94 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 4, APRIL 1999
A Canonical Space–Time Characterization
of Mobile Wireless Channels
Akbar M. Sayeed, Member, IEEE, Eko N. Onggosanusi, Student Member, IEEE,
and Barry D. Van Veen,
Senior Member, IEEE
Abstract—A canonical space–time characterization of mobile
wireless channels is introduced in terms of a fixed basis that is in-
dependent of the true channel parameters. The basis captures the
essential degrees of freedom in the received signal using discrete
multipath delays, Doppler shifts, and directions of arrival. This
provides a robust representation of the propagation dynamics
and dramatically reduces the number of channel parameters
to be estimated. The resulting canonical space–time receivers
deliver optimal performance at substantially reduced complexity
compared to existing designs.
Index Terms— Dimension reduction, dispersive channels,
space–time sampling.
I. INTRODUCTION
T
HE USE OF antenna arrays for enhancing the capacity
and quality of multiuser wireless communication systems
has spurred significant interest in space–time signal processing
techniques [1]. A key consideration in space–time receiver
design is modeling the complex time-varying multipath prop-
agation environment. Most existing receiver designs employ
an “ideal” front-end processing matched to all the dominant
multipaths and corresponding direction of arrivals (DOA’s).
In addition to suffering from high computational complexity
in a dense multipath environment, such receivers rely heavily
on accurate estimation of the delay and DOA parameters of
dominant scatterers.
We introduce a canonical characterization of the received
signal in terms of the essential degrees of freedom in the
channel that are observable at the receiver. These degrees of
freedom are captured by a fixed underlying basis corresponding
to certain discrete multipath delays, Doppler shifts, and DOA’s
of the signaling waveform. In addition to eliminating the need
for estimating arbitrary delays, Doppler shifts and DOA’s
of dominant scatterers, the canonical space–time receivers
dictated by our signal model deliver optimal performance at
substantially lower complexity compared to existing “ideal-
ized” receivers, especially in dense multipath environments.
Section II briefly derives the canonical channel characteriza-
tion and Section III discusses some implications and illustrates
its utility with a simple coherent receiver example.
Manuscript received October 26, 1998. The associate editor coordinating
the review of this letter and approving it for publication was Prof. N. C.
Beaulieu.
The authors are with the Department of Electrical and Computer Engineer-
ing, University of Wisconsin–Madison, Madison, WI 53706 USA (e-mail:
akbar@engr.wisc.edu; eko@cae.wisc.edu; vanveen@engr.wisc.edu).
Publisher Item Identifier S 1089-7798(99)04261-1.
Fig. 1. Signal reception geometry.
II. CANONICAL CHANNEL CHARACTERIZATION
Consider a sensor array with aperture We develop the
signal model using continuous aperture, as illustrated in Fig. 1.
denotes the angular spread of the scatterers associated with
the desired signal. For clarity of exposition, we characterize
the channel effects on a single symbol. The complex baseband
signal received at location
in the aperture is
(1)
where
denotes the signal arriving from direction
and denotes the carrier wavelength. The signal is
related to the transmitted symbol waveform
via the angle-
dependent time-varying channel impulse response
or, equivalently, the multipath-Doppler spreading function
[2], [3]
(2)
where
and denote the multipath and Doppler spreads,
respectively.
1
Due to the finite duration and essentially finite bandwidth
of the signal exhibits only a finite number of
temporal degrees of freedom that are captured by a set of
uniformly spaced discrete multipath delays and Doppler shifts
[4]. Similarly, the finite aperture dictates that
possesses
only a finite number of spatial degrees of freedom that are
1
We note that
T
m
and
B
d
denote the maximum spreads—the variation of
spreads with
is captured by
H
(
;
)
:
1089–7798/99$10.00 1999 IEEE
SAYEED et al.: MOBILE WIRELESS CHANNELS 95
captured by certain discrete DOA’s. The following canonical
space–time characterization of
identifies the essential
spatio-temporal degrees of freedom in the channel that are
observable at the receiver.
Canonical Channel Representation
The signal
admits the equivalent representation
(3)
in terms of the space–time basis waveforms
(4)
The coefficients
are uniformly spaced sam-
ples of the smoothed spreading function
(5)
where
and
The number of terms
in (3) are determined by
and
where
Proof (sketch) [5]: Via a change of variables
in (1), can be written as
(6)
where
is the Fourier transform of and
(7)
In general,
has infinite support in and How-
ever, due to the finite array aperture
and finite duration
and approximately finite bandwidth of only a
corresponding “gated” version
of matters in
(6), which we express as the Fourier series
(8)
It follows from (7) that the series coefficients are given by the
samples of
in (5)
(9)
Fig. 2. A schematic depicting the canonical space–time coordinates.
Substituting (8) and (9) in (6) yields (3). The constants
, and correspond to the
normalized multipath Doppler and angular spreads.
