.2 2.4 10

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Example: R = 2/3 with 8-PSK

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Example: R = 2/3 with 8-PSK

1.76 But uncoded QPSK has min2 = 2...

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Example: R = 1/2 with 8-PSK

1.76

Parallelbranches 4.0

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Example: R = 1/2 with 8-PSK

4.59

Best possible

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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

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On uniform and nonuniform 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 (GU) mappings Even easier to analyze Most systems are not GU