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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

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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 Euclidean distance

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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 power Es

• 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 2k+1 different signal points, and scaled down so that the average signal power is Es

• Spectral efficiency: k

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Coded modulation : Constellations

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Coded modulation : Energy per symbol

• Assume in all cases that the Euclidean distance between 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

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

• 8-CROSS: Es = 4( ((+1)2 + (+1)2 )+((0)2 + 8) )/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((dmin

2/2N0)1/2) Amin/2 e-dmin2/4N0

• Amin = Number of nearest neighbour points• dmin

2 = minimum squared Euclidean distance (MSE) between signal points (= 2Es in QPSK)

• Coded modulation in general

• Pe (y) Adfree/2 e-dfree2/4N0

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Coding gain• Now the uncoded reference system and the coded system have the

same spectral efficiency

• Asymptotic coding gain = (dfree:coded / Ecoded )/(dfree:uncoded / Eunoded )

= (Eunoded / Ecoded ) (dfree:coded /dfree:uncoded ) = c

-1 d = constellation expansion factor 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 point in the signal constellation

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Average Euclidean WEF• Constellation with 2k+1 points • Let: e{0,1}k+1, vs, (v is label of s) v’ = (ve)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 weight enumerating function ise

2(X) = 2-k-1 v Xd(s,s’)

• For a specified constellation and for each error vector e, the minimum Euclidean ”weight enumerating function” ise

2(X) = Xminvd(s,s’)

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Computing the WEFs

• The error trellis• A modified Viterbi algorithm on the error trellis with the

branch labels XwH(e) can compute the Hamming WEF• The same algorithm applied to the error trellis with branch

labels e2(X) can compute the minimum free Euclidean

distance of the system• The same algorithm applied to the error trellis with branch

labels e2(X) can compute the AEWEF of the system,

provided the coded-bit-to-signal mapping is uniform

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

• A 1-1 mapping f: vs is uniform iff e,02(X) = e,1

2(X) =,e

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Uniform mappings: Example

a2 = (1/20.5)2+(1- 1/20.5)2 0.586

b2 = 2

c2 3.414

d2 = 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

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Nonuniform mappings: ExampleNo isometry

between Q(0) and Q(1)

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Nonuniform mappings: 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 mappings

• Lemma: Consider a (k+1,k) binary convolutional code whose output is blockwisely and uniformly mapped to a 2k+1-ary signal constellation. Then the SE distance between two code sequences y(D) and y’(D) is

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 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

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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

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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 on 8-PSK

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Example: R=2/3 on 8-PSK

1.76 But uncoded QPSK has min

2=2...

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Example: R=1/2 on 8-PSK

1.76

Parallel

branches4.0

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Example: R=1/2 on 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 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

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Suggested exercises• 18.1-18.10