Optimal Selection of Analyzing Window of Arbitrary Shape for S-transform using PSO Algorithm

A. Ashrafian, M. Agha Mirsalim, A. A. Suratgar Electrical Engineering Department

Amirkabir University of Technology Tehran, Iran

Email: {Ashrafian & Mirsalim & A-suratgar}@aut.ac.ir

Abstract—In this paper a method for optimal selection of analyzing window for generalized S-transform has been proposed. The S-transform is applied to the signal and the S-matrix is computed, firstly. Then, a concentration measure is introduced which can be calculated according to S-matrix. At last, the Particle Swarm Optimization (PSO) algorithm is employed to select an analyzing window which presents the best time-frequency resolution by minimizing the concentration measure. To evaluate the effectiveness of the proposed method, it has been implemented in Matlab environment and has been tested on the two most popular Gaussian and hyperbolic windows. Then, the improved S-transform has been applied to two sample signal cases and time-frequency contours have been plotted, accordingly. Also, the standard non-optimized S-transform has been implemented. The time-frequency resolutions of optimized Gaussian and hyperbolic windows have been compared to the time-frequency resolution presented by standard S-transform. It is found that both optimized hyperbolic as well as Gaussian widows provide a better time-frequency representation in comparison to non-optimal standard window.

Keywords—wavelet transform; s-transform; signal processing; optimization

I. INTRODUCTION Time-frequency analysis is used to extract both time and

frequency information of a signal. This analysis is especially useful in the case of non-stationary signals whose behaviors change with time. Several time-frequency techniques have been introduced in the area of signal processing, such as Short Time Fourier Transform (STFT), Wavelet Transform (WT) [1] and S-transform. The STFT suffers from the drawback of having a fixed analyzing window. The wavelet transform is based on moving and scaling a window, called mother wavelet. Like the wavelet transform, the standard S-transform [2] has a moving scalable localizing Gaussian window whose shape is governed by frequency. In other words the standard S-transform has a Gaussian window whose height and width scale directly and inversely with the frequency, respectively. Unlike the wavelet transform, the S-transform localizes the phase spectrum, referenced to the time origin, in addition to the amplitude spectrum. This phase provides supplementary information about spectra which is not given by WT [2]. Also, the S-transform is more robust to noise in comparison to wavelet transform [2]-[5]. Then, the S-transform has been

developed to accept an arbitrary window [6]. Having this generalized S-transform, bi-Gaussian [7] and hyperbolic windows [8] are commonly used in addition to the Gaussian window of the standard S-transform. Though, the S-transform is a powerful tool used in a wide range of areas, the performance of the S-transform is dependent on the shape of the window. The Gaussian window has only one parameter, but the shape of hyperbolic window is governed by a set of parameters (i.e., forward tapper, backward tapper and positive curvature parameters). The energy concentration of the S-transform will be poor if these parameters are not selected judiciously. This can result in a weaker time and frequency resolutions and smearing of signal components in both time and frequency. A tradeoff between time and frequency resolution can be acquired by selecting an optimal width for analyzing window. Some energy concentration measures have been proposed in [9]-[12]. To address the S-transform energy concentration problem, some researches have been conducted to find an optimal width for analyzing window [13]-[16]. In [13]-[14], this has been performed through the introduction and the optimal selection of an extra parameter in the standard S-transform relation. The methods proposed in [15]-[16] are based on optimal selection of Gaussian window parameters, considering a concentration measure. Since hyperbolic window has three parameters which define its shape in addition to its width, the optimal selection of hyperbolic window parameters is trickier in comparison to a Gaussian window which has only one parameter. Optimal selection of hyperbolic window parameters has not been studied in [13]-[16].

The main idea of this paper is to propose a method for optimal selection of analyzing window for generalized S-transform (can have an arbitrary window such as hyperbolic or Gaussian windows) which provides the most energy concentration and the best time-frequency resolution using the PSO algorithm. The method is applicable in a wide range of areas such as seismic signal analysis [2], [6]-[7], transformer internal fault [3]-[5], [17] and internal incipient fault [8], [18]-[20] detection and pattern recognition [21].

