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Hacettepe University
Robust Channel Shortening Equaliser Design
Cenk Toker and Semir AltınişHacettepe University, Ankara, Turkey
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Outline
• Why channel shortening?– MLSE, MCM
• MMSE channel shortening equaliser• Robust equaliser design
– Stochastic
– Worst case
• Results and Conclusions
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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 (M=4 for QPSK)
– can easily become infeasible with increasing channel length.
21
0
1
0ˆ
ˆmin
N
n
L
llnln
x
xhyn
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• 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.
MCM
prefix
N samplesv samples
prefixsymbol n symbol n+1
v samples N samples
1Lv
vN
N
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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.)
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Channel Shortening Equalisation
• MMSE criterion is considered:– The receiver filter w, – the target impulse response b and – the delay are designed in order to minimise
H + w +
nk
xk zk
bz -
zk
^ k
2 kEJ
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Channel Shortening Equalisation
• Error:
• Receiver filter coefficients:
• Target Impulse Response:
H + w +
nk
xkzk
bz - zk
^ k
1
1
0
wnw
w
w
w
1 nn
Hxx
Hxx
HH RHHRHRbw
1
1
0
bnb
b
b
b
nwHwb HHk z )(
1 s.t. , min bbRbb HHJ
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0 50 100 150 200 250 300 3500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
taps
magnitude of the equaliser output
Channel Shortening Equalisation
0 5 10 15 20 25 30 35 40 45 500
0.05
0.1
0.15
0.2
0.25
0.3
0.35
taps
magnitude of the channel impulse response
0 50 100 150 200 250 3000
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
taps
magnitude of the equaliser impulse response
Channel(50 taps)
Equalised Channel(10 taps)
Equaliser IR
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• 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
Estimation Error
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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.
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Robust Equalisation
• Channel model:
• H is known at the receiver (estimated)• Elements of H are
– zero mean Gaussian rv.s with variance .
ΔHHH ~
estimatedchannel
actualchannel
uncertainty
2H
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Robust Equalisation
• Error becomes
• Problem optimised by the receiver:
• and Target Impulse Response
where
nwΔHHwb HHk z ))
~((
1
,
nnx
Hxx
Hxx
HH RRHHRHRbw ΔH
1 s.t. , min bbbRb HHJ
HHx ER ΔHΔHxxΔH , I2
HHn ( for i.i.d. x[n] and h[i]. )
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Simulations
• A single carrier scenario with MLSE is considered. • Original channel of length 6 is shortened to 2 taps.
– Viterbi Algorithm has 41=4 states instead of 45=1024 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.
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Simulations
• No noise is included.
• Robust scheme can withstand 3 dB more uncertainty than the nominal CSE at BER=10-2.
• Not as good at high uncertainty, other methods may be tried.
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Simulations
• Gaussian noise is included.
• Uncertainty:
• In the low SNR region, uncertainty due to noise dominates -> both schemes have similar performances.
• In the high SNR region nominal CSE cannot compensate the uncertainty -> robust CSE outperforms nominal CSE.
• Transition occurs at SNR=20 dB.
dBHH 20/ 22
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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.
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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.