A multipath channel is a channel in which the received signal contains both the signal of interest and attenuated time delayed reflections of the signal of interest. In many cases, these reflections take the form of a small number of clusters of signals, where the clusters are sparse in the time domain.
To correctly demodulate and decode a signal, it is important to have knowledge of this channel. In a real system, this corresponds to estimating the channel characteristics. Usually, this is done by probing the channel with a predetermined waveform and estimating the impulse response.
Assume that the channel is modeled by
|y = x * B + z = XB + z||(1)|
where x, y, B and z represent the signal of interest, the received signal, the channel coefficients and white noise, respectively and X is the Toeplitz-structured convolution matrix of x. a * b represents the convolution of a and b. Generally, the channel estimate B̂ is obtained by solving a variation of the least squares (LS) problem given by B̂LS = (XHX)-1XHy. However, observing that (1) is very similar to the standard sampling problem in compressed sensing (CS) with X as the sampling matrix, we could potentially use standard CS sampling and reconstruction techniques to find the channel coefficients (assuming that X obeys the restricted isometry property, which is easy to do in practice by choosingas x a pseudo-random sequence). Further, it has been shown that given this type of system, ℓ1 minimization results in a more accurate representation of the signal than can be obtained by ℓ2 minimization.