CAPON-LIKE_SPATIAL_SPECTRUM_ESTIMATOR

CAPON-LIKE_SPATIAL_SPECTRUM_ESTIMATOR
A CAPON-LIKE SPATIAL SPECTRUM ESTIMATOR FOR CORRELATED SOURCES
Richard Abrahamsson , Andreas Jakobsson† and Petre Stoica
Dept. of Information Technology, Uppsala University, Box 337, SE-751 05 Uppsala, Sweden
Phone: +46 (0)18 471 7840 , Email: [email protected], [email protected]
† Dept. of Electrical Engineering, Karlstad University, SE-651 88 Karlstad, Sweden
Phone: +46 (0)54 700 2330, Email: [email protected]
ABSTRACT
In this paper, we propose a data-adaptive method for estimating the spatial spectrum for correlated and/or coherent
sources. The method is reminiscent of the MVDR/Capon
beamformer, but yields a multidimensional spatial response
with a dimension for each examined source. The estimated
directions of arrival (DOA) are given as the points of energy
concentration in the multidimensional spatial response. Numerical examples indicate a significant improvement over the
standard MVDR/Capon beamformer.
1. INTRODUCTION
The area of sensor array signal processing has received a considerable interest in the recent literature, and numerous algorithms addressing different aspects of the topic have been
proposed (see, e.g., [1, 2, 3, 4] and the references therein).
Typically, these algorithms exploit the difference in propagation delay recorded at the different sensor array elements
to form a parametric or non-parametric spatial spectral estimate. Lately, non-parametric spatial spectrum estimators
have received renewed interest, mainly because of their benefit of not explicitly assuming an a priori known signal model,
which leads to more robustness to variations in the measured
signal than parametric counterparts.
A traditional non-parametric method for spatial spectral
estimation is to apply a simple beamformer for all directions
of interest and use the beamformer outputs to estimate the
spatial power spectral distribution. However, such a nonadaptive beamformer suffers from either low resolution or
high leakage [3]. The minimum variance distortionless response (MVDR) or Capon beamformer [3, 5] has several desirable properties, and is often applied as a bank of spatial
filters when high spatial resolution is desired. The Capon
method weighs the array elements in an adaptive manner so
as to minimize the beamformer output power while passing
signals from a given direction of interest undistorted. This
effectively places deep nulls canceling interferences from
sources in directions other than the one of interest, resulting
in a very high resolution spatial estimate. However, when the
impinging wavefronts are correlated or coherent, the Capon
beamformer suffers from signal cancellation, causing severe
distortions in the spatial spectral estimate [6, 7, 8]. This happens, e.g., when there are multipaths or when smart jammers
retrodirect a transmitted signal.
In this paper, we present an extended version of the
Capon beamformer that handles correlated and/or coherent
This work was partially supported by the Swedish Research Council
and Saab Bofors Dynamics. Please address all correspondence to Richard
Abrahamsson ([email protected]).
signals. The method forms a multidimensional spatial spectral estimate, with a separate dimension for each examined
source. Like the Capon beamformer, the proposed method
does not require point-like sources or a known noise covariance model in order to perform well; it is only needed that
the selected number of examined sources is at least as large
as the number of correlated and coherent sources.
2. PRELIMINARIES
Let s p t denote the pth narrowband signal (narrowband in
the sense that the signal bandwidth is much less than the reciprocal of the time it takes for the signal to propagate across
the array) impinging from direction θ p on an M-element sensor array. The array output can be expressed as
P
t ∑
p 1
θp sp
t t where θ denotes the array manifold
vector for a generic di
rection of arrival (DOA), θ , t is a possibly colored additive measurement noise, and P is the number of examined sources. Assuming possibly correlated and/or coherent
sources, each signal, s j t , can be decomposed as
s j t E s j t sk t E sk t 2
sk t s j k t (1)
where E denotes the expectation operator, the complex
conjugation, and s j k t is the part of s j t which is uncorre
lated with another impinging signal (or interference), sk t .
Alternatively, assuming constant correlation coefficients
be
tween signals over the observation interval, s j t can be written as a linear combination of all interfering signals and a
residual term
s j t P
∑
k 1k j
α j k sk t s̃ j t (2)
for some set of scalars α j k . Here, s̃ j t is the component of
s j t which is uncorrelated with all the interfering sources,
i.e., sk t , k j. Hereafter, for simplicity, we limit our interest to pairwise correlated and/or coherent signals, assuming
these pairs to be uncorrelated with the remaining source signals. We note that this case is of interest in its own right; pairwise correlated impinging wavefronts occur, e.g., in low angle tracking over sea [9] and in certain GPS applications [10].
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However, the following derivation is easily extended to handle the more general case considered in (2). For pairwise
correlated/coherent sources, (2) simplifies to, cf. (1),
α j k sk t s j k t (3)
Note that, if s j t and sk t are uncorrelated, then α j k 0.
