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Journal ArticleDOI

A canonical space-time characterization of mobile wireless channels

01 Apr 1999-IEEE Communications Letters (IEEE)-Vol. 3, Iss: 4, pp 94-96

TL;DR: A canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent of the true channel parameters that provides a robust representation of the propagation dynamics and dramatically reduces the number of channel parameters to be estimated.

AbstractA canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent 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.

Summary (1 min read)

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] .
  • 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 their signal model deliver optimal performance at substantially lower complexity compared to existing "idealized" receivers, especially in dense multipath environments.

II. CANONICAL CHANNEL CHARACTERIZATION

  • The authors develop the signal model using continuous aperture, as illustrated in Fig. 1 . denotes the angular spread of the scatterers associated with the desired signal.
  • 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 angledependent time-varying channel impulse response or, equivalently, the multipath-Doppler spreading function [2] , [3].
  • Similarly, the finite aperture dictates that possesses only a finite number of spatial degrees of freedom that are 1.
  • 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 samples of the smoothed spreading function (5) where and The number of terms in (3) are determined by and where Proof [5] :.
  • 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.
  • Even if joint angle-delay estimation frameworks [1] are employed, a large number of observations and relatively complex algorithms are necessary to obtain accurate parameter estimates for the "idealized" conventional receivers.

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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
AbstractA 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
EFERENCES
[1] A. J. Paulraj and C. B. Papadias, “Space-time processing for wireless
communications,” IEEE Signal Processing Mag., pp. 49–83, Nov. 1997.
[2] J. G. Proakis, Digitial Communications, 3rd ed. New York: McGraw
Hill, 1995.
[3] P. A. Bello, “Characterization of randomly time-variant linear channels,”
IEEE Trans. Commun. Syst., vol. COM-11, pp. 360–393, 1963.
[4] A. M. Sayeed and B. Aazhang, “Joint multipath-Doppler diversity in
mobile wireless communications,” IEEE Trans. Commun., vol. 47, pp.
123–132, Jan. 1999.
[5] E. N. Onggosanusi, A. M. Sayeed, and B. D. Van Veen, “Canoni-
cal space-time processing in wireless communications,” IEEE Trans.
Commun., vol. 47, Apr. 1999.
[6] A. M. Sayeed, “Canonical multipath-Doppler coordinates in wireless
communications,” in Proc. 36th Annu. Allerton Conf. on Communication,
Control and Computing, 1998, pp. 536–545.
[7]
, “Canonical space-time processing in CDMA systems,”
in Proc. 1999 IEEE Int. Conf. on Acoust., Speech, and Signal
Processing—ICASSP ’99, vol. 5, pp. 2611–2614.
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Journal ArticleDOI
TL;DR: A canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent of the true channel parameters and provides a robust representation of the propagation dynamics.
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References
More filters

Journal ArticleDOI
TL;DR: Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath, and a model called the Quasi-WSSUS channel is presented to model the behavior of such channels.
Abstract: This paper is concerned with various aspects of the characterization of randomly time-variant linear channels. At the outset it is demonstrated that time-varying linear channels (or filters) may be characterized in an interesting symmetrical manner in time and frequency variables by arranging system functions in (timefrequency) dual pairs. Following this a statistical characterization of randomly time-variant linear channels is carried out in terms of correlation functions for the various system functions. These results are specialized by considering three classes of practically interesting channels. These are the wide-sense stationary (WSS) channel, the uncorrelated scattering (US) channel, and the wide-sense stationary uncorrelated scattering (WSSUS) channel. The WSS and US channels are shown to be (time-frequency) duals. Previous discussions of channel correlation functions and their relationships have dealt exclusively with the WSSUS channel. The point of view presented here of dealing with the dually related system functions and starting with the unrestricted linear channels is considerably more general and places in proper perspective previous results on the WSSUS channel. Some attention is given to the problem of characterizing radio channels. A model called the Quasi-WSSUS channel is presented to model the behavior of such channels. All real-life channels and signals have an essentially finite number of degrees of freedom due to restrictions on time duration and bandwidth. This fact may be used to derive useful canonical channel models with the aid of sampling theorems and power series expansions. Several new canonical channel models are derived in this paper, some of which are dual to those of Kailath.

2,331 citations


"A canonical space-time characteriza..." refers background in this paper

  • ...The signal is related to the transmitted symbol waveform via the angledependent time-varying channel impulse response or, equivalently, the multipath-Doppler spreading function [2], [3]...

    [...]


