Sparse Channel Separation using Random Probes. (arXiv:1002.4222v1 [math.NA])

Sparse Channel Separation using Random Probes. (arXiv:1002.4222v1 [math.NA]): "

This paper considers the problem of estimating the channel response (or
Green's function) between multiple source-receiver pairs. Typically, the
channel responses are estimated one-at-a-time: a single source sends out a
known probe signal, the receiver measures the probe signal convolved with the
channel response, and the responses are recovered using deconvolution. In this
paper, we show that if the channel responses are sparse and the probe signals
are random, then we can significantly reduce the total amount of time required
to probe the channels by activating all of the sources simultaneously. With all
sources activated simultaneously, the receiver measures a superposition of all
the channel responses convolved with the respective probe signals. Separating
this cumulative response into individual channel responses can be posed as a
linear inverse problem.

We show that channel response separation is possible (and stable) even when
the probing signals are relatively short in spite of the corresponding linear
system of equations becoming severely underdetermined. We derive a theoretical
lower bound on the length of the source signals that guarantees that this
separation is possible with high probability. The bound is derived by putting
the problem in the context of finding a sparse solution to an underdetermined
system of equations, and then using mathematical tools from the theory of
compressive sensing. Finally, we discuss some practical applications of these
results, which include forward modeling for seismic imaging, channel
equalization in multiple-input multiple-output communication, and increasing
the field-of-view in an imaging system by using coded apertures.

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