Sparse filter design for channel shortening
In many discrete-time signal processing systems, the distortion caused by signals passing through the environment is modeled as a discrete-time finite impulse response (FIR) filter, typically referred to as a channel. An important property of the channel is its delay spread, defined as the duration in time (or samples for discrete-time systems) within which most of the channel impulse response energy is contained. This delay spread may be caused by the inherent frequency selectivity of the communication medium or by multi-path propagation of the signal through the medium. In many signal processing and communication applications, a large delay spread in the channel causes impairments in analyzing signals that have passed through this channel. For example, a large delay spread manifests as reverberation in acoustic channels, causes inter-symbol interference (ISI) in communication systems and increases complexity of sequence estimators such as the Viterbi decoders.
Channel shortening equalizers are used in acoustics to reduce reverberation, in error control decoding to reduce complexity, and in communication receivers to reduce inter-symbol interference. The goal of channel shortening is to design an equalizer to reduce the delay spread of the channel impulse response. The cascade of a channel and channel shortening equalizer ideally produces an overall impulse response that has most of its energy compacted into fewer adjacent samples. Once designed, channel shortening equalizers filter the received signal on a per-sample basis and need to be adapted or re-designed if the channel impulse response changes significantly. Channel shortening equalizers have shown significant improvement in data rate performance of discrete multitone modulation (DMT) receivers and ISI reduction in underwater acoustic communication systems.
Conventional channel shortening equalizers are designed as dense FIR filters with contiguous non-zero coefficients and attempt to optimize some metric of shortening performance [9]. The channel shortening design tradeoff arises from the fact that a longer equalizer typically has better shortening performance at the cost of increased complexity of the filter implementation. In this study, I evaluated sparse filters as channel shortening equalizers. Unlike conventional dense filters, sparse filters have a small number of non-contiguous non-zero coefficients. By obviating the need for arithmetic operations corresponding to zero-valued coefficients, sparse filters with a large delay spread have the same runtime computational complexity as a dense FIR filter with equal number of non-zero taps. Sparse filters can also translate to fewer circuit components compared to dense FIR filters, consequently reducing area or power consumption. My contributions include (1) proposing optimal and sub-optimal low complexity algorithms for sparse shortening filter design, and (2) evaluating impulse response energy compaction vs. design and implementation stage computational complexity tradeoffs for the proposed algorithms.
The proposed equalizer design procedures were applied to (1) asymmetric digital subscriber line channels and (2) underwater acoustic communication channels. Simulation results using measured channel impulse responses show that sparse filters are able to achieve the same channel energy compaction with half as many coefficients as dense filters.