II. BACKGROUND THEORY

A. The Standard S-transform The S-transform provides features of both wavelet

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The 22nd Iranian Conference on Electrical Engineering (ICEE 2014), May 20-22, 2014, Shahid Beheshti University

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transform and STFT. The standard S-transform of time series x(t), can be obtained by the following relation [2]:

( ) ( )22

( , ) ( ) exp exp 222

f f tS f x t ift dt

ττ π

π

+∞

−∞

⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫ (1)

where, S is the S-transform of time series x(t), f denotes the frequency and the parameter τ translates the Gaussian window on the time axis. On the other hand the Continuous Wavelet Transform (CWT) of time variable signal x(t) can be defined as a series of correlation of the signal with mother wavelet

( , )w t aτ − , as follows [2]:

( ) ( , )CWT x t w t a dtτ+∞

−∞

= −∫ (2)

where, a is dilation parameter. So, the standard S-transform can be presented as a CWT, multiplied by a phase factor [19]:

( , ) ( , )exp( 2 )S f CWT a i ftτ τ π= − (3)

where, the mother wavelet is [2], [21]: 2 2( )( , ) exp

22

f f tw t f ττπ

⎛ ⎞− −− = × ⎜ ⎟⎝ ⎠

(4)

B. Generalized S-transform The generalized S-transform is derived from the standard S-

transform by substituting Gaussian window of (1) with a generalized window as in the following relation [6]:

( , , ) ( ) ( , , )exp( 2 )gS f p x t w t f p ift dtτ τ π+∞

−∞

= − −∫ (5)

where, p denotes a set of parameters which govern the shape of the analyzing window. Having this generalized formulation, the generalized window (i.e., ( , , )gw t f pτ − ) can be substituted by an arbitrary window. For example the Gaussian window of (4) or a hyperbolic window can be used as analyzing window. If a hyperbolic window is used, the analyzing window will be as in below [19]-[20]:

{ } 22 2

( t, , , , )

( , , , )2exp

22 ( )

h f b

f b

f b

w f

f V tf

τ γ γ λ

τ γ γ λ

π γ γ

− =

⎡ ⎤⎡ ⎤− −⎢ ⎥⎣ ⎦× ⎢ ⎥+ ⎢ ⎥⎣ ⎦

(6)

where, , ,f bγ γ λ are forward tapper, backward tapper and positive curvature parameters, respectively. These parameters control the shape of hyperbolic window. V is defined by the following relation [19]-[20]:

{ }2

2 2

( , , , ) ( )2

( )2

f bf b

f b

b f

f b

V t t

t

γ γτ γ γ λ τ ζγ γ

γ γ τ ζ λγ γ

⎡ ⎤+− = − − +⎢ ⎥⎣ ⎦

⎡ ⎤− − − +⎢ ⎥⎣ ⎦

(7)

V is a hyperbola in ( )tτ − and the translation factor ζ is used to ensure that the peak of hyperbolic window occurs at ( ) 0tτ − = . ζ is defined by the following relation:

2 2( )4

f b

f b

γ γ λζγ γ−= (8)

The most important feature of a mathematical transform is that it should be invertible. In the other words, the user is able to reconstruct the analyzed signal. The window of S-transform should satisfy the following condition [6]:

( , , ) 1w t f p dτ τ+∞

−∞

− =∫ (9)

Then, it is easy to show that the S-transform is invertible. According to (9), averaging of S-transform over all values of τ is the Fourier transform of the original signal, from which x(t) can be obtained and ensures that the S-transform is fully invertible, as follows [6], [19]:

( , , )

x( )exp( 2 ) ( , , ) ( )

S f p d

t ift w t f p d dt X f

τ τ

π τ τ

+∞

−∞+∞ +∞

−∞ −∞

=

− × − =

∫

∫ ∫ (10)

Also, the S-transform can be performed through use of the convolution theorem by the following equation:

2 2

22( , ) ( )exp exp( 2 )S f X f i dfπ ατ α π ατ α

+∞

−∞

⎛ ⎞−= + +⎜ ⎟⎜ ⎟⎝ ⎠

∫ (11)

α is the convolution variable and has the same unit as f.

