Sensor Array Processing
for Scattered Sources
Mats Bengtsson
TRITAS3SB-9729
ISSN 1103-8039
ISRN KTH/SB/R - - 97/29 - - SE
Signal Processing
Department of Signals, Sensors and Systems
Royal Institute of Technology
Abstract
In the search of improved capacity and performance in wireless communication systems, antenna arrays have emerged as a promising technique.
In these applications, as well as radar and other sensor array applications, it is important to have a good model of the propagation channel.
This thesis deals with one such model an environment where the signal
of each source is scattered by a large number of reections close to the
source as an example to study some techniques that can be used in
more realistic environments.
Two main problems are studied, estimation of parameters characterizing the channel and estimation of the signal transmitted at each source,
the so-called Signal Copy problem. Since the optimal solutions require
high computational power, several algorithms with suboptimal performance but low complexity are presented. For the particular choice of
model, an approximation of each spatially scattered source by two point
sources is shown to perform well.
For the signal copy problem, the rate of change of the channel has large
impact on the theoretical treatment, but makes almost no dierence on
the results. It is shown that in the search of high signal to interference
and noise ratio as well as low probability of outage, it is no loss to assume
that the channel variations are rapid.
A new low complexity algorithm is presented for estimation of Direction of Arrival (DOA) and spread angle of scattered sources. The main
computational step is performed using a standard DOA estimation algorithm for point sources, such as root-MUSIC or MODE. The estimates
are shown to be consistent and the asymptotic variance of the estimation
errors is derived. As an alternative for future algorithm development, the
idea of subspace tting is extended to estimation of parameters in full
rank data models.
All results are exemplied and veried with numerical simulations.
Acknowledgments
Leaving an interesting and safe job in Karlstad for a new life as a Ph.D.
student in Stockholm, was certainly a big step and a hard decision. Now
2 12 years later, I certainly do not regret my choice. I would like to express
my gratitude to my supervisor Professor Björn Ottersten for giving me
the opportunity to spend these years at KTH and for all his encouragement, ideas, proof reading and advice. I want to thank all my colleagues
at the department for the nice atmosphere and all discussions at the lunch
table and in the corridors.
A special thanks to my department at Ericsson Telecom AB (now at
Ericsson Infotech AB) for letting me take a leave to spend so many years
in the academic world.
I want to thank Professor Holger Broman for acting as opponent at
the seminar.
Finally, I would like to thank all my fellow musicians in dierent orchestras and ensembles who ll my spare time with so much joy and my
parents who always support me.
Mats Bengtsson
Stockholm, November 1997
Contents
1 Introduction
1.1 Contributions and Outline . . . . . . . . . . . . . . . . . .
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Data Models
2.1
2.2
2.3
2.4
2.5
Background . . . . . . . . . . . . . . . . . . . . . .
A Physical Model . . . . . . . . . . . . . . . . . . .
An Approximative Model . . . . . . . . . . . . . .
Further Approximations . . . . . . . . . . . . . . .
Assumptions and Properties . . . . . . . . . . . . .
2.5.1 Common Assumptions . . . . . . . . . . . .
2.5.2 Assumptions for the Physical Model . . . .
2.5.3 Assumptions for the Approximative Model
2.5.4 Comparison . . . . . . . . . . . . . . . . . .
2.A Formulas for Gaussian Distributed Scattering . . .
2.B Formulas for Uniformly Distributed Scattering . .
3 Signal Waveform Estimation
3.1
3.2
3.3
3.4
3.5
3.A
Background . . . . . . . . . . . . . . . . . . . . . .
Rapidly Time Varying Channels . . . . . . . . . .
Slowly Time Varying Channels . . . . . . . . . . .
Numerical Examples . . . . . . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . .
Numerical Optimization of the Outage Probability
4 Parameter Estimation
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1
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37
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Review of Some Point Source Algorithms . . . . . . . . . 38
vi
Contents
4.3
4.4
4.5
4.6
4.7
4.8
4.A
Low Complexity Algorithms . . . . . . . . . . . . .
Performance Analysis . . . . . . . . . . . . . . . .
Numerical Examples . . . . . . . . . . . . . . . . .
A Theoretical Curiosity . . . . . . . . . . . . . . .
Subspace Fitting Algorithms . . . . . . . . . . . .
Conclusions . . . . . . . . . . . . . . . . . . . . . .
Miscellaneous Results . . . . . . . . . . . . . . . .
4.A.1 Pseudo-Signal and Pseudo-Noise Subspaces
4.A.2 Useful Lemmas . . . . . . . . . . . . . . . .
4.B Proofs for root-MUSIC . . . . . . . . . . . . . . . .
4.C Proofs for MODE . . . . . . . . . . . . . . . . . . .
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40
45
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56
62
63
63
67
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72
5 Signal Estimation Using Estimated Channel Parameters 79
5.1 Background . . . . . . . . . . . . . . . . .
5.2 Two-Point Approximations Revisited . . .
5.3 Numerical Examples . . . . . . . . . . . .
5.3.1 A Rapidly Time Varying Scenario
5.3.2 A Slowly Time Varying Scenario .
5.4 Conclusions . . . . . . . . . . . . . . . . .
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79
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82
84
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6 Conclusions and Future Research
87
Bibliography
Index
91
97
6.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . 87
6.2 Directions for Future Research . . . . . . . . . . . . . . . 89
Chapter 1
Introduction
With the increasing popularity of radio based transmission for mobile
telephony and data transmission, the use of antenna arrays has emerged
as a promising technique to improve the capacity and performance of
existing and future systems.
The basic idea is very simple use several antennas instead of a single
one. Since the transmitted signal from the dierent antennas interferes
constructively at some receiver locations and destructively at others, it
it possible to transmit most energy in a specic direction or to direct different signals towards dierent receivers. Conversely, using the antenna
array as a receiver, it is possible to lter out an incoming signal from a
specic transmitter but also to locate the directions of several incoming
signals.
One advantage is the increased coverage. The signal at each antenna
can be weaker, since the total contribution from several antennas is added.
However, in this thesis we are mainly concerned with the use of antenna
arrays to improve the spectrum eciency of the system. Since signals can
be received/transmitted in certain directions and suppressed in others,
it should be possible to use the same carrier frequency for more users
than in current mobile systems. This concept is sometimes called Spatial
Division Multiple Access (SDMA).
We will concentrate on uplink transmission, i.e., the antenna is used
as a receiver, but many of the results can readily be translated to a
downlink situation, where the antenna array is used to transmit signals.
The most commonly used model for the transmission channel between
mobile and antenna array is the point source model, where each source
2
1 Introduction
is seen as a point source, is located in the far eld and all the received
signal power arrives along the line of sight. This will result in a plane
wavefront arriving at the array. If the sources are narrowband, it is easy
to derive the following baseband relation between the transmitted and
received signals from purely geometrical considerations, see e.g. [KV96].
x(t) = A()s(t) + n(t) :
(1.1)
Here x(t) = [x1 (t); : : : ; xm (t)]T is a complex valued data vector with the
baseband signals received at the m antenna elements, the signals transmitted at the d sources are collected in the vector s(t) = [s1 (t); : : : ; sd(t)]T ,
n(t) = [n1(t); : : : ; nm(t)]T is the sensor noise and in the matrix A,
each column is the array response vector for the corresponding source,
A() = [a(1 ) : : : a(d)]. If the antenna elements are placed along a
straight line with constant element separation, a so-called Uniform Linear Array (ULA), it is easy to see that the array response vector of a
source at direction relative to the broadside of the array is given by
a() = [1; ej2 sin ; : : : ; ej2(m?1) sin ]T , where is the element separation measured in wavelengths.
The point source model is very popular and has resulted in a large
collection of algorithms, see [VB88, KV96] for good surveys. However
in many practical situations it is not fully appropriate. Buildings, trees,
mountains and other obstacles can block the direct path or cause reections. A few strong reections, so-called specular multipath, can easily
be handled with (1.1), but scenarios with a large number of reections,
with reections on rough surfaces or with rapidly changing conditions,
all fall outside what can be described by the simple point source model.
In this thesis we use a model where each source is surrounded by
a large number of scatterers and there is no strong direct wave, so the
sources appear as spatially scattered. This model of local scattering is
fairly simplistic and can by no means describe all real scenarios, still it has
its merits. It is a generalization of the point source model, since a point
source is the special case of a scattered source with zero spread angle.
It is also a generalization of the well known Rayleigh fading channel to
the multidimensional case. Although simple, the model is still complex
enough to illustrate some diculties that must be handled in a real world
system. Note, however, that no time dispersion is included in the model,
i.e., we assume that the time delay is almost the same in all the reected
paths, so the resulting fading is frequency independent. Also, the long
term variations due to large scale changes in the environment, so-called
shadow fading, are not included in the model.
1 Introduction
3
Figure 1.1: Point source versus scattered source.
In the model of local scattering, each source is described by two parameters, the nominal Direction of Arrival (DOA) and the spread angle
(precise denitions are given in Chapter 2). In contrast to the point
source model, these parameters give a stochastic characterization of the
channel and the channel will typically change from time to time even
when the parameters remain constant.
One main goal is to estimate the transmitted waveform as well as possible. In this as well as in many other applications, it is important to have
good estimates of the parameters describing the channel. Thus, the thesis
focuses on two main problems, suggesting algorithms and characterizing
the performance for each of the problems:
How to estimate the nominal DOA and spread angle of each source.
How to utilize this knowledge to estimate the signal actually transmitted from each source.
The results will depend on the rate of change of the channel realization.
If the channel varies slowly, it is much better to track the instantaneous
array response, using some other channel parameterization, instead of
using only the stochastic characterization. However this is not always
possible, especially in the downlink situation where the actual channel
is not directly observable at the array, but also in some uplink scenarios. The two main problems are rst treated independently and then in
combination.
These problems are of interest also in other applications. A good
characterization of the propagation environment is crucial in all sensor
array applications. Several dierent phenomena can be described by the
data model presented here or variations thereof. One scenario is when
the radiating source itself is physically distributed over an area, where
the principles presented in Chapter 4 can be used to estimate both the
4
1 Introduction
location and the extent of the target. Even when the source itself is relatively small, it can appear to be spatially extended, not only because of
reections in surrounding objects but also because of a refracting propagation medium, for example in underwater sonar applications. The
beamforming weight vectors derived in Chapter 3 to estimate the transmitted waveform are also useful in sonar and radar applications for target
detection algorithms.
One main theme in the thesis is to nd solutions, algorithms, with
reasonably low computational complexity. Another common theme, as
a tool to reduce the complexity, is the approximation of each scattered
source with two closely separated point sources, as is illustrated in Figure 1.2.
Figure 1.2: A scattered source can be approximated by two point
sources.
1.1 Contributions and Outline
The optimal solutions to most of the problems treated herein are known
before. We have mainly concentrated on the design and analysis of suboptimal solutions with lower computational complexity and on comparisons
with the optimal results.
1.1 Contributions and Outline
5
Chapter 2
This chapter introduces two versions of the baseband channel model for
local scattering, one expressed in terms of physical parameters and one
approximation thereof expressed in terms of spatial frequencies. With the
approximative model, the behavior of any source is easily related to that
of a source at broadside. The same and similar models are well known in
the literature, see for example [Ban71, AFWP86, Zet97]. However, most
previous authors have assumed a specic distribution in azimuth, here
we generalize the expressions and approximations in an obvious way to
handle any kind of azimuthal distribution.
Chapter 3
Here we concentrate on estimation of the signals transmitted from the
sources. Signal to Interference and Noise Ratio (SINR) and probability
of outage are used as quality measures and a clear distinction is made
between slowly and rapidly changing environments. The optimal SINR
solution for a rapidly changing environment has appeared previously in
a downlink formulation in [ZO95], whereas the optimal solutions for a
slowly changing environment appear to be new. These solutions can not
be expressed in closed form, but we show that the results from the rapidly
changing case can be used with good approximation. All these results
hold for general Rayleigh fading vector channel models.
It is also shown that a simple two-point approximation of each scattered source gives near optimal performance whereas the ordinary point
source model performs much worse.
Part of this material has been presented in
Mats Bengtsson. The impact of local scattering on signal copy algorithms for antenna arrays. In Proceedings of Nordiskt radioseminarium 1996 (NRS96), pages 2427, August 1996.
Mats Bengtsson and Björn Ottersten. Signal waveform estimation from array data in angular spread environment. In Proc.
30th Asilomar Conf. Sig., Syst.,Comput., pages 355359, November 1996.
Chapter 4
Previously published algorithms for estimation of spread angle and nominal DOA have mostly been useful for o-line batch computations because
6
1 Introduction
of the high computational complexity. This could be useful in a test measurement setup, but for an on-line receiving algorithm, a solution with
low complexity is desirable.
Here we show how the spread angle and nominal DOA can be estimated using a standard algorithm such as root-MUSIC, MODE or ESPRIT followed by a simple table lookup. The estimates are shown to be
consistent for the approximative model. The variance of the estimation
errors are analyzed theoretically and by simulations. As a by-product,
general expressions are given for the performance of MODE and rootMUSIC in colored noise.
Finally, we introduce the concept of subspace tting for full rank models as a possible path for future algorithm development, and provide some
basic results on the statistical distribution of eigenvalues and eigenvectors
of general covariance matrices.
Part of this material has been submitted as
Mats Bengtsson and Björn Ottersten. Low complexity estimation
for distributed sources. Submitted to IEEE Transactions on Signal
Processing, 1997.
and has also appeared in
Mats Bengtsson and Björn Ottersten. Rooting techniques for estimation of angular spread with an antenna array. In Proceedings of
VTC'97, pages 11581162, May 1997.
Mats Bengtsson and Björn Ottersten. Low complexity estimation of angular spread with an antenna array. In Proceedings of
SYSID'97, pages 535540. IFAC, July 1997.
Chapter 5
Here we study the combined eects of estimating both channel parameters and transmitted signals. The idea of two-point approximations is
developed even further and a couple of case studies are carried out by
simulations.
Chapter 6
Gives some concluding remarks and ideas for future research.
1.2 Notation
7
1.2 Notation
Throughout the thesis uppercase boldface letters denote matrices, lowercase boldface letters denote (column) vectors and italics denote scalars.
X; X^ ; X Nominal value, estimated value and estimation error, respectively, of a quantity. X = X^ ? X.
X First order term in the Taylor expansion of X, see Appendix 4.A.
XT ; X; Xc Matrix transpose, conjugate transpose (Hermitian) and complex conjugate, respectively.
Xy The Moore-Penrose pseudoinverse of X. Xy = (X X)?1 X if X is
full rank.
X; ?X The projection and perpendicular?projection
onto the column
space of X, respectively. X = I ? X = XXy .
kXk Any norm of X.
kXkF The Frobenius norm of X, kXk2F = Tr[XX ]
Xkl ; [X]kl Element k; l of a matrix.
vec [X] The vec-operator. vec[X] = [xT1 : : : xTn ]T if X = [x1 : : : xn ].
X Y Schur Hadamard product, i.e., elementwise product, [X Y]kl =
Xkl Ykl .
2X Y X Y3
11
1n
.. 75
X Y Kronecker product, X Y = 64 ...
.
Xm1Y Xmn Y
(: : : ) The previous expression conjugated and transposed.
? ? O g(X) Big ordo, f (X) = O g(X) if f (X)=g(X) is bounded in a neighborhood of X = 0.
?
? o g(X) Small ordo, f (X) = o g(X) if f (X)=g(X) ! 0 as X ! 0.
Chapter 2
Data Models
2.1 Background
A good model of the propagation between a mobile and a base station
is crucial in the design of a communication system employing antenna
arrays. The point source model often used in the sensor array processing
literature may be useful for environments with open areas and direct line
of sight between the mobile and the base station. However, for many
situations, natural and man made reectors and obstacles result in a
much more complex propagation environment.
