Comparison and Application of Time-Frequency Analysis Methods for Nonstationary Signal Processing - Advanced Research on Computer Education, Simulation and Modeling - page 289

Imposing Hilbert transform on each IMF component, the Hilbert spectrum of x(t)
can be obtained by taking the real part of the sum of the Hilbert transform results. Thus,
the instantaneous frequency representation of the signal is,

n n

∫

jftdt ()

∑ ∑

j θ () t

xt f ( , ) = e a te ( ) = e a te ( )

i

i

i i

.

(9)

i = 1 i = 1

In the expression (9), r n is ignored; Re denotes the real part of the complex signals.

3 Simulation and Comparison of Time-Frequency Analysis

To compare the characteristics of the above mentioned time-frequency methods, a
typical nonstationary signal x(t) is designed as that in equation (10). Note that the x(t) is
a multi-component signal constituted with a cosine component (frequency: 50Hz), a
linear frequency modulation component (fundamental frequency: 200Hz) and a sine
frequency modulation component (fundamental frequency: 100Hz, modulation
frequency: 15Hz). The x(t) in time domain and its frequency spectrum are showed in
Figures 2 and 3, respectively.

2

xt ( ) = 1.5 cos100 π t + 2sin(400 π t + 100 π t ) + cos(200 π t + sin 30 π t )

(10)

.

Fig. 2. The designed signal x(t) Fig. 3. Frequency spectrum of x(t)

The STFT, WT, WVD and PWVD are performed to analyze the nonstationary signal
x(t) , and the time-frequency maps are shown in Figure 4. It can be seen that, the STFT
can recognize the linear frequency component and the cosine component, but the time
and frequency resolutions are low. The WT with multiresolution characteristic can
improve the resolutions; it has a good frequency resolution in the low frequency region
and a good time resolution in the high frequency region. However, the STFT and WT
can't exactly recognize the sine frequency modulation component. In contrast, the
WVD has higher resolution and can describe the sine frequency modulation, but there
are a lot of cross terms appeared in the distribution. The useful informations in the
distribution are disturbed seriously by the cross terms. Although the PWVD can