From sinusoids in noise to blind deconvolution in communications

Equalization for digital communications constitutes a very particular blind deconvolution problem in that the received signal is cyclostationary. Oversampling (OS) (w.r.t. the symbol rate) of the cyclostationary received signal leads to a stationary vector-valued signal (polyphase representation (PR)). OS also leads to a fractionally-spaced channel model and equalizer. The multichannel formulation also arises in mobile communications, when multiple receiving antennas are used. In the multichannel case, channel and equalizer can be considered as an analysis and synthesis filter bank. Zero-forcing (ZF) equalization corresponds to a perfect-reconstruction filter bank. We show that in the multichannel case FIR ZF equalizers exist for a FIR channel. The noise-free multichannel power spectral density matrix has rank one and the channel can be found as the (minimum-phase) spectral factor. The multichannel linear prediction of the noiseless received signal becomes singular eventually, reminiscent of the single-channel prediction of a sum of sinusoids. As a result, a ZF equalizer can be determined from the received signal second-order statistics by linear prediction in the noise-free case, and by using a Pisarenko-style modification when there is additive noise. Due to the singularity and the FIR assumption, the spectral factorization reduces to the triangular factorization of a finite covariance matrix. In the given data case, Music (subspace) or ML techniques can be applied. We present these developments by drawing the parallel with existing techniques for the sinusoids in noise subspace problem.