In this report, we use a simple model of multipath propagation caused
by local scattering around the mobile, where each source contributes
with a large number L of independent rays from directions randomly distributed around the nominal DOA. This model was not chosen because it
is a particularly good model a more thorough validation against propagation measurements still remains to be done but since it is reasonably
simple and still complex enough to illustrate some complications that
must be handled in a realistic environment. Since a point source is the
special case of a scattered source with zero spread, the scattering model
introduces more degrees of freedom and can never perform worse than the
point source model when appropriately applied to a real environment. It
is also worth noting that the model is a generalization of the at Rayleigh
fading scalar channel [Pro95].
This model of local scattering was rst, to our knowledge, reported in
[AFWP86] and has later been used in, for example, [ZO95] and [TO96]. A
thorough development of this and other models for dierent propagation
10
2 Data Models
gain: n
~n
Figure 2.1: Local scattering.
environments both in uplink and downlink, is given in [Zet97]. Similar
models appeared already much earlier [Ban71] and result if each source is
assumed to have a certain spatial distribution, as in [VCK95, MSW96],
or if the propagation medium introduces refraction, as for example in
underwater sonar applications [PK88].
Most authors have assumed a specic angular density function of the
incoming rays, such as a Gaussian or uniform distribution. Here, a more
general setup is used where any angular density function can be used.
Also, two dierent versions of the model are used. The physical model
described in Section 2.2 is used in all examples and simulations and is
formulated in terms of azimuth angles, whereas the approximative model
described in Section 2.3 forms the basis for all the analytical results. The
latter model is described in terms of spectral frequencies. Section 2.4
discusses some other possible approximations to the channel model and
the chapter concludes with a summary, Section 2.5, of all assumptions
used for the models.
2.2 A Physical Model
The physical channel model is described for the case of a single source
signal. Please refer to [Zet97] for a more thorough derivation.
Assume a single source that contributes with a large number of wavefronts originating from reections near the source, see Figure 2.1. Each
incoming ray has a complex random gain n and a random angular de-
2.2 A Physical Model
11
viation ~n from the nominal DOA , of the source. We assume that
the dierence in time delay of the dierent rays is small compared to
the inverse signal bandwidth and can be included as a phase shift in the
baseband model. Assume that the angular deviations ~n of the source are
zero-mean and distributed according to the probability density function
(PDF) p(~; ) parameterized by the standard deviation , the spread
angle. The gain factors n are independent from ray to ray, zero-mean
and circularly symmetric, i.e.,
E[n k ] = 0; 8n; k
E[n ] = 0
(2.1)
E[n k ] = 0; n 6= k
E[jn j2 ] < 1 :
The baseband signals received at the antenna array are collected in a
vector x(t) = [x1 (t); : : : ; xm (t)]T , which is modeled as
x(t) = s(t)
L
X
n=1
n (t)a( + ~n (t)) + n(t) , s(t)v(t; ; ) + n(t) (2.2)
where s(t) is the signal transmitted from
the source. Assuming that we
j
have a uniform linear array, a() = 1; e 2 sin ; : : : ; ej(m?1)2 sin T
is the array response vector for a point source at direction where is
the element separation in wavelengths. Note that v(t; ; ) is a random
vector drawn from a distribution parameterized by and . The sensor noise n(t) is zero-mean complex Gaussian and is spatially as well as
temporally white, i.e.,
E[n(t)] = 0
E[n(t1 )nT (t2 )] = 0
(2.3)
2
E[n(t1 )n (t2 )] = n I (t1 ? t2 ) :
The temporal correlation of v(t) may dier in dierent scenarios. For
simplicity, only two extreme cases are studied in this report,
Rapidly time varying channels, where v(t) is temporally white, i.e.,
independent from sample to sample.
Slowly time varying channels, where v(t) is constant during an
entire data burst and but is uncorrelated from burst to burst.
Since in most practical cases, the truth is somewhere between these extreme cases, the results will indicate what performance could be expected
in a real situation.
12
2 Data Models
2.3 An Approximative Model
Introduce the spatial frequency as ! = 2 sin and the corresponding array response vector a(!) = 1; ej! ; : : : ; ej(m?1)! T (with an abuse
of notation). Each stochastic distribution of + ~ will correspond to
a distribution of ! + !~ , see [Zet95], but a symmetric distribution in
DOA around will not exactly correspond to a symmetric distribution
in spatial frequency around !. However, if is small, 2 sin( + ~) 2(sin + ~ cos ) , ! + !~ , and a PDF p(~; ) on ~ will approximately correspond to an !~ with PDF p(~!; ! ) and standard deviation
! = 2 cos . Although, conceptually, all deviations in DOA or
spatial frequency should stay within [?; ], for computational simplicity we allow the PDF p(~; ) to have innite support both in the physical
and the approximative model. All angles are counted modulo 2.
Since L, the number
P of incoming rays, is large, the complex random
vector v(t; !; ! ) = Ln=1 n (t)a(! + !~n (t)) is approximately Gaussian,
by the central limit theorem. Also, because of the large L, the discrete
rays can be approximated by a continuous spatial distribution. Since n
is zero-mean and circularly symmetric, the same holds for v(t; !; ! ), i.e.,
v(t; !; ! ) 2 N (0; Rv (!; ! )) with
(Rv (!; ! ))kl = E[vk (t; !; ! )vl (t; !; ! )]
Z1
p(~!; ! )ej(k?l)(!+~!) d!~
?1 Z
1 1 !~
j
p( ; 1)ej(k?l)~! d!~
= e (k?l)!
!
!
?1?
j
(
k
?
l
)
!
=e
!~ (k ? l)!
=
(2.4)
where !~ ( ) is the characteristic function [Pap91] corresponding to p(~!; 1).
This result can be compactly written in matrix form as
Rv (!; ! ) = Da (!)B(! )Da(!)
(2.5)
where
Da(!) = diag[a?(!)]
[B(! )]kl = !~ (k ? l)! :
Equation (2.5) can also be written in the form
Rv (!; ! ) = ?a(!)a (!) B(! ) :
(2.6)
(2.7)
(2.8)
2.3 An Approximative Model
13
The Toeplitz matrix B(! ) typically has full rank but only a few
dominating eigenvalues, see Figure 2.2. Note that B(! ) = Rv (0; ! )
and can be interpreted as the covariance corresponding to a source at
broadside, i.e., = 0. This means that it often suces to analyze, for
example, how an algorithm behaves for a source at broadside and then
generalize the results to other directions using (2.5).
2
10
1
0
10
2
−2
10
3
−4
Eigenvalues
10
4
−6
10
5
−8
10
6
−10
10
7
−12
10
8
−14
10
0
1
2
3
4
5
6
σ , degrees
7
8
9
10
θ
Figure 2.2: Magnitude of the 1st, 2nd, : : : , 8th eigenvalue of B(2 )
for dierent . Uniformly distributed angular deviations, m = 8,
= 1=2.
For the specic choices of Gaussian and uniformly distributed angular
deviations, full expressions for B(! ) can be found in Appendix 2.A
and 2.B, respectively.
The resulting data model is
x(t) = s(t)v(t; !; ! ) + n(t)
(2.9)
14
2 Data Models
and the covariance of the received data is
Rx = E[x(t)x (t)] = S Rv (!; ! ) + n2 I :
(2.10)
Because of the implicit normalization of n in (2.4), the source signal
power S = E[js(t)j2 ] also includes the eects of path gain and shadow
fading.
The corresponding model for d independent sources is
x(t) =
d
X
i=1
si (t)v(t; !i ; !i ) + n(t)
(2.11)
and the covariance of the received data is
Rx = E[x(t)x (t)] =
d
X
i=1
Si Rv (!i ; !i ) + n2 I :
(2.12)
2.4 Further Approximations
One disadvantage of the models presented above is the need to decide
on a specic distribution of the angular deviations. An alternative is to
make a Taylor expansion of a( + ~).
~2
a( + ~) = a() + ~d() + 2 h() + (2.13)
2
where d() = @ a@() and h() = @ @a(2) .
For slowly time varying channels, v(t; ; ) is approximatively constant during one data burst and
v
L
X
n=1
n (a() + ~n d()) / a() + d()
(2.14)
for some complex constant . Thus, the covariance matrix of the noisefree data from one burst has rank one and is parameterized by and ,
see [AOS97] for more information.
2.5 Assumptions and Properties
15
For rapidly time varying channels,
Rv (; ) =
i h
L h
X
n=1
E jn j2 E
2
i
~2
a() + ~d() + 2 h() : : :
a()a() + 2 a()h() + h()a() + 2d()d()
2
2
a() + 2 h() a() + 2 h() + 2 d()d() (2.15)
P
using the same normalization, Ln=1 E[jn j2 ] = 1 as was used in (2.4).
This shows that for small , Rv (; ) essentially has rank two (compare
to Figure
2.2)2 and that the two-dimensional signal subspace is spanned
?
by a() + 2 h() and d(). The third expression of (2.15) was used in
[MSS95] to analyze how the bias of DOA estimation algorithms is aected
by local scattering.
A closely related result can be derived from (2.7). Assume that
p(~!; ! ) is an even function of !~ , then B(! ) is real valued and for
small ! , its elements are given by
B( ) = 1 ? 1 (k ? l)22 + O(4 )
(2.16)
! kl
!
!
2
which follows directly from (2.7) since p(~!; 1) is a PDF with unit variance.
The two rst terms of (2.16) can be used as an alternative model which
is independent of the specic spatial distribution.
Using (2.16) it is shown in [BO97a] that the two principal eigenvectors,
suitably scaled, of B(! ) tend to
e1 = [1; 1; : : : ; 1]T
(2.17)
e2 = [? m 2? 1 ; ? m 2? 3 ; : : : ; m 2? 1 ]T
(2.18)
as ! ! 0. However, this is a straightforward corollary of the more
general result (2.15) since a(0) = e1 and d(0) = e2 + m2?1 e1 . The proof
using (2.16) is considerably more involved and is omitted here since the
result is weaker.
2.5 Assumptions and Properties
All important assumptions for the dierent models used, are collected
here for easy reference. First the assumptions common to both the phys-
16
2 Data Models
ical and the approximative model, then the additional assumptions for
each specic model.
2.5.1 Common Assumptions
The antenna array is linear with omnidirectional elements, a uniform element separation and is perfectly calibrated.
The source signals are narrowband.
The source signals are uncorrelated.
The dierence in time delay between the incoming rays is small
relative to the inverse signal bandwidth, i.e., the delay dierences
are included in the gain factors n as complex phase shifts.
The path gains from two dierent directions are statistically independent.
The spread angle, or ! , respectively, is relatively small.
The sensor noise n(t) is independent of the signal, zero-mean, tem-
porally and spatially white, complex Gaussian, E[n(t1 )n (t2 )] =
n2 I (t1 ? t2 ).
For the temporal characteristics, we use one of the two following assumptions.
Rapidly time varying channels, E[v(t1 )v (t2 )] = 0 for t1 6= t2 .
Slowly time varying channels, where v(t) = v is constant during
each data burst.
2.5.2 Assumptions for the Physical Model
Each source contributes with a large number L of discrete rays.
The probability density function p(~; ) of the angular deviations
~n is a known symmetric function in ~ parameterized by its standard
deviation .
2.5 Assumptions and Properties
Physical Model
x(t) =
Rx =
d
X
Approximative Model
si (t)v(t; i ; i ) + n(t)
x(t) =
Si Rv (i ; i ) + n2 I
Rx =
i=1
d
X
i=1
17
d
X
i=1
d
X
i=1
si (t)v(t; !i ; !i ) + n(t)
Si Rv (!i ; !i ) + n2 I
Rv (; ) Da()B(2 cos )Da () Rv (!; ! ) = Da (!)B(! )Da(!)
Table 2.1: Summary of the two models.
2.5.3 Assumptions for the Approximative Model
Each source gives a spatially continuous contribution.
The probability density function p(~!; ! ) of the deviations in spatial
frequency is a known symmetric function in !~ parameterized by its
standard deviation ! .
2.5.4 Comparison
Table 2.1 shows a comparison of the two models. Note that ! = 2 sin and ! = 2 cos .
The approximation that a symmetric distribution in DOA corresponds
to a symmetric distribution in spatial frequency, holds with good accuracy
unless the DOA is close to 90 or the spread angle is very large. This
approximation causes a bias if ordinary DOA estimation algorithms are
used to estimate the nominal DOA of a scattered source, see [MSS95].
However, in typical cellular applications the DOA is limited to a sector
of, say, [?60; 60] and the corresponding bias is very small.
18
2 Data Models
Appendix 2.A Formulas for Gaussian Distributed Scattering
!~ 2 N(0; ! ) gives
B( ) =e? ((k?l2)! )2
! kl
@ B( ) = ? (k ? l)2 e? ((k?l2)! )2
! kl
!
@!
(2.19)
(2.20)
and for the physical model
R (; ) ej2(k?l) sin e? (2(k?l)2 cos )2
v
kl
@ R (; ) (j 2(k ? l) ? (2(k ? l) )2 sin )
@ v kl
cos Rv (; ) kl
@ R (; ) ? (2(k ? l) cos )2 R (; ) :
@
v
kl
v
kl
(2.21)
(2.22)
(2.23)
See also [TO96] for an exact formula for Rv (; ).
Appendix 2.B Formulas for Uniformly Distributed Scattering
p
!~ 2 Rect[! ; ! ] gives (! = 3! ).
B( ) = sin((k ? l)! )
! kl
(k ? l)!
cos((
k ? l)! ) ? B(! ) kl
@ B( ) =
!
@!
kl
!
(2.24)
(2.25)
2.B Formulas for Uniformly Distributed Scattering
19
and for the physical model
R (; ) ej2(k?l) sin sin(2(k ? l) cos )
v
kl
2(k ? l) cos ?
@ R (; ) j 2(k ? l) cos @ v
kl
(2.26)
+ 2(k ? l) sin cot(2(k ? l) cos )
(2.27)
? tan Rv (; ) kl
?
@ cot(2(k ? l) cos ) ? 1
@ Rv (; ) kl 2(k ? l) cos
Rv (; ) kl :
(2.28)
Chapter 3
Signal Waveform
Estimation
3.1 Background
We study the problem of estimating the waveform transmitted from a
mobile source. This problem, also called the Signal Copy problem, has
been studied extensively for several decades, see for example [MM80] and
[VB88] for a good introduction to the subject. Here, we study the class
of direction based algorithms (i.e., algorithms that use only parameters
in a parametric data model to form the beamformer) for environments
with local scattering.
The performance can be measured in many ways. In this analysis, we use Signal to Interference and Noise Ratio (SINR) and outage
probability as cost functions in the design and comparison of the algorithms. The common Minimum Mean Square Error (MMSE) estimate
s^(t) = Rsx R?xx1 x(t) is not directly applicable on our data model since
E[v(t)] = 0 and consequently Rsx = E[v(t)]Rss + Rsn = 0 which would
give the unacceptable solution s^(t) = 0. The maximum SINR formulation has been used previously for other channel models, for example in
[YS95].
The results depend upon the rate of change of the angular spread.
Therefore the analysis is divided into two separate cases, rapidly time
varying and slowly time varying environments, respectively.
For rapidly time varying environments, the beamformer giving opti-
22
3 Signal Waveform Estimation
mal SINR is readily derived. In interference limited scenarios, the optimal
algorithm outperforms the traditional methods that are based on a point
source model. In terms of the beamforming diagram, the problem is not
to point a main beam in the right direction, but to suppress the interferers. Point source based algorithms give a very narrow zero at each
interferer. One ad-hoc remedy is to try to widen the zeros, inserting several closely separated zeros or using derivative constraints. These simple
solutions appear to give near optimal performance at a low computational
cost.
For environments with slow time variations, it is more interesting to
study the instantaneous SINR, its distribution (i.e., the outage probability) and mean value (the average SINR). Closed form expressions are
derived both for the outage probability and the average SINR. The optimal beamformers can be found using numerical optimization techniques.
