SIViP (2014) 8:533–541DOI 10.1007/s11760-013-0551-1

ORIGINAL PAPER

The S-transform using a new window to improve frequencyand time resolutions

Kamran Kazemi · Mohammadreza Amirian ·Mohammad Javad Dehghani

Received: 6 May 2012 / Revised: 26 August 2013 / Accepted: 26 August 2013 / Published online: 20 September 2013© Springer-Verlag London 2013

Abstract The S-transform presents arbitrary time seriesas localized invertible time–frequency spectra. This trans-formation improves the short-time Fourier transform andthe wavelet transform by merging the multiresolution andfrequency-dependent analysis properties of wavelet trans-form with the absolute phase retaining of Fourier trans-form. The generalized S-transform utilizes a combination of aFourier transform kernel and a scalable-sliding window. Thecommon S-transform applies a Gaussian window to provideappropriate time and frequency resolution and minimizesthe product of these resolutions. However, the Gaussian S-transform is unable to obtain uniform time and frequency res-olution for all frequency components. In this paper, a novelwindow based on the t student distribution is proposed forthe S-transform to achieve a more uniform resolution. Sim-ulation results show that the S-transform with the proposedwindow provides in comparison with the Gaussian windowa more uniform resolution for the entire time and frequencyrange. The result is suitable for applications such as spectrumsensing.

Keywords S-transform · Wavelet transform · Short-timeFourier transform · Spectral localization · t student

1 Introduction

The spectral localization methods utilizing standard Fouriertransform are unable to provide adequate time resolutionfor event recognition in nonstationary time series. Thereare numerous applications that challenge nonstationary time

K. Kazemi (B) · M. Amirian · M. J. DehghaniDepartment of Electrical and Electronics Engineering,Shiraz University of Technology, Shiraz, Irane-mail: kazemi@sutech.ac.ir

series such as electrocardiograms and human speech record-ing. In order to overcome the shortcomings of Fourier trans-form and provide appropriate information on time resolu-tion, the short-time Fourier transform (STFT) was proposedby Gabor [1]. STFT applies a sliding window to the signaland computes the Fourier transform of the windowed part.Since STFT applies a fixed-width window, it cannot track thesignal dynamics properly for nonstationary signals. In orderto overcome the fixed window in STFT, the wavelet trans-form (WT) [2,3] has been proposed, which is based on smallwavelets with limited duration. The scaled version waveletsallow analyzing the signal in different scales. Therefore, itis an appropriate transform to deal with nonstationary timeseries. However, it produces time-scale plots that are unsuit-able for intuitive visual analysis [4]. Moreover, it is sensitiveto noise.

Stockwell et al. [5] proposed a superior method for time–frequency analysis of time series known as S-transform.Conceptually, it is a combination of the STFT and WT.It has frequency-dependent resolutions similar to waveletand retains the absolute phase of time series componentlike STFT. Furthermore, it provides time–frequency rep-resentation of the signal so that the components of thesignal can be isolated and processed independently in thetime–frequency plane. The S-transform has found manyapplications in different fields such as geophysics [6],oceanography [7], atmospheric physics [8], medicine [9,10],hydrogeology [11], and mechanical engineering [12] for theevent recognition and interpretation of time series. Further-more, the generalized S-transform is invertible and can beused for band-limited filtering.

The original S-transform utilizes a Gaussian windowwhose amplitude and width scale linearly and inversely withthe frequency, respectively [13]. The window scaling overfrequency dimension provides frequency-dependent resolu-

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tion for the S-transform. However, the scaling variations aretoo intense to obtain a uniform frequency resolution. Thefrequency resolution of the S-transform decreases on high-frequency components. The adjustable parameter in the S-transform with Gaussian window can be used to alter timeand frequency resolutions. By increasing the adjustable para-meter to obtain more frequency resolution in high-frequencycomponents, the time resolution is decreased especially inlow-frequency components.

Thus, in order to provide a more uniform resolution in allfrequency components and time domains at the same time,a t student distribution [14] based window in the frequencydomain is proposed for the S-transform. The width and ampli-tude of the proposed window change with the square root offrequency unlike the Gaussian window. This softer variationin the window provides a more uniform resolution in all timeand frequency ranges and enhances noise immunity in high-frequency components for the S-transform.

