A multiwindow time-frequency approach based on the concepts of robust estimate theory

Abstract

An approach to multiwindow time-frequency analysis that provides robust performance in noisy environment is proposed. The concept of robust estimates of instantaneous frequency is used to define the optimal weighting coefficients for the multiwindow spectrogram. The proposed form of multiwindow spectrogram provides improved instantaneous frequency estimation for nonstationary signals in the presence of additive Gaussian noise. The efficiency of the proposed approach is tested in the experiments.

Full text

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A MULTIWINDOW TIME-FREQUENCY APPROACH BASED ON THE CONCEPTS OF
ROBUST ESTIMATE THEORY

Irena Orovi?1, Nikola Žari?1, Srdjan Stankovi?1, Moeness Amin2

1Faculty of Electrical Engineering, University of Montenegro, 81000 Podgorica, Montenegro
2University of Villanova, Villanova, PA 19085


ABSTRACT

An approach to multiwindow time-frequency analysis that
provides robust performance in noisy environment is
proposed. The concept of robust estimates of instantaneous
frequency is used to define the optimal weighting
coefficients for the multiwindow spectrogram. The proposed
form of multiwindow spectrogram provides improved
instantaneous frequency estimation for nonstationary signals
in the presence of additive Gaussian noise. The efficiency of
the proposed approach is tested in the experiments.

Index Terms— time-frequency analysis, instantaneous
frequency estimation, Hermite functions, multiwindow
approach

1. INTRODUCTION

Time-frequency analysis has been used in applications
where the information, carried by the signal, is contained in
its time-varying frequency content. Various time-frequency
distributions have been proposed for the instantaneous
frequency (IF) estimation [1]-[4]. One of the commonly used
quadratic distributions is the spectrogram, defined as the
square module of the short-time Fourier transform. It is
simple for realization, and thus suitable for practical
applications. However, the spectrogram has a low time-
frequency resolution. In order to improve concentration in
the time-frequency plane, other quadratic distributions are
considered, such as the Wigner distribution, the S-method,
etc. For instance, the Wigner distribution provides ideal
representation when IF is a linear function of time. For non-
linear IF, the estimation accuracy can be improved by using
the multiwindow approaches [5]-[8]. The multiwindow
spectrogram [5] is defined as a weighted sum of
spectrograms calculated with Hermite functions of different
order. The weighting coefficients for the Hermite window-
based spectrograms are typically set according to desired
constraints related to signal power localization in the time-
frequency domain.
The signals used in the time-frequency analysis are usually,
more or less, corrupted by the noise, which consequently
affects the time-frequency representation. In order to reduce
the noise influence, in this paper we propose a modified
form of multiwindow spectrogram. The modification is
related to the weighting coefficients, which are derived by
using the concept of robust estimates [9]. The proposed
modified form of multiwindow spectrogram provides good
time-frequency representation and high accuracy of IF
estimation even for low signal to noise ratio. It outperforms
other standard distributions,
multiwindow spectrogram form.
The paper is organized as follows. Section II presents the
theoretical background related to the standard form of
multiwindow spectrogram. The multiwindow spectrogram
based on the optimal weighting coefficients in noisy
conditions is proposed in Section III. The efficiency of the
proposed approach is tested experimentally in Section IV.
Concluding remarks are given in Section V.

2. THEORETICAL BACKGROUND –
MULTIWINDOW TIME-FREQUENCY ANALYSIS

2.1. Optimal windows selection

In order to improve the analysis of stationary and non-
stationary spectrum over the standard methods, various
multiwindow approaches have been introduced [5]-[7], [10]-
[11]. In stationary spectrum estimation, the multiple
windows are usually chosen as discrete prolate spheroidal
sequences (DPSS) [10], being a family of orthonormal
functions that maximize the energy in the given frequency
band. Further, the minimum bias windows, having close
behavior to the DPSS, have been used in spatial filtering for
beamspace processing [11]. However, in the non-stationary
cases, it is preferred to use functions that are optimally
concentrated in the time-frequency domain with elliptic
symmetry [12]. Such functions are orthogonal Hermite
functions. They minimize the variance and simultaneously
provide low bias estimate. The k-th order Hermite function
is defined as:
including the former

222
/2
1
( )( 1)
?
2!
k
tk t
e
t
k
k
d
dt
h tee
k
?
??
?
?
?
??
??
?
?
?
?
(1)
3584978-1-4577-0539-7/11/$26.00 ©2011 IEEEICASSP 2011

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The recursive realization of the Hermite functions, which
is considered attractive in practical applications, is given by:


22
/2 /2
01
44
12
12
?
( ), ( ),
2
k
1
( ) ( )
t
( ),
t
2.
tt
kkk
t
h teh te
k
h tthhk
k
?
??
??
??
?
?? ? ?
(2)