However, tight bounds on these cost functions can be expressed in terms
of the SINR for rapidly time varying channels, so all the results for rapidly
time varying channels can be used with good approximation also for the
slowly time varying case.
All the results are formulated for an uplink scenario, where a direction based beamformer often is far from optimal, since the instantaneous
channel can be better estimated using training sequences or blind techniques. This means that the results are mainly relevant in uplink situations where the channel can not be estimated rapidly enough (because
of limited computational power or other reasons), but the results can
also be translated into a downlink situation, i.e., where a signal is transmitted from the antenna array, see for example [ZO95] where the same
optimal SINR beamformer for rapidly changing environments is derived
for a downlink scenario.
The theoretical results are veried by numerical simulations.
3.2 Rapidly Time Varying Channels
Assume that d dierent signals are received at the antenna array and that
the signal of interest is number 1. The signal estimate is formed as a linear
combination of the data received at the array, s^1 (t) = w x(t). Similarly
to [YS95], we divide the estimate into three
P terms, s^1 = cS + cI + cN
where cN = w n(t) is the noise, cI = w dk=2 vk sk emanates from the
interfering signals and cS = w v1 s1 is the contribution from the signal
3.2 Rapidly Time Varying Channels
23
of interest. Dene the signal to interference and noise ratio as
jcS j2 ]
:
(3.1)
SINR = E[jc jE[
I 2 ] + E[jcN j2 ]
For rapidly time varying channels, it follows directly from Section 2.3
that
(3.2)
SINRrapid = Pdw S1 Rv (1 ; 1 )w w k=2 Sk Rv (k ; k ) + n2 I w
if w is treated as a deterministic quantity. The weight vector w that
maximizes SINRrapid is given by the eigenvector corresponding to the
largest eigenvalue of the following generalized eigenvalue problem [GL96]
S1 Rv (1 ; 1 )wopt = max
d
X
k=2
!
Sk Rv (k ; k ) + n2 I wopt :
(3.3)
The resulting SINR is given by SINRopt = max . We call this Optimal
algorithm for Rapidly time varying angular Spread, the ORS algorithm.
When the angular spread is zero, i.e., for point sources, it is well
known that the optimal SINR algorithm coincides with the Minimum
Variance Beamformer [MM80]
w / R?x 1a(1 ) :
(3.4)
Figure 3.1 demonstrates a couple of beampatterns from the ORS algorithm. Note that the beampattern contains two closely separated nulls
around the interferer. This is intuitive, as the spatially distributed interferer cannot be suppressed by a single narrow null and the pair of nulls
in eect forms a wider null in the beampattern.
If the main problem (at least in a interference limited scenario) is to
suppress signals from a wider range of angles around each interferer, there
are several well known beamforming techniques available. One possibility
is to place two deep zeros at angles k for each interferer [BRK88].
(3.5)
w = I ? A (A A )?1 A a(1 ) = ?A a(1 )
where
A = [a(2 ? ); a(2 + ); : : : ; a(d ? ); a(d + )] :
(3.6)
24
3 Signal Waveform Estimation
DOA of interferer: 10
10
Gain, dB
0
−10
−20
−30
−100
−80
−60
−40
−20
0
20
Angle (degrees)
40
60
80
100
−20
0
20
Angle (degrees)
40
60
80
100
DOA of interferer: 25
10
Gain, dB
0
−10
−20
−30
−100
−80
−60
−40
Figure 3.1: Sample beampatterns from the ORS algorithm. The source
of interest is held at 0. The full parameter setting is described in Section 3.4.
Another interpretation of this ad-hoc solution is that a scattered source,
as seen from an antenna array, can be well approximated by two closely
separated point sources.
A second possibility is to widen the zeros, using a constrained beamformer where both the beampattern and its rst derivative is zero at the
nominal direction of each interferer.
This can be done similarly to (3.5):
where
w = I ? X(X X)?1 X a(1 ) = ?Xa(1 )
(3.7)
X = [a(2 ); d(2 ); : : : ; a(d); d(d)]
(3.8)
3.3 Slowly Time Varying Channels
25
or using a Linearly Constrained Minimum Variance beamformer (LCMV)
[VB88]:
w = arg Cmin
(3.9)
w=f w Rx w
with the solution
where
w = Rx?1C(C R?x 1C)?1 f
(3.10)
C = a(1); a(2 ); d(2); : : : ; a(d); d(d )
f = [1; 0; 0; : : :; 0; 0]T :
(3.11)
(3.12)
However, numerical calculations show that the LCMV solution gives very
poor performance. This could be explained as signal cancellation eects
since the signal of interest is spatially distributed, not concentrated to 1 .
One limitation of all these ad-hoc solutions is that the available number of antenna elements may not provide enough degrees of freedom, since
2(d ? 1) m ? 1 must hold.
3.3 Slowly Time Varying Channels
If the channel realization is constant v(t; k ; k ) = vk 2 N (0; Rk ) (Rk
is shorthand for R(k ; k )) during an entire data burst, then the instantaneous SINR during the burst is given by
SINRburst =
w S1v1v1 w :
P
d
w k=2 Sk vk vk + n2 I w
(3.13)
As a benchmark for comparisons, let us rst study what can be
achieved if the instantaneous channel is perfectly known. Then the
SINRburst is maximized by the minimum variance beamformer
w = R?1v1 /
x
d
X
k=2
!?1
v + 2 I
Sk vk k
n
and the resulting optimal SINR is
SINRburst, opt = S1 v
1
d
X
k=2
Sk vk k
!?1
v + 2 I
n
v1
(3.14)
v1 :
(3.15)
26
3 Signal Waveform Estimation
This is the best possible performance in a system where the channel can
be estimated in real time. However, in this study, we limit ourselves to
the class of signal copy algorithms that are based only on a statistical
characterization of the channel, where only the nominal DOA and the
spread angle are known for each source. Thus, we seek the weight vector
w that maximizes the average SINR, i.e.,
SINRslow = E[SINRburst] :
(3.16)
In a communications application it is important to have a good performance on the average but even more important that the worst case
performance is still acceptable or at least that the probability of unacceptable performance is very low. One such characterization is the
probability of outage, i.e., the probability that the SINRburst falls below
a certain threshold , in other words the cumulative distribution function of SINRburst . It turns out that the outage probability also makes it
possible to derive a closed form expression for SINRslow .
Introduce the variables , n2 w w and Zk , Sk w vk vk w and note
that Zk is 22 distributed with E[Zk ] = Sk w Rk w , %k , i.e.,
fZk (z ) = %1 e? %k :
z
k
(3.17)
Now the outage probability is
3
2
S1 v1 v1 w
w
< 5
F ( ) = Pr [SINRburst < ] = Pr 4 Pd
w k=2 Sk vk vk + n2 I w
"
! #
d
X
2
= 1 ? Pr w S1 v1 v1 w > w
Sk vk vk + n I w (3.18)
k=2
"
!#
d
X
= 1 ? Pr Z1 > k=2
Zk + 3.3 Slowly Time Varying Channels
which gives
Z
1 ? F ( ) =
Z
Yd
Z1 > (Pdk=2 Zk +) k=1
Zk >0; k=2;:::;d
= Qd
Z
1
Z
= e? %1
= e? %1
k=2 %k
Yd
1 e? %zkk dz dz : : : dz
1 2
d
%
e?
k=2 %k Zk >0
Z1
d
Y 1
0
27
k
Pd
zk
k=2 %k
e? %1 (+
Pd
k=2 zk )
dz2 dz3 : : : dzd
1 + % zk
?
%
e k 1 dzk
(3.19)
1
1
+
%%k1
k=2
and, going back to the original notation
2 w w
n
F ( ) = 1 ? e? S1 w R1 w
Yd
1
Sk w Rk w :
1
+
S1 w R1 w
k=2
(3.20)
Similar results are derived in e.g. [SW88].
It is possible to nd a simple upper bound on the outage probability,
using the inequality ex 1 + x.
i 1
0 hPd
2I w
w
S
R
+
k
k
n
k=2
A
F ( ) 1 ? exp @?
wS1R1w
(3.21)
= 1 ? e?=SINRrapid :
We see that ORS, the optimal weight vector for rapidly changing channels also minimizes this upper bound. Since the bound is very tight
when w Rk w w R1 w, i.e., when F ( ) 1 which is the interesting region, we could expect the ORS beamformer to give near optimal
performance in terms of outage probability also for a slowly time varying
channel.
The exact expression (3.20) can be minimized numerically, using standard multivariable optimization techniques or the algorithm described in
Appendix 3.A, but in practice, the gain compared to the ORS solution
is almost negligible, see Section 3.4.
Using the outage probability, we can nd an expression for SINRslow .
Recall that if X is a random variable, FX (x) = Pr[X x] and FX (0) = 0,
28
3 Signal Waveform Estimation
R
then E[X ] = 01 (1 ? FX (x))dx [Chu74]. Thus, the mean of SINRburst is
given by
SINRslow = E[SINRburst ] =
=
=
Z1
2 Z1
0
n w w
e? S1 w R1 w
0
Z1
(1 ? F ( ))d
Yd
2 w w X
n
e? S1 w R1 w
d
k
Sk w Rk w
k=2 1 + S1 w R1 w
0
!
1
S
wRk w d
k
k=2 1 + S1 w R1 w
!
d
(3.22)
2 w w d
R1 w ? n2 w w
X
n
Sk w Rk w E
k SS1 w
e
1 S w R w
Rk w
w
k
k
k
k=2
R
where E (x) = 1 e?t dt is the exponential integral [AS64], and the =
1
x
t
are dened by the partial fractions expansion of the product.
k
Yd
d
X
1
k
=
:
(3.23)
S
w
R
w
S
k
k
k w Rk w
k=2 1 + S1 w R1 w k=2 1 + S1 w R1 w
The weight vector that maximizes SINRslow can be found from (3.22)
using numerical optimization. We call the resulting w the Optimal algorithm for Slowly time varying angular Spread, the OSS algorithm. Unfortunately the computational complexity is very high. A more attractive
alternative is to use (3.21) to get a tight lower bound on SINRslow
SINRslow Z 1
0
e?=SINRrapid d = SINRrapid
(3.24)
which shows that the ORS solution and low complexity approximations
thereof give near optimal performance also in terms of SINRslow .
Note that all the results (3.19)(3.24) hold in general for any Rayleigh
fading channel model.
3.4 Numerical Examples
A scenario with two sources has been analyzed using the theoretical results as well as simulations. The antenna array has 8 elements separated
half a wavelength and is uniform and linear. The signal of interest is xed
3.4 Numerical Examples
29
at = 0 and is disturbed by a 10 dB stronger signal. The background
noise power is 10 dB less than the signal of interest. The angle between
the two sources is varied between 1 and 40. Both sources have a Gaussian distributed angular spread with standard deviation = 3. In the
simulations, L = 50 dierent rays contributed for each source and the
SINR and outage levels for each case were calculated from 500 dierent
data bursts of 100 samples each. The true parameter values have been
used as input to the algorithms, i.e., no eects of estimation errors on
the parameters are included, see Chapter 5.
The comparison covers the following algorithms.
ORS (3.3), the optimal solution for rapidly time varying channels.
Two-ray approximation (3.5)(3.6) with = .
Derivative constraints (3.7)(3.8).
The LS solution for a point source model,
w = ?Aa(1 ) where A = [a(2 ); : : : ; a(d)] :
(3.25)
This is a typical representative of traditional point source algorithms. This solution is conveniently formulated for the simultaneous estimation of all source signals as ^s = Ay x.
The algorithm derived in [YS95] that gives maximum SINR for
a model of random perturbations to the array response for point
sources, A = A0 +A~ where the columns of A~ are zero-mean random
vectors with E a~k a~l = a2 I(k ? l). a2 = 0:2 was used in the
comparisons.
In addition, the following algorithms were used for the slowly time varying
channel.
OSS, the beamformer that gives optimal SINRslow calculated by
numerical optimization of (3.22).
The optimal outage probability solution, see Appendix 3.A.
The minimum variance beamformer when the channel is completely
known (3.14). Of course this can only be done in simulations, but
still provides an interesting benchmark.
30
3 Signal Waveform Estimation
20
15
SINR (dB)
10
5
0
ORS
Two−ray approximation
LS
Optimal for array perturbations
−5
−10
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 3.2: Simulated SINR for dierent beamformers. Rapidly time
varying channel.
The performance on a rapidly time varying channel is shown in Figures 3.2 (simulated results) and 3.3 (theoretical results calculated using (3.2)). The simulation results agree well with the theory (the only
approximations, are the model approximations in Section 2.3). As seen,
much can be gained compared to the point source solutions, whereas the
ad-hoc solutions perform reasonably well.
It could be expected that robustness against array perturbations would
give some gain also for local scattering, but somewhat surprisingly, this is
not the case. A realistic system should probably include robustications
both to array perturbations and scattered sources.
For slowly time varying channels, the average SINR is shown in Figure 3.4 which includes a plot of the best possible results from an adaptive
algorithm, which could be calculated from (3.15) in the simulations. The
results agree well with the theoretical expression (3.22) as can be seen
in Figure 3.5, which also shows that the lower bound given in (3.24) is
very tight. This is also apparent since the ORS solution is so close to the
optimum.
The probability of outage was estimated from the same simulations for
a source separation of 8 , see Figure 3.6 where, again, the best possible
3.5 Conclusions
31
20
15
SINR (dB)
10
5
0
ORS
Derivative constraints
Two−ray approximation
LS
Optimal for array perturbations
−5
−10
−15
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 3.3: Theoretically calculated SINR for dierent beamformers.
Rapidly time varying channel.
performance of an adaptive algorithm is included. The results are in
good correspondence with the theoretical results from (3.20) shown in
Figure 3.7. A numerically calculated optimum of (3.20) is included but
is virtually identical to the ORS performance. Also the OSS algorithm
gives near optimal probability of outage.
3.5 Conclusions
We have studied the problem of signal waveform estimation under dierent assumptions on the time variations of the channel and using dierent
criteria of optimality. The conclusion, shown both analytically and numerically, is that the optimal solution is almost the same for all the
dierent cases, if the beamformer is calculated based only on the statistical characterization of the channel, not on the instantaneous realization.
Since the assumptions of rapid versus slow time variations were chosen
as being extreme cases, the same conclusion should hold regardless of the
rate of the time variations.
The assumption of a rapidly time varying channel gives simple expressions for the SINR and the optimal beamformer is easily found as
32
3 Signal Waveform Estimation
20
15
SINR (dB)
10
5
0
ORS
OSS
LS
Optimal SINR when v is known
−5
−10
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 3.4: Simulated SINR for dierent beamformers. Slowly time
varying channel.
20
15
SINR (dB)
10
5
0
ORS
OSS
LS
Lower bound, ORS
−5
−10
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 3.5: Theoretically calculated SINR for dierent beamformers.
Slowly time varying channel.
3.5 Conclusions
33
0
10
−1
γ
F (γ)
10
−2
10
ORS
OSS
LS
Two−ray approximation
Optimal SINR,
when v known
−3
10
−4
10
−20
−10
, (dB)
0
10
20
30
Figure 3.6: Simulated outage probability for dierent beamformers.
Slowly time varying channel. Source separation 8 .
0
10
−1
γ
F (γ)
10
−2
10
ORS
OSS
LS
Optimal CDF,
when v unknown
−3
10
−4
10
−20
−10
, (dB)
0
10
20
30
Figure 3.7: Theoretically calculated outage probability for dierent
beamformers. Slowly time varying channel. Source separation 8 .
34
3 Signal Waveform Estimation
the solution of a generalized eigenvalue problem. Even though it is possible to derive expressions for both outage probability and average SINR
for a slowly time varying channel, the tight bounds expressed in terms of
the performance of a rapidly varying channel are more useful in practice.
