Random channel coding and blind deconvolution

Blind deconvolution arises naturally when dealing with finite multipath interference on a signal. In this paper we present a new method to protect the signals from the effects of sparse multipath channels — we modulate/encode the signal using random waveforms before transmission and estimate the channel and signal from the observations, without any prior knowledge of the channel other than that it is sparse. The problem can be articulated as follows. The original message x is encoded with an overdetermined m × n (m > n) matrix A whose entries are randomly chosen; the encoded message is given by Ax. The received signal is the convolution of the encoded message with h, the S-sparse impulse response of the channel. We explore three different schemes to recover the message x and the channel h simultaneously. The first scheme recasts the problem as a block l1 optimization program. The second scheme imposes a rank-1 structure on the estimated signal. The third scheme uses nuclear norm as a proxy for rank, to recover the x and h. The simulation results are presented to demonstrate the efficiency of the random coding and proposed recovery schemes.