Exploring the complexity limits of joint data detection and channel estimation

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Exploring the complexity limits of joint data
detection and channel estimation

• Achilleas Anastasopoulos
• EECS Department, University of Michigan, Ann
  Arbor, MI

University of Parma, Italy May 3, 2004
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Overview

• Motivation
• Theory
• Exact detection/estimation in less than
  exponential complexity for a class of problems
• Specific example sequence and symbol-by-symbol
  detection in highly correlated fading
• Application
• the family of ultra-fast decoders
• Extensions arbitrary correlation, space-time
  codes, etc.
• Conclusions

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Motivation a simple problem
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Motivation a harder problem
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Motivation a harder problem

• Solution more difficult because we cannot
• decompose it into 3 smaller problems

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Motivation a communication problem

• Data detection in correlated fading (unknown to
  the receiver)
• Maximum Likelihood Sequence Detection (MLSqD)

Complex Gaussian random process (fading),
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Motivation a communication problem

• Since the transition metric depends on the entire
  sequence, no dynamic programming (e.g., V.A.)
  solution is available ? complexity of optimal
  solution is exponential in N (i.e., test every
  possible sequence of length N)
• However if the channel coherence time is
  approximately L
• Conclusion Complexity of approximate algorithms
  is roughly exponential in L (counterintuitive
  the slower the channel, the more complex the
  decoding !?!)
• Why is this problem relevant today?

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Coding in channels with memory
Channel Constraints

Code Constraints e.g., parity-check equations

• According to the traditional belief, generation
  of the exact messages for decoding has
  exponential complexity w.r.t. channel coherence
  time

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Questions

• How accurate is the conventional wisdom that
  exact joint detection and estimation requires
  exponential complexity with respect to the
  channel coherence time?
• What is the connection with the problem of
  decoding turbo-like codes at low-SNR?
• What is the impact of the above question on the
  design of near-optimal approximate algorithms
  suited for ultra-fast integrated circuit
  implementation?

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The basic problem

• In order to present all the ideas, lets look at
  the simple problem of MAPSqD of an uncoded
  sequence in highly correlated fading
• All results generalize to the case of
  symbol-by-symbol soft metric generation
  (MAPSbSD).
• A concrete example will be used throughout the
  talk.

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Working example

• Uncoded M-PSK data sequence in complex Gaussian
  fading (fading affects both amplitude and phase).
• Fading remains constant over N symbols (time
  selective fading with long memory)

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Perfect CSI case
which can be decomposed into N simple,
symbol-by-symbol minimum distance problems
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MAPSqD Solution (no CSI)

• Complexity of maximizing seems
  exponential w.r.t. N (metric cannot be
  decomposed)
• For M-PSK, each of the MN sequences needs to be
  tested explicitly.

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Approximations

• Approximate solutions (developed over the last 15
  years)
• Memory truncation
• Linear predictive receiver LoMo90,
  YuPa95, etc.
• Non-exhaustive search PSP, M-algorithm,
  T-algorithm RaPoTz95, SeFi95, etc.
• Expectation-Maximization GeHa97
• They are all effective (especially for small
  channel memory)

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Basic contribution of this work

• The exact MAPSqD solution for this problem (and
  other problems of interest in communications) can
  be obtained with only polynomial complexity
  w.r.t. N
• Contrary to traditional belief, the slower the
  channel, the smaller the complexity
• The proof of this statement hints at approximate
  solutions with linear (and very small) complexity
  w.r.t. N

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Sketch of proof

• First, transform the MAPSqD problem to a more
  complicated double-maximization problem
• This is an exact equality
• Average likelihood ? generalized likelihood

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More definitions

• Sequence-conditioned parameter estimate
• (Least Squares solution)
• Parameter-conditioned sequence estimate
• (linear complexity w.r.t. N )
• Order of maximization two possible approaches

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Approach A Estimator-correlator
Obviously exponential complexity w.r.t. N
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Approach B Parameter space scan
Unfortunately, this method has infinite complexity
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Key idea
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Key idea
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Almost there

• Sufficient set size T N2
  
• AND
• There is a recursive algorithm to find T with
  complexity N2
• THUS
• can be found with polynomial
  complexity w.r.t. N

QED
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Parameter space partitioning example

• For each a parameter space
• boundary is defined (for BPSK) by the
• equation
• which represents a line in the complex plain

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Parameter space partitioning example
Complex plane
l2

l2

l1
l1
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Connection with Sphere Decoding

• No connection whatsoever with sphere decoding
• Sphere decoding worst case complexity is
  exponential, but average complexity (at high SNR)
  is polynomial (for sufficiently small N)
• This approach proves (worst-case/average-case)
  polynomial complexity irrespective of SNR
• However, sphere decoding is applicable to a much
  wider class of problems
• Possible research direction combine the two
  approaches

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Symbol-by-Symbol Detection

• What if we need to generate SbS reliability
  information (e.g., for turbo detection) ?
• Define a suitable metric
• Need to marginalize the sequence metric over
  nuisance parameters.
• If you choose max as the marginalization operator
  (over sequences)
• the problem becomes very similar to MAPSqD,
  and can be solved with polynomial complexity.

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Symbol-by-Symbol Detection

• Set T is no longer sufficient
• Sufficient set can be found by expanding T
• i.e., flip one bit at a time in each sequence
  in T
• Exact version of the bit flipping
• (or toggle/swap) approximate algorithms

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Practical Implications the ultra-fast receiver

• Previous results have mostly conceptual value
• However, the optimal algorithm hints at some
  ultra-fast approximate solutions
• Instead of finding the optimal partition, use an
  arbitrary partition of the parameter space
• This implies an approximate set T
• The rest of the decoder remains as in the exact
  case (i.e., expansion of T by bit flipping,
  etc)
• No multiplication operations required

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Example
Block-independent, flat, complex fading channel
(L11)
Length 4000 uniform LDPC code with variable and
check node degrees 3 and 6, respectively.
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Generalization arbitrary correlation

• Memoryless fading ? linear complexity
• Constant fading ? polynomial complexity
• What happens in the general case ?
• The answer depends on both the rank of the
  covariance matrix and its shape in a
  straightforward way

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Extension multiple antennae

• Can this receiver principle be extended to MIMO
  channels, i.e., space-time codes?
• Extension to multiple Rx antennae is
  straightforward
• Extension to multiple Tx antennae is trickier.
  Possible for
• Alamouti-type space-time codes
• other orthogonal space-time codes (ongoing
  research)

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Example 1Tx 2Rx
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Other applications

• Joint data detection and forward phase and
  frequency acquisition/tracking
• The parameter space is
• and is partitioned by straight lines (simple
  polygon processing algorithm complexityN3)
• Algorithm remains exact for small and large
  frequency offsets
• Also 2-state trellis codes

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Conclusions

• Exact MAP Sq or SbS detection in channels with
  memory is not necessarily an NP-hard problem.
• The proof of the above statement leads to new
  receiver structures (ultra-fast)
• Performance has been verified for several
  applications
• Extensions to certain classes of space-time codes
  is possible
• Several other joint data detection/estimation
  problems can be put under the same framework