Robust Channel Shortening Equali s er Design

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

2. Outline • Why channel shortening? • MLSE, MCM • MMSE channel shortening equaliser • Robust equaliser design • Stochastic • Worst case • Results and Conclusions Robust Channel Shortening Equaliser Design

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 (M=4 for QPSK) • can easily become infeasible with increasing channel length. Robust Channel Shortening Equaliser Design

4. prefix symbol n prefix symbol n+1 v samples N samples v samples N samples 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. Robust Channel Shortening Equaliser Design

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.) Robust Channel Shortening Equaliser Design

6. nk ^ ek xk zk H w + + zk z - d b 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 Robust Channel Shortening Equaliser Design

7. Channel Shortening Equalisation ^ nk ek zk xk • Error: • Receiver filter coefficients: • Target Impulse Response: H w + + zk z - d b Robust Channel Shortening Equaliser Design

8. Channel Shortening Equalisation Channel (50 taps) Equaliser IR Equalised Channel (10 taps) Robust Channel Shortening Equaliser Design

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 Robust Channel Shortening 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. Robust Channel Shortening Equaliser Design

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 Robust Channel Shortening Equaliser Design

12. Robust Equalisation • Error becomes • Problem optimised by the receiver: • and Target Impulse Response where ( for i.i.d. x[n] and dh[i]. ) Robust Channel Shortening Equaliser Design

13. 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. Robust Channel Shortening Equaliser Design

14. 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. Robust Channel Shortening Equaliser Design

15. 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. Robust Channel Shortening Equaliser Design

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. Robust Channel Shortening Equaliser Design

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. Robust Channel Shortening Equaliser Design