If the computational complexity is still too high in an implementation,
simple ad-hoc techniques were shown to give near optimal performance.
Note, however, that if the channel varies slowly enough that the instantaneous channel realization can be tracked using an adaptive algorithm, then much can be gained compared to the limited class of beamformers studied here.
3.A Numerical Optimization of the Outage Probability
35
Appendix 3.A Numerical Optimization of the
Outage Probability
The optimum weight vector for a given SINR level 0 should minimize
F (0 ), the outage probability or, equivalently, minimize
l(0 ; w) , ? log(1 ? F (0 ))
d
Sk w Rk w : (3.26)
w w + X
log
1
+
= 0 n2 S w
0 S w R w
1 R1 w k=2
1
1
@ = @ + j @ , is
The derivative, dened similarly to [Bra83] but with @x
@xR
@xI
w
w wS1R1w
@l(0 ; w) = 2
(3.27)
0 n S w R w ? (S w R w)2
@w
1
1
1
1
Sk Rk w Sk wRk wS1R1w ? (S w R w)2 :
Sk w Rk w
1
1
k=2 1 + 0 S1 w R1 w S1 w R1 w
P
Let k = 1 + 0 SSk1 ww RRk1 ww and Q = n2 I + dk=2 SkRk k , then
d
@l(0 ; w) = 0
2 I + X Sk Rk w
n
@w
S1 w R1w
k=2 k
!
w n2 I + Pdk=2 SkRk k w
?
S1 R1 w (3.28)
S1 w R1 w
Qw
w
0
= S w R w Qw ? w R w R1w :
1
1
1
+
d
X
0
The minimum, i.e., the zero of (3.28), can be found iteratively. Given an
initial weight vector w0 , for example the OSR solution, calculate the k
and Q and assume for a moment that these quantities are constant. Now
note that (3.28), apart from a multiplicative constant, is the derivative
of
w Qw
wR1w
(3.29)
which is minimized by the generalized eigenvector with minimal eigenvalue of Qw = R1 w. This gives a new weight vector w1 which is used
to update the k . The procedure is iterated until convergence. Typically
only a few iterations are necessary.
Chapter 4
Parameter Estimation
4.1 Background
It is clear from Chapter 3 that good estimates of the spread angle k
of the sources are crucial in the design of good receiver algorithms for
a channel with local scattering, see also e.g. [ZO95]. Even if only the
nominal DOA k needs to be estimated, it is not obvious how to design
an estimation algorithm.
Recently, a few algorithms have been published that address this problem. The ML estimator is derived in [TO96], together with a weighted
covariance matching algorithm. When optimally weighted, the covariance matching algorithm is shown to be asymptotically ecient, as is the
ML algorithm. Modications of the classical MUSIC algorithm [KV96]
have lead to the algorithms DSPE [VCK95], DISPARE [MSW96] and vecMUSIC [WWMR94], the latter using fourth order moments of the data.
The disadvantage of all these algorithms is the computational complexity, as a numerical optimization must be performed with a numerically
heavy cost function. DOA estimation in the presence of local scattering
is addressed also in earlier references, such as [Jän92] and [PK88].
Here, we take a slightly dierent approach. First, we study how
some standard high resolution DOA estimation algorithms (mainly rootMUSIC [Bar83] and MODE [SS90a], even though the results hold for a
large class of algorithms) behave if they are used to estimate the parameters of one or several point sources when there is only a single scattered
source present. These results are exploited to suggest a new algorithm for
estimation of both DOA and spread angle. Actually, it is a kind of meta
38
4 Parameter Estimation
algorithm, since it can be applied using one of several standard DOA
estimation algorithms in the main computational step. The computational complexity is signicantly lower than that of previously published
algorithms for the problem. The asymptotic variance of the parameter
estimates is derived analytically and the results are veried by simulations.
The analysis also shows that the standard algorithms at hand do
actually give consistent estimates of the nominal DOA.
Since the signal part of the data covariance matrix is full rank, it is
not obvious that a subspace algorithm, such as MUSIC, can be applied
to the problem, even though the covariance is almost low rank in the
sense that only a few eigenvalues are dominant. In Section 4.7, we sketch
a general framework for the use of subspace tting methods on full rank
data models. The algorithm developed in Section 4.3 as well as the
previously referred MUSIC variations, DSPE and DISPARE, can all be
explained as approximate solutions within this framework.
4.2 Review of Some Point Source Algorithms
Several algorithms have been devised for DOA estimation based on the
traditional point source model (1.1). A couple of these methods, MUSIC,
root-MUSIC and MODE, are briey reviewed here since they will be used
in the sequel. A more thorough treatment can be found in [KV96] and
the references cited therein.
Since x(t) = A()s(t) + n(t),
Rx = E[x(t)x (t)] = ASA + n2 I
(4.1)
where S = E[s (t)s (t)]. If d sources are present, ASA has rank d and
the eigenvalue decomposition of Rx can be written
Rx = EssEs + n2 EnEn
(4.2)
where s is a diagonal matrix with the d principal eigenvalues. Since
span[Es ] = span[A], Es is called the signal subspace and En is called
the noise subspace.
P Estimate the covariance matrix from sampled data
using R^ x = N1 Nt=1 x (t)x (t) and perform a corresponding eigenvalue
decomposition R^ x = E^ s ^ s E^ s + E^ n ^ n E^ n .
The MUSIC [Sch81] algorithm nds the d array response vectors a()
that are most orthogonal to E^ n , in the sense that
ka ()E^ n k2 = a ()E^ n E^ n a ()
(4.3)
4.2 Review of Some Point Source Algorithms
39
is minimized.
For Uniform Linear Arrays (ULAs), let a(z ) = [1; z; : : : ; z m?1]T and
note that a() = a(z )jz=ej2 sin and a () = aT (z ?1 )jz=ej2 sin . Thus,
an ecient implementation to minimize (4.3), the root-MUSIC algorithm
[Bar83], is to root the polynomial
g(z ) = aT (z ?1 )E^ n E^ n a(z )
(4.4)
zk ]
and estimate the DOAs by ^k = arcsin arg[
2 for the d zeros of g (z )
inside the unit circle that are closest to the unit circle (by construction,
all roots will appear in mirror pairs, z and 1=z ).
Instead of tting a single array response vector to the estimated noise
subspace, it is possible to match the whole array response matrix A()
to the estimated signal subspace, using
^ = arg min Tr[E^ s ?A ()E^ s W] :
(4.5)
The weighting matrix W can be chosen to give the same large sample accuracy as the maximum likelihood estimate, i.e., this Weighted Subspace
Fitting (WSF) method gives optimal performance [VO91].
For ULAs, dene the m (m ? d) Toeplitz matrix G by
2g0 g1 : : : gd 0 : : : 0 3
6
. . .. 7
G = 666 0.. .g.0 .g.1 :. :. : gd . . . . 777
(4.6)
4. . . .
. 05
0 : : : 0 g0 g1 : : : gd
P
Q
where dk=0 gk z k = dk=1?(z ? ej2 sink ) and note that G() ? A().
Thus, ?A () = G () G ()G () ?1 G () and the minimization
of (4.5) can be performed with the following algorithm, MODE [SS90a,
SS90b], also called root-WSF:
1. Solve the quadratic optimization problem
^ ^
g^ = arg min
(4.7)
g2 Tr[Es GG Es W]
to get consistent estimates of g = [g0 ; : : : ; gd]T .
2. Solve the quadratic optimization problem
^ ^ ^ ?1 ^
g^ = arg min
g2 Tr[Es G (G G ) G Es W] :
(4.8)
40
4 Parameter Estimation
zk ]
3. Let ^k = arcsin arg[
2 where zk are the roots of the polynomial
Pd g zk = 0.
k=0 k
The optimization constraints are given by = fgj Re[g0] = 1; gk = gd?k g.
The rst constraint avoids the all-zero solution and the second is necessary, but not sucient, for the roots to stay on the unit circle.
^ n]
In the examples below we have used the weighting W = ^ s ? Tr[
m?d I
given in [SS90a] which gives the same asymptotic performance as deterministic ML, but the theoretical results derived in Appendix 4.C hold for
any diagonal weighting matrix.
4.3 Low Complexity Algorithms
The algorithm is derived for the case of a single source. Generalizations
to several sources are discussed at the end of the section. First, we will
explore some properties of DOA estimation algorithms.
Assume for a moment that
R1 = A(!1; : : : ; !d)SA (!1; : : : ; !d) + n2 I
where A(!1 ; : : : ; !d) = [a(!1 ) : : : a(!d )], i.e., R1 is the covariance matrix
for d point sources. Since, for a ULA
[a(! + !)]k = ej(k?1)(!+!) = [a(!)]k [a(!)]k
a translation of all spatial frequencies by ! corresponds to
A(!1 + !; : : : ; !d + !) = Da(!)A(!1; : : : ; !d)
and a corresponding covariance matrix
R2 = A(!1 + !; : : : ; !d + !)SA (!1 + !; : : : ; !d + !) + n2 I
= Da (!)R1Da (!) :
Now if we have an algorithm F (R^ ; d) that gives DOA estimates (in
terms of spatial frequencies)
f!^1 ; : : : ; !^ dg = F (R^ ; d)
(4.9)
of d point sources from a sample covariance matrix R^ , then clearly
f^1 ; : : : ; ^d g = f!^1 + !; : : : ; !^ d + !g
(4.10)
4.3 Low Complexity Algorithms
41
if
f!^1 ; : : : ; !^ dg = F (R1 ; d) and
(4.11)
?
f^1 ; : : : ; ^d g = F (R2 ; d) = F Da (!)R1Da (!); d
(4.12)
provided that F (R^ ; d) gives consistent estimates of !k as N ! 1. As will
be explained below, it is desirable that (4.10) holds, not only for covariance matrices corresponding to d point sources, but for any covariance
matrices related through R2 = Da (!)R1 Da . This property is indeed
true for most estimators, for example root-MUSIC, see Appendix 4.B.
Similar proofs can easily be found for e.g. ESPRIT [RK89] and ML
[KV96] (both in the deterministic and stochastic signal formulations).
Note that the property is not true for all DOA estimation algorithms,
one counterexample is MODE in its original formulation [SS90a]. However, with a slight modication of the algorithm, the property holds also
for MODE, see Appendix 4.C and in the sequel, the term MODE will
refer to the modied version of the algorithm.
Recall from Chapter 2 that the covariance of the received data vector
x(t) from a single scattered source can be written
Rx = E[x(t)x (t)] = S Da (!)B(! )Da(!) + n2 I :
(4.13)
The rotational invariance property (4.10)(4.12) shows that it suces to
study the behavior of F (B(! ); d), i.e., of a single scattered source at
broadside, since the general case F (Rx(!; ! ); d) is easily obtained by a
translation of all estimates by !.
Let us rst study the case d = 1. Since B(! ) = Rv (0; ! ) corresponds to a scattered source symmetrically distributed around the origin, we would expect to get the estimate F (B(! ); 1) = 0 for any reasonable estimation algorithm. Proofs of this property can be found in
Appendix 4.B for root-MUSIC and in Appendix 4.C for MODE. This, together with the rotational invariance, shows that F (Rx (!; ! ); 1) yields
a consistent estimate of the nominal spatial frequency ! as N ! 1, since
F (Rx (!; ! ); 1) = F (Da (!)B(! )Da (!); 1) = ! + F (B(! ; 1) = ! :
(4.14)
Furthermore, if we look for two point sources, i.e., we set d = 2, it is
reasonable to expect the algorithm to give a pair of estimates symmetrically placed around the origin, i.e., F (B(! ); 2) = f(! ); ?(! )g for
some function (! ). It can also be expected that a larger spread angle
42
4 Parameter Estimation
would give a larger separation between the estimates, i.e., that (! ) is
monotonically increasing in ! . These properties can be proven for e.g.
root-MUSIC (see Appendix 4.B) and MODE (see Appendix 4.C). An
example of the function (! ) is shown in Figure 4.1.
0.6
0.5
(! )
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
!
0.8
1
1.2
Figure 4.1: (! ) for an 8 element ULA, uniformly distributed angular
spread and the root-MUSIC algorithm.
Note that the mean of the two estimates is zero a consistent estimate
of the nominal spatial frequency of B(! ) = Rv (0; ! ). Because of the
rotational invariance, the same will be true for all values of !. Similarly,
an estimate of (! ) is easily obtained from the distance between the
two values obtained from F (R^ x ; 2) and ! can be interpolated from a
pre-computed table over (! ). This suggests the following algorithm
for estimation of ! and ! .
f^1 ; ^2 g = F (R^ x ; 2)
(4.15)
^
+
^
!^ = 1 2 2
(4.16)
^1 ? ^2 ^! = ?1
:
(4.17)
2
When the underlying DOA estimation algorithm F (R^ ; d) is root-MUSIC
or MODE, we will use the name Spread root-MUSIC and Spread MODE
respectively for this new algorithm.
4.3 Low Complexity Algorithms
43
To summarize, we have shown the following theorem.
Theorem 4.1. Suppose that the algorithm f!^1; : : : ; !^dg = F (R^ ; d) obeys
the following three properties,
P1 Rotational
invariance. If f!^ 1 ; : : : ; !^d g = F (R; d) and f^1 ; : : : ; ^dg =
?
F Da (!)RDa (!); d then f^1 ; : : : ; ^d g = f!^1 + !; : : : ; !^ d + !g,
for any covariance matrix R.
P2 The estimate does not change if the signal is scaled or if spatially
white noise is added, i.e., F (R; d) = F (S R + n2 I; d).
P3 For all ! in some range [0; max ] of useful values of the spread angle,
F (B(! ); 1) = 0
F (B(! ); 2) = f(! ); ?(! )g
(4.18)
(4.19)
for some monotonically increasing function (! ).
P
Suppose further that R^ x = N1 Nt=1 x (t)x (t), where x(t) was generated
from (2.9), then
^ ; 1) is a consistent estimate of ! as N ! 1.
1. !^ = F (R
2. The algorithm (4.15)(4.17) gives consistent estimates of ! and !
when N ! 1 as long as 0 ! max .
In particular, the theorem holds when the algorithm is root-MUSIC or
MODE (modied as described in Theorem 4.14).
Going back to the physical model, the parameters are estimated as
^ = arcsin
^ =
^!
!^ 2
(4.20)
:
(4.21)
2 cos ^
Because of the approximations made in the transition between the physical and the approximative model, as described in Section 2.3, the estimates ^ and ^ will not be exactly consistent, but with a good approximation.
If the approximations in Section 2.3 are used to express the algorithm directly in terms of the physical parameters, using a version of the
44
4 Parameter Estimation
DOA estimation algorithm that returns the DOA values, not the spatial
frequencies, the following modication of the algorithm is obtained.
n^ ^ o
(4.150)
#1 ; #2 = F (R^ x ; 2)
^ ^
^ = #1 +2 #2
(4.200)
!
^1 ? #^2
#
?
1
^ = (4.210)
2
where ( ) is dened by
f ( ); ? ( )g = F (Rv (0; ); 2) :
(4.190)
It is not obvious which version of the algorithm is to prefer. The asymptotic variance is the same but (4.200) might give less bias than (4.20)
when is large, since the same approximation is used both going from
to ! and back again in the derivation of the algorithm. However, in
the analysis and most of the numerical examples, the rst version of the
algorithm will be used.
A couple of robustications are necessary in an implementation of the
algorithm. If the angular spread is zero, then one of the estimates ^1 ; ^2
will correspond to the true DOA whereas the other will be random which
causes the algorithm to fail. One solution is to use an algorithm such as
MDL [WK85] to estimate the number of point sources. If only a single
source is detected, set ^ = 0 and use the standard DOA algorithm to
estimate ^. The same should be done if j^1 ? ^2 j is larger than some
threshold.
One problem occasionally experienced using MODE, is that if the
sample covariance because of both source spread and nite sample eects
is too dierent from a point source model, then the roots of the MODE
polynomial will not stay on the unit circle but give a pair of mirror points,
resulting in ^1 = ^2 .
