Blind equalisation has been an intensive field of research leading to different successful algorithms of practical interest. Most of these algorithms are realised using adaptive transversal filters (FIR). The objective of this project is to investigate the alternative employment of recursive filter structures for equalisation of severely distorted channels, motivated by the achievements of conventional non-blind adaptive IIR filters.
Established equalisation methods with FIR filters using higher-order statistics (HOS) implicitly and explicitly are reviewed. An equaliser architecture including a recursive structure is pursued by separating the equaliser into a cascade of a recursive prewhitening filter adapted with second-order statistics (SOS) and a phase equaliser adapted with HOS.
A gradient descent algorithm implementing the described concept has been published recently by Labat et al. [1]. An improved version of this algorithm is proposed to address a problem of unstable poles in the recursive filter. The recursive prewhitening approach is also extended to block adaptive algorithms. A novel algorithm is presented, that uses a cascade of a block adaptive prewhitening filter and the block adaptive eigenvector approach (EVA) [2]. Simulation results for FIR and IIR equalisers are provided.
This project has shown that blind equalisation with IIR filters is an important alternative to traditional FIR equalisers for equalisation of severely distorted channels. The obtained results suggest that employment of recursive structures in the equaliser can provide significant performance improvements, in terms of convergence speed and complexity. This could be a contribution that makes blind equalisers more suitable for high-speed applications where nonblind equalisers are used today. Recursive equaliser structures is a concept that should be investigated further.
It is shown that the MMSE solution of the equaliser can be decomposed into two parts, where the first can be implemented as a recursive prewhitening filter that decorrelates the received signal. The recursive filter is adapted with second-order statistics and should theoretically be inherently stable. The second filter in the cascade must be implemented as an FIR filter, due to its unstable poles. The FIR filter must resort to higher-order statistics to remove the intersymbol interference that remains after the prewhitening operation.
The recursive prewhitening approach can be implemented both with gradient descent algorithms that calculates an equaliser estimate at the receival of each symbol, and with block adaptive algorithms that calculate equaliser estimates from a block of received symbols. The first approach is demonstrated by Labat et al [1]. The problem of the algorithm described here is that it adapts unstable poles.
An improved version of the mentioned algorithm is proposed in this report. The modification consists of using a feedback lattice realisation instead of a direct-form realisation in the recursive part. Simulations show that the resulting algorithm has a higher steady state error and slightly lower convergence speed. However, it ensures stability and maintains the key feature of gradient descent algorithms, which is their low complexity. Furthermore, the inclusion of the recursive part provides a convergence rate which approaches the speed of block adaptive FIR equalisers with high complexity.
The recursive prewhitening approach is also pursued for block adaptive algorithms. Block adaptive FIR filters are capable of equalising severely distorted channels, but such performance is associated with a computational cost that makes them impractical for real time implementations. A new algorithm is suggested here that uses a block adaptive recursive prewhitening filter in front of a traditional block adaptive equaliser. The effect is to reduce the required number of filter taps in the transversal filter, and hence the complexity. Equalisation performance of the new algorithm is shown to be superior to block adaptive equalisers, even if the latter is implemented with an impractically large number of filter taps.
[1] LABAT, J., O. MACCHI and C. LAOT (1998). "Adaptive decision feedback equalization: Can you skip the training period?" IEEE Transactions on Communications, vol. COM-46, pp. 921-930.
[2] JELONNEK, B., D. BOSS and K.D. KAMMEYER (1997). "Generalized eigenvector algorithm for blind equalization," EURASIP Signal Processing, vol. 61, pp. 237-264.