IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
4189
Kinematics of Complex-Valued Time Series
Patrick Rubin-Delanchy and Andrew T. Walden
Abstract—The contribution to a stationary complex-valued time
series at a single frequency magnitude takes the form of a random
ellipse, and its properties such as aspect ratio (which includes rotational direction) and orientation are of great interest in science. A
case when both the aspect ratio and orientation are fixed is found,
and their variability, in general, results from the additional influence of an orthogonal ellipse. It is shown how a magnitude squared
coherence coefficient controls both the relative influences of these
components and the variation of both the orientation and aspect
ratio of the resultant ellipse. Realizations of random ellipses are
recovered very accurately from simulated time series. The mean
orientation of the random ellipse is formally derived.
Index Terms—Aspect ratio, azimuth, coherence, complex-valued
processes, random ellipse, simulation.
I. INTRODUCTION
. Combined with covariance stationwith
arity, this condition makes
second-order stationary (SOS),
and such a process is assumed here.
Older seminal studies (e.g., [17, p. 42]) of complex-valued
,
series considered the relation sequence to be null, i.e.,
, in which case the process is said to be proper. Recent
statistical studies of complex-valued time series have not assumed the relation sequence to be null [22], [26]. Interestingly,
analysts investigating complex-valued time series formed from
vector components in oceanography and atmospheric science
(e.g., [6], [11], [19], and [28]) have for decades exploited this
more general approach, following the highly influential work
of Gonella and Mooers [8], [18], (together cited more than 300
times according to the ISI Web of Knowledge, February 2008).
A zero-mean covariance stationary complex-valued random
sequence is harmonizable, so
T
HE scientific value of complex-valued time series analysis
occurs when the real and imaginary parts are geometrically related through the physics governing the data, most notably when the real and imaginary parts are orthogonal vector
components for motion taking place in a plane. For example,
in oceanography currents are typically resolved into eastward
(zonal) and northward (meridional) components [6] and in meteorology a similar decomposition may be made for wind vectors [11].
In such cases, e.g., [15], it can be useful to construct a com,
plex-valued time series from the two real-valued series
. Denoting the discrete complex-valued random process by
and the real series by
and
we thus have
; without loss of generality we assume the processes to have means of zero.
and
at lag in
Define the autocovariance between
the usual way as
, (where deis said to be covariance stanotes complex conjugate).
is a function of only,
tionary if and only if (iff)
, with
and we obtain the autocovariance sequence
. Let us also define the relation [22] (or
and
at lag
complementary covariance [26]) between
as
; iff this is a
,
function of only we obtain the relation sequence
Manuscript received August 6, 2007; revised April 1, 2008. First published
June 6, 2008; last published August 13, 2008 (projected). The associate editor
coordinating the review of this manuscript and approving it for publication was
Dr. Lucas Parra. The work of P. Rubin-Delanchy is supported in part by the
EPSRC (UK).
The authors are with the Department of Mathematics, Imperial College
London, London SW7 2BZ, U.K. (e-mail: patrick.rubin-delanchy@imperial.ac.uk; a.walden@imperial.ac.uk).
Color versions of one or more of the figures in this paper are available at
http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2008.926106
(1)
where
is a random function of with uncorrelated increments, and zero mean [14, pp. 482–483], [25, pp. 18–19]. At
the contribution to
is
frequencies and
(2)
is the parametric equation of a random ellipse, comand
prising the addition of two oppositely rotating circular motions
with random amplitudes and phases [6, pp. 428–429]. (Practically, a negative-frequency complex exponential can be seen as
a counter-clockwise rotation evolving in the opposite time direction—or just a clockwise rotation.) In this paper, we study
the statistical properties of this random ellipse. It is the properties of elliptical motion at a particular frequency that are of
such widespread scientific interest where quantities such as the
aspect ratio (the ratio of the lengths of the axes, with sign indicating counterclockwise/clockwise rotation), and ellipse orientation, are important. For reasons of statistical precision we use
(2) as the fundamental quantity of interest, rather than the discrete Fourier transform of finite portions of signal.