Journal ArticleDOI
TL;DR: This article focuses largely on the receive (mobile-to-base station) time-division multiple access (TDMA) (nonspread modulation) application for high-mobility networks and describes a large cell propagation channel and develops a signal model incorporating channel effects.
Abstract: Space-time processing can improve network capacity, coverage, and quality by reducing co-channel interference (CCI) while enhancing diversity and array gain. This article focuses largely on the receive (mobile-to-base station) time-division multiple access (TDMA) (nonspread modulation) application for high-mobility networks. We describe a large (macro) cell propagation channel and discuss different physical effects such as path loss, fading delay spread, angle spread, and Doppler spread. We also develop a signal model incorporating channel effects. Both forward-link (transmit) and reverse-link (receive) channels are considered and the relationship between the two is discussed. Single- and multiuser models are treated for four important space-time processing problems, and the underlying spatial and temporal structure are discussed as are different algorithmic approaches to reverse link space-time professing with blind and nonblind methods for single- and multiple-user cases. We cover forward-link space-time algorithms and we outline methods for estimation of multipath parameters. We also discuss applications of space-time processing to CDMA, applications of space-time techniques to current cellular systems, and industry trends.

1,059 citations


"A canonical space-time characteriza..." refers background in this paper

  • ...Conventional coherent space–time receivers, such as those proposed in [1], are based on the “ideal” test statistic...

    [...]

  • ...Even if joint angle-delay estimation frameworks [1] are employed, a large number of observations and relatively complex algorithms are necessary to obtain accurate parameter estimates for the “idealized” conventional receivers....

    [...]

  • ...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]....

    [...]


Proceedings ArticleDOI
21 Apr 1997
TL;DR: This paper reviews space-time signal processing in mobile wireless communications and focuses on antenna arrays deployed at the base stations since such applications are of current practical interest.
Abstract: This paper reviews space-time signal processing in mobile wireless communications. Space-time processing refers to the signal processing performed in the spatial and temporal domain on signals received at or transmitted from an antenna array, in order to improve performance of wireless networks. We focus on antenna arrays deployed at the base stations since such applications are of current practical interest.

686 citations


Journal ArticleDOI
TL;DR: Performance analysis shows that even the relatively small Doppler spreads encountered in practice can be leveraged into significant diversity gains via the new approach to diversity in spread-spectrum communications over fast-fading multipath channels.
Abstract: We introduce a new approach for achieving diversity in spread-spectrum communications over fast-fading multipath channels. The RAKE receiver used in existing systems suffers from significant performance degradation due to the rapid channel variations encountered under fast fading. We show that the Doppler spread induced by temporal channel variations in fact provides another means for diversity that can be further exploited to combat fading. We develop the concept of Doppler diversity and propose a framework that exploits joint multipath-Doppler diversity in an optimal fashion. Performance analysis shows that even the relatively small Doppler spreads encountered in practice can be leveraged into significant diversity gains via our approach. The framework is applicable in several mobile wireless multiple access systems and can provide substantial performance improvement over existing systems.

372 citations


"A canonical space-time characteriza..." refers background in this paper

  • ...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]....

    [...]

  • ...The accuracy of (12) can be improved by increasing although at the expense of losing orthogonality [4], [5]....

    [...]

  • ...3The special case of canonical multipath-Doppler coordinates in time-only processing is discussed in [4] and [6]....

    [...]

  • ...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]....

    [...]


Journal ArticleDOI
TL;DR: A canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent of the true channel parameters and provides a robust representation of the propagation dynamics.
Abstract: A canonical space-time characterization of mobile wireless channels is introduced in terms of a fixed basis that is independent 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 (DOA). The canonical representation provides a robust representation of the propagation dynamics and eliminates the need for estimating delay, Doppler and DOA parameters of different multipaths, Furthermore, it furnishes a natural framework for designing low-complexity space-time receivers. Single-user receivers based on the canonical channel representation are developed and analyzed, It is demonstrated that the resulting canonical space-time receivers deliver near-optimal performance at substantially reduced complexity compared to existing designs.

35 citations


"A canonical space-time characteriza..." refers background or methods in this paper

  • ...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]....

    [...]

  • ...Furthermore, the canonical matched filter outputs can be efficiently computed via a space–time RAKE receiver structure [5]....

    [...]

  • ...While the canonical representation (12) is quite general, it is particularly advantageous in the context of spread spectrum signaling [5], [6]....

    [...]

  • ...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...

    [...]

  • ...The choice can generate an approximately orthonormal basis, albeit at the expense of a loss of accuracy in (12) [5], [7]....

    [...]


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This provides a robust representation of the propagation dynamics and dramatically reduces the number of channel parameters to be estimated.