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SPARSE MULTIPATH CHANNELS: MODELING AND ESTIMATION
Waheed U. Bajwa, Akbar Sayeed, and Robert Nowak
Electrical and Computer Engineering, University of Wisconsin-Madison
ABSTRACT
Multipath signal propagation is the defining characteristic of terrestrial
wireless channels. Virtually all existing statistical models for
wireless channels are implicitly based on the assumption of rich
multipath, which can be traced back to the seminal works of Bello
and Kennedy on the wide-sense stationary uncorrelated scattering
model, and more recently to the i.i.d. model for multi-antenna channels
proposed by Telatar, and Foschini and Gans. However, physical
arguments and growing experimental evidence suggest that physical
channels encountered in practice exhibit a sparse multipath structure
that gets more pronounced as the signal space dimension gets large
(e.g., due to large bandwidth or large number of antennas). In this
paper, we formalize the notion of multipath sparsity and discuss applications
of the emerging theory of compressed sensing for efficient
estimation of sparse multipath channels.
1. INTRODUCTION
Multipath—signal propagation over multiple spatially distributed
paths—is the most salient feature of wireless channels and necessitates
statistical channel modeling due to the large number of
propagation parameters involved. Multipath propagation is both a
curse and a blessing from a communications viewpoint [1]. On the
one hand, multipath propagation leads to signal fading—fluctuations
in received signal strength—that severely impacts reliable communication.
On the other hand, research in the last decade has shown
that multipath is also a source of diversity—multiple statistically independent
modes of communication—that can increase the rate and
reliability of communication. Multipath diversity manifests itself in
various forms, including delay, Doppler, spatial and multiuser. The
impact of multipath fading versus diversity on performance critically
depends on the the amount of channel state information (CSI) available
to the system. For example, knowledge of instantaneous CSI at
the receiver (coherent reception) enables exploitation of diversity to
combat fading. Further gains in capacity and reliability are possible
if (even partial) CSI is available at the transmitter as well.
Statistical characteristics of a wireless channel depend on the
interaction between the physical multipath propagation environment
and the signal space of the wireless transceivers. For modern
wideband, multi-antenna transceivers, this interaction happens in
multiple dimensions of time, frequency and space. Technological
advances in RF front-ends, including frequency- and bandwidthagility
and reconfigurable antenna arrays, are enabling sensing and
exploitation of CSI at varying resolutions afforded by the spatiotemporal
signal space. Accurate channel modeling and characterization
in time, frequency and space, as a function of multipath and
signal space characteristics, is thus critical for studying the impact
and potential of such emerging agile wireless transceivers. In particular,
while most existing models for wireless channels assume a
rich multipath environment, there is growing experimental evidence
that physical channels exhibit a sparse structure even with a small
number of antennas and especially at wide bandwidths [2, 3].
In this paper, we use a virtual representation of physical multipath
channels that we have developed in the past several years
to present a framework for modeling sparse wireless channels and
to study the implications of multipath sparsity for channel estimation.
The virtual channel representation, discussed in Sec. 2, samples
the physical multipath in angle-delay-Doppler at a resolution
commensurate with the signal space parameters, and the dominant
non-vanishing virtual channel coefficients characterize the statistically
independent degrees of freedom (DoF) in the channel. Sparse
channels, discussed in Sec. 3, exhibit fewer DoF compared to channels
induced by rich multipath. We also introduce the concept of
channel sparsity pattern in Sec. 3 that captures the configuration of
the sparse DoF in the angle-delay-Doppler domain and constitutes
the most important element of CSI. In Sec. 4, we discuss estimation
of sparse channels using the emerging theory of compressed sensing.
For the sake of this exposition, we focus only on estimation
of time- and frequency-selective single-antenna channels and blockfading
narrowband multi-antenna channels. Our discussion focusses
on the nature of the waveforms used by the transmitter for probing
the channel, the algorithms used at the receiver for learning the
sparse channel, and quantification of the mean-squared-error in the
resulting channel estimate.
An important application of the modeling and estimation framework
proposed in this paper is the emerging area of cognitive radio in
which wireless transceivers sense and adapt to the wireless environment
for enhanced spectral efficiency and interference management.
In particular, the channel estimation strategies discussed in this paper
can be leveraged for learning the network CSI—a critical element of
cognitive radio. In a related work [4], we have also studied how accurate
knowledge of the CSI of a sparse channel can be exploited by
agile wireless transceivers for improved link performance.
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