The algorithm can be extended to the case of several scattered sources.
The simplest solution is to look for twice the number of scattered sources,
pair the estimates together, two by two and use the scheme (4.16)(4.21)
for each pair. This will perform well as long as the signal power and
the angular spread is of the same magnitude for all the sources and the
source separation is reasonably large, see Section 4.5.
As an alternative, some kind of iterative scheme could be devised,
which estimates one source at a time, projecting or subtracting away the
4.4 Performance Analysis
45
impact of the source before the next source is estimated. Similar ideas
can for example be found in [SHN95].
Estimation of the number of sources is an interesting topic but falls
beyond the scope of this study.
4.4 Performance Analysis
In order to make the performance analysis more tractable, we will assume
that the channel varies rapidly, i.e., that the instantaneous array response
vector realizations v(t) are independent from sample to sample. We
will also require x(t) to be Gaussian. This is not true in general, but
a sucient requirement is that the transmitted signal s(t) has constant
modulus, since the product of a complex Gaussian variable and a constant
modulus variable is still Gaussian. None of these assumptions are vital
for the algorithms themselves.
The Cramér-Rao lower bound (CRB) on the estimation error variance
is given by
E (^ ? )(^ ? )T FIM?1
(4.22)
where the Fisher Information Matrix (FIM) is given by Bangs' formula
[Ban71]
FIMij = N Tr
R?x 1 @@Rix R?x 1 @@Rjx
(4.23)
see also [TO96], where an optimally weighted covariance matching algorithm, WLS, is derived, that asymptotically reaches the CRB. The WLS
algorithm is simply given by
[^; ^ ] = arg min Tr
;2;
S;n
(Rx (; ; S; 2 ) ? R^
n
x )W
2
(4.24)
where the optimal weighting W = R^ ?x 1 gives asymptotically ecient
estimates.
The algorithms presented here cannot be expected to give the same
performance, the main purpose has been to reduce the computational
complexity. In order to obtain the asymptotic variance of the estimation
error, we need the asymptotic variance of the underlying DOA estimation algorithm and an expression for the derivative of the function (! ),
46
4 Parameter Estimation
which also depends on the underlying algorithm used. Unfortunately, the
standard results found in the literature cannot be used, since the true covariance matrix in this case does not correspond to point sources in white
noise. This raises two major problems in the analysis. First of all, the
standard results on the statistical distribution of the signal eigenvectors
cannot be used since not all noise eigenvalues are equal. Secondly, the
estimated array response vectors will not be exactly orthogonal to the
noise subspace. The root-MUSIC and MODE algorithms are analyzed in
Appendices 4.B and 4.C, respectively, using some general tools developed
in Appendix 4.A.
Note that these results also hold for the problem of DOA estimation
of point sources in colored noise and therefore can be of more general
interest. Similar results have been published previously for some other
algorithms, see e.g. [Vib93].
The performance analysis of our algorithm can be reduced to the
special case of a signal at broadside since if x(t) 2 N (0; Rx) then let
y(t) = Da (!)x(t) and note that y(t) 2 N (0; R0) where
R0 = Da (!)RxDa (!)
= Da (!)(S Rv (!; ! ) + n2 I)Da (!)
= S B(! ) + n2 I = S Rv (0; ! ) + n2 I :
(4.25)
From (4.16), it follows that the error variance of !^ is given by
2 )2 ] E[j1 j2 + 1 2 ]
=
:
E[j!j2 ] = E[(1 +
4
2
(4.26)
The last equality follows from the symmetry relationship between the two
estimates, imposed by (4.19). Next, from (4.17)
2 1 2 ]
2
E[j! j2 ] = E[(?@1 (?!) 22 ) ] = E[j?1 j@?
4 @!
2 @(!! ) 2
(4.27)
where the derivative depends on the underlying algorithm and is given in
!)
, which in its turn depends on
the appendices as a function of @ Rv@(!;
!
the angular distribution and can be found in the appendix of Chapter 2.
For the parameters of the physical model,
!j2 ] = E[j1 j2 + 1 2 ]
E[jj2 = (2Ejcos
)2
2(2 cos )2
(4.28)
4.5 Numerical Examples
47
and if the two DOA estimates are uncorrelated (which mostly holds with
good approximation) then it can be shown that
E[j1 j2 ] 1
2
+
(
tan
)
E[j j2 ] = 2(2
?
cos )2 @(! ) 2
@!
E[j1 j2 ]
?
@
(2
cos ) 2
2
(4.29)
@
since the second term can be neglected.
4.5 Numerical Examples
Simulations have been performed using a basic scenario with an 8 element ULA with half wavelength element separation, a single source located at broadside, i.e = 0 , SNR 10 dB and ~ uniformly distributed
over [? ; ] with = 3 (i.e., 5:2). The number of incoming
wavefronts was set to L = 50. Each estimate was calculated from a burst
of N = 100 data samples.
The plots in Figures 4.34.6 show the theoretical and estimated RMS
values of ^ and ^ , when the dierent parameters are varied one at a
time, calculated from 500 trials for each test case. The gures show
the performance of Spread root-MUSIC, i.e., root-MUSIC was used as
the underlying point source algorithm, except for Figure 4.4 that shows
the performance of Spread MODE. As a comparison, the performance of
ordinary root-MUSIC/MODE (for the DOA estimation) and WLS (4.24)
have been included as well as the Cramér-Rao lower bound.
Figure 4.7 illustrates that the alternative version of the algorithm
(4.150)(4.210) gives slightly less bias for large . The bias was calculated
from an average of 5000 experiments, with a few outliers removed.
The eect of the robustications mentioned at the end of Section 4.3
is clearly visible in Figure 4.3 for 1, where the source has been
detected as a point source for most of the trials.
The theoretical performance results coincide well with the empirical results as long as the spread is not too large, as was anticipated in
Section 4.4. As expected, the theoretical results for MODE are more
accurate than those for root-MUSIC, since a more exact derivation could
be performed.
In general, the algorithm performs best in situations when exactly
two eigenvalues of Rx are signicantly larger then the noise level. This
48
4 Parameter Estimation
is no surprise, since a rank 2 approximation is used. This conclusion can
also be drawn from the theoretical expressions.
There is almost no dierence between using root-MUSIC or MODE as
underlying point source algorithm. No results from the ESPRIT version
of the algorithm are shown here, but it generally gives slightly larger
estimation errors.
Max spread angle , degrees
34
32
30
28
26
24
22
20
18
4
6
8
10
12
14
Number of sensors
16
18
20
Figure 4.2: Maximum spread angle that the algorithm theoretically
can handle for dierent size of the array. Uniformly distributed angular
spread. Half wavelength element separation, = 0 , 100 snapshots.
The maximum spread angle that could theoretically be handled by
Spread root-MUSIC is shown in Figure 4.2 as a function of m, the number
of antenna elements. The maximum is calculated as the maximum spread
angle where ( ) in (4.190) is still a monotonously increasing function
of .
The generalization of the algorithm to handle several sources was
simulated on a scenario with an 8 element ULA, 20 dB SNR, N=100,
with two scattered sources, one xed at 1 = 5 and the other varied
between 8 and 30. Both sources had a uniformly distributed angular
spread with = 3 . The RMS error of the parameters of the xed source
is shown i Figure 4.8. The results for the second source are similar. The
strange behavior compared to the Cramér-Rao bound is explained in
4.5 Numerical Examples
49
6
Spread root−MUSIC sim.
RMS error of ^, degrees
5
Spread root−MUSIC theory
root−MUSIC sim.
root−MUSIC theory
4
WLS
CRB
3
2
1
0
0
1
2
3
4
5
6
True , degrees
7
8
9
10
7
8
9
10
3
RMS error of ^ , degrees
Spread root−MUSIC sim.
2.5
Spread root−MUSIC theory
WLS
CRB
2
1.5
1
0.5
0
0
1
2
3
4
5
6
True , degrees
Figure 4.3: RMS values of ^ and ^ for dierent angular spread, .
= 0 , 8 sensors, 10 dB SNR, 100 snapshots.
Section 4.6. As a robustication, MDL was used to estimate the number
of point sources and the ordinary root-MUSIC algorithm was used if
the the number was less than four. The eect can be seen in the gures
when the source separation is less than 5 . The algorithm does not give
50
4 Parameter Estimation
5
RMS error of ^, degrees
Spread MODE sim.
4.5
Spread MODE theory
4
MODE sim.
MODE theory
3.5
CRB
3
2.5
2
1.5
1
0.5
0
0
1
2
3
4
5
6
7
8
9
10
4
5
6
7
8
9
10
True , degrees
1.6
RMS error of ^ , degrees
1.4
Spread MODE sim.
Spread MODE theory
1.2
CRB
1
0.8
0.6
0.4
0.2
0
0
1
2
3
True , degrees
Figure 4.4: RMS values of ^ and ^ using MODE instead of rootMUSIC. . = 0 , 8 sensors, 10 dB SNR, 100 snapshots. Compare to
Figure 4.3.
consistent estimates when more than one source is present and the error
is mainly due to the bias in the estimates, at least when the source
separation is small.
4.5 Numerical Examples
51
0.8
RMS error of ^, degrees
Spread root−MUSIC sim.
Spread root−MUSIC theory
root−MUSIC sim.
0.6
root−MUSIC theory
WLS
CRB
0.4
0.2
0
0
5
10
15
20
25
30
SNR, dB
35
40
45
50
40
45
50
2.5
RMS error of ^ , degrees
Spread root−MUSIC sim.
2
Spread root−MUSIC theory
WLS
CRB
1.5
1
0.5
0
0
5
10
15
20
25
30
SNR, dB
35
Figure 4.5: RMS values of ^ and ^ for dierent SNR. = 0, = 3,
8 sensors, 100 snapshots.
52
4 Parameter Estimation
2.5
RMS error of ^, degrees
Spread root−MUSIC sim.
Spread root−MUSIC theory
2
root−MUSIC sim.
root−MUSIC theory
1.5
WLS
CRB
1
0.5
0
0
10
20
30
40
50
60
70
80
40
50
60
70
80
Nominal , degrees
4
Spread root−MUSIC sim.
RMS error of ^ , degrees
3.5
Spread root−MUSIC theory
WLS
3
CRB
2.5
2
1.5
1
0.5
0
0
10
20
30
Nominal , degrees
Figure 4.6: RMS values of ^ and ^ for dierent nominal DOA .
= 3 , 8 sensors, 10 dB SNR, 100 snapshots.
4.5 Numerical Examples
53
0.1
Bias of ^, degrees
0
−0.1
−0.2
−0.3
Spread root−MUSIC 1
−0.4
Spread root−MUSIC 2
root−MUSIC
−0.5
0
10
20
30
40
50
60
70
80
30
40
50
60
70
80
Nominal , degrees
0.5
Bias of ^ , degrees
0
−0.5
−1
−1.5
−2
−2.5
Spread root−MUSIC 1
Spread root−MUSIC 2
−3
0
10
20
Nominal , degrees
Figure 4.7: Bias of ^ and ^ for the two variants of the algorithm, Spread
MUSIC 1 (4.15)(4.21) and Spread MUSIC 2 (4.150)(4.210). = 3,
8 sensors, 10 dB SNR, 100 snapshots.
54
4 Parameter Estimation
RMS error of ^, degrees
3
Spread root−MUSIC
root−MUSIC
2
WLS
CRB
1
0
0
5
10
3
RMS error of ^ , degrees
15
20
Source separation, degrees
25
Spread root−MUSIC
WLS
CRB
2
1
0
0
5
10
15
20
Source separation, degrees
25
Figure 4.8: RMS values of ^ and ^ of one source in a two source
scenario. 1 = 5 , 1 = 2 = 3 , 8 sensors, 20 dB SNR, 100 snapshots.
4.6 A Theoretical Curiosity
55
4.6 A Theoretical Curiosity
The CRB curve for estimation of two sources in Figure 4.8 looks strange,
especially compared to the simulation results of our algorithm. At about
11 source separation, the Fisher Information Matrix is singular and since
the CRB for a biased estimator is given by [Por94]
varf^ g (I + @@g(T ) ) FIM?1 (I + @@g(T ) )T
(4.30)
g() = E[^] ? (4.31)
where
any estimator, biased or unbiased, should give innitely large estimation
variance for this specic choice of scenario. Of course, one assumption
for the CRB to hold is that the FIM is non-singular for all parameter
values, however we could seemingly exclude a small interval where the
singularity occurs and still get a CRB that is much larger than the actual
simulation result.
What happens in this specic situation is that when two sources with
uniform spatial distribution end up edge to edge, they cannot be distinguished from a single source of the double width. This explains why
the specic scenario is not identiable and the FIM is singular. Our
algorithm matches the estimated data covariance to four point sources,
pairs them together two and two and estimates the spread angle based
on the separation within each pair. Thereby, the algorithm implicitly
assumes that both the sources have the same spread angle. Locally,
E ^1 = E ^2 = 1 +2 2 which shows that the rst and the last factor
of (4.30) will be almost singular and this singularity will apparently cancel the singularity of the FIM, thus the results shown in Figure 4.8 do
not contradict the theory.
This robustness that was implicitly built into the algorithm also shows
a weakness, namely the bias in the estimates if two sources of dierent
spatial width appear close to each other.
Compare also to the use of Bayesian estimators for data models where
the original model does not give identiability.
The term supereciency is sometimes used for these kind of phenomena, see [IH81, SO96]. Actually, our algorithms exhibits also another
supereciency phenomenon. When a single point source is present, the
robustied version of the algorithm uses MDL to detect that this indeed
56
4 Parameter Estimation
is a point source and sets ^ = 0 which at = 0 gives an unbiased
estimate with zero variance. This is more like the standard examples of
supereciency, see [SO96].
4.7 Subspace Fitting Algorithms
Let us step back and view the problem at a slightly higher level of abstraction. We have a data model that in a noise free environment gives
a data covariance matrix R() as a function of a parameter vector .
With added noise the data vector x has covariance
Rx(; n) = R() + n2 I :
(4.32)
For simplicity, we have assumed that the additive noise is spatially white.
In contrast with the data models traditionally used for subspace tting
algorithms, the signal covariance matrix R() has full rank but, at least
in the actual case, a number of the eigenvalues are small compared to n2 .
Even if there are no true signal and noise subspaces, it is still possible to perform an eigenvalue decomposition of Rx = EE , pick the d
principal eigenvectors of Rx as a pseudo-signal subspace and write the
covariance matrix as Rx = Es s Es + En n En . Here d, the dimension
of the pseudo-signal subspace is a parameter that can be chosen by the
user.
Denote estimated values of Rx, Es and En with R^ x , E^ s and E^ n , respectively. Generalizing the ideas of traditional subspace tting methods,
^ as orthogonal as
we would like to nd an estimate ^ that makes Es ()
possible to E^ n .
Two possible cost functions are
f1 () = kEs ()E^ n k
(4.33)
^
f2 () = kEn()Es k
(4.34)
representing pseudo-noise and pseudo-signal subspace tting approaches,
respectively. These can be generalized to
f3 () = kS()E^ n k
(4.35)
^
f4 () = kN ()Es k
(4.36)
where S() and N() are functions such that spanfS()g = spanfEs()g
and spanfN()g = spanfEn ()g. The idea of this generalization is, if
4.7 Subspace Fitting Algorithms
57
15
1=f1()
10
10
10
5
10
5
4
3
0
2
1
−5
0
Figure 4.9: Cost function of the pseudo-noise subspace tting criterion,
1=f1()). d = 2, 0 = 0 , 0 = 2, m = 8, no noise, true covariance
matrix.
possible, to nd S() or N() that are easier to compute than the eigenvectors. Any norm can be used in (4.33)(4.36) and further research is
necessary for example to nd optimal weighting matrices if a weighted
Frobenius norm is used.