, then
(with denoting
Let
conjugate transpose) has Fourier transform
, say,
1053-587X/$25.00 © 2008 IEEE
(3)
4190
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
We show how the statistics of the random ellipse depend on the
rotationally invariant magnitude squared coherence
(4)
is discussed later.) Some
(The particular case
background statistical ideas are developed in Section II. In
Section III we show that the random ellipse can be decomposed
into two ellipses which are oriented orthogonally but have the
same aspect ratio, the aspect ratio that would be obtained if
was set to unity, so that
is singular, while
,
and
were unchanged. As
, the
modification of the ‘singular’ ellipse by the orthogonal ellipse
grows towards the two ellipses having equal influence.
In Section IV, we discuss statistical distributions for the estimators of ellipse parameters and, under Gaussianity, formally
prove a heuristic claim [8], [18] that the mean orientation of the
. Section V describes how we
random ellipse is
very accurately simulated the kinematics of some random ellipses, something we have not been able to find elsewhere in the
. Elliterature. The plots illustrate the effect of changing
lipse parameters derived directly from these simulated ellipses
are shown to fit the theoretical distributions extremely closely.
as the “stability
Gonella [8] had previously interpreted
of the ellipse orientation” but the probability density functions
(pdfs) of azimuth (orientation) and aspect ratio, and the simulalikewise
tions show that this is only half the story, and that
controls the aspect ratio. Results are summarized in Section VI.
(7)
, so
is distributed as a proper complex Gaussian vector which may
be denoted
(8)
The covariance matrix must be non-negative definite and
hence it follows that [22]
and
(9)
Now
, and
, and
, and so
of (4) follows and by
the magnitude squared coherence
(9),
is symmetric and bounded by 0 and 1. The influence
of the coherence on the increments is now considered.
B. General Case
II. BACKGROUND STATISTICS
A. Orthogonal Increments
If
is complex Gaussian, then
, with
, is Gaussian as well [4, p. 483]. To see this
,
more clearly we note, using [3, p. 151], that as
in mean-square (and hence in distribution), so that by choosing
and
,
may
be written as a weighted sum of complex Gaussian RVs and
is therefore also complex Gaussian, i.e., its real and imaginary
is also complex
parts are jointly Gaussian. Likewise
. Since all such RVs are derived from
Gaussian, as is
, the vector
the jointly Gaussian real and imaginary parts of
is jointly complex Gaussian.
Now, since
is SOS, we know that
if
if
This follows because the expectations in (5) and (6) correspond
to two definitions of the Loève spectrum, and the right-hand
sides follow from the corresponding stationary manifolds (
and
, respectively), [27].
and
, which
So
is proper, with indepenmeans the random variable (RV)
dent and equivariable real and imaginary parts.
has a covariance
The vector
matrix given by, say,
(5)
and
are jointly proper Gaussian, the
Since
optimal estimator of
in terms of
is the linear
minimum mean-square error (LMMSE) estimator [23], given by
. The estimation mean-square error (MSE)
achieved is
so that
controls the accuracy with which
.
be estimated from
can
C. Singular Case
then the estimation MSE vanishes so that
, with probability one
then one or both of
(w.p.1). Additionally, if
and
is zero w.p.1;
is defined to be unity.
In both situations
and
, are perfectly linearly
is singular. But if
then
related and hence
, and
, where
, so that, w.p.1,
If
(10)
and
if
if
(6)
when
Section III-B.
. We shall exploit this result in
RUBIN-DELANCHY AND WALDEN: KINEMATICS OF COMPLEX-VALUED TIME SERIES
4191
Fig. 2. An example continuous-time random ellipse Z (f ) and its discrete
equivalent Z (f ) shown by points. A(f ) and B (f ) denote the semi-major and
semi-minor axes, respectively, while 2(f ) denotes the azimuth (orientation) of
the ellipse. (Dependence on f is suppressed.)
