Efficient Statistical Pruning for Maximum Likelihood Decoding

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Efficient Statistical Pruning for Maximum
Likelihood Decoding

• Radhika Gowaikar
• Babak Hassibi
• California Institute of Technology
• July 3, 2003

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Outline

• Integer Least Squares Problem
• Probabilistic Setup, Complexity as Random
  Variable
• Sphere Decoder
• Modified Algorithm
• Statistical Pruning, Expected Complexity
• Results
• Analysis
• Conclusions and Future Work

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Integer-Least Squares Problems

• Search space is discrete, perhaps infinite
• Given a skewed lattice
• Given a vector
• Find closest lattice point

Known to be NP-hard
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Applications in ML Decoding

• ML detection leads to integer least-squares
  problems
• Signal constellation is a subset of a lattice
  (PAM, QAM)
• Noise is AWG
• Eg. Multi-antenna systems

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Approximate Solutions

• Zero forcing cancellation
• Nulling and canceling
• Nulling and canceling with optimal ordering

But Bit Error Rate suffers
BER comparison ML vs. Approximate
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Exact Methods

• Sphere Decoding search in a hypersphere
  centered at (Fincke-Pohst Viterbo,
  Boutros Vikalo, Hassibi)

How do we find the points that are in the
hypersphere?
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Sphere Decoder

• To find points without exhaustive search
• When , this is an interval
• Use this to go from a -dimensional point to a
  (k1) dimensional point.
• Search over spheres of radius r and
• dimensions 1,2,, N.
• Use to facilitate this

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Sphere Decoder How it Works
Call
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How it Works contd.

• depends only on

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Search Space and Tree
Solve these successively --- get a
tree Complexity depends on the size of the tree
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Reducing Complexity
Not ML decoding any more
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Results
Complexity exponent and BER for N20 with QPSK
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Probability of Error

• Let e be the probability that the transmitted
  point s is not in the search space
• Can be shown that

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Finding epsilon

• can be determined exactly in terms of s

Theorem
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Computational Complexity

• is the search region at dimension

is the constellation
Need to find
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Finding
s are independent. Hence
Also, can be determined exactly
Yet have to employ approximations
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Upper Bound

• For , it needs to satisfy
  conditions.
• For upper bound, just the -th condition.

is the incomplete gamma function.
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Approximations

• Can be shown that

where and are functions of
The complexity can now be determined by Monte
Carlo simulations
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Simulation Results
Complexity exponent and BER for N20 with QAM
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Simulation Results
Complexity Exponent and BER for N50 with QAM
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Conclusions and Future Work

• Significant reduction in Complexity
• BER can be made close to optimal
• Quantify trade-off between BER and Complexity
• Compare with other decoding algorithms
• Analyze for signaling schemes with coding
• Other applications for these techniques?

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How it Works contd.
Solve these successively