IEEE Signal Processing - March 2018 - 147

Applications

chosen to use a relatively low number of complex filter coefficients, giving only the main features of the power spectrum.
We note that similar problems occur with other methods. In the
EMD and VMD algorithms, the number of modes must be estimated or chosen.

To test the performance of the three CSTAR techniques, we
have applied them to two synthetic noise-free signals taken
from the recent literature [4], [18]. In the first example, they are
compared with the results for STAR [4], [32] and the maximum
entropy Wigner-Ville (MEWV) time-frequency distribution
[30]. The results show that the methods of [35] and [36] have
better resolution than the other methods. A further comparison
shows that the method in [35] is stabler, and it was used in the
next data examples (a second synthetic data example [18], seismic field data, and the LIGO gravitational wave data [37]). In
all examples, we use a sliding time window that is moved one
sample forward in each step.
A problem with the AR method for power spectrum estimation is the choice of the dimension of the prediction error operator. Reference [32] uses the final prediction error criterion in
[45] to estimate the order of the AR model. Reference [40] minimizes a weighed least-square fit between STFT and the STAR
spectrum with real filter coefficients. This results in fairly long
filters. When the filter coefficients are complex, only half the
number of coefficients is needed to resolve a certain number of
spectral peaks (complex poles in the power spectrum). We have

Frequency (Hz)
1.8

60

80

100

120

140

Synthetic data 1
In this first example from [4], the signal is a sum of five elements:
s 1 ^ t h = 0.8 cos ^30rt h
s 2 ^ t h = 0.6 cos ^70rt h
s 3 ^ t h = 0.7 cos ^130rt + 5 sin ^2rt hh
t /8
s 4 ^ t h = sin c 8r100 m
log ^100h
2
s 5 ^ t h = 3e -1250 (t - 2) cos ^710 ^t - 2hh

1.8

60

80

100

120

160

1.8

2.0

2.0

2.2

2.2

2.2

60

80

0 # t # 10s. (27)

Frequency (Hz)
140

2.0

1.8

6s 1 t # 10s

This signal is composed of two harmonic components with
frequency of 15 and 35 Hz, s 1 and s 2, a frequency-modulated
harmonic around 65 Hz, s 3, a sliding harmonic from 35 to
158 Hz, s 4, and a Morlet wavelet with central frequency of
approximately 113 Hz, s 5 . A sampling interval of 0.003 s was
used to generate a discrete signal shown in Figure 1(a) from

Frequency (Hz)
160

0 # t # 6s
0 # t # 6s
4s # t # 8s

60

80

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120

140

(a)

(b)

(c)

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

100

120

140

160

1.8

60

80

100

120

140

160

1.8

2.0

2.0

2.0

2.2

2.2

2.2

(d)

(e)

60

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160

(f)

CSTAR

Figure 3. Time-frequency distributions around the Morlet wavelet peak for the six methods (zoom) from Figures 1 and 2; (a) STFT, (b) MEWV, (c) STAR,
(d) Burg, (e) Marple, and (f) Morf.

IEEE Signal Processing Magazine

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March 2018

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147
Table of Contents for the Digital Edition of IEEE Signal Processing - March 2018
