Krylov Subspace Algorithms and Circulant-Embedding Method for Efficient Wideband Single-Carrier Equalization

Wider bandwidth allows higher data rate by transmitting narrower pulses. However, doing so would also increase the discrete channel memory length. For single-carrier communication systems this results in higher computational burden at the receiver. We are concerned with single-carrier nonblock transmission schemes with receiver oversampling, as they can provide higher spectral efficiency than block transmission schemes in the presence of large delay spreads. We first propose a simple finite-impulse-response (FIR) equalizer that is based on the circulant-embedding (CE) method and analyze its performance by investigating the relationship between solutions of various finite-dimensional models and the original infinite-dimensional problem. We show that under proper conditions the CE FIR equalizer converges exponentially fast to the IIR equalizer. We then focus on the conjugate gradient (CG) algorithm as an efficient means for equalization that is specifically well suited for dealing with large-delay-spread channels. We discuss the importance of stopping the iterations for the CG algorithm at the right time in the presence of noise and present several reliable low-cost stopping criteria. It turns out that the CG algorithm equipped with appropriate stopping criteria can outperform MMSE equalizers. Since both the CE and the CG methods can be efficiently implemented via fast Fourier transforms, equalization complexity is only in the order of N log(N) for N data symbols. Several numerical experiments demonstrate the performance of the proposed methods.