Robust Channel Shortening Equaliser Design

1
Robust Channel Shortening Equaliser Design

• Cenk Toker and Semir Altinis
• Hacettepe University, Ankara, Turkey

2
Outline

• Why channel shortening?
• MLSE, MCM
• MMSE channel shortening equaliser
• Robust equaliser design
• Stochastic
• Worst case
• Results and Conclusions

3
MLSE

• MLSE is a very effective tool to combat ISI.
• Minimises the following metric
• Viterbi Algorithm can efficiently solve this
  problem
• Complexity Number of states M L (M4 for
  QPSK)
• can easily become infeasible with increasing
  channel length.

4
MCM

• Another efficient method to combat multipath
  channel.
• Popular candidate for next generation systems.
• Requires a cyclic prefix of length at least as
  long as the channel to maintain orthogonality (
  ).
• Throughput efficiency decreases as the length of
  the channel increases.

5
Long Channel Impulse Response

• Length of the multipath channel affects the
  performance and complexity of both a
  single-carrier and multi-carrier system, i.e.
• SC Complexity of Viterbi algorithm increases
  exponentially,
• MC Throughput efficiency and BER performance
  decreases.
• Solution
• Channel Shortening Equalisation The effective
  length of the channel after linear equalisation
  is shortened to an allowable level.
• ( Not to a single spike as in total
  equalisation.)

6
Channel Shortening Equalisation

• MMSE criterion is considered
• The receiver filter w,
• the target impulse response b and
• the delay d are designed in order to minimise

7
Channel Shortening Equalisation

nk
e k
zk
xk
H
w


zk
b
z - d

• Error
• Receiver filter coefficients
• Target Impulse Response

8
Channel Shortening Equalisation
Channel (50 taps)
Equaliser IR
Equalised Channel (10 taps)
9
Estimation Error

• MMSE CSE assumes perfect knowledge of the
  channel, i.e. H,
• In reality, channel is estimated at the receiver,
• Estimates may include uncertainty due to
• Estimation error,
• Noise,
• Quantization, etc.
• Under these uncertainties, performance of MMSE
  CSE may degrade.
• Solution Robust equaliser design

10
Robust Equalisation

• Two main approaches
• Worst-case min-max problem
• Equaliser is designed to minimise the cost
  function under the maximum uncertainty condition.
• how often worst case uncertainty occurs?
• Stochatic approach
• Uncertainty is modeled as a random variable whose
  only statistics are known (mean, variance)
• Equaliser is designed to minimise the cost
  function by considering these statistics.

11
Robust Equalisation

• Channel model
• H is known at the receiver (estimated)
• Elements of DH are
• zero mean Gaussian rv.s with variance .

uncertainty
estimatedchannel
actualchannel
12
Robust Equalisation

• Error becomes
• Problem optimised by the receiver
• and Target Impulse Response
• where

( for i.i.d. xn and dhi. )
13
Simulations

• A single carrier scenario with MLSE is
  considered.
• Original channel of length 6 is shortened to 2
  taps.
• Viterbi Algorithm has 414 states instead of
  451024 states.
• i.i.d. channel coefficients and equal variance
  uncertainty taps are assumed.
• It is assumed that the variance on the
  uncertainty is known.
• To minimise the effect of the equaliser length, a
  50 tap filter is utilised.
• Nominal MMSE CSE Assumes only estimated channel,
• Robust MMSE CSE Takes uncertainty into account
  also.

14
Simulations

• No noise is included.
• Robust scheme can withstand 3 dB more uncertainty
  than the nominal CSE at BER10-2.
• Not as good at high uncertainty, other methods
  may be tried.

15
Simulations

• Gaussian noise is included.
• Uncertainty
• In the low SNR region, uncertainty due to noise
  dominates -gt both schemes have similar
  performances.
• In the high SNR region nominal CSE cannot
  compensate the uncertainty -gt robust CSE
  outperforms nominal CSE.
• Transition occurs at SNR20 dB.

16
Conclusions

• We proposed a channel shortening equaliser which
  is robust in the stochastic sense.
• If the uncertainty is modelled as zero mean
  Gaussian r.v.s, only the variance is required and
  the channel uncertainty appears to have similar
  effect the the additive noise.
• Calculation of the robust equaliser is very
  similar to the nominal one and introduce
  negligible computation complexity.
• It was demonstrated that the proposed equaliser
  significantly outperforms the nominal one in the
  medium-to-high SNR region.

17
Future work

• Although a significant gain is achieved with the
  proposed equaliser, there may still be some room
  for improvement when an Hinf equaliser is used.
• MIMO channel shortening may be a part of the next
  generation telecommunication systems. Since the
  channel will still have to be estimated, the
  extension of the proposed algorithm to MIMO
  channels may be sought.