Fig. 1. (a) 0(f ), (b) R(f ) (real part solid line, imaginary part dotted), and (c)
realization of CAR(2) process.
Note also that in the singular case
, given by
we denote by
takes the form, which
(11)
are the lengths of the semiand
is the azimuth
major and semi-minor axes, respectively.
(orientation), the angle which the major axis of the ellipse makes
. The
with the -direction, and
where the sign is
(signed) aspect ratio is
, and is positive/negative for counterclockwise/clockwise rotation. The intensity of the ellipse is
. All the geometrical properties of the
ellipse, except for the center and frequency of rotation, depend
and
.
on the RVs
A. Singular Case
III. THE RANDOM ELLIPSE
Consider the complex SOS autoregressive process
order 2, denoted CAR(2)
For the singular case with
, (10) gives
of
(12)
and
are complex-valued parameters such that all
where
are outthe roots of the -polynomial
is doubly white noise [22], i.e.,
side the unit circle, and
and
, where
denotes the Kronecker delta. Fig. 1 shows
,
, and the sample points on
the complex plane for a realisation of such a process of length
5000. Most power in the process occurs in narrow bands cen; also
at all frequencies
tered about
(for more details see Section V-D). The sample points fill out an
elliptical shape. Fig. 2 illustrates the random ellipse (2) which is
centred at the origin, with a full rotation being performed with
. Since is discrete,
only describes
a period
a subset of the points of the ellipse, say
(shown by the
elliptical curve in Fig. 2) that would be traced if were contin, then
uous. This subset is finite iff is rational; e.g., if
takes only
values (points in the complex plane)
as traverses the integers. These points are shown by the black
dots in Fig. 2 and the sample points in Fig. (1c) are an aggregate
of such elliptically distributed points over the narrow bands of
.
frequencies around
Fig. 2 also illustrates our notation for the geometrical proper.
ties of the random ellipse
and the random intensity
mean intensity is given by
follows. The
Note that
so that the shape of the ellipse is fixed, while its size is random.
is therefore
The aspect ratio
(13)
and so is fixed with value
Now from (10)
and therefore
.
4192
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
so that
Rearranging terms, we have
(17)
(14)
. Hence
and we see that the azimuth is also fixed, with value
in the singular case the only aspect of the random ellipse which
is random is its size.
and
being perfectly linWe emphasize that
early related, (10), is a sufficient condition for the aspect ratio
and azimuth to be fixed with values (13) and (14), respectively.
B. General Case
Likewise
(18)
where
In the general case
and
are not perfectly
linearly related, and we will show that the random ellipse can be
decomposed into two orthogonal ellipses with the same aspect
, and
ratio. We firstly introduce a zero-mean “error term”
modify (10) to
(15)
Since the error term is the difference of two complex Gaussian
RVs, it too is complex Gaussian. Note that
accounts for
both the suboptimality of (10) in the non-singular case and the
inherent estimation error even with the optimum LMMSE filter.
From (15), we have that
so that
where
Now
and
are each parametric equations of a
random ellipse, centred at the origin, with a full rotation being
, and describe a subset of the points
performed with a period
and
say, that would be
of the ellipses, respectively
traced if were continuous.
It is straightforward to show that
and
for
, and hence, since the terms have a proper complex
Gaussian distribution, the ellipses (17) and (18) are independent.
and
Now consider the vectors
. After some manipulation we find
that
in (2) can be written
where
From (15) we also have
so that
where we met
previously in (11), and [see the equation
in (7) is
at the bottom of the next page], and we note that
given by
in (2) can also be written
(19)
In addition, we find that
. Both
and
have determinants
of zero.
has covariance matrix proportional to
,
Since
with determinant of zero, and relation matrix of zero, we know
and
are perfectly
from Sections II-C and III-A that
linearly related, i.e., w.p.1,
where
Hence
(20)
(16)
say, where
.
and that therefore the aspect ratio is fixed with value (13) and
the azimuth is fixed with value (14). We shall say that
defines the singular ellipse.
