Optimal Finite Alphabet Sources Over Partial Response Channels

1
Optimal Finite Alphabet Sources Over Partial
Response Channels

• Deepak Kumar
• Advisors Scott L. Miller Krishna R. Narayanan
• Texas AM University

2
Outline of Presentation

• Problem Definition
• Background Tools
• Survey of MIRC
• Spectrum Matching Codes
• Conclusions

3
Channel Model

• Capacity

4
Perspective
5
Background Tools

• Shannon-McMillan-Breiman Theorem
• BCJR Algorithm

6
Kavcics Algorithm

• Computes the optimal probability distribution of
  a Markov source over AWGN channel

7
Kavcics algorithm (contd.)

• Initialize P
• Expectation Simulate and estimate T
• Maximization

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Kavcics Algorithm (contd.)
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Kavcics Algorithm (Discussion)

• Special Cases
• Noiseless Entropy
• Memoryless case
• Extension to ISI channels

10
Super Source
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Channel Capacity (Lower Bounds)

• Kavcics Algorithm on Channel Extensions
• IID Capacity

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Channel Capacity (Upper Bounds)

• Waterfilling Upper Bound
• Vonotobel Upper Bound

13
Calvins algorithm for effective presentation

• Check if the audience are awake.
• It was not me but spaceman Spiff!!

14
Matched Information Rate Codes
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Design of inner code

• Idea
• Optimal Markov Source (nth extn.)
• k/n Trellis encoder

1
1/2
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Design of inner code (contd.)
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Inner code (contd.)

• Tradeoff between complexity and performance
• Large number of permutations possible

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Design of outer code results

• Target rate k/n x rout
• Tools of density evolution used for optimizing
  the k outer LDPC codes
• Threshold 0.17 dB away from trellis code
  capacity. Gain of 0.25 dB from IID capacity.

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Matched Spectrum Codes

• Idea
• Designing optimal trellis encoders
• Information rate?
• Spectrum? Why?

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Why to match spectrum?

• Water filling
• PSD of optimal Markov source
• Shannons low rate limit

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Cavers-Marchetto coding technique
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Cavers Marchetto coding technique (contd.)
Rate ½ code matched to 1-D
-1
-1
1
1
-1
1/11
-1/1-1
-1/-11
-1
1
1/-1-1
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Cavers-Marchetto technique results discussion

• Is the technique optimum?
• Generalization for any rate k/n?

Performance of codes matched to 1-D when
transmitted over
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Generalization of Cavers-Marchetto technique

1. Form a trellis with states.
2. Assign input k-tuple to each branch.
3. Select the best out of possible ouptut
   n-tuples.
4. Terminate the branches.
5. Discard redundant states.

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Energy spectrum of various filters
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Performance of rate 1/3 codes over 1-D/sqrt(2)
channel
Match filter p.i.
1-D 0.83
1-D2 0.53
1D 0.17
IID 0.5
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Performance of rate 1/3 codes over 1-D0.8D2
channel
Match filter p.i.
1-D0.8D2 0.71
1-D2 0.23
1D 0.1
IID 0.34
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Performance of matched codes over 1-D/sqrt(2)
channel
rate p.i.
2/3 0.75-
1/2 0.75
1/3 0.83
1/4 0.87
1/5 0.90
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Performance of matched codes over 1-D0.8D2
channel
rate p.i.
2/3 0.58
1/2 0.63
1/3 0.71
1/4 0.82
1/5 0.85
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Design of outer irregular LDPC codes

• EXIT chart technique to design outer LDPC codes
  over pseudo channel

Channel LLR
Extrinsic Info.
Extrinsic Info.
A priori Info.
VND
CND
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EXIT charts (contd.)
1
IE,VND
IA,CND
0
1
IA,VND
IE,CND
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EXIT chart for 1-D channel Threshold-2.6 dB
rate0.3
i
2 0.2617
3 0.3566
5 1
6 0.0236
8 0.1655
18 0.1926
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EXIT chart for 1-D0.8D2 channel Threshold-3.55
dB rate0.3
i
2 0.2621
3 0.3592
5 1
6 0.0084
8 0.1831
18 0.1872
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Conclusions and results

• Design of low complexity encoders (2 or 4 states)
• A gain of 1.4 dB (1-D channel) and 2.59 dB
  (1-D0.8D2 channel)
• Asymptotically optimal
• Performance index as a reliable asymptotic measure

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Fun stuff

• Concavity of Information rate
• Extension of Kavcics algorithm for Markov
  sources with inter dependent transition
  probabilities
• Designing optimal Markov sources to match the PSD
• Different results for different channels?

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THANK YOU !!!

• I do not suffer from insanity I enjoy every
  moment of it.