1

Coded modulation: Design issues

• Signal set selection

• Labeling of the signal set

• Code selection

2

Signal set selection

• Want spectral efficiency of k bits/symbol

• Select constellation with (at least) 2k+1 points

• QAM constellations give a good tradeoff between power efficiency and implementation complexity

• Square constellations less efficient, but easier to implement than spherical ones

• Some applications require constant amplitude modulations. Then, use PSK

3

Labeling of the signal set: Set partitioning

• Idea: The selected signal constellation is split into successively finer subsets

• When a constellation is split into further subsets, the minimum SE distance between points in the new subsets are at least as large as, but sometimes strictly larger than in the original subset

• At the lowest level, the subsets contain just one point

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Coded modulation: Set partitioning• (Optimum) set partitioning of 8-PSK

Isometry Q(0)↔Q(1)

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Example: Choice of code

• Need a rate 2/3 code. Here: Rate ½ code plus an uncoded bit

Parallel branches labeled with points within a level 2 subset

Butterflies labeled with level 2 subsets

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Codes used with set partitioning

Select trellis state transition

Select among parallel branches

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Another example: 16-QAM MSSD doubles

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Code design• Want a (k’+1,k’) encoder with the following properties:

● v(0) the same for all branches leaving a given state● v(0) the same for all branches entering a given state

• A (k’+1,k’) CC is best described in terms of a parity check matrix; here a systematic feedback one

• For any codeword v: v(D) HT(D) = 0(D)

• H(D) = (h(k’)(D) / h(0)(D), ..., h(1)(D) / h(0)(D), 1)

• Requirements:● h0

(i) = 0 for i = 1, ..., k’

● hν(i) = 0 for i = 1, ..., k’

• Maximize δfree2 = min

e(D) ≠ 0(D) ∑

l ∆2(e

l)

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Code design

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The set partitioning lemma• The set partitioning lemma: Let q(e) = number of trailing

zeros in the (k’+1)-dimensional error vector e. Then∆2(e) ≥ ∆q(e)

2

• Proof:

• Consider two trellis branch labels v and v’ = v ⊕ e• These will coincide in the trailing q(e) positions

• In other words, they will belong to the same level-q(e) subtree

• Holds for any v

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The set partitioning lemma• Note: The inequality ∆2(e) ≥ ∆q(e)

2 is almost always satisfied with equality

• Exception: 8-PSK and e = 101

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The set partitioning lemma• More exceptions: 16-QAM, e = 1001, e = 1101, e = 1111

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Set partitioning: Remarks

• For e = 0, we use ∆q(e)2 = 0

• Then, the SE free distance δfree2 ≥ min

e(D) ≠ 0(D) ∑

l ∆

q(el)2

• Usually satisfied with equality

• Usually many paths in the error trellis that correspond to the minimum value min

e(D) ≠ 0(D) ∑

l ∆

q(el)2

• Usually some of these avoid error patterns e for which the inequality ∆2(e) ≥ ∆q(e)

2 is not satisfied

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Tables of ”best” TCM codes

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TCM performance analysis

• Can compute the average weight (distance) enumerating function (AWEF) Aav(X) and average input/output weight (distance) enumerating function (AIOWEF) Aav(W,X) by labeling branches in the state diagram by their AEWEs

• No parallel branches: Error probabilities bounded by

• P(E) ≤ f (δfree2 ⋅ Es/4N0) Aav(e-Es/4N0)

• Pb(E) ≤ k-1f (δfree2 ⋅ Es/4N0) ∂Aav(W,X)/∂W|W=1, X=e-Es/4N0

• f(x) = exQ((2x)1/2)

• Average signal energy = 1 when computing MSE distance

• Similar to error probabilities for binary codes with BPSK

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TCM performance analysis: Parallel branches

• Parallel branches: Error probabilities bounded by

• P(E) ≤ f (δmin2 ⋅ Es/4N0) Aav

P(e-Es/4N0)

+ f (δfree2 ⋅ Es/4N0) Aav

T(e-Es/4N0)

• Pb(E) ≤ k-1f (δmin2 ⋅ Es/4N0) ∂Aav

P(W,X)/∂W|W=1, X=e-Es/4N0

+ k-1f (δfree2 ⋅ Es/4N0) ∂Aav

T(W,X)/∂W|W=1, X=e-Es/4N0

• Superscripts P and T represent the AWEF/AIOWEF for parallel transitions and trellis paths, respectively

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TCM performance analysis: Example

• Rate ½ encoder for 4-AM natural mapping, AEWEs on branches. η = 1 bit/symbol

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TCM performance analysis: Example

• On average: Aav(W,X) = WX7.2 + 1.25W2X8.0 +...

Multiplicities are average values, and fractional (due to finite constellation and nonregular mapping

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TCM performance analysis: Example

• Rate ½ encoder for 8-PSK natural mapping, one uncoded bit, AEWEs on branches. η = 2 bits/symbol

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TCM performance analysis: Example• On average: Aav

T(W,X) = (W2+2W3+W4)X4.586

+ [(W+W2)2+0.25(W+W2)4]X5.172 +...; AavP(W,X) = WX4.0

At high SNR, most errors occur in parallel transitions

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Repetition lectures• You will give the repetition lectures

• Approximately 20-25 minutes each

• Thursday November 16

• Chapters 9, 10, 11, 12, 14, and 15