1. Hacettepe University Robust Channel Shortening Equaliser Design Cenk Toker and Semir Altıniş Hacettepe University, Ankara, Turkey

2. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design2/17 Outline Why channel shortening? –MLSE, MCM MMSE channel shortening equaliser Robust equaliser design –Stochastic –Worst case Results and Conclusions

3. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design3/17 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.

4. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design4/17 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 prefix symbol nsymbol n+1 v samplesN samples

5. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design5/17 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. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design6/17 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 + nknk xkxk zkzk bz -  zkzk ^ kk

7. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design7/17 Channel Shortening Equalisation Error: Receiver filter coefficients: Target Impulse Response: H + w + nknk xkxk zkzk bz -  zkzk ^ kk

8. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design8/17 Channel Shortening Equalisation Channel (50 taps) Equalised Channel (10 taps) Equaliser IR

9. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design9/17 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

10. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design10/17 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. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design11/17 Robust Equalisation Channel model: H is known at the receiver (estimated) Elements of  H are –zero mean Gaussian rv.s with variance. estimated channel actual channel uncertainty

12. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design12/17 Robust Equalisation Error becomes Problem optimised by the receiver: and Target Impulse Response where ( for i.i.d. x[n] and  h [i]. )

13. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design13/17 Simulations A single carrier scenario with MLSE is considered. Original channel of length 6 is shortened to 2 taps. –Viterbi Algorithm has 4 1 =4 states instead of 4 5 =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.

14. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design14/17 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.

15. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design15/17 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.

16. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design16/17 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. Hacettepe University 13 July, 2006Robust Channel Shortening Equaliser Design17/17 Future work Although a significant gain is achieved with the proposed equaliser, there may still be some room for improvement when an H inf 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.