Fig. 2 illustrates the canonical space–time coordinates de-
fined by the multipath-Doppler-angle sampling in the above
representation. For a discrete
-dimensional sensor array, the
canonical space–time basis functions (4) become
(10)
where
are the canonical array manifold vectors, which
take the following form for a uniform linear array with spacing
(11)
The angles
corresponding to the canonical spatial sam-
pling in (4) are governed by the relationship
The corresponding canonical signal characterization
for an
-sensor array is
(12)
for some canonical channel coefficients
III. DISCUSSION
While the canonical representation (12) is quite general, it
is particularly advantageous in the context of spread spectrum
signaling [5], [6]. First, it provides a robust
and maximally parsimonious characterization of space–time
propagation effects in terms of the fixed basis (4). It eliminates
the need for estimating arbitrary delays, Doppler shifts and
DOA’s of dominant scatterers.
2
Second, the representation
provides a versatile framework for channel modeling—both
deterministic and stochastic. In particular, the
dimensional canonical coordinates defined by
the basis (10) characterize the inherent diversity afforded
by a wide-sense stationary uncorrelated scatterer (WSSUS)
channel [5], [4].
3
Finally, the representation (12) induces a
2
Up to synchronization to a “global” delay, Doppler offset and DOA to
“align” the basis, which is required in all receivers.
3
The special case of canonical multipath-Doppler coordinates in time-only
processing is discussed in [4] and [6].
96 IEEE COMMUNICATIONS LETTERS, VOL. 3, NO. 4, APRIL 1999
canonical subspace structure that can be fruitfully exploited
for interference suppression in multiuser scenarios [5], [6].
The main source of error in (3) is due to the bandlimited
approximation, which can be made arbitrarily small by suffi-
cient oversampling [5]. For a direct sequence CDMA system,
is inversely proportional to the chip duration and in the
following we use
where is the oversampling
factor, typically 2, 4, or 8. The choice
can generate
an approximately orthonormal basis, albeit at the expense of a
loss of accuracy in (12) [5], [7]. The accuracy of (12) can be
improved by increasing
although at the expense of losing
orthogonality [4], [5].
We illustrate the advantage of the canonical space–time co-
ordinates with a simple example of single-user coherent BPSK
signaling over a slow fading channel (negligible Doppler
effects). An
element uniform linear array is assumed
with half-wavelength spacing. The
-dimensional complex
baseband signal for one symbol is given by
where is the data symbol and denotes a zero-
mean complex white Gaussian noise process. The signal
is modeled as where and are the
DOA and delay corresponding to the
th path. denotes the
total number of multipaths, and the fading coefficients
are uncorrelated.
Conventional coherent space–time receivers, such as those
proposed in [1], are based on the “ideal” test statistic
(13)
which performs matched-filtering to all the multipath com-
ponents and requires estimates of
and for each
multipath. The canonical representation (12) provides an new
equivalent characterization of the ideal statistic (13) that
eliminates the need for DOA and delay estimates
(14)
Note that the canonical receiver only requires estimates of
the fading coefficients
and the number of canonical
coordinates,
can be substantially smaller than
the number of physical coordinates,
especially in dense
multipath environments. For comparison purposes, we assume
perfect parameter estimates in both receivers. The coefficients
are obtained by projecting the noise-free signal
onto the canonical subspace [5].
A length-31
sequence is used as the spreading
code in the following numerical example. A dense
multipath environment is simulated using a total of
21
16 scatterers distributed evenly over
The canonical representation
samples at DOA’s
radians, and delays
with Fig. 3
compares the performance of the conventional (ideal) and
canonical receivers. At a symbol error probability
the
Fig. 3. Space–time processing: conventional (ideal) versus canonical for
different
O
s.
canonical receiver is within 1 dB of the ideal receiver for
or 8, and this gap decreases with increasing [5].
Note that the canonical receiver delivers this near-optimal
performance at a substantially reduced complexity. The ideal
receiver requires 21
16 3 estimates of and
computation of 21
16 matched space–time filter outputs.
In contrast, the canonical receiver for
only requires
estimates of 4
9 coefficients and computation of 4 9
matched filter outputs. Furthermore, the canonical matched
filter outputs can be efficiently computed via a space–time
RAKE receiver structure [5].
All practical receivers have limited operational bandwidth
which limits the accuracy of closely spaced delay estimates in
a dense multipath environment and also limits the benefits of
oversampling. Even if joint angle-delay estimation frameworks
[1] are employed, a large number of observations and relatively
complex algorithms are necessary to obtain accurate parame-
ter estimates for the “idealized” conventional receivers. The
canonical channel characterization introduced here dictates
receivers that have dramatically reduced complexity and are
likely to be far more robust to channel estimation errors
associated with limited data and the presence of noise.
R
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