C. Numerical Procedure

The discrete S-transform can be computed by taking the advantage of convolution and Discrete Furrier Transform (DFT) theorems. The sampled DFT of the analyzed signal x(t) is presented as follows:

1

0

2expN

k

n i πnkX x kT ( )NT N

−

=

−⎡ ⎤ =⎢ ⎥⎣ ⎦∑ ( ) (12)

where, T is the sampling interval, nfNT

= and t kT= . So,

the discrete S-transform with generalized window can be obtained by the following relation:

2 1

2

2, )expN

m N

n m n m n i πmjS jT X W ( )NT NT NT NT NT

−

=−

+ +⎡ ⎤ =⎢ ⎥⎣ ⎦∑ ( ) ( , (13)

W is the sampled DFT of the general window. j is a non-negative integer time index, jTτ = and m

NTα = . The generalized window can be substituted by any window such as Gaussian and hyperbolic windows.

D. Particle Swarm Optimization Algorithm

Particle Swarm Optimization (PSO) algorithm is a population-based stochastic optimization method that models the behavior of organisms such as fish schooling, and bird flocking [22]. The position of each particle in the search space, corresponding to a candidate response, is adjusted considering the best position visited by itself called pbest (i.e. its own experience) and the position of the best particle in the search

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space called gbest. The pbest of the ith particle at time step t is updated as:

( ) (z ( 1)) ( ( ))( 1)

( 1) ( ( 1)) ( ( ))i i i

ii i i

y t if cf t cf y ty t

z t if cf z t cf y t+ ≥⎧ ⎫⎪ ⎪+ = ⎨ ⎬+ + <⎪ ⎪⎩ ⎭

(14)

where, zi and yi are the current and personal best position of the particle, respectively. cf denotes the cost function. The gbest is defined by selecting the best pbest.

{ }

{ }0 1

0 1

( ) , ,...,

( ( )) min (y (t),c (y (t)),...,c (y (t))

s

s

y t y y y

cf y t cf f f

∗

∗

∈

= (15)

where, ( )y t∗

is gbest at time step t and s denotes the size of the swarm. If the optimization problem has more than one dimension, the position of particle i is updated by the following relation:

,d ,d 1 1, , ,

2 2, ,

( 1) ( ) ( )( ( ) ( ))

( )( ( ) ( ))

i in i d i d i d

d d i d

v t w v t c r t y t z t

c r t y t x t∗

+ = + − +

− (16)

where, win is the inertia weight, c1 and c2 are acceleration constants, 1, ( ) (0,1)dr t R∼ , and 2, ( ) (0,1)dr t R∼ .This updating process is repeated until a specified number of iterations is exceeded or velocity updates are close to zero.

III. THE PROPOSED METHOD The aim is the optimal selection of width and shape of the

analyzing window which gives the best time as well as frequency resolutions. The method can be applied to any kind of window such as Gaussian or hyperbolic windows.

A. Gaussian Window

To improve the energy concentration of Gaussian S-transform, an additional parameter u is introduced which controls the width of analyzing window. So, (1) is modified, as follow [13]-[14]:

( ) ( )22

( , ) ( ) exp exp 222

u uuk

f f tS f x t ift dt

ττ π

π

+∞

−∞

⎡ ⎤⎛ ⎞− −⎢ ⎥⎜ ⎟= −⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦∫

(17)

When, 1u < the window is wider in the time domain and narrower in the frequency domain in comparison to original window (i.e., 1u = ) . For 1u > , the window is narrower in the time domain and wider in the frequency domain. The opposite is true when 0 1f< < . In (17) the standard deviation

is 1( )u

f fσ = instead of 1( )f fσ = in (1).

B. Hyperbolic Window

As it can be seen in (6) and (7), the shape of hyperbolic window is defined by forward tapper, backward taper and positive curvature parameters. By optimal selection of these parameters the best energy concentration and resolution is achieved. Totally, if two windows have the same width in the time domain, the symmetrical window (i.e., Gaussian window) provides a better frequency resolution compared to an

asymmetrical window (i.e., a hyperbolic window). The degree of asymmetry of hyperbolic window varies. At low frequencies, where the window is wide and frequency resolution is good, hyperbolic window is the most asymmetrical. But at high frequencies, where the window is narrow and time resolution is less critical, its shape converges to a symmetrical window [19]-[20].

C. Concentration Measure

The optimization can be performed with respect to time or frequency. For signals with time varying amplitude, information on signal amplitude is destroyed when integration is performed over frequency in the time-frequency domain. But, integration over the time retains information of the spectral content. Hence, optimization is performed in the frequency domain [13]. The optimal selection of window parameters is performed by the concentration measure proposed in [9].

