IEEE SIGNAL PROCESSING LETTERS, VOL. 12, NO. 12, DECEMBER 2005 827
A Generalized MVDR Spectrum
Jacob Benesty, Senior Member, IEEE, Jingdong Chen, Member, IEEE, and Yiteng (Arden) Huang, Member, IEEE
Abstract—The minimum variance distortionless response
(MVDR) approach is very popular in array processing. It is also
employed in spectral estimation where the Fourier matrix is used
in the optimization process. First, we give a general form of the
MVDR where any unitary matrix can be used to estimate the
spectrum. Second and most importantly, we show how the MVDR
method can be used to estimate the magnitude squared coherence
function, which is very useful in so many applications but so few
methods exist to estimate it. Simulations show that our algorithm
gives much more reliable results than the one based on the popular
Welch’s method.
Index Terms—Capon, coherence function, cross-spectrum, min-
imum variance distortionless response (MVDR), periodogram,
spectral estimation, spectrum.
I. INTRODUCTION
S
PECTRAL estimation is a very important topic in signal
processing, and applications demanding it are countless
[1]–[3]. There are basically two broad categories of techniques
for spectral estimation. One is the nonparametric approach,
which is based on the concept of bandpass filtering. The other
is the parametric method, which assumes a model for the
data, and the spectral estimation then becomes a problem of
estimating the parameters in the assumed model. If the model
fits the data well, the latter may yield a more accurate spectral
estimate than the former. However, in the case that the model
does not satisfy the data, the parametric model will suffer sig-
nificant performance degradation and lead to a biased estimate.
Therefore, a great deal of research efforts are still devoted to
the nonparametric approaches.
One of the most well-known nonparametric spectral estima-
tion algorithms is the Capon’s approach, which is also known
as minimum variance distortionless response (MVDR) [4], [5].
This technique was extensively studied in the literature and is
considered as a high-resolution method. The MVDR spectrum
can be viewed as the output of a bank of filters, with each filter
centered at one of the analysis frequencies. Its bandpass filters
are both data and frequency dependent, which is the main dif-
ference with a periodogram-based approach where its bandpass
filters are a discrete Fourier matrix, which is both data and fre-
quency independent [3], [6].
The objective of this letter is twofold. First, we generalize the
concept of the MVDR spectrum. Second and most importantly,
Manuscript received June 2, 2005; revised July 14, 2005. The associate editor
coordinating the review of this manuscript and approving it for publication was
Dr. Hakan Johansson.
J. Benesty is with the Université du Québec, INRS-EMT, Montréal, QC,
H5A 1K6, Canada (e-mail: benesty@emt.inrs.ca).
J. Chen and Y. Huang are with Bell Laboratories, Lucent Technologies,
Murray Hill, NJ 07974 USA (e-mail: jingdong@research.bell-labs.com;
arden@research.bell-labs.com).
Digital Object Identifier 10.1109/LSP.2005.859517
we show how to use this approach to estimate the magnitude
squared coherence (MSC) function as an alternative to the pop-
ular Welch’s method [7].
II. G
ENERAL
FORM OF THE
SPECTRUM
Let
be a zero-mean stationary random process that is the
input of
filters of length
where superscript denotes transposition.
If we denote by
the output signal of the filter , its
power is
(1)
where
is the mathematical expectation, superscript de-
notes transpose conjugate of a vector or a matrix
(2)
is the covariance matrix of the input signal
, and
In the rest of this letter, we always assume that is positive
definite.
Consider the unitary matrix
with . In the proposed generalized MVDR
method, the filter coefficients are chosen so as to minimize the
variance of the filter output, subject to the constraint
(3)
Under this constraint, the process
is passed through the
filter
with no distortion along and signals along other
vectors than
tend to be attenuated. Mathematically, this is
equivalent to minimizing the following cost function:
(4)
where
is a Lagrange multiplier. The minimization of (4) leads
to the following solution:
(5)
We define the spectrum of
along as
(6)
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