The rest of the paper is organized as follows: in Sect. 2,a brief review on generalized S-transform is given. The S-transform with the proposed window is introduced in Sect. 3.Detailed simulation results for time–frequency represen-tation of time series based on the proposed method andother time–frequency representation methods are providedin Sect. 4. Concluding remarks and discussion are given inSect. 5.

2 The generalized S-transform

The STFT is the most commonly used method for time–frequency representation introduced by Gabor [1]. The STFTof a time series h(t) is as follows:

STFT(τ, f ) =+∞∫

−∞h(t)w(τ − t) exp(−2π i f t)dt (1)

where f and t are frequency and time variables, respectively,and τ determines the position of the sliding window w in thetime domain. The STFT cannot provide appropriate time andfrequency resolutions in the whole frequency range becauseof fixed shape and width window. This disadvantage of STFToccurs as a consequence of fading high-frequency signals ina very short time, while low-frequency components remainlonger in time. Accordingly, although applying a narrowwindow can provide good time resolution of high-frequencycomponents, frequency resolution in low-frequency compo-nents is missed as a result and vice versa.

In order to overcome these shortcomings, the S-transformwhich is a generalization of STFT was proposed by Stockwell[5]. The S-transform applies a window in which its amplitudeand width change by frequency. The generalized S-transformof continues time series h(t) is as follows:

S(τ, f, p) =+∞∫

−∞h(t)w(τ − t, f, p) exp(−2π i f t)dt (2)

where f and t represent frequency and time, respectively,and τ is an additional time variable with the same unit ast , which determines the position of window w over time.The shape of modulating window w is function of f anda set of parameters p to define the window. The general-ized S-transform and the generalized window are denotedby S and w, respectively. The kernel of the S-transform isconsisting of product of w with the kernel of the Fouriertransform with the same integral transformation.

According to the convolution theory, the S-transform canbe written as a function of Fourier transform of time series,as follows:

S(τ, f, p) =+∞∫

−∞H(α + f )W (α, f, p) exp(2π iτα)dα

(3)

where H and W are the Fourier transform of h and w, respec-tively. The additional frequency variable α is defined with thesame dimension as f, since w and W are both functions off .

Typically, Gaussian function is applied as the modulat-ing window in the S-transform [5]. The Gaussian functionminimizes the product of time and frequency resolution. TheGaussian window in the frequency domain WGS is as follows:

WGS (α, f, {γGS}) = exp

(−2π2α2γ 2

GS

f 2

)(4)

where γGS is the adjustable parameter, which has beendefined to tune the time and frequency resolutions of theS-transform. It presents the number of Fourier sinusoidalperiods within one standard deviation of the Gaussian win-dow. Raising γGS increases the frequency resolution and con-sequently decreases the time resolution of the S-transformand vice versa. The S-transform with Gaussian window isobtained by substitution of WGS for W in (3).

3 S-transform with proposed window

Small value of γGS improves the resolution of the S-transformin time direction; however, as a result of the uncertaintyprinciple, loss of resolution in the frequency direction onS-transform is seen. On the other hand, resolution of S-transform does not distribute uniformly in all frequencies,i.e., the S-transform has a lower frequency resolution inhigh-frequency components compared with low frequencies.Thus, the adjustable parameter γGS cannot adjust time andfrequency resolutions simultaneously because of different

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frequency resolutions in the whole frequency range. Thus,low-time resolution for low-frequency components is a resultof tuning the parameter γGS to obtain a suitable frequencyresolution in high frequencies.

Uniform time and frequency resolutions are required inapplications such as spectrum sensing in cognitive radio [15]in an arbitrary frequency band. Because of the quick andintense variations of the Gaussian window over frequency,the S-transfrom with Gaussian window is not appropriate insuch applications. Thus, a window with gradual changes overfrequency may improve the S-transform and make it suitablefor applications such as spectrum sensing.