For Gaussian window the concentration measure is as below:

( , ) ( ,f, )q

uG kCM f u s u dτ τ

+∞

−∞= ∫ (18)

where, 0 0.25q< ≤ . The best value for u which gives the most energy concentration will offer the minimum value of CM. So, the optimization problem is to find a value for u which minimizes the cost function (18).

In the case of hyperbolic S-transform, the concentration measure is as follows:

, ,( , , , ) ( ,f, , , )f bq

h f b f bkCM f s dγ γ λγ γ λ τ γ γ λ τ+∞

−∞= ∫ (19)

In the optimization problem for hyperbolic S-transform, optimal values for fγ , bγ and λ should be discovered which minimize the cost function (19). The optimal values of window parameters can be selected using particle swarm optimization algorithm. The procedure can be summarized as follows:

1- Assign random values to parameters for each particle, considering the number of parameters of analyzing window.

2- Compute S-transform for each particle.

3- Compute the cost (concentration measure) for each particle using cost function (18) or (19) according to the employed window.

4- Define gbest and pbests.

5- Update parameter values for particles considering gbest and pbests.

6- Repeat steps two to five until the stopping criterion is satisfied.

7- Introduce parameters of gbest as optimal parameters and the respect S-transform as optimal S-transform.

Since the method is independent of the type of window, it can be used for optimal selection of any window with arbitrary shape.

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IV. EVALUATION OF THE METHOD An application of the S-transform is the detection of

incipient faults in transformers, suggested in [8] and [19]. In this method, the impulse voltage waveform is applied to transformer windings using an impulse generator. Then, a digital oscilloscope is employed for measuring the input current as shown in Fig.1. Afterward, the S-transform is applied to the measured current and then, faulty and healthy windings are distinguished, accordingly. More details are available in [8] and [19]. Also, the suggested method can be applied to recently developed TT-transform based fault detection method proposed in [23].

To appraise the effectiveness of the suggested method, it has been implemented in Matlab environment. Two sample signals have been analyzed using the standard S-transform (u=1), the optimal Gaussian and optimal hyperbolic S-transforms and the results have been compared.

A. Case study 1 The signal, shown in Fig. 2 has been analyzed by standard,

Gaussian and hyperbolic S-transform. The analytical expression of this signal is as follows:

h(1:70)=cos(2pi (1:70) 7/256)h(71:128)=0h(129:256)=cos(2pi (1:128) 25/256)h(30:60)=h(30:60)+0.5cos(2pi (30:60) 65/256)

× ×

× ×× ×

This signal contains three frequency components of 7 Hz, 25 Hz and 65 Hz. The analyzed signal, the time-frequency contours for the non-optimal standard S-transform, the time-frequency representation of the optimal Gaussian S-transform and the time-frequency contours for the optimal hyperbolic S-transform have been presented in Figs.2-5, respectively. As it can be seen in Fig.3, although all three components have been detected, but there are smears and additional spikes in the time-frequency plane. In the case of optimal Gaussian window, the smears and spikes have been omitted, as it is obvious in Fig.4. According to Fig.5, similar to the optimized Gaussian window, the optimal hyperbolic window has a reasonable performance and the three frequency components have been detected properly with acceptable resolution.

Fig.6 shows the results of the PSO algorithm during the selection of parameter u for the optimal Gaussian window. The number of the population and maximum iteration are 300 and 15, respectively. PSO algorithm has converged to u=0.9029. Fig.7 shows the values of the cost function (concentration measure) during optimization. Values for forward and backward tappers during hyperbolic window optimization have

Fig 1. Test setup for detection of incipient fault in transformers

Fig 2. Time domain analyzed signal

Fig 3. Time-frequency representation for Standard S-transform

Fig 4. Time-frequency representation for optimal Gaussian window

Fig 5. Time-frequency contours for optimal hyperbolic window

been shown in Fig.8. The number of population is 600, and the maximum iteration of 15 has been selected in this case. The optimal values for forward tapper and backward tapper are 1.3591 and 1.3724, respectively. Variations of the positive curvature parameter and concentration measure during optimization process have been depicted in Fig.9 and Fig.10, respectively. The optimal value for the positive curvature is 27.8962 and the concentration measure converges to 83.76.

