Channel sensing and estimation is an important component of modern
wireless communication systems, and it is a crucial element of
emerging systems such as cognitive radio.  Wireless channels are
multidimensional; delay, Doppler, and spatial diversity provides rich
signaling environments that can be exploited to enhance communication
capabilities.  The identification of such channels is usually carried
out by fully exciting or illuminating all dimensions and then applying
linear estimation techniques.  For high-dimensional channels, this is
prohibitive and often ineffective.  Linear estimation methods are
ineffective because they ignore a key feature of such channels.
Because of their high-dimensionality and the physical causes
underlying them, wireless channels are often quite sparse, with a
small number of dominant modes characterizing their structure.
Estimation methods that do not capitalize on this sparsity do not
perform well.  This paper addresses channel sensing and estimation
from a new perspective that exploits the inherent sparsity in
high-dimensional wireless communication channels.  Adapting ideas from
the theory of compressed sensing, we make two key observations: 1)
sparse channels can be identified from a small number of properly
chosen excitations or illuminations, and exhaustive probing is
unnecessary; 2) nonlinear, sparse estimation methods such as the Lasso
yield superior channel estimates compared to linear techniques.  One
of the especially novel mathematical aspects of our work is that we
show that Toeplitz-structured sensing matrices, which naturally arise
when pseudorandom excitations are employed, are very well suited to
the identification of sparse channels.