In contrast to the case of point sources, subspace tting algorithms
can never give optimal performance for full rank models, since spanfE^ s g
is not a sucient statistic. Still, for many problems, we believe that
the performance is near optimal and could have computational advantages compared to ML or covariance matching techniques. Note that
the algorithms give consistent estimates as long as is uniquely determined by spanfEs ()g. As an example, the noise subspace tting cost
function (4.33) is shown in Figure 4.9. Figure 4.12 shows the result of
58
4 Parameter Estimation
4
10
3
1=fDSPE()
10
2
10
1
10
0
10
5
4
3
0
2
1
−5
0
Figure 4.10: Cost function of DSPE (1=fDSPE()), same test case as
Figure 4.9.
a numerical simulation of this algorithm, using the same scenario as in
Figures 4.34.4, for three dierent choices of signal subspace dimension,
d = 1; 2; 3. As could be expected, the algorithm gives best performance
when d is chosen as the number of eigenvalues of Rx that are signicantly
larger than the background noise. In this application, d = 2 gives the
best overall performance for the parameter range of interest.
Let us review a few algorithms in terms of this framework. First of all,
the root-MUSIC version of the algorithm presented in Section 4.3 can be
seen as an approximation of (4.35) since the algorithm nds the rank two
matrix A = [a(! ? ! ) a(! + ! )] that gives the best approximation of
spanfRv g. Similarly, the MODE version can be seen as an approximation
of (4.36).
The DSPE algorithm derived in [VCK95] uses a cost function (in our
4.7 Subspace Fitting Algorithms
59
3
1=fDISPARE()
10
2
10
1
10
0
10
−1
10
5
4
3
0
2
1
−5
0
Figure 4.11: Cost function of DISPARE (1=fDISPARE()), same test
case as Figure 4.9.
notation)
h
fDSPE() = Tr E^ n Rv ()E^ n
i
(4.37)
where = [; ]T . A similar algorithm, DISPARE, is derived in [MSW96]
with the cost function
i
h
(4.38)
fDISPARE() = kE^ n Rv ()k2F = Tr E^ n R2v ()E^ n :
In order to study the consistency, assume that the true covariance
matrix is inserted and see what happens at the true parameter value 0 .
fDSPE(0 ) = Tr [En Rv En ]
(4.39)
= Tr [En (Es s Es + En n En )En ] = Tr [n (0 )] :
When the parameters are varied around 0 , it may very well happen that
the second term decreases and that the decrease is larger than the increase
60
4 Parameter Estimation
of the rst term, i.e., 0 is not even a local minimum of FDSPE(). This
phenomenon is illustrated in Figure 4.10, where there is a local peak at
[; ] = [0; 1:6 ] and two global peaks at [; ] = [2:7; 0 ], whereas
the true source is at [0 ; 0 ] = [0 ; 2 ]. It is clear from (4.39) that a
normalization, such as f () = kE^ nRv ()k2F =kRv ()k will not make
any signicant dierence in this respect.
Similarly, for the DISPARE cost function
fDISPARE(0 ) = Tr En R2v En
= Tr En (Es 2s Es + En 2n En )En
= Tr 2n (0 ) :
(4.40)
Again, this can lead to inconsistencies, but the proportion between the
rst and the second term is squared compared to (4.39) which should
decrease the risk of inconsistent estimates. Figure 4.11 illustrates the
DISPARE cost function, which in the same example gives a peak at
[; ] = [0 ; 2:9]. Seen from another point of view, R2v () is an approximation of spanfEs ()g for some small signal subspace dimension
h ^ dk, which
i
makes fDISPARE() f3 (). Using the same argument, Tr En Rv ()E^ n
could be expected to give better and better approximations of f3 () with
d = 1, when k is increasing. Compare to the method of iterated powers
to calculate the principal eigenvector of a matrix [GL96].
Note that the Figures 4.104.11 also illustrate how well a scattered
source can be approximated by two point sources.
4.7 Subspace Fitting Algorithms
61
6
NSF, d=1
NSF, d=2
NSF, d=3
CRB
RMS error of ^, degrees
5
4
3
2
1
0
1
2
3
4
5
6
7
True , degrees
8
9
10
8
9
10
6
NSF, d=1
NSF, d=2
NSF, d=3
CRB
RMS error of ^ , degrees
5
4
3
2
1
0
1
2
3
4
5
6
7
True , degrees
Figure 4.12: RMS values of ^ and ^ using the pseudo-noise subspace
tting algorithm (4.33) with dierent values of d. = 0, 8 sensors, 10
dB SNR, 100 snapshots. Compare to Figures 4.34.4.
62
4 Parameter Estimation
4.8 Conclusions
We have introduced a new algorithm for estimation of DOA and spread
angle of spatially distributed sources. The presented algorithm shows
that it indeed is possible to get reasonable estimation performance for
this problem with low computational complexity. The algorithm can
be based on most existing DOA estimation algorithms for ULAs, and
is shown to give consistent estimates for a single scattered source if the
used DOA estimation algorithm fullls certain properties. In particular,
we have shown that root-MUSIC and MODE obey these properties and
have derived expressions for the asymptotic variance of the estimated
parameters for these specic versions of the algorithm. The numerical
simulations conrm the performance analysis and show that the performance is good within a range of parameter values.
When the algorithm is extended to handle more than a single source,
the estimates are no longer consistent, but the estimation error is still
comparable to optimal methods.
Finally, we have introduced the concept of pseudo-subspace tting
for full rank models. This is in general a suboptimal method since a low
rank approximation of the sample covariance matrix is not a sucient
statistic if the noise free signal covariance is full rank, but in this application where the model is almost low rank, the performance loss is small.
The low complexity algorithm can be interpreted as an approximative
version of a pseudo-noise subspace tting algorithm, using a two-point
approximation of the scattered source. A couple of previously published
attempts to perform pseudo-subspace tting were shown to give inconsistent parameter estimates.
4.A Miscellaneous Results
63
Appendix 4.A Miscellaneous Results
4.A.1 Pseudo-Signal and Pseudo-Noise Subspaces
Perform Singular Value Decompositions (SVDs) on Rx and B, then
from (4.13) it is clear that
B = EB B EB
(4.41)
2
Rx = ERRER = DaEB (S B + nI)EB Da
(4.42)
which gives the following simple relations for the eigenvalues and eigenvectors of Rx and B
R = S B + n2 I
(4.43)
ER = Da EB :
(4.44)
Since, in general, B has full rank, it is impossible to make the standard
separation into a true signal subspace and noise subspace. However, we
can still pick the d principal eigenvectors of Rx as a pseudo-signal subspace and decompose the covariance matrix as Rx = Es;R s;R Es;R +
En;Rn;REn;R. Using (4.44), the pseudo-signal and pseudo-noise subspaces of Rx and B are related by
Es;R = DaEs;B En;R = Da En;B :
(4.45)
Using the assumptions of Section 4.4, x(t) is a random Gaussian vector with zero-mean and E[x(t1 )x (t2 )] = Rx (t1 ? t2 ). The algorithms
use an estimate of the covariance matrix
N
X
(4.46)
R^ = 1 x(t)x (t) :
x
N t=1
Denote the estimation error by Rx = R^ x ? Rx.
For a low rank data model with added white noise, the statistical
properties of E^ s and E^ n are well known [SN89]. For general covariance
structures, some results can be found in [Gup65], however, we provide
an alternative derivation that, together with Lemma 4.9, gives results
directly in a matrix form. Note that the problem is dicult, since if
several eigenvalues are equal or are closely separated compared to the
magnitude of the disturbances, then the corresponding eigenvectors are
not uniquely dened.
The fundamental result which we will use is given by the following
theorem by Rellich [Rel69] and Kato [Kat82].
64
4 Parameter Estimation
Theorem 4.2. Suppose that Rx() = Rx + Rx is Hermitian for real .
Suppose that is an eigenvalue of nite multiplicity h of Rx and suppose
that there is a positive number such that the interval [?; +] contains
no other eigenvalue. Then there exists power series 1 (); : : : ; h () and
e1(); : : : eh() all convergent in a neighborhood of = 0, such that
1. Rx()ek () = k ()ek () and ei ()ek () = ik .
2. For each 0 < , the spectrum of Rx() in [ ? 0 ; + 0 ] consists
exactly of the points 1 (); : : : ; h () for real with kRx kF < 0 .
Proof. See [Rel69, Chapter I:1]. A proof of the bound on can be found
in [Kat82, Chapter II:3].
When it comes to explicit expressions for these power series, a partial
result is given by the following theorem by Krim [KF96] (only the rst
order term is quoted here, the reference gives a recursive formula for all
higher order terms).
Theorem 4.3. Let = EsEs and ? = EnEn be the projection matrices into the pseudo-signal and pseudo-noise subspaces, respectively. Then
the corresponding matrices calculated from the sample covariance matrix
are given by
^ = + + O(kRxk2)
(4.47)
?
?
2
^ = ? + O(kRxk )
(4.48)
where
?
=Es (Es RxEn ) MTns En
+ En ((En Rx Es ) Mns ) Es
(4.49)
and
1
:
(4.50)
[Mns ]kl = ?
nk sl
Proof. See [KF96] and Corollary 4.5. Note that the theorem does not hold
with certainty if the smallest eigenvalue of s and the largest eigenvalue
of n are closer than 2kRxkF .
The rst order terms for the eigenvalues and eigenvectors are given
by the following theorem.
4.A Miscellaneous Results
65
Theorem 4.4. If all eigenvalues (single or multiple) of Rx are separated
at least kRxk, then there exists an E such that
Rx = EE
(4.51)
Rx + Rx = E^ ^ E^
(4.52)
where
^ = + + O(kRxk2)
E^ = E + E + O(kRxk2)
(4.53)
(4.54)
= I (E RxE)
E = E(M (E Rx E) + )
( 1
if k 6= l
[M]kl = k ?l
0
if k = l
(4.55)
(4.56)
and
(4.57)
and is some diagonal purely imaginary matrix of the same magnitude
as Rx.
Note that in general, E is a function both of Rx and Rx. It can be
determined solely from Rx only if all eigenvalues of Rx are single. The
undetermined appears since each eigenvector is only determined up to
a multiplication by ej! for some !1 .
Proof. For notational simplicity, equality means equality only up to rst
order terms in this proof.
First, note that E E = ?EE since
I = (E + E)(E + E) = I + EE + EE :
Since Rx + Rx = (E + E)( + )(E + E)
+ ERxE = E(Rx + Rx)E = (I + EE)( + )(I + EE)
= + E E + E E + which gives
ERxE = ? EE + EE = + L (EE) (4.58)
1 An alternative is to enforce uniqueness, choosing ! such that
This criterion is tacitly assumed in most literature on the subject.
e^k ek > 0
[SS97].
66
4 Parameter Estimation
where Lkl = k ? l .
Since is diagonal and L is zero on the diagonal, (4.55) follows
immediately. All values of E E except for the diagonal are determined
solely by (4.58), but since E E is skew-Hermitian, its diagonal elements
are purely imaginary, say . Thus (4.56) is necessary for E + E to be
eigenvectors of Rx + Rx. Straightforward calculations show that up to
rst order (4.55)(4.56) gives (E + E) (E + E) = I and (E + E)( +
)(E + E) = Rx + Rx which proves suciency.
Divide the matrices into blocks corresponding to the pseudo-signal
and pseudo-noise subspaces.
0 = 0s n
E = Es En
0 M ?MT ss
ns
= 0s n :
M = Mns Mnn
Then we have the following corollary.
Corollary 4.5. Suppose that in addition to the assumptions of Theorem 4.4, all pseudo-signal eigenvalues are single, then
^ s = s + s + O(kRxk2)
(4.59)
^Es = Es + Es + O(kRxk2 )
(4.60)
where
s =I (Es Rx Es )
(4.61)
Es =Es (Mss (Es Rx Es ))
+ En (Mns (En RxEs )) + Es s :
(4.62)
Proof. First note that Es does not depend on Rx since all eigenvalues
are simple. If En0 is a block of En corresponding to a multiple eigenvalue
n0 , then Mn0 s (En0 RxEs ) = En Rx Es DM 0 where DM 0 is the diagonal matrix [DM 0 ]kk = n0 ?1 sk . Thus, the corresponding block of the
second term of (4.62) is given by
En0 Mn0s (En0 RxEs ) = En0 EnRxEsDM 0 = n0 RxEsDM 0
which is independent on the specic choice of eigenvectors corresponding
to n0 .
It is an easy exercise to prove Theorem 4.3 using this corollary. Note
that in general, no similar result can be given for En .
4.A Miscellaneous Results
67
4.A.2 Useful Lemmas
Lemma 4.6. Some results for Schur Hadamard products and Kronecker
products.
vec [A B] = diag [vec A] vec B
= diag [vec B] vec A
?
Tr [AB] = vec AT T vec B
vec(ABC) = (CT A) vec B
(A B)(C D) = (AC) (BD)
(4.63)
(4.64)
(4.65)
(4.66)
Proof. These standard results can be found for example in [Gra81]
Lemma 4.7.
Tr [A (B C)] = Tr (A BT ) C
Tr [(A B)(C D)] = Tr (A DT )(BT C)
Proof.
?
(4.67)
(4.68)
Tr [A (B C)] = vec AT T vec [B C]
?
= vec AT T diag [vec B] vec C
?
= vec AT T (diag [vec B])T vec C
?
= diag [vec B] vec AT T vec C
? = vec AT B T vec C = Tr (A BT ) C
(4.68) is an immediate corollary.
Lemma 4.8. If R = R^ ? R where R^ is the sample covariance matrix
formed from N independent samples of a Gaussian random vector with
covariance R, then
E [vec(ARB)(vec(CRD)) ] = N1 (BT Rc Dc ) (ARC ) :
(4.69)
Proof. It is shown in [WF93] that E[(vec R)(vec R) ] = N1 Rc R, thus
68
4 Parameter Estimation
using some results from Lemma 4.6
E vec(ARB)(vec(CRD))
=(BT A) E [(vec R)(vec R) ] (Dc C )
= 1 (BT A)(Rc R)(Dc C )
N
= N1 (BT Rc Dc ) (ARC ) :
Lemma 4.9. With the same prerequisites as Lemma 4.8,
i
h X((En REs) MT ) Tr Y((En REs) NT )
= N1 Tr [(X M)n (Y N )s ]
i
h
E Tr [X((Es REn ) M)] Tr [Y((Es REn) N)]
= N1 Tr (X MT )s (Y Nc)n
i
h
E Tr [X((Es REs ) M)] Tr [Y((Es REs ) N)]
= N1 Tr (X MT )s (Y Nc)s
i
h E Tr X((En REs ) MT ) Tr [Y((Es REn ) N)] = 0
i
h E Tr X((Es REs ) MT ) Tr [Y((Es REn ) N)] = 0 :
E Tr
(4.70)
(4.71)
(4.72)
(4.73)
(4.74)
Proof. We prove (4.70), the other results are proved similarly. Using
4.B Proofs for root-MUSIC
69
results from Lemmas 4.64.8,
h i
E Tr X((En REs ) MT ) Tr Y((En REs ) NT ) h
i
= E Tr [(X M)(En REs )] Tr [(Y N)(En REs )]
= vec(XT MT )T E[vec(En REs )(vec(En REs)) ] vec(Y N )
?
= vec(XT MT )T 1 (ET Rc Ec ) (E RE ) vec(Y N )
N
s
s
n
n
= vec(XT MT )T N1 (s n ) vec(Y N )
= 1 vec(XT MT )T vec( (Y N ) )
n
N
s
= N1 Tr [(X M)n (Y N )s ] :
Appendix 4.B Proofs for root-MUSIC
Theorem 4.10. Property P1 holds for root-MUSIC.
Proof. Let a(z ) = [1; z; : : :; z m?1]T and note that Da (!)a(z ) = a(ze?j! ).