2.2. Multiwindow spectrogram

The multiwindow spectrogram has been introduced as an
improved quadratic time-frequency
analyzing signals with time-varying spectra. It has been
defined as a weighted sum of spectrograms calculated for
different window functions. By using the Hermite functions,
as an optimal choice, the multiwindow spectrogram is
defined as [5]:

K
MWS t
?? ?
distribution for

1
0
( , ) ( )
t S t
( , ),
?
kk
k
d
?
?
(3)
where
k-th order Hermite function as follows:

( , )
t ? represents a spectrogram calculated using the
k
S

2
1
?
( , )
t
( )
s t h t
().
2
j
kk
Sed
??
???
?
??
?
(4)

Let us observe the signal in the form:


( )
t
( )
s t
( )
j
A t e?
?
(5)

where A(t) is signal’s amplitude, while
function. Then, the conditional mean frequency obtained by
using the spectrogram
( , )
k
St ? is defined as [5]:

2
( , )
k
kt
k
S td
??
?
?

Consequently, the conditional mean frequency
( ) t
?
is a phase
2
22
()( ) '(
? ?
)
( , )
()( )
?
k
k
S tdA thtd
A thd
??????
?
??
??
??
?
??
(6)
t
?
of the
multiwindow spectrogram is given by:


1
0
( )
t
K
kk
tt
k
d
??
?
?
? ?
. (7)

For signals with the K-th order polynomial phase:


0
( )
t
K
n
n
n
a t
?
?
??
, (8)
the conditional mean frequency can be expressed as:

KK
?
??

( )( )
t
?
denotes the n-th derivative of the phase,
while the n-th order moment of the spectrogram is:

2
(
( )
n
Mt
A t
?

In order to recover the instantaneous frequency from (9), the
following equation holds:

1
( )( )
kn
k
?

Hence, to satisfy the instantaneous frequency constraints, the
weights
( )
k
dt are obtained by using at least K
spectrograms.
Furthermore, as it is derived in [5], a number of 2K-1
windows is required to satisfy the instantaneous bandwidth
requirements. It means that, to satisfy both the instantaneous
frequency and instantaneous bandwidth requirements, the
weights are calculated by using 2K-1 windows:

22
( )
kn
k
?

This method is asymptotically unbiased in the noise-free
case. However, as shown in [5], it outperforms standard
distributions, such as the single window (standard)
spectrogram, in noisy conditions as well.

3. OPTIMAL WEIGHTING COEFFICIENTS FOR
NOISY ENVIRONMENT

The peak frequency at each time is generally less affected
by noise than the conditional mean frequency. Thus, in order
to increase the performance of multiwindow distribution in
the presence of noise, the concept of robust estimates is used
in the sequel. Instead of conditional mean frequency, one
may use the M-estimate of the instantaneous frequency [9].
It is obtained as a solution of the optimization problem given
by:
?
argmin ( ).
k
ω=

11
(1)
00
( )
t
( )
t
( )
t
!
k
n
n
n
k
t
kn
M
d
?
??
?
? ? ?
, (9)

where
n

2
k
h
22
k
)( )
? ?
() ( )
?
n
k
A thd
d
??
??
?
?
?
?
. (10)

0
10
00
K
?
k
n
n
dt Mt
?
?
??
? ?
?
?
???
. (11)

0
10
( )
t
.
00
K
?
k
n
n
dt M
?
?
??
? ?
?
?
???
(12)

J
ω
ω
(13)
where,

12
( )
ω
( )( ( )
F e
τ
())
JF ed
τ
=+
??
(14)

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The function F is convex non-negative loss function,
while e1 and e2 are, respectively, real and imaginary part of
error function [9].
In order to determine optimal weights in the presence of
noise, we may consider argmin ( ) ( , ).
t
ω
k
JS
ω
=
The
necessary minimum conditions in (13) and (14) can be given
by the following complex valued equations:

( )
( )
h
τ
ωω
∂∂
?
∂∂
=
∂∂

By applying the linearization with respect to small IF bias
ω
Δ
and small influence of higher phase derivatives [9]:

12
(1)(1)
1122
12
(
ω
∂
)
0,
( )
F e
ω
(
ω
∂
)
( )
e
,()
F e
∂
F e
∂
J
e
ω
F e
∂
e
ω
∂
FFe
?
?
?
?
?
∂
=+=
∂
=
?
(15)
k

( )
n
2
( )( )
t
,
!
n
n
n
τ
δϕ τ ϕ
∞
=
=?
(16)
the bias for the spectrogram with the k-th order window
function is defined as:
2
(
k
E
A t
?