Fig 6. The optimal selection of parameter u for the Gaussian window

Fig 7. Concentration measure variations during the selection of parameter

u for the Gaussian window

Fig 8. The optimal selection of forward tapper and backward tapper for the

hyperbolic window

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Fig 9. The optimal selection of the positive curvature for the hyperbolic

window

Fig 10. Variations of the concentration measure during the selection of the

hyperbolic widow parameters

B. Case study 2

To conduct more investigations, a signal with continuous frequency variations has been analyzed. The analytic expression of this signal is as follows:

h(1:256)=cos(2pi (10+(0:255)/7) (0:255)/256)+ cos(2pi (256/2.8-(0:255)/6) (0:255)/256)

× ×× ×

Fig.11 shows the time domain representation of the analyzed signal. Fig.12 depicts the time-frequency contours for the non-optimal standard window. It is seen that this window does not provide proper frequency resolution, and that components are smeared in frequency. The result for the optimal Gaussian window has been presented in Fig.13. It is safe to say that the frequency resolution has been improved by the optimal selection of the window. The time-frequency representation for the hyperbolic S-transform has been presented in Fig.14. It is found that the optimized hyperbolic window has a good resolution similar to the optimal Gaussian window. Fig.15 shows variations of Gaussian window parameter during optimization using the PSO algorithm. The number of population of 250 has been selected and the maximum

Fig 11. Time domain representation of the analyzed signal

Fig 12. Time-frequency contours for the Standard S-transform

Fig 13. Time-frequency representation for the optimal Gaussian window

Fig 14. Time-frequency contours for the optimal hyperbolic window

iteration is 15. The optimal value of the Gaussian window parameter u is 0.7904 in this case. Variations of concentration measure have been depicted in Fig.16. The concentration measure for optimal parameter is 99.3626, as it can be seen in Fig.16.

Values for the forward tapper as well as the backward tapper during the optimal parameter selection for the hyperbolic window have been shown in Fig.17. The number of population is 600 and maximum iteration has been 15 in this case. The optimal values for forward and backward tappers are 2.4109 and 2.3085, respectively. The optimal value for positive curvature has converged to 28.5203, as it can be seen in Fig.18. Fig.19 shows the variations of concentration measure during optimization. The cost for the optimal parameters is 100.7.

The parameters for the three types of windows as well as the corresponding time-frequency concentration measure for the two presented cases have been summarized in Table I. It can be said that the concentration measure for the Gaussian and hyperbolic windows are close to each other, and are different from the concentration measure of the standard window. Furthermore, it is found that the optimal selection of parameters has more influence on the energy concentration of

Fig 15. The optimal selection of parameter u for the Gaussian window

Fig 16. The cost function during the selection of parameter u for the

Gaussian window

Fig 17. The optimal selection of the forward tapper and the backward

tapper for the hyperbolic window

Fig 18. The optimal selection of the positive curvature for the hyperbolic

window

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Fig 19. The cost during the selection of the hyperbolic widow parameters

TABLE I. Analogy of standard and optimal windows Case study 1 Case study 2

Window parameters

Concentration measure

Window parameters

Concentration measure

Standard Stransform u=1 85.2848 u=1 121.4791

Optimal Gaussian

S-transform u=0.9029 83.3231 u=0.7904 99.3626

Optimal hyperbolic S-transform

1.3591

1.3724

27.8962

f

b

γ

γ

λ

=

=

=

83.76 2.30

2

8

.4

5

109

28.5203

f

b

γ

γ

λ

=

=

=

100.7

the time-frequency representation for the second case, which has more frequency variation during the time.

According to the simulation results it is found that the optimal selection of the window can improve the time-frequency resolution of the S-transform. The proposed method is especially helpful in the case of hyperbolic windows where the search space has three dimensions (three parameters) and it is not easy to find the optimal values of these parameters, simultaneously.

V. CONCLUSION To address the weak energy concentration problem of the

S-transform, this paper suggested a PSO based method for the optimal selection of the analyzing window for the generalized S-transform. The S-transform has been applied to the analyzed signal and the energy concentration index has been computed. The best parameters for the analyzing window have been selected by minimizing the energy concentration index. Then, the optimized S-transform has been employed for analyzing the signal. The effectiveness of the optimized and the standard S-transforms in the detection of frequency components of signals were compared. According to the simulation results, the optimized windows provide a better time-frequency resolution in comparison to the standard window due to the improvement in the energy concentration. The method is more helpful for hyperbolic windows which have three parameters and for analyzing signals in which frequency changes continually.

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