On the other hand, if s j t and sk t are fully coherent, then
s j t α j k sk t . To simplify the notation, we hereafter replace α j k with just α .
sj t
3. THE EXTENDED CAPON BEAMFORMER
The Capon beamformer estimates the spatial spectrum for
each direction of interest by forming a data-adaptive beampattern focused in the examined direction, say θ , such that
deep nulls are placed at directions different from θ that contain power. More specifically, the Capon beamformer minimize the array output power
σθ2
E
2 t
(4)
with respect
to the M-tap array sensor weight vector (the spatial filter), , subject to the constraint that a signal in the direction of interest should pass undistorted. In (4),
E t t is the array output covariance matrix and denotes the
conjugate transpose. The Capon beamformer design criterion is usually written as
min
subject to
θ
1
is used in lieu of
(6)
1
ˆ 1
θ
σθ2 θ
θ 1 2
σ 2 1 2 θ θ
1
1 σ 2
θ
σ2 θ
θ
0
0
0
1
1
θ
θ
where 0 denotes the positive semidefiniteness of , the
design objective of the Capon beamformer for spatial power
spectrum estimation can equivalently be written as [11, 12]
max σθ2
σθ2
subject to
σθ2 θ 1
1 α
θ2
1
1
θp 1 θq
θ1
θ2
p q for
1
α
(11)
1 2 enables us to write (11) as in (12), on the top of the last page
of this paper. Thus, the α minimizing (12) (for a particular
θ1 θ2 ) is given as
α
σθ2
1
(7)
when (6) is inserted in (4). Noting that
j
b θ1 θ2 b θ 2 θ2 (13)
Insertion of (12) in (11), using (13), concentrates the optimization in (10) to
, yielding
σ̂θ2
1
t
1 N 1 t
N t∑
0
σθ2
b θ p θq In practice, the expectation in (4) is replaced by a sample
average, and the sample covariance matrix
ˆ
Introducing
θ
θ
nal sk t and hence sk t will appear in the covariance data as
if it did not come from the examined direction, θk . Thus, the
data-adaptive filter steered to θk will treat the resulting signal
(first term on right hand side of (9)) as an interference and
will attempt to cancel it out.
Using (8) and (9), we now form the extended Capon
beamformer as (10), on the top of next page, where θ1 denotes the DOA whose power content we are interested in
and θ2 is the DOA of a possible correlated impinging signal. Similarly to (7) that maximizes (8), the solution to (10),
for a given α and θ2 , is found as
(5)
The solution to (5) is easily found as (see, e.g., [3])
1 1 θ The formulation in (8) is intuitively appealing: Finding the
maximum power of a signal component at direction θ such
that the residual covariance matrix, after removing this signal
component, is still positive semi-definite. In the following,
we will exploit this formulation to enable estimation of the
spatial spectrum for correlated and/or coherent sources.
Under the assumption of pairwise correlated sources, a
generic term of the
matrix from one such pair
covariance
of signals, say s j t and sk t , can be written as (9), on the top
of next page, where the signal s j t has been decomposed as
in (3), and σk2 and σ 2j k are the powers of sk t and the com
ponent of s j t which is uncorrelated with sk t , respectively.
Note that (9) illustrates why the Capon beamformer fails for
correlated sources; the coherent part of an interference s j t will create an offset, α θ , to the steering vector for the sig-
θ θ
0
(8)
θ2
b θ 2 θ2 b θ 1 θ1 b θ 2 θ2 b θ 1 θ2 2
max (14)
Note that as θ1 approaches θ2 , i.e., when two possible correlated signals get closer together, σθ2 in (11) will yield very
1
large estimates. This is easily seen from (14), or from the
constraint in (10) (when θ gets close to θ , the optimal α
1
2
will be such that θ α θ becomes very small). Thus, care
1
2
needs to be taken not to evaluate the multidimensional spatial
response for a narrow region close to θ1 θ2 . If available,
prior knowledge of source separation can be exploited to design the width of this narrow region. Worth noting is the fact
that one may determine from the cost function in (14) which
of the source signals are uncorrelated and which are coherent.
The coherent components will appear as peaks mirrored with
respect to the line θ1 θ2 , whereas the uncorrelated parts,
with α 0, are independent of θ2 and show up as ridges parallel to the θ2 axis. See Section 5 for an illustration of this.
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E θk
sk t θj
s j t max σθ2
σθ2
1
α θ2
1
θk
sk t θj
s j t subject to
σk2 σθ2
1
θ1
θk α
α
θ2
σ2 α
j k
θk
θj
j
α 0
θ1
θ2
θ θj θj
(9)
(10)
4. COMPUTATIONAL COMPLEXITY
For a Uniform Linear Array (ULA), the array manifold vector can be written as
where
θ
1 e iω
e iω M
1
T
(15)
T denotes the transpose,
ω
d
2π sin θ λ
is the spatial frequency, d is the interelement spacing of the
array and λ is the operating wavelength. By sampling the
cost function in (14) uniformly on a grid in ω1 , ω2 (instead
of on a grid in θ1 , θ2 ), the Fourier-vector form of the array steering vectors, as shown in (15), can be exploited for
efficient implementation. Using
of K K spatial fre a grid
1
quency points, the values of θ p
θq , for p q 1 2, can
be computed by first applying
inverse Fast Fourier trans the
1 and then applying the FFT
form (FFT) to each row of
to each column of the result. This results in a total
computational complexity of the order O M K K log2 K which
should be compared
to O M 2 K 2 if direct multiplication is
used (or O MK M K if intermediate data is stored). We
note that the computational cost of forming and inverting the
sample covariance matrix is of the order of O M 3 MN ,
which also is the main computational burden for the Capon
beamformer if the number of points on the DOA grid, K, is
of the same order as M.
