Stoica, P.; Viberg, M.; Wong, M. & Wu, Q.
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
c

1999byCRCPressLLC
“A Unified Instrumental Variable Approach to Direction Finding in Colored Noise Fields”Ó
64
A Unified Instrumental Variable
Approach to Direction Finding in
Colored Noise Fields
1
P. Stoica
Uppsala University
M. Viberg
Chalmers University of Technology
M. Wong
McMaster University
Q. Wu
CELWAVE
64.1 Introduction
64.2 Problem Formulation
64.3 The IV-SSF Approach
64.4 The Optimal IV-SSF Method
64.5 Algorithm Summary
64.6 Numerical Examples
64.7 Concluding Remarks
References
Appendix A: Introduction to IV Methods
The main goal herein is to describe and analyze, in a unifying manner, the spatial and

brief introduction is given in the appendix of this chapter. Computationally simple IVMs for array
signal processing appeared in [10, 11]. These methods perform poorly in difficult scenarios involving
closely spaced DOAs and correlated signals.
More recently, the combined Instrumental Variable Signal Subspace Fitting (IV-SSF) technique
has been proposed as a promising alternative to array signal processing in spatially colored noise
fields [12, 13, 14, 15]. The IV-SSF approach has a number of appealing advantages over other DOA
estimation methods. These advantages include:
• IV-SSF can handle noises with arbitrary spatial correlation, under minor restrictions on
the signals or the array. In addition, estimation of a noise model is avoided, which leads
to statistical robustness and computational simplicity.
• The IV-SSF approach is applicable to both non-coherent and coherent signal scenarios.
• The spatial IV-SSF technique can make use of the information contained in the output of
a completely uncalibrated subarray under certain weak conditions, which other methods
cannot.
Depending on the type of “instrumental variables” used, two classes of IV methods have appeared
in the literature:
1. Spatial IVM, for which the instrumental variables are derived from the output of a (pos-
sibly uncalibrated) subarray the noise of which is uncorrelated with the noise in the main
calibrated subarray under consideration (see [12, 13]).
2. Temporal IVM, which obtains instrumental variables from the delayed versions of the
array output, under the assumption that the temporal-correlation length of the noise
field is shorter than that of the signals (see [11, 14]).
The previous literature on IV-SSF has treated and analyzed the above two classes of spatial and
temporal methods separately, ignoring their common basis. In this contribution, we reveal the
common roots of these two classes of DOA estimation methods and study them under the same
umbrella. Additionally, we establish the statistical properties of a general (either spatial or temporal)
weighted IV-SSF method and present the optimal weights that minimize the variance of the DOA
estimation errors. In particular, we point out that the optimal four-weight spatial IV-SSF of [12, 13]
(called UNCLE there, and arrived at by using canonical correlation decomposition ideas) and the
optimal three-weight temporal IV-SSF of [14] are asymptotically equivalent when used under the

m×1
is a noise term, and
A =[a(θ
1
) ···a(θ
n
)]
(64.2)
Hereafter, θ
k
denotes the kth DOA parameter.
The following assumptions on the quantities in the array equation, (64.1), are considered to hold
throughout this section:
A1. The signal vector x(t) is a normally distributed random variable with zero mean and a possibly
singular covariance. The signals may be temporally correlated; in fact the temporal IV-SSF approach
relies on the assumption that the signals exhibit some form of temporal correlation (see below for
details).
A2. The noise e(t ) is a random vector that is temporally white, uncorrelated with the signals and
circularlysymmetric normally distributed with zeromean and unknown covariancematrix
2
Q > O,
E [e(t )e
∗
(s)]=Q δ
t,s
; E [e(t )e
T
(s)]=O
(64.3)
A3. The manifold vectors {a(θ)}, corresponding to any set of m different values of θ, are linearly