RUBIN-DELANCHY AND WALDEN: KINEMATICS OF COMPLEX-VALUED TIME SERIES
4193
Now
has covariance matrix proportional to
, with
determinant of zero, and relation matrix of zero. It thus follows
that w.p.1
(21)
Then, following the same arguments as in Sections II-C and
III-A, we find that the ellipse (18) has again the aspect ratio with
is
value (13), but the azimuth
(22)
i.e., the ellipse corresponding to (18) is orthogonal to that corresponding to (17). We shall say that
defines the orthogonal
ellipse.
is the sum
The mean intensity of the ellipse specified by
and
. This mean intensity
of the mean intensities of
, just as in
in the general case is found to be
the singular case.
C. Interpretation of the Representation in (16)
can be decomposed into ellipses,
and
as
has a fixed aspect ratio and orientation given by
in (16).
(13) and (14), respectively, so that, apart from its size, it is statistically identical to the ellipse found in the singular case.
also has these parameters fixed, but whilst the aspect ratio is the
controls the
same, the ellipse is oriented orthogonally.
,
relative sizes of the two ellipses—when unity,
and
reduces to
, but as
, so
, and
grows towards the ellipses having equal
influence.
To illustrate the summation of two orthogonally oriented
ellipses (in continuous time), we consider an example where
has
, a semi-major axis length of 4, a
semi-minor axis length of 2, (aspect ratio 0.5), and azimuth
, while
has an azimuth of
of
and the same aspect ratio. We took
Here the terms
and
reflect the fact that
can lead/lag
—
is a proper complex
random vector and so its statistical properties are unchanged by
which feeds into
and
. The term
a rotation
reflects the relative sizes of the two ellipses. Fig. 3(a) shows the
and
and Fig. 3(b) shows
two ellipses when
their sum. The dot marks a common time and the star marks a
common later time, so that the direction of motion is clear. We
Fig. 3. (a), (c), and (e) each show the two ellipses P (f ), Q (f ) and (b), (d),
and (e) show the resultant sums, respectively. The dot marks a common time
and the star marks a common later time. See the text for parameter values for
each of the three cases.
see that two counter-clockwise-rotating ellipses may combine
to give an ellipse—in fact rectilinear—with the same orientation as the singular ellipse, but with a very different aspect ratio.
and
Fig. 3(c) shows the two ellipses when
and Fig. 3(d) shows their sum. Two counter-clockwise-rotating
ellipses combine to give a clockwise-rotating ellipse with the
, as the singular ellipse, but with an assame orientation,
pect ratio which is quite different. Fig. 3(e) shows the two eland
and Fig. 3(f) shows their sum.
lipses when
Here two counterclockwise-rotating ellipses combine to give a
counterclockwise-rotating ellipse with a different orientation to
the singular ellipse, but with an aspect ratio which is the same
as for the singular ellipse.
are possible; see, e.g., [2] for
Other decompositions of
a discussion in the context of optics. Our decomposition has a
very attractive form—two statistically independent orthogonal
ellipses with identical aspect ratios, with relative contributions
; this adds to our
controlled by the size of the coherence
in the analysis of
understanding of the role played by
complex-valued sequences.
4194
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
IV. DISTRIBUTIONS
A. Aspect Ratio
Let us write
,
on frequency is undersay, where the dependence of ,
has the proper complex Gaussian distribution
stood.
