hacettepe university robust channel shortening equaliser design cenk toker and semir altıniş...

Hacettepe University

Robust Channel Shortening Equaliser Design

Cenk Toker and Semir AltınişHacettepe University, Ankara, Turkey

13 July, 2006 Robust Channel Shortening Equaliser Design 2/17


Outline

• Why channel shortening?– MLSE, MCM

• MMSE channel shortening equaliser• Robust equaliser design

– Stochastic

– Worst case

• Results and Conclusions



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



• 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



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.)



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




• 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



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


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



• 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



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.



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



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]. )



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.



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.



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



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.



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.

hacettepe university robust channel shortening equaliser design cenk toker and semir altıniş...

Documents

equalisation channel

equaliser design617

equaliser design717

equaliser design817

multipath channel

channel increases

equalised channel

mcm mmse channel