Williams, D.B. “Detection: Determining the Number of Sources”
Digital Signal Processing Handbook
Ed. Vijay K. Madisetti and Douglas B. Williams
Boca Raton: CRC Press LLC, 1999
c

1999byCRCPressLLC
67
Detection: Determining the
Number of Sources
Douglas B. Williams
Georgia Institute of Technology
67.1 Formulation of the Problem
67.2 Information Theoretic Approaches
AIC and MDL
•
EDC
67.3 Decision Theoretic Approaches
The Sphericity Test
•
Multiple Hypothesis Testing
67.4 For More Information
References
The processing of signals received by sensor arrays generally can be separated into two problems:
(1) detecting the number of sources and (2) isolating and analyzing the signal produced by each
source. We make this distinction because many of the algorithms for separating and processing array
signals make the assumption that the number of sources is known a priori and may give misleading
results if the wrong number of sources is used [3]. A good example are the errors produced by many
high resolution bearing estimation algorithms (e.g., MUSIC) when the wrong number of sources
is assumed. Because, in general, it is easier to determine how many signals are present than to
estimate the bearings of those signals, signal detection algorithms typically can correctly determine

m
(t) =
N
s

i=1
s
i
(
t − 
i
(m)
)
+ n
m
(t) .
The noise observed at the mth sensor is denoted by n
m
(t). The propagation delays, 
i
(m),are
measured with respect to an origin chosen to be at the geometric center of the array. Thus, s
i
(t)
indicates the ith propagating signal observed at the origin, and s
i
(t − 
i
(m)) is the same signal
measured by the mth sensor. For a plane wave in a homogeneous medium, these delays can be found

ω
o

=
N
s

i=1
S
i

ω
o

e
−jω
o

i
(m)
+ N
m

ω
o

.
The detection algorithms then exploit the form of the spatial correlation matrix, R, for the array.
The spatial correlation matrix is the M × M matrix formed by correlating the vector of the Fourier
transforms of the sensor outputs at the particular frequency of interest

+ SCS

,
where K
n
is the correlation matrix of the noise, S is the matrix whose columns correspond to the
vector representations of the signals, S

is the conjugate transpose of S, and C is the matrix of the
correlations between the signals. Thus, the matrix S has the form
S =



e
−jω
o

1
(0)
··· e
−jω
o

N
s
(0)
.
.
.

m
, C has full rank, and the
form of R is
R = σ
2
n
I
M
+ SCS

.
(67.1)
c

1999 by CRC Press LLC
We will assume that the columns of S are linearly independent when there are fewer sources than
sensors, which is the case for most common array geometries and expected source locations. As C
is of full rank, if there are fewer sources than sensors, then the rank of SCS

is equal to the number
of signals incident on the array or, equivalently, the number of sources. If there are N
s
sources, then
SCS

is of rank N
s
and its N
s
eigenvalues in descending order are δ

n
+ δ
N
s
, σ
2
n
,···, σ
2
n
.The
correlation matrix is generally divided into two parts: the signal-plus-noise subspace formed by the
largest eigenvalues (σ
2
n
+ δ
1
, ···,σ
2
n
+ δ
N
s
) and their eigenvectors, and the noise subspace formed
by the smallest, equal eigenvalues and their eigenvectors. The reason for these labels is obvious as
the space spanned by the signal-plus-noise subspace eigenvectors contains the signals and a portion
of the noise while the noise subspace contains only that part of the noise that is orthogonal to the
signals [3]. If there are fewer sources than sensors, the smallest M − N
s
eigenvalues of R are all equal

o
; S
i
(ω
o
), i = 1, ..., N
s
; are assumed to be zero mean Gaussian random processes
that are statistically independent of the noise and have a positive definite correlation matrix C.We
also assume that the N samples taken when forming

R are statistically independent of each other.
With these assumptions, the spatial correlation matrix is still of the same form as in (67.1), except
that now we can more easily derive statistical tests on the eigenvalues of

R.
67.2 Information Theoretic Approaches
We will see that the source detection methods to be described all share common characteristics.
However, we will classify them into two groups—information theoretic and decision theoretic
approaches—determined by the statistical theories used to derive them. Although the decision
theoretic techniques are quite a bit older, we will first present the information theoretic algorithms
as they are currently much more commonly used.
67.2.1 AIC and MDL
AIC and MDL are both information theoretic model order determination techniques that can be
used to test the eigenvalues of a sample correlation matrix to determine how many of the smallest
eigenvalues of the correlation matrix are equal. The AIC and MDL algorithms both consist of
minimizing a criterion over the number of signals that are detectable, i.e., N
s
= 0, ..., M− 1.
c

“A” designated it as the first such test, but it is now more commonly considered an acronym for the
“Akaike Information Criterion.” If we have N independent observations of a random variable with
probability density g(Y) and a family of models in the form of probability densities f(Y|θ)where θ
is the vector of parameters for the models, then Akaike chose his criterion to minimize
I(g; f(·|θ))=

g(Y) ln g(Y)dY−

g(Y) ln f(Y|θ)dY
(67.2)
which is known as the Kullback-Leibler mean information distance.
1
N
AI C(θ) is an estimate of
−E{

g(Y) ln f(Y|θ)dY} and minimizing AI C(θ) over the allowable values of θ should minimize
(67.2). The expression for AI C(θ ) is
AI C(θ) =−2ln

f

Y|
ˆ
θ
(
N
s
)


s
signals theparameters
that completely parameterize the correlation matrix R are {σ
2
n
,λ
1
, ···,λ
N
s
, v
1
, ···, v
N
s
} where
λ
i
and v
i
, i = 1, ..., N
s
, are the eigenvalues and their respective eigenvectors of the signal-plus-noise
subspace of the correlation matrix. As the vector of sensor outputs is a Gaussian random vector with
correlation matrix R and all the samples of the sensor outputs are independent, the log-likelihood
function of f(Y|θ)is
ln f

Y|σ
2