coded modulation: design issues
TRANSCRIPT
![Page 1: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/1.jpg)
1
Coded modulation: Design issues
• Signal set selection
• Labeling of the signal set
• Code selection
![Page 2: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/2.jpg)
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
![Page 3: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/3.jpg)
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
![Page 4: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/4.jpg)
4
Coded modulation: Set partitioning• (Optimum) set partitioning of 8-PSK
Isometry Q(0)↔Q(1)
![Page 5: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/5.jpg)
5
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
![Page 6: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/6.jpg)
6
Codes used with set partitioning
Select trellis state transition
Select among parallel branches
![Page 7: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/7.jpg)
7
Another example: 16-QAM MSSD doubles
![Page 8: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/8.jpg)
8
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)
![Page 9: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/9.jpg)
9
Code design
![Page 10: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/10.jpg)
10
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
![Page 11: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/11.jpg)
11
The set partitioning lemma• Note: The inequality ∆2(e) ≥ ∆q(e)
2 is almost always satisfied with equality
• Exception: 8-PSK and e = 101
![Page 12: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/12.jpg)
12
The set partitioning lemma• More exceptions: 16-QAM, e = 1001, e = 1101, e = 1111
![Page 13: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/13.jpg)
13
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
![Page 14: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/14.jpg)
14
Tables of ”best” TCM codes
![Page 15: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/15.jpg)
15
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
![Page 16: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/16.jpg)
16
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
![Page 17: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/17.jpg)
17
TCM performance analysis: Example
• Rate ½ encoder for 4-AM natural mapping, AEWEs on branches. η = 1 bit/symbol
![Page 18: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/18.jpg)
18
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
![Page 19: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/19.jpg)
19
TCM performance analysis: Example
• Rate ½ encoder for 8-PSK natural mapping, one uncoded bit, AEWEs on branches. η = 2 bits/symbol
![Page 20: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/20.jpg)
20
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
![Page 21: Coded modulation: Design issues](https://reader031.vdocuments.mx/reader031/viewer/2022012512/618ad0de45644043630c008a/html5/thumbnails/21.jpg)
21
Repetition lectures• You will give the repetition lectures
• Approximately 20-25 minutes each
• Thursday November 16
• Chapters 9, 10, 11, 12, 14, and 15