T student distribution [14] has more gradual changes overfrequency in comparison with sharp changes of Gaussian dis-tribution. Thus, in this paper, we propose the t student basedwindow Wts over x with the freedom degree of ν(F(x |ν) )

for the S-transform as follows:

Wts(x, ν, p)= F(x |ν) = �(

ν+12

)�

(ν2

) 1√νπ

1(1+ x2

v

) ν+12

(5)

where �(.) is the Gamma function. In order to obtain a moreuniform time and frequency resolutions, we applied a win-dow with t student distribution over α2/ f with one degreeof freedom F(α2/ f |1). Furthermore, an adjustable parame-ter γPr is added to the proposed window to tune the timeand frequency resolutions of the S-transform. The proposedwindow in the frequency domain is expressed as follows byneglecting the constant coefficients:

WPr (α, f, {γPr}) = 1

1 +(

α2

f γPr

)2 (6)

Thus, the S-transform with the proposed window can be writ-ten by substituting (6) in (3), as follows:

S (τ, f, {γPr}) =+∞∫

−∞H(α + f )

1

1 +(

α2

f γPr

)2

× exp(+2π iατ)dα (7)

The amplitude of the proposed window scales with the squareroot of frequency unlike other windows previously used inthe S-transform in which the amplitude is scaled linearly withfrequency [13]. The width of the all windows scales with theinverse of their amplitude. Variation of the proposed windowwith the square root of frequency instead of linear scalingprovides more uniform spectral representation because ofmore gradual changes in the window.

Figure 1 shows the proposed window in the time and fre-quency domains according to the different adjustable para-meters γPr and different frequencies. As shown in Fig. 1a,rising γPr increases the width of the window in the timedomain and leads to a narrow window in the frequency

domain (Fig. 1c) which is equivalent to more resolution in thefrequency domain. The variation of the proposed window inthe time domain according to different frequencies with fixedγPr is illustrated in Fig. 1b. Raising the frequency parameterf is resulting in a narrow window in the time domain andwide window in the frequency domain Fig. 1d. Consequently,a lower frequency resolution is obtained in high frequencies.

As shown in ‘Example 1’ (Fig. 2), the time and fre-quency resolutions of the S-transform with proposed win-dow are more uniform than the S-transform with Gaussianwindow. Moreover, the proposed window can be used inthe time–time transform (TT transform) [16] deriving fromthe S-transform. It maps the primary time series into a setof secondary time-limited, local, constituent time series.According to the invertibility of the TT transform, it is usedfor filtering and improving the signal to noise in the timedomain [17].

4 Simulation results

In this section, the S-transform using the proposed win-dow is compared with other localized windows such as theGaussian, hyperbolic [13], and bi-Gaussian [4] windows.Furthermore, the proposed method is compared with clas-sical tools: reassigned spectrogram [18], S-method (pro-posed by Stankovic [19]), pseudo Wigner-Ville distribution(PWVD) [20], and smoothed pseudo Wigner-Ville distribu-tion (smoothed PWVD) [20].

Example 1 In this example, we consider a signal with con-stant amplitude, and with a linear frequency modulation vary-ing from 0 to 0.4 in normalized frequency (ratio of the fre-quency in Hertz to the sampling frequency, with respect tothe Shannon sampling theorem). Figure 2 illustrates the timeseries in the time domain (Fig. 2a) and its S-transform withdifferent windows: Gaussian, hyperbolic, bi-Gaussian, andthe proposed window. The adjustable parameters of the S-transform with hyperbolic and bi-Gaussian windows are thesame as proposed in [13] and [4], respectively. As can beseen, the S-transform with Gaussian (Fig. 2b), hyperbolic(Fig. 2c), and bi-Gaussian (Fig. 2d) windows cannot provideuniform frequency resolution in all the frequency ranges, andfrequency resolution declines significantly in high-frequencycomponents. However, S-transform with the proposed win-dow (Fig. 2e) provides an approximately uniform resolutionfor all the frequency ranges.