Thus, the total bias for the multiwindow spectrogram with K
window functions can be defined in the form:
1
K
E
ω
=

where ck(t) are optimal weights for noisy conditions. Thus,
by using (16) and (18), the total bias can be written as:

2
1
( )
k
nk
A t
==
?
Furthermore, the estimated instantaneous frequency can be
written as follows:

K
ttE
ωωω
=+Δ=?

{}
22
)( )
τ τδϕ τ
( )
()( )
τ τ
k
k
A td
d
ττ
ω
ττ
+
Δ=
+
?
?
?
. (17)

{}
{}
0
( )
kk
k
c t E
ω
−
Δ=Δ
?
(18)

{}
1
( )
n
22
20
()( )
τ τ
( )
t
,
!
()( )
τ τ
n
K
k
?
k
A td
Ec t
n
d
τ τϕ
ω
ττ
+
∞
? ?
−
+
Δ=
+
?
?
(19)

?
( )
n
1
01
( )
t
( )( ){} ( )
t
.
!
k
kn
kn
c
n
ϕ
−∞
??
==
(20)

where
21
22
()( )
τ τ
()( )
τ τ
n
k
?
k
n
k
A t
?
constant
1
n
d
τ τ
d
A td
ττ
ττ
+
+
=
+
?
?
?
and
1
1
k=
?
holds.
For the
?
?
amplitude case we have:
2
( )
( )
τ τ
k
k
n
k
d
τ
τ
+
=?
?
?
.
Finally, the weights ck can be obtained by solving the linear
set of equations:

1
0
1,1
( )
0,1
K
?
k
kn
k
n
n
c t
−
=
=
?
?
=?
>
?
(21)
Based on the optimal weighting coefficients, the
multiwindow spectrogram is defined as:

K
k
k
?
?

Note that the windows

2
1
0
1
( , ) ( )( )
s t
().
2
j
k
MWS t c tted
?
??
???
?
?
??
?
?
(22)
k ? should satisfy a number of
conditions [9], e.g., they should be even and non-negative
functions. Thus, we use the windows in the form:
2/4
( )( )
kk
teh t
=
?
, where
function. Although, there are some of the required
conditions that are not satisfied, the proposed form of
windows, combined with the coefficients ck, provides
improved results in comparison
multiwindow approach. The optimal windows choice can be
a topic for some further research.

4. EXPERIMENTAL RESULTS

Consider the noisy signal in the form:

(5cos(5.5 ) 5sin(2.5 ))
( )
x te
?
?????
,
?
noises (zero mean with variance equal to 1). The time
interval t∈[-1,1], with sampling rate T=2/128, is used. The
spectrogram, the Wigner distribution and the S-method are
shown in Fig 1.a-c, respectively. Note that these
distributions suffer from the noise and interference terms
and do not provide satisfactory results. Further, the standard
multiwindow spectrogram (MWS) is calculated with weights
dk (Fig 1.d). The proposed multiwindow spectrogram with
coefficients ck is shown in Fig 1.e. Three Hermite functions
are used (K=3). Namely, it has been shown experimentally
that this number of functions provides efficient results.
Clearly, the proposed approach with optimal weighting
coefficients ck, provides the best results for the considered
noisy signal.
Based on the distributions from Fig 1, the IF is estimated
for different values of SNR. The mean square errors (MSE)
of IF estimations are calculated and presented in Table 1.
The results are given for the spectrogram (SPEC) with
rectangular window, the Wigner distribution (WD), the
multiwindow spectrogram with coefficients dk (MWS_d, for
K=1 and K=3), and the proposed multiwindow spectrogram
MWS_c (for K=1 and K=3). The case with K=1 (one
window function) is considered only for the comparison
purpose. It can be observed that the proposed approach
(MWS_c, K=3) provides high accuracy of IF estimation for
signals with fast varying IF, even for low values of SNR.
t
−
( )
k h t is the k-th order Hermite
with the former

( )
t
jtt
??
?
?
?

where
12
( )
t
( )
t
( )
tj
1( ) t
and
2( ) t
?
are Gaussian
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a) b) c)

d)
e)
Fig 1. a) spectrogram, b) Wigner distribution, c) S-method, d) multiwindow spectrogram with coefficients dk , e)
multiwindow spectrogram with coefficients ck . SNR=10dB


Table 1. MSE for different distributions

Distribution SNR=5dB
SPEC 210
WD 158
MWS_d, K=1 12.4
MWS_c, K=1 6.7
MWS_d, K=3 5.8
MWS_c, K=3 3


SNR=10dB
207
150
11.2
6.1
3.7
2
SNR=15dB
200
140
10.4
5.8
3
1.6

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