The root-MUSIC polynomial is given by g(z ) = aT (z ?1)En En a(z ), see
[KV96]. From (4.45), E0n = Da (!)E if En and E0n are the pseudo-noise
subspaces for R and R0 = Da (!)RDa (!), respectively. The corresponding root-MUSIC polynomials g(u) and g0(z ) are related through
g0 (z ) = aT (z ?1 )E0n E0n a(z ) = aT (z ?1 )Da (!)En En Da (!)a(z )
= aT (z ?1 ej! )En En a(ze?j! ) = g(ze?j! ) :
The only dierence between the root loci of g0 (z ) and g(u) is a rotation
z = uej! , see Figure 4.13. Especially, for the d roots within the unit
circle with magnitude closest to one, arg[z ] = ! + arg[u].
Theorem 4.11. Property P3 holds for root-MUSIC.
Proof. Note that it is not sucient to assume that B is real and Toeplitz. In
fact
2
5
6
1
B = 642
3
1
2
5
1
1
5
3
27
7
15
2
1
5
3
(4.75)
70
4 Parameter Estimation
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−0.4
−0.4
−0.6
−0.6
−0.8
−0.8
−1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
−1
−1
!
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Figure 4.13: Root loci from root-MUSIC applied to B (left) and Rv
(right) respectively. Only the roots inside the unit circle are shown, d = 2.
is one example where root-MUSIC does not give a symmetrical pair of estimates
when d = 2.
Since B(! ) is real, En is also real and by construction, if z0 is a zero
of the root-MUSIC polynomial g(z ) then so are z0 , 1=z0 and 1=z0. If, in
addition, jz0 j = 1 then
g(z0 ) =
m
X
k=d+1
jek a(z0 )j2 = 0
(4.76)
and consequently ek a(z0 ) = 0, k = d + 1; : : :; m. Suppose that d = 1 and
! = 0. Then, by (2.15), a(z )jz=1 is the principal eigenvector of B(0)
and consequently, z0 = 1 is a root of g(z ) with multiplicity 2 and there
can be no other root on the unit circle. By continuity and because of the
symmetry relations between the roots, the root loci corresponding to the
two roots at z0 = 1 will stay on the real axis when ! grows and no other
roots are closer to the unit circle as long as ! is suciently small. This
proves that (4.18) holds for root-MUSIC.
If d = 2 then by (2.15), spanfe1; e2 g = spanfa(z )jz=1 ; d(z )jz=1 g when
! = 0, thus En a(1) = En d(1) = 0 which shows that z0 = 1 is a root
of g(z ) with multiplicity 4 and no other root is on the unit circle. As !
grows, the root loci will form a quadruple of conjugate mirror points or
possibly stay as double roots on the real axis, In either case, the roots
used by root-MUSIC will be of the form z1;2 = ej for all suciently
4.B Proofs for root-MUSIC
71
small ! which proves (4.19).
Theorem 4.12. For root-MUSIC, the asymptotic variance is given by
?
Re Tr (Es ak dk En ) MTns n ((En dk ak Es ) Mns ) s
E[j! j2 ] k
=
2N (dk ? dk )2
d mX
?d
X
1
2N (dk ? dk )2 i=1
jes;i ak j2 jen;l dk j2 ( s;i?n;l )2
s;i
n;l
l=1
(4.77)
where Mns is dened in (4.50).
Proof. If zk = rk ej!k is a root found by root-MUSIC from Rx and
z^k = (rk + rk )ej(!k +!k ) is the corresponding estimate obtained from
R^ x, then using a second order Taylor expansion of the root-MUSIC polynomial, it is proved in [KFP92] that
!k
= dk ak +?ak dk
2dk dk
(4.78)
@ a(! )j!=! .
if the root zk is on the unit circle. Here, ak = a(!k ) and dk = @!
k
In our application, zk will not be exactly on the unit circle but still
stay close enough that (4.78) gives a good approximation, at least for
small ! . The derivation in [KFP92] is dicult to generalize since the
Taylor expansion in !k and rk only decouples in a nice way if the
unperturbed root is on the unit circle.
Applying Theorem 4.3 on (4.78), neglecting the terms involving En ak
(the same kind of approximation as in the derivation of (4.78)) we obtain
!k
1
Tr Es ak dk En ((En Rx Es ) Mns )
?
2dk dk
?
+E dk a Es (E Rx En ) MT
n
k
s
(4.79)
ns
which, using Lemma 4.9, gives the desired result.
The cross covariance between the estimates of root-MUSIC is dicult
to derive analytically, but simulations have shown that it is negligible
compared to the covariance term in the expressions (4.26) and (4.27).
72
4 Parameter Estimation
Theorem 4.13. For root-MUSIC, the derivative of (! ) is given by
@(! ) 1
ak d En (E @ Rv Es ) M
Tr
E
n @!
s
k
ns
@! 2dk ? dk
+ En dk ak Es (Es @@Rv En) Mns
(4.80)
!
= 1? Re Tr Es ak dk En (En @@Rv Es ) Mns :
dk dk
!
Proof. Insert Rx = B(! + ! ) ? B(! ) = B( )! in (4.79), divide
@B .
by ! and let ! ! 0. This gives (4.80) with @@R!v replaced by @
!
Finally note that a translation of the spatial frequency by ! gives the
more general expression (4.80), since all factors Da (!) cancel out.
Appendix 4.C Proofs for MODE
Theorem 4.14. Property P1 holds for MODE if the algorithm is modied such that the constraint Re[g0 ] = 1 used in the minimization of (4.7)
and (4.8) is replaced by the constraint kgk2 = 1 which means that the
minimizing g is given as the singular vector with smallest singular value
of a certain matrix.
Proof. Dene the m (m ? d) matrix G by
2g g : : : g 0 : : :
0 1
d
66 0 g g : : : g . . .
G = 66 .. . .0 . .1 . . d . .
4. . . .
.
0 ::: 0
3
0
.. 77
.7:
7
05
(4.81)
g0 g1 : : : gd
Given coecients g0; : : : ; gd, let gk0 = gk e?jk! and collect the gk0 in a
similar matrix G0 . Then
G = (!)G0 Da(!)
(4.82)
where (!) = diag[1; ej! ; : : : ; ej(m?d?1)! ].
In MODE, the coecients gk are found through minimization of the
^ ], where V = I in the rst step
cost function f (g) = Tr[E^ s GVG E^ s W
?
^ G
^ ) 1 . Since W
^ only depends on the
and in the second step V = (G
4.C Proofs for MODE
73
eigenvalues, it is the same for both R and R0 = Da (!)RDa (!). Now
if V0 = (!)V (!)
h
i
f 0 (g0 ) = Tr E0s G0 V0 G0 E0s W
h
= Tr Es Da G0 V G0 Da Es W
= Tr [Es GVG Es W] = f (g)
i
(4.83)
and
0 0
min f (g) = jgmin
0 j=1 f (g ) :
jgj=1
(4.84)
Note that the same kind of relationship cannot be established in the
original formulation of MODE.
In the rst step of MODE,
V = V0 = I = (!)I(!), so (4.84)
^ 0 Da . In the second step V0 = (G
^0 G
^ 0 )?1 = V
^ = G
holds with G
which again gives estimates according to (4.84).
The spatial frequency estimates
P are obtained as the argument of the
roots of the polynomial g(z ) = d0 g^k z k . Since g^k0 = g^k e?jk! ,
g0(z ) =
X
g^k z k =
X
g^k (ze?j! )k = g(ze?j! )
(4.85)
which proves the relation for MODE.
Theorem 4.15. Property P3 holds for MODE.
Proof. Similarly to the proof for Theorem 4.11, note that since B(! ) is
real, also Es , W and thereby G are real. By construction, if z0 is a zero
of the root-MUSIC polynomial g(z ) then so are z0, 1=z0 and 1=z0.
For d = 1, e1 ? G and through (2.15) a(z )jz=1 ? G for ! = 0.
Thus, g(1) = 0 and z0 = 1 is a single root and there is no other root on
the unit circle. When ! grows, continuity and the symmetry restrictions
show that the root will stay at z0 = 1 and the spatial frequency estimate
is arg[z0 ] = 0.
For d = 2, fe1 ; e2 g ? G. As above, when ! = 0, (2.15) shows that
fa(z )jz=1 ; d(z )jz=1 g ? G, thus g(z )jz=1 = @g@z(z) jz=1 = 0. Consequently,
z0 = 1 is a double root and there is no other root on the unit circle.
When ! grows, the two roots must stay on the unit circle or on the real
axis. Numerical experiments show that they will stay on the unit circle
and arg[z1;2 ] = (! ) for some function (! ).
74
4 Parameter Estimation
In the analysis of root-MUSIC, certain approximations were necessary, since the estimated array response vectors were not exactly orthogonal to the pseudo-noise subspace. For MODE, it is possible to perform
an exact rst order perturbation analysis. The standard results on
MODE are derived for the signal subspace parameterized version of the
algorithm and the noise subspace parameterization, which we use here,
is shown to be asymptotically equivalent, see [SS90b]. With the data
model for scattered sources, this asymptotic equivalence is not true since
the true pseudo-signal subspace is not exactly parallel to the estimated
array response vectors, so the asymptotic performance must be analyzed
specically for the algorithm formulation with parameterized noise subspace. Also, in the analysis, the algorithm has to be viewed not as a
two-step algorithm, but as an iterative algorithm that solves
!^ = arg min fMODE (!)
(4.86)
where
h
i
^
fMODE(!) = Tr E^ s G(!)(G (!)G(!))?1 G (!)E^ s W
i
h
(4.87)
^
= Tr E^ s G (!)E^ s W
P
Q
and G(!) is dened by (4.81) with dk=0 gk z k = dk=1 (z ? ej!k ) (the
scaling of the coecients is dierent in the actual algorithm, but that
does not aect (4.87)).
Theorem 4.16. For MODE, the asymptotic covariance is given by
E[!!T ] = H?1 CH?1
(4.88)
where
h
C = 1 Re Tr 2((WE @ G E )MT ) ((E @ G E W)M )
ns s
ns n n @!l s
s @!k n
N
G Es) Mss Q)s((E @ G Es) Mss Q)si (4.89)
+ ((Es @@!
s @!l
k
kl
Qkl = Wkk ? Wll
(4.90)
@ 2G
Hkl = Tr Es @!k @!l EsW
(4.91)
(note that W is diagonal)
4.C Proofs for MODE
75
and the derivatives are given by [GP73]
@ G = ? @ G Gy + (: : : )
G @!k
@!k
(4.92)
@ 2 G = ? ? @ G Gy @ G Gy ? Gy @ G ? @ G Gy + ? @ 2 G Gy
G @!l @!k
G @!k @!l
@!k @!l
@!l G @!k
@ G (G G)?1 @ G ? ? ? @ G Gy @ G Gy + (: : : ) :
+ ?G @!
@!l G G @!k @!l
k
(4.93)
G(!) and its derivatives are calculated from the denition.
for d = 1
2?e?j! 1 : : : 03
G(!) = 64 ... . . . . . . ... 75
?j!
0
21 : :0: ?: e: : 031
@ 2 G = ?j @ G
@ G = je?j! 6 .. . . . . .. 7
4
5
.
.
.
.
@!
@!2
@!
0 :::
Especially,
(4.94)
1 0
and for d = 2
2 e?j(!1 +!2) ?(e?j!1 +e?j!2 ) 1
:::
03
.. 75
...
...
...
G(!1; !2) = 64 ...
.
0
:::
e?j(!1 +!2 ) ?(e?j!1 +e?j!2 ) 1
2 ?je?j(!1 +!2) je?j!k
1
::: 03
@G = 6
..
...
...
. . . .. 75
(4.95)
.
.
@!k 4
?j(! +! ) ?j!
0
21 : 0: : ?0je : :1 : 203je k 1
@ 2 G = ?j @ G :
@ 2 G = e?j(!1 +!2 ) 6 .. . . . . . . .. 7
5
4
. . . . .
@!1 @!2
@!2
@!k
0 :::
1
0 0
k
Proof. Standard Taylor expansion arguments show that
^ ? ! = ?H?1 @fMODE(!) + o(j!j)
! = !
@!
(4.96)
76
4 Parameter Estimation
where
@ 2fMODE(!) ! H when N ! 1 w.p. 1
(4.97)
@ !@ !T
which clearly gives (4.91). Since ! is a minimum of fMODE (!) when R
is used,
@fMODE(!) = Tr E @ G E W + E @ G E W :
s @!k s
s @!k s
@!k
(4.98)
Use Corollary 4.5 (the pre-conditions can be justied, since all eigenvalues
of Rx that are close compared to kRxk can be replaced by their average
value resulting in a small change compared to kRx k) and note that the
Es s terms cancel out since both s and W are diagonal and commute.
This gives
@fMODE(!) = TrhE @ G E (M (E R E ))W
@!k
s
@!k s
ss
s
x s
G E W
? (Mss (Es RxEs ))Es @@!
s
k
+ Es @ G En(Mns (En Rx Es ))W
@!k
G E W
+ (MTns (Es Rx En ))En @@!
s
i
k
G Es(Q Mss (E RxE ))
= Tr ?Es @@!
s
s
k
(4.99)
WEs @@!kG En(Mns (En RxEs))
+ Tr E @ G E W(MT (E RxE ))
+ Tr
n
@!k s
ns
s
which together with Lemma 4.9 shows that
@f (!) @fMODE(!) C = E MODE
@!
@ !T
n
(4.100)
is given by (4.89).
Theorem 4.17. For MODE, the derivative of (! ) is given by
@(! ) = ?H?1
(4.101)
@!
4.C Proofs for MODE
where
77
i
h
@ Rv
@ G
k = Tr ?Es @!k Es (Q Mss (Es @! Es ))
h
G En(Mns (E @ Rv E ))i :
+ 2 Re Tr WEs @@!
n @ s
k
(4.102)
!
Proof. Similarly to the proof of Theorem 4.13, the result follows directly
from (4.96) and (4.99).
Chapter 5
Signal Estimation Using
Estimated Channel
Parameters
5.1 Background
In Chapters 3 and 4, the following two problems were studied separately.
Estimating parameters of a channel.
Estimating the transmitted data, using knowledge of the channel.
What happens when the two parts are put together?
In both chapters, we have used two-ray approximations of the channel.
If the same approximation is used in both steps, then the nominal DOA
and spread angle, and are uninteresting as parameters, instead the
direction of the two rays should be estimated in a way that gives the best
performance of the beamformer. This idea is developed in Section 5.2
and interestingly enough, the result leads to another interpretation and
motivation of the algorithm from Section 4.3.
The rest of the chapter, Section 5.3, is devoted to numerical studies
of a few scenarios, since a theoretical analysis of the complete system is
very dicult.
80 5 Signal Estimation Using Estimated Channel Parameters
5.2 Two-Point Approximations Revisited
The purpose of this section is not to give a rigorous derivation but to
motivate one possible choice of algorithm.
Consider a rapidly time varying channel with local spread. We approximate each scattered source by two point sources with directions
fi1 ; i2 g and wish to use the LS beamformer
^s = A y x
(5.1)
where A = [a(11 ) a(12 ) : : : a(d1 ) a(d2 )] and s = [s11 ; s21 ; : : : ; s1d; s2d ]T contains two estimates for each scattered source. Using the matrix inversion
lemma, it is easy to show that
s^1
1
? ?1 ?
(5.2)
s^21 = (A1 A A1 ) A1 A x
where A 1 contains the two rst columns of A and A is the rest. The
natural estimate of the signal actually transmitted is some linear combination of s^11 and s^21 , i.e.,
?
(5.3)
s^1 = 1 s^11 + 2 s^21 = 01 a (11 ) + 02 a (12 ) ?A x = w x :
Since
x ? S1 Rv (1 ; 1 ))w
w Rx w
1=SINR = w (SRw
=
S1 w Rv (1 ; 1 ))w ? 1 (5.4)
1 Rv (1 ; 1 ))w
and 1 , 1 are unknown, a robust procedure to nd the direction estimates of the interferers that gives maximum SINR is to minimize w Rxw.