Example 2 Figure 3 illustrates a time series with low-,middle-, and high-frequency components. The time seriesis similar, but not exactly the same as that used in [4,13].As shown, the frequency resolution decreases in middle andespecially high-frequency bursts. On the other hand, fre-

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Fig. 1 a The proposed window, wPr, in time domain at f =10 for γPr = 0.5 (solid line), γPr = 1 (dashed line) and γPr = 1.5 (dot-ted line). b The proposed window, wPr, in time domain at f = 7 (solidline), f = 10 (dashed line) and f = 15 (dotted line) for γPr = 1. c The

proposed window, wPr, in frequency domain at f = 10 for γPr = 0.5(solid line), γPr = 1 (dashed line) and γPr = 1.5 (dotted line). d Theproposed window, wPr, in frequency domain at f = 7 (solid line),f = 10 (dashed line) and f = 15 (dotted line) for γPr = 1

quency resolution is more uniform in S-transform with theproposed window.

The lack of frequency resolution in middle- and high-frequency components (Fig. 3b) of the S-transform withthe Gaussian window is result of linear scaling of theGaussian window with f in the frequency domain. Increas-ing the width of the window leads to more noise effects onmiddle- and high-frequency ranges. Figures 3d, e show the S-transform with Gaussian and proposed window, respectively,for the same time series which is corrupted by additive whiteGaussian noise (AWGN) with zero mean and signal-to-noiseratio (SNR) of 10 dB. As can be seen, noise disturbance ismore effective on middle- and high-frequency components ofthe S-transform with Gaussian window in comparison withthe proposed window.

Example 3 To obtain a thorough comprehensive compari-son of the S-transform using the proposed window withother time–frequency representation methods, we consid-ered a multi-component signal with nonlinear chirps. Thefirst test signal in this example consists of two logarith-mic chirps. The frequency of the first chirp increases intime, while the frequency decreases in the other chirpsignal. These signals can compare tracing ability of dif-

ferent approaches to follow violent and slow frequencychanges. Moreover, multi-component and chirp signalsclarify the cross terms issue in time–frequency meth-ods. Figure 4 shows the comparison results of apply-ing the S-transform with the proposed window with reas-signed spectrogram, S-method, pseudo Wigner-Ville distrib-ution (PWVD), smoothed pseudo Wigner-Ville distribution(smoothed PWVD), and S-transform with Gaussian window.

The reassigned spectrogram with window length of 128and 120 samples overlap (Fig. 4b) provides high-frequencyresolution in low-frequency components. However, it haspoor time resolution for violent changes in high frequencies.As shown in Fig. 4c, where the rate of frequency changeis low, the S-method provides better time–frequency resolu-tion in comparison with sharp frequency variations. PWVDdistribution provides a significant time–frequency resolu-tion (Fig. 4d). However, its drawback is that it producescross terms. They grow when the frequency components areclose. The smoothed PWVD and S-method apply windows toreduce the cross terms of PWVD. Using window in smoothedPWVD declines the cross terms of PWVD. However, thesame shortcoming of time resolution of the spectrogram forviolent frequency changes appears in smoothed PWVD. On

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Fig. 2 a Sinusoidal signal with linearly increasing frequency, b S-transform with Gaussian window, c S-transform with hyperbolic window, dS-transform with bi-Gaussian window, e S-transform with the proposed window

the other hand, the S-method performs as well as smoothedPWVD.

As shown in Fig. 4, the S-transform with Gaussian win-dow provides appropriate time resolution for all frequencyranges. However, the frequency resolution of this methoddecreases for high frequencies. In comparison with apply-ing Gaussian window in the S-transform, using the proposedwindow (Fig. 4g) presents a more uniform representation.On the other hand, in case of inverted chirps, i.e., shallowat high frequencies and steep at low frequencies, the perfor-mance of the S-transform is declined specially in the caseof S-transform with Gaussian window. However, in compar-

ison with applying Gaussian window, the S-transform withproposed window improves the results by adjusting the para-meter γPr (Fig. 5).

In order to statistically evaluate the results obtained inthis example, the instantaneous bandwidths of the estimatedfrequencies using different methods were compared for therising logarithmic chirp signal. Figure 6 shows the instanta-neous bandwidth over each estimated frequency using differ-ent methods. The instantaneous bandwidth were calculatedat each time based on the range of frequencies that corre-spond to coefficients no less than −3 