5. NUMERICAL EXAMPLE
In the following example, data from a 15-element ULA with
half wavelength interelement spacing have been
simulated.
Wavefronts impinge on the array from angles 55 , 40 ,
0 , 10 , 30 , and 60 (as measured from broadside). The impinging signals are simulated as temporally white zero-mean
circularly symmetric complex Gaussian sequences. Those
from 0 and 10 are of unit power and uncorrelated, both
mutually and with the other signals, whereas the signals from
30 and 60 are partially correlated with the covariance matrix
54 1
1 54 and uncorrelated with the rest of the signals.
The remain
ing two signals from the sources located at 40 and 55 are fully coherent with each other (identical up to a constant
phase factor), uncorrelated with the other signals, and each
having a signal power twice that of the signal impinging from
broadside. The noise power in each sensor is 4, i.e., 6 dB
above the signal at θ0 0o , with the noise being spatially and
temporally white and zero-mean circularly symmetric complex Gaussian distributed. The array output is observed for a
time period of N 100 samples.
Figure 1: Two dimensional representation of the cost function of the proposed method.
Figure 1 shows the cost function in equation (14) that is
to be maximized with respect to the unknown correlated interference direction θ2 . Note how the uncorrelated signals at
0 and 10 appear as ridges at θ1 0o and θ1 10o , both independent of θ2 (except along the diagonal θ1 θ2 ) as there
is no interference that is correlated with these two signals.
The true powers of the signals at 0o and 10o are indicated
by two vertical lines with circles at θ2 90 and their respective
DOAs
in θ1 . The mutually fully coherent signals at
40 and 55 have their locations and true powers indicated by vertical lines marked with x. Note how the coherent
signals appear as peaks at the coordinates of their DOAs in
θ1 and the coherent interference direction in θ2 . Further, the
partially correlated signals at 30 and 60 , with the true locations and powers denoted by vertical lines with + signs, appear as superpositions of a θ2 -independent ridge (the uncorrelated component) and peaks at the correlated interference
direction in θ2 (the fully coherent component). In this way,
the amount of correlation between signals can be found by
observing the peak height in relation to the height of the θ2 independent ridges. Since the observations are noisy and are
made during a limited time, also the fully coherent signals
appear to have small uncorrelated components. We note that
to enable visualization in Figure 1, the ridge along θ1 θ2
has been truncated.
In Figure 2, the result of the proposed method after maximizing the objective function with respect θ2 , is compared to
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1 α
b θ1 θ1 b θ1 θ2 b θ 1 θ2 b θ 2 θ2 1
α
α b θ1 θ2 α 2b θ2 θ2 b θ1 θ2 2 b θ1 θ2 2
α b θ2 θ2 b θ2 θ2 b θ1 θ1
b θ 2 θ2 b θ1 θ1 α b θ1 θ2 2
Beamformer
Proposed Method
Capon MVB
true impinging signals
1.8
1.6
REFERENCES
1.2
Power
(12)
higher than that of the Capon method. To simplify notation,
only pairwise correlated and/or coherent sources have been
considered, with the more general case of several mutually
correlated/coherent sources following similarly.
1.4
1
0.8
0.6
0.4
0.2
0
−80
−60
−40
−20
0
20
DOA θ1 [Degrees]
40
60
80
Figure 2: Proposed method (with the ridge at θ1 θ2 removed) as compared to the standard Capon method and the
traditional beamformer
the traditional Capon beamformer estimate. To enable the
maximization, the ridge along θ2 θ1 has been excluded
prior to the optimization with respect to θ2 . As seen in the
figure, the proposed method significantly improves the estimation of the power distribution over different DOAs as compared to the classical Capon beamformer. For reference, the
spatial spectral estimate of the traditional data-independent
beamformer is also shown. In the figure, the true DOAs and
powers of the signals are shown with vertical dotted lines
marked with x. Finally, we remark that by using the proposed method we retain the highly desirable resolution and
leakage properties of the Capon beamformer even for correlated and/or coherent sources.
6. CONCLUSIONS
In this paper, a Capon-like spatial spectrum estimator has
been presented. The proposed algorithm is based on a multidimensional extension of the covariance fitting formulation
of the MVDR/Capon beamformer. The extension shows a
significant improvement over the MVDR/Capon beamformer
when correlated and/or coherent wavefronts are present,
while the excellent resolution and leakage properties are retained. Moreover, the multidimensional spatial spectral estimate shows the amount of correlation between signals from
different directions. For uniform linear arrays, the computational complexity of the proposed method is not too much
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