Statistics, Vol. 40, No. 2, April 2006, 129–147
On discrete-domain multidimensional sinusoidal models
DEBASIS KUNDU*† and SWAGATA NANDI‡
†Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India
‡Statistics and Mathematics Division, Indian Statistical Institute, 7 SJS Sansanwal Marg,
New Delhi 110016, India
(Received 29 March 2005; in final form 5 November 2005)
We consider three-dimensional sinusoidal frequency model in a random field. Three-dimensional
frequency model has wide applications in statistical signal processing. In this article, we mainly con-
sider the usual least squares estimators and the estimators that can be obtained by maximizing the
periodogram function. We obtain consistency and asymptotic normality property of both the estima-
tors. It is observed that they are asymptotically equivalent. Finally we generalize the results to the
multidimensional case.
Keywords: Multidimensional sinusoidal signals; Least squares estimators; Strong consistency
AMS Subject Classification: 62F12; 62M10
1. Introduction
In this paper, we consider the following three-dimensional (3-D) frequency model:
y(n
1
,n
2
,n
3
) =
p
k=1
A
0
k
cos(n
1
λ
0
k1
+ n
2
λ
0
k2
+ n
3
λ
0
k3
) + B
0
k
sin(n
1
λ
0
k1
+ n
2
λ
0
k2
+ n
3
λ
0
k3
)
+ X(n
1
,n
2
,n
3
), (1)
for n
j
= 1,...,N
j
, j = 1, 2 and 3. Here, A
0
k
and B
0
k
are unknown real amplitudes and
λ
0
k1
, λ
0
k2
and λ
0
k3
are unknown frequencies, λ
0
k1
,λ
0
k2
,λ
0
k3
∈ (0,π). The error random variable
X(n
1
,n
2
,n
3
) is from a stationary random field, and it satisfies Assumption 1 (will be defined
in the next section). The number of components ‘p’ is assumed to be known. Given a sample
{y(n
1
,n
2
,n
3
), n
j
= 1,...,N
j
, j = 1, 2 and 3}, the problem is to estimate A
k
’s, B
k
’s and
λ
kj
’s.
This is an important problem in statistical signal processing. See, for example, the site
http://www.mddsp.enel.ucalgary.ca/people/bruton/enel699/chap1_2002.pdf of Professor Len
Bruton, where the author has demonstrated several applications of this particular model in
*Corresponding author. Email: kundu@iitk.ac.in; Tel.: +91-512-2597141; Fax: +91-512-2597500
Statistics
ISSN 0233-1888 print/ISSN 1029-4910 online © 2006 Taylor & Francis
http://www.tandf.co.uk/journals
DOI: 10.1080/02331880500520730