This is expected to give a good solution since it can be shown that
?A Rv (1 ; 1 ), and thereby the denominator of (5.4), is fairly constant,
as long as no DOA in A is too close to 1 (no ki within one lobe width
of 1 ). Now, w Rx w = a^1 ?A Rx?A a^1 , where a^=1 01 a(11 ) + 02 a(12 ) and
since the parameters of source 1 are unknown, a suboptimal procedure
is to minimize k?A Rx?A k for some suitable norm. We do not wish
to reestimate all DOA values for each source and it is natural to extend the idea and estimate all the 2d directions, not only the 2(d ? 1)
corresponding to the interferers. Finally, if the Frobenius norm is used,
the algorithm reduces to the familiar Deterministic Maximum Likelihood
(DML) estimate [KV96]
(5.5)
[^11 ; ^12 ; : : : ^d1 ; ^d2 ] = arg mink?A R^ xk :
5.3 Numerical Examples
81
Since the true model is not within the model set of 2d point sources,
this is of course not a ML estimate, which also explains the tedious and
somewhat vague derivation. Since MODE (with the weighting given in
[SS90a]) is asymptotically equivalent to DML for point source models
[SS90b], MODE should give very good performance also here. The same
holds for (root)MUSIC, since for uncorrelated point sources, also (root
)MUSIC is asymptotically equivalent to DML [SN89].
For reasons given in Section 4.3, we mainly use root-MUSIC in the
numerical studies below.
In Section 3.2, the direction of the two rays modeling each interferer
were chosen as k k , with the somewhat arbitrary choice k = k
in the simulations. In light of this section and Chapter 4, it is clear that
l = ( ) is a preferable choice, where MODE or root-MUSIC could
be used in the calculation of ( ) via (4.190).
How should a^1 , i.e., 01 and 02 , be chosen? One possibility is to
use a good estimate of a(1 ), for example a^1 = a(^1 ) as in (3.5) or
a^1 = a(^11 ) + a(^12 ). This corresponds to modeling the source
of interest
as a single point source. Another possibility is to estimate s^11 (t) and s^21 (t)
for a sequence of samples, using (5.2), collect the samples in vectors si1 =
[si1 (1); : : : ; si1 (N )]T and nd the 01 and 02 that maximizes k01 s11 + 02 s21 k,
i.e., chose [01 ; 02 ] as the right singular vector of the matrix [s11 s21 ] with
largest singular value. This is equivalent to ^s1 = s11 + s21 where is the
Total Least Squares [GL96] solution of s21 s11 .
5.3 Numerical Examples
Dierent combinations of algorithms have been tested on a scenario with
two scattered sources with uniformly distributed angular deviations. The
signal of interest is xed at 1 = 3 and has spread angle 1 = 2:5. The
second signal is moved in the interval [4 ; 43], is 10 dB stronger and has
a spread angle of 3 = 3:5 . The noise level is 10 dB below the signal of
interest. The antenna array has 8 elements and an element separation of
half a wavelength. In the simulations, the number of rays contributing
from each source was L = 10 and each data burst was N = 100 samples
long.
Basically, the following strategies were compared.
The ORS beamformer using parameter estimates from Spread rootMUSIC. The signal and noise powers were estimated using a least
82 5 Signal Estimation Using Estimated Channel Parameters
squares t.
n^
; ^ 2
Sk n
o
d
2
X
= arg min Si Rv (^i ; ^i ) + n2 I ? R^ x :
F
i=1
(5.6)
This sometimes gives negative power estimates, but we used them
anyway.
The ORS beamformer using the same strategy, except that WLS,
the weighted covariance matching was used for the parameter estimation. For numerical reasons, a regularized weighting matrix,
W = (R^ x + I)?1 was used in (4.24), with = 1.
The LS solution for a point source model, (3.25). The DOAs were
estimated using root-MUSIC.
Two-point approximation. Four directions were estimated using
root-MUSIC and the beamformer discussed in Section 5.2, w =
?A a^1 , was used with a^1 = a(^11 ) + a(^12 ).
The Minimum variance or Capon beamformer, w = R^ ?x 1 a(^1 ),
where 1 was estimated using ordinary root-MUSIC.
A couple of minor modications were necessary in the slowly time varying
scenario as described below.
5.3.1 A Rapidly Time Varying Scenario
The assumptions we have made for the rapidly time varying channel are
convenient in the theoretical analysis and as a benchmark, being an extremal case. However, in a communication application, the channel is
almost useless since all phase information is lost from sample to sample. A more realistic assumption would be that the channel varies slowly
compared to the symbol rate but rapidly compared to the burst rate.
Nevertheless, we kept to the simplistic assumptions in the simulations
and even set the signal constant, since a constant signal will not give any
dierent statistical performance than any constant modulus signal (x(t)
will still be Gaussian). It is still possible to estimate the SINR in the
simulations.
Figure 5.1 shows the average, for a rapidly time varying channel, of
1000 experiments for each value, together with the theoretical performance of ORS if all parameters are known. Each data burst was pro-
5.3 Numerical Examples
83
20
15
SINR (dB)
10
5
0
Theoretical optimum
ORS+Spread root−MUSIC
ORS+WLS
LS, single ray
LS, double ray
Capon
−5
−10
−15
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 5.1: SINR for dierent algorithms. Rapidly time varying channel.
cessed separately, forming a sample covariance matrix, beamformer and
then estimating the data.
WLS gives good parameter estimates and the performance is close to
the optimum regardless of the source separation. Also, the two-point approximation performs very well for most cases. The Spread root-MUSIC
estimator does not perform satisfactory. When the sources are closely separated (up to about 10), MDL mostly detects only three point sources
and the robustied Spread root-MUSIC algorithm gives point source estimates, see Section 4.3. A more sophisticated multiuser version of the
algorithm could estimate one of the sources as a point source and the
other one as scattered, using the remaining pair of direction estimates
to calculate nominal DOA and spread angle of the second source. This
would help in this scenario, where it is most critical to treat the strong
interferer as a scattered source.
Using a single point source approximation for each scattered source
clearly gives inferior performance, both with the LS and the Capon beamformer.
84 5 Signal Estimation Using Estimated Channel Parameters
20
15
Theoretical optimum
ORS+Spread root−MUSIC
ORS+WLS
LS, single ray
LS, double ray
MVDR
SINR (dB)
10
5
0
−5
0
5
10
15
20
25
Source separation, degrees
30
35
40
Figure 5.2: SINR for dierent algorithms. Slowly time varying channel.
5.3.2 A Slowly Time Varying Scenario
With a slowly time varying channel, the sample covariance matrix from
a single burst will contain a low rank signal subspace (the rank d instantaneous channel realization) plus noise. In order to use the Spread
root-MUSIC and WLS algorithms, we averaged the data from the 10 most
recent bursts to form R^ x . Another possibility would have been to rst
estimate the signal subspace for each burst and then average these over
a number
P of bursts to get a virtually noise free, but low sample estimate
of di=1 Sd Rv (d ; d ).
The single point, two-point and Capon solutions were calculated for
each burst. In this scenario, the two-point solution really amounts to
approximating the instantaneous array response with a linear combination of two ULA steering vectors (compare to the model (2.14) mentioned
in Section 2.4). Since MUSIC is ill-suited to handle correlated sources,
MODE was used instead of root-MUSIC for DOA estimation of the twopoint approximation.
Random QPSK signals were used for both sources. The resulting
SINR is shown in Figure 5.2 together with the theoretical performance
of the OSS solution if the channel is perfectly known (note that this is
5.4 Conclusions
85
the optimum only within the class of beamformers based solely on the
statistical channel characterization).
Somewhat surprisingly, none of the solutions beat the theoretical OSS
limit, except when the sources are very closely spaced where Capon gives
better performance. From Figure 3.4 we know that there is a potential
for large improvements. The two-point approximation but also the single
point approximation show good performance. The parameter estimates
from both Spread root-MUSIC and WLS are clearly worse in the slowly
than in the rapidly time varying scenario, which is not surprising since
in eect only 10 snapshots of the channel are used. For source separations up to 17 the robustication of Spread root-MUSIC gives mainly
point source estimates, however it is not clear what the number of samples should be set to in the MDL algorithm. Here, 10, the number of
bursts, was used. The Capon solution, even though it is good for closely
separated sources, gives very poor performance when the sources are further apart. This is due to signal cancellation, since v1 , the instantaneous
channel for the signal of interest is too far from a(^1 ).
The probability of outage for the dierent algorithms is illustrated
in Figure 5.3. The theoretical optimum within the class of beamformers
based on the true and is included for comparison.
5.4 Conclusions
The combined eects of estimating both the channel parameters and the
transmitted signal could to some extent be forecast from the results of
Chapters 3 and 4, but the simulation studies performed here provide some
additional insight into the problem. However, there are many possibilities
to combine dierent algorithms, of which only a few have been covered
here.
The point source based algorithms fail when the channel variations
are rapid, but give reasonable performance when the variations are slow.
However, this is not true for the Capon beamformer, which is useful
only when the sources are closely spaced. With the simple extension
of Spread root-MUSIC to handle several scattered sources, described in
Chapter 4, there is a threshold eect so that the sources are detected as
point sources when they are too closely spaced. Thus, the signal estimation performance is signicantly worse than when covariance matching
is used for the channel parameter estimation, unless the sources are well
separated.
86 5 Signal Estimation Using Estimated Channel Parameters
0
10
−1
Fγ(γ )
10
ORS+Spread root−MUSIC
ORS+WLS
LS, single ray
LS, double ray
Capon
Optimal CDF,
when v is unknown
−2
10
−3
10
−30
−20
−10
0
γ, (dB)
10
20
0
10
−1
Fγ(γ )
10
−2
ORS+Spread root−MUSIC
ORS+WLS
LS, single ray
LS, double ray
Capon
Optimal CDF,
when v is unknown
10
−3
10
−30
−20
−10
0
γ, (dB)
10
20
Figure 5.3: Outage probability for the dierent algorithms when the
source separation is 10 (top) and 25 (bottom).
Interestingly enough, a two-point approximation of each scattered
source gives, in general, very good performance, which also gives an indication that a more sophisticated version of Spread root-MUSIC could
work well even for closely separated sources.
Chapter 6
Conclusions and Future
Research
6.1 Concluding Remarks
We have studied a seemingly very specic kind of scenario, namely sensor
array signal processing using a perfectly calibrated uniform linear array in
an environment that introduces spatial scattering but no time dispersion.
One not very surprising conclusion is that many of the algorithms
and results that are based on the even more specic assumption of point
sources give unsatisfactory performance in this more dicult environment. Since the optimal algorithms often require high computational
power, one main issue has been the development and evaluation of suboptimal solutions of lower complexity. One recurring theme to reduce
the complexity was to approximate each scattered source by two point
sources, which allows the use of existing point source algorithms as part
of the solutions.
Studying Signal Copy algorithms, the inuence of the rate of the
channel time variations has not always been stressed enough in the array processing literature. We have used two extreme cases, independent
channel realizations from sample to sample versus a channel that is constant during each burst, to obtain results that should be relevant for a
large range of conditions. Closed form expressions were derived for both
SINR and outage probability and the optimal beamformers were developed. Even though the results dier between the two models, we showed
88
6 Conclusions and Future Research
that the assumption of rapid channel variations can be used with good
approximation even for a slowly time varying channel, which simplies
the mathematical treatment. This is true for beamformers based on the
statistical channel characterization, but we also showed that much can be
gained with an adaptive spatial lter that tracks the channel variations.
The other main topic of the thesis was estimation of channel parameters based on second order statistics of the received data. We used a
simple estimation principle apply a function to the sampled data and
nd the model parameters that give the same value of the function. Under certain conditions, this gives consistent estimates, see [IH81]. Here,
the function used was simply a standard DOA estimation algorithm told
to look for twice the true number of sources. Exploiting the specic data
model, the parameters separate and the DOA estimates can be found
from an average of two direction estimates, whereas the spread angle
estimate is found through a one dimensional interpolation in a precomputed table. This gives consistent estimates for an approximative data
model that closely coincides with the physical model when the sources
are kept within a sector of, say, [-60,60].
General expressions were derived for the asymptotic performance of
root-MUSIC and MODE for point sources in colored noise and the result
was used to calculate the performance of two specic versions of the suggested algorithm, Spread root-MUSIC and Spread MODE. Simulations
were used to illustrate that the algorithms perform well for a reasonable
range of parameter values.
We also gave some general ideas on how to design pseudo-subspace
tting algorithms for this kind of data model where the signal contribution to the covariance matrix is full rank. The low complexity algorithm
was shown to be a two-point approximation of pseudo-subspace tting
and the ideas were also used to show why two previously suggested algorithms, DPSE [VCK95] and DISPARE [MSW96], give inconsistent estimates. Some general results were given for the statistical properties of
pseudo-signal subspaces of general covariance matrices, which were used
in the analysis of root-MUSIC and MODE but can also be useful for a
large class of pseudo-subspace tting algorithms.
When the two main blocks, channel parameter estimation and signal
copy, are to be combined, there are many possibilities. We tried a few
combinations in a couple of simulated scenarios. When the interferer is
close to the source of interest, the suboptimal solutions from previous
chapters did not really compare to the optimal algorithms. However,
the idea of two-point approximations, rened even further, turned out to
6.2 Directions for Future Research
89
perform very well both in rapidly and slowly time varying conditions.
To conclude. Even though the assumptions may seem very specic,
we have illustrated a number of problems that appear with more realistic
channel models and have given a number of solutions and ideas that can
be used to tackle these problems.
6.2 Directions for Future Research
There are many topics that call for further investigations. Here we only
list a few of them.
Develop the idea of subspace tting algorithms for full rank models. What are the advantages and disadvantages compared to other
methods? Find more application areas. Analyze the asymptotic
performance and nd the optimal weighting.
Apply the algorithms to measured data, in order to verify the channel model and nd practically useful versions of the algorithms.
Find good beamformers for the downlink scenario. This problem
has received relatively little interest in the literature, even though
many results are presented in [Zet97] and the works referenced
therein.
How can the the number of sources be detected? For use together
with the algorithms developed in Section 4.3, it is also necessary to
divide between point sources and scattered sources.
Develop more realistic channel models based on empirical data.
Investigate the combined eects of calibration errors and a scattering environment. The problems have been treated separately but
the combination may give raise to additional diculties.
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Index
Approximative model, 10
CRB, Cramér-Rao Bound, 45
DISPARE, 37, 59
DML, Deterministic Maximum Likelihood, 40, 80
DOA, Direction of Arrival, 3, 11
DSPE, 37, 58
Physical model, 10
Point source model, 1, 3840
Pseudo-noise subspace, 63
Pseudo-signal subspace, 56, 63,
66
Rapidly time varying channels,
11, 16, 2225, 8283
Root-MUSIC, 37, 39, 6972
LCMV, Linearly Constrained MinSINR, Signal to Interference and
imum Variance beamformer,
Noise Ratio, 5, 21, 23,
25
25
Local scattering, 3
Slowly time varying channels, 11,
LS beamformer, 29, 80
16, 2528, 8485
Spatial frequency, 12
MDL, Minimum Description Length, Spread angle, 3, 11, 37, 42
44, 49, 55, 83, 85
Two-point approximation, 4, 24,
MODE, 37, 39, 7277
7986
MUSIC, 37, 38
ORS, Optimal algorithm for Rapidly
varying channels, 23, 27
33, 8186
OSS, Optimal algorithm for Slowly
varying channels, 28, 29
33, 84
Outage probability, 5, 21, 26, 30,
33, 35, 8586
PDF, Probability density function, 11
ULA, Uniform Linear Array, 2
WLS, Weighted covariance matching, 45, 47, 82