¯n = rank () ≤¯m.
(64.7)
It is assumed that no row of  is identically zero and that the inequality
¯n>2n − m
(64.8)
holds (note that a rank-one  matrix can satisfy the condition (64.8)ifm is large enough, and hence
the condition in question is rather weak). Owing to its (partial) uncorrelatedness with {e(t )}, the
vector {z(t)} can be used to eliminate the noise from the array output equation (64.1), and for this
reason {z(t)} is called an IV vector. Below, we briefly describe three possible ways to derive an IV
vector from the available data measured with an array of sensors (for more details on this aspect, the
reader should consult [12, 13, 14]).
EXAMPLE 64.1:
Spatial IV
Assume that the n signals, which impinge on the main (sub)array under consideration, are also
receivedbyanother (sub)array thatis sufficiently distancedfromthe mainone so thatthe noise vectors
in the two subarrays are uncorrelated with one another. Then z(t ) can be made from the outputs of
the sensors in the second subarray (note that those sensors need not be calibrated) [12, 13, 15].
EXAMPLE 64.2:
Temporal IV
When a second subarray, as described above, is not available but the signals are temporally corre-
lated, one can obtain an IV vector by delaying the output vector: z(t ) =[y
T
(t −1) y
T
(t −2) ···]
T
.
Clearly, such a vector z(t) satisfies (64.4) and (64.5), and it also satisfies (64.8) under weak conditions
on the signal temporal correlation. This construction of an IV vector can be readily extended to cases
where e(t ) is temporally correlated, provided that the signal temporal correlation length is longer

integer-valued parameters by means of IV/SSF-based methods, we refer to [24, 25]).
64.3 The IV-SSF Approach
Let
ˆ
R =
ˆ
W
L

1
N
N

t=1
z(t)y
∗
(t)

ˆ
W
R
( ¯m × m)
(64.10)
where
ˆ
W
L
and
ˆ
W

OO

S
∗
?

= US
∗
(64.13)
where U
∗
U = S
∗
S = I ,  ∈ R
¯nׯn
is diagonal and nonsingular, and where the question marks
stand for blocks that are of no importance for the present discussion.
The following key equality is obtained by comparing the two expressions for R in Eqs. (64.11)
and (64.13)above:
S = W
R
AC
(64.14)
where C

= 
∗
W
L
U

ˆ
 contains the ¯n largest singular values. Note that
ˆ
U,
ˆ
, and
ˆ
S are consistent estimates of
U, , and S in the SVD of R.
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1999 by CRC Press LLC
The SSF step
— Compute the DOA estimate as the minimizing argument of the following
signal subspace fitting criterion:
min
θ
{min
C
[vec (
ˆ
S −
ˆ
W
R
AC)]
∗
ˆ
V [vec (
ˆ

ˆ
S)]}
(64.17)
where
ˆ
W is a positive definite weight, and B ∈ C
m×(m−n)
is a matrix whose columns form a basis
of the null-space of A
∗
(hence, B
∗
A = 0 and rank (B) = m − n). The alternative fitting criterion
above is obtained from the simple observation that Eq. (64.14) along with the definition of B imply
that
B
∗
W
−1
R
S = 0
(64.18)
It can be shown [27] that the classes of DOA estimates derived from Eqs. (64.16) and (64.17),
respectively, are asymptotically equivalent. More exactly, for any
ˆ
V in Eq. (64.16) one can choose
ˆ
W in Eq. (64.17) so that the DOA estimates obtained by minimizing Eq. (64.16) and, respectively,
Eq. (64.17) have the same asymptotic distribution and vice-versa.
In view of the previous result, in an asymptotical analysis it suffices to consider only one of the

ˆ
W
Define
g(θ) = vec (B
∗
ˆ
W
−1
R
ˆ
S)
(64.19)
and observe that the criterion function in Eq. (64.17) can be written as,
g
∗
(θ)
ˆ
Wg(θ)
(64.20)
In [27] it is shown that g(θ) (evaluated at the true DOA vector) has, asymptotically in N, a circularly
symmetric normal distribution with zero mean and the following covariance:
G(θ) =
1
N
[(W
L
U
−1
)
∗