(8). Then we can write
,
, and
.
where
is the hypergeometric function with 2
Here
and , and scalar argument , which
and 1 parameters, ,
may be written explicitly as
(26)
Now
defines the angular distribution of a random vari. Following [16] the mean orientation
able over
of is given by the value satisfying
The pdf of
is (e.g., [13, eq. (36)])
(27)
Now
where
the PDF for , for
is given by
. Therefore, a change of variable gives
,
(23)
Let us consider the first part of the integral. If we set
can be written
By a reparamerization using
this
this PDF can be written as
(24)
which integrates to zero due to the integrand being odd. If we set
and expand
according to (26) then
for the second part of the integral we get terms proportional to
which is the PDF of the (instantaneous) aspect ratio given by
Barakat [1, eq. 5.9] (as corrected in [5, p. 2885]) in his study
of a partially polarized quasi-monochromatic plane wave field
in the time domain. In that study is the degree of polarization defined by the “coherency matrix” (time-averaged covariance matrix), whereas here is the degree of polarization de. The parameter is exactly what oceanographers
fined by
[6], [8], [28] call the “rotary coefficient” which ranges from 1
for clockwise motion, to zero for unidirectional flow, to 1 for
counterclockwise motion. When is negative the PDF has its
peak at a negative value and is skewed to the right, and vice
. The distribution
versa; the skew is less pronounced as
.
is symmetric when
for
, and each such term is zero again because
is the value satisfying (27)
the integrand is odd. Hence,
and thus
in (14) does indeed define the mean orientation.
C. Transformation From Real Variables
Consider jointly Gaussian stationary time series
and
in the and directions, respectively, with Hermitian
spectral matrix
B. Azimuth
The azimuth (orientation), the angle which the major axis
of the ellipse makes with the -direction is given by
. Now
can be written as
, a phase difis given in [13,
ference. A convenient form for the PDF of
to
eq. (18)] and if we make the transformation from
we obtain
(25)
where
and
are the power spectra for
and
and
is the cross-spectrum. Then if
,
the vector-valued time series
has the spectral
matrix
given in (3). Further, [26],
,
where
RUBIN-DELANCHY AND WALDEN: KINEMATICS OF COMPLEX-VALUED TIME SERIES
and
with value
response located at
extracts
plied to
; the result is the same as
4195
which when apand then enlarges it by
. Hence,
(28)
(31)
(29)
The right-hand side of (31) will form the basis of a simula, and the latter will correspond
tion method for
to simulating
with a mere rescaling of the real and imagi. In practice we will need to make
very
nary axes by
small, and use a good (non-ideal) bandpass filter.
Equations (14) and (28) together give (e.g., [15])
which is the (fixed) azimuth for the singular case. Since we have
proven that
in (14) does indeed define the mean orientation under the Gaussian model, this formally proves a statement
in [8] also used elsewhere in the oceanographic literature (e.g.,
given in the form (29) is also the mean ellipse
[6]), that
orientation in the general case.
V. SIMULATION
A. A Necessary Rescaling
of the conSuppose we wish to simulate a realization
from a realization
of
. We know that
tribution
, so that
, where
denotes order in probability [7, p. 220]. (Priestley [24, p.
244] says that “
over an inifinitesimal interval
is infiniitself”.)
tesimal, but of a much larger order of magnitude than
, and we will need to have a physThus,
if we are to do useful
ically sensible way of rescaling
computations. One approach is to work with
initself. We can write
stead of
(30)
and
. Clearly,
and
.
with an ideal narrowband unit-norm filter
If we filter
with frequency response function given by
where
B. The Filter
The filter of length
which is most concentrated in the
is the zeroth-order discrete prolate
frequency interval
spheroidal sequence (dpss) or Slepian sequence (e.g., [20]),
which is an eigenvector of an eigenproblem. As we shall require
a very-narrow-band filter the length will necessarily be very
large. Hence, we need a robust method for calculating the filter
such as that given in [29] where it was demonstrated that the
, say,
filter coefficients,
odd, could be very accurately approximated, within a
with
constant of proportionality, as the sequence
for
, where is the modified Bessel function
of the first kind and 0th order. The filter coefficients were scaled
,
,
to have a norm of unity. The choice of
.
gave an effective bandwidth [30] of the filter of about
To achieve filters with energy at
and
we simply used
and
.
complex demodulation,
Since the filters are so narrow and have very low sidelobes, inwas chosen close to
terference would only be a problem if
zero. Hence, at
, we approximate
in (31)
by
(32)
if
otherwise,
we obtain
C. Start-Up Values
which is the same as
. Filtering
and then enlarges it by
can define an ideal narrowband filter,
by
extracts
. Similarly, we
say, with frequency
For process
we use the CAR(2) process of (12) which,
as seen in Fig. 1 is highly concentrated around
. Suppose
consisting of values. Since the
we require a sequence
is symmetric about
, then we need to
filter
values of
so that the central values
generate
are unaffected by end-effects. So for
50 001 and
we generated 50 100 values of the CAR(2) process. We show in
the Appendix how we generated the first two RVs, namely
and , with the correct second-order (covariance and relation)
structure so that the iterative procedure could be run without any
start-up transients.
4196
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
Fig. 4. Independent realizations of the elliptical paths Z (f )[df ]
, at f = f = 1=10, including fitted ellipses. The dotted line shows the orientation for the
singular case ( = 1), i.e., the mean orientation in general, while the dashed line shows the orientation for the individual ellipses. (a)–(c) are for = 1, (d)–(f)
are for = 0:6, and (g)–(i) are for = 0:2.
D. Parameter Values
We chose
to be
nomial
and took the roots of the CAR(2) poly,
so that
and
. For the CAR(2)
model
(33)
(34)
The choice of
for
the value
,
means that
. From (34)
takes
(the term
is dominant here), and thus from (14) the thefor the singular case is given by
oretical azimuth value at
.
as in (2), rather than as the aggregate from multiple frequency
magnitudes. An ellipse was fitted using least squares [9], and
calculated; the dashed line shows
the azimuth estimate
the orientation corresponding to the estimated azimuth. (Note
we can disregard the error in fitting the ellipse by least squares
since the generated points have minuscule departures from elliptical form). The dotted line shows the theoretical azimuth value
at for the singular case. In Section III-A we concluded that in
the singular case the only aspect of the random ellipse which is
random is its size. Fig. 4(a)–(c) shows three independent real. As predicted, the estimated azimuth
izations for which
for the singular
agrees with the theoretical azimuth value at
case in all three cases and only the scale of the ellipse varies.
Fig. 4(d)–(f) shows three independent realizations for which
and Fig. 4(g)–(i) shows the same for
. Here
we see that the estimated azimuths now vary from the singular
case, as well as the size of the ellipses changing.
and 0.2, we repeated such independent realFor
izations 250 times and calculated the sample aspect ratios and
and
azimuth values of the fitted ellipses. (Note that
are invariant to a rescaling of
so that our
will not affect the distributions of the samrescaling by
ples.) The resulting histograms are shown in Fig. 5, along with
the theoretical PDFs (23) and (25). The fits are excellent and
validate our theoretical results.
F. Corollary
E. Kinematics
Each plot in Fig. 4 shows the 100 simulated filtered AR(2)
values of the form (32). There are 10 points on each revolution of
the ellipse and the values from the 10 revolutions approximately
overlay each other, indicating that the band is narrow enough
to be considered the contribution of one frequency magnitude
The simulations, PDFs, and Section III-C show that as
decreases, the variability in both ellipse orientation (azimuth)
and aspect ratio increases. Gonella [8] had previously interas the “stability of the ellipse orientation” but we
preted
likewise controls the
see that this is only half the story, as
aspect ratio.
RUBIN-DELANCHY AND WALDEN: KINEMATICS OF COMPLEX-VALUED TIME SERIES
4197
Then
(36)
where
..
.
..
.
..
.
..
.
and
..
.
..
.
..
.
and we have used the fact that
and
From (36) we can write [10, 10.2.15],
are SOS.
Fig. 5. Empirical histograms and theoretical PDFs for = 0:6 for (a) aspect
ratio, and (b) azimuth, and for = 0:2 for (c) aspect ratio and (d) azimuth.
VI. SUMMARY
We have shown how the magnitude squared coherence
controls (i) the relative influences of the two orthogonal random
ellipses with the same aspect ratio which add to give the resultant random ellipse, and (ii) the variation of both the orientation
and aspect ratio of the resultant ellipse, whereas previously it
has been associated only with the ‘stability’ of the orientation.
Statistical analysis of the random azimuth proved that, under
is the mean orientation of
Gaussianity,
the random ellipse. We also showed how the combination of
an accurately realized very narrow-band filter, a complex autoregressive process and a rescaling allow random ellipses to be
simulated, confirming the theoretical conclusions.
where
denotes the stacking of the columns of the matrix
one underneath the other to form a single vector. But, (e.g.,
, where denotes
[12])
the usual Kronecker product of matrices. Hence,
and so
Therefore, the first elements of the first column of
are precisely
.
Next, we write
(37)
where
APPENDIX
..
.
..
.
..
.
..
.
A. Start-up Values for CAR(p) Simulation
Consider the complex SOS autoregressive process
order , denoted
of
and
..
.
(35)
is doubly
where the are complex-valued parameters and
and
. Given values of
white noise,
, for which all the roots of the -polynomial
are outside the unit circle, and values of
and , for which
, [22], we first find
and
.
and rewrite it in the form
Let
, where
..
.
..
.
..
.
..
.
again using the fact that
From (37) we have
But
and
..
.
..
.
are SOS.
,
, and so
Therefore the first elements of the first column of
are precisely
.
The covariance matrix of
[21]
..
.
where
.
is
4198
Define
and
where
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008
and
, for
. Then [21]
,
,
and
The covariance matrix of
is given by
.
such that
We can calculate a Cholesky decomposition of
. If we let
where the
’s
are independent standard normal random variables, then if
, we know that the covariance matrix of is given by
.
with the corHence, to obtain our start-up sequence
rect second-order (covariance and relation) structure we simply
, and take the first entries.
compute
ACKNOWLEDGMENT
P. Rubin-Delanchy would like to thank EPSRC (UK) for financial support. The authors would like to also thank the anonymous reviewers for their constructive comments and suggestions which are greatly appreciated.
REFERENCES
[1] R. Barakat, “The statistical properties of partially polarized light,” Opt.
Acta, vol. 32, pp. 295–312, 1985.
[2] M. Born and E. Wolf, Principles of Optics, 4th ed. Cambridge, U.K.:
Cambridge Univ. Press, 1970.
[3] P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods,
2nd ed. New York: Springer-Verlag, 1991.
[4] J. L. Doob, Stochastic Processes, Wiley Classics ed. New York:
Wiley, 1990.
[5] D. Eliyahu, “Vector statistics of correlated Gaussian fields,” Phys. Rev.
E, vol. 47, pp. 2881–2892, 1993.
[6] W. J. Emery and R. E. Thomson, Data Analysis Methods in Physical
Oceanography. New York: Pergamon, 1998.
[7] W. A. Fuller, Introduction to Statistical Time Series, 2nd ed. New
York: Wiley, 1996.
[8] J. Gonella, “A rotary-component method for analysing meteorological and oceanographic vector time series,” Deep-Sea Res., vol. 19, pp.
833–846, 1972.
[9] R. Halif and J. Flusser, “Numerically stable direct least squares fitting
of ellipses,” in Proc. 6th Int. Conf. Central Europe Computer Graphics
Visualization (WSCG), Plzeň, Czech Republic, 1998, pp. 125–132.
[10] J. D. Hamilton, Time Series Analysis. Princeton, NJ: Princeton Univ.
Press, 1994.
[11] Y. Hayashi, “Space-time spectral analysis of rotary vector series,” J.
Atmospher. Sci., vol. 36, pp. 757–766, 1979.
[12] H. V. Henderson and S. R. Searle, “Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics,” Canad.
J. Statist., vol. 7, pp. 65–81, 1979.
[13] J.-S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity
and phase statistics of multilook polarimetric and interferometric
SAR imagery,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 5, pp.
1017–1028, Sep. 1994.
[14] M. Loève, Probability Theory, 3rd ed. Princeton, NJ: Van Nostrand,
1963.
[15] L. R. M. Maas and J. J. M. van Haren, “Observations on the vertical
structure of tidal and inertial currents in the central North Sea,” J. Mar.
Res., vol. 45, pp. 293–318, 1987.
[16] K. V. Mardia, Statistics of Directional Data. New York: Academic.
[17] K. S. Miller, Complex Stochastic Processes. Reading, MA: AddisonWesley, 1974.
[18] C. N. K. Mooers, “A technique for the cross spectrum analysis of pairs
of complex-valued time series, with emphasis on properties of polarized components and rotational invariants,” Deep-Sea Res., vol. 20, pp.
1129–1141, 1973.
[19] M. Orlić, B. Penzar, and I. Penzar, “Adriatic sea and land breezes:
Clockwise versus anticlockwise rotation,” J. Appl. Meteorology, vol.
27, pp. 675–679, 1988.
[20] D. B. Percival and A. T. Walden, Spectral Analysis for Physical Applications. Cambridge, U.K.: Cambridge Univ. Press, 1993.
[21] B. Picinbono, “Second-order complex random vectors and normal
distributions,” IEEE Trans. Signal Process., vol. 44, no. 10, pp.
2637–2640, Oct. 1996.
[22] B. Picinbono and P. Bondon, “Second-order statistics of complex signals,” IEEE Trans. Signal Process., vol. 45, no. 2, pp. 411–420, Feb.
1997.
[23] B. Picinbono and P. Chevalier, “Widely linear estimation with complex
data,” IEEE Trans. Signal Process., vol. 43, no. 8, pp. 2030–2033, Aug.
1995.
[24] M. B. Priestley, Spectral Analysis and Time Series. New York: Academic, 1981.
[25] M. Rosenblatt, Stationary Sequences and Random Fields. Boston,
MA: Birkhäuser, 1985.
[26] P. J. Schreier and L. L. Scharf, “Second-order analysis of improper
complex random vectors and processes,” IEEE Trans. Signal Process.,
vol. 51, no. 3, pp. 714–725, Mar. 2003.
[27] P. J. Schreier and L. L. Scharf, “Stochastic time-frequency analysis
using the analytic signal: Why the complementary distribution matters,” IEEE Trans. Signal Process., vol. 51, no. 12, pp. 3071–3079, Dec.
2003.
[28] H. van Haren, “On the polarization of oscillatory currents in the Bay
of Biscay,” J. Geophys. Res., vol. 108, no. C9, p. 3290, 2003, 10.1029/
2002JC001736.
[29] A. T. Walden, “Accurate approximation of a 0th order discrete prolate
spheroidal sequence for filtering and data tapering,” Signal Process.,
vol. 18, pp. 341–348, 1989.
[30] A. T. Walden, E. J. McCoy, and D. B. Percival, “The effective bandwidth of a multitaper spectral estimator,” Biometrika, vol. 82, pp.
201–214, 1995.
Patrick Rubin-Delanchy received the M.Sci. degree
in mathematics and computing from Imperial College, London, U.K., in 2004. He is currently working
towards the Ph.D. degree at Imperial College.
His research interests include multichannel time
series and spectral analysis.
Andrew T. Walden (AM’86–M’07) received the
B.Sc. degree in mathematics from the University
of Wales, Bangor, in 1977, and the M.Sc. and
Ph.D. degrees in statistics from the University of
Southampton, U.K., in 1979 and 1982, respectively.
He was a Research Scientist at BP from 1981 to
1990, specializing in statistical signal processing.
He joined the Department of Mathematics at Imperial College, London, U.K., in 1990, where he is
currently a Professor of Statistics. He is coauthor
(with D. Percival) of the texts Spectral Analysis for
Physical Applications (Cambridge Univ. Press, 1993) and Wavelet Methods for
Time Series Analysis (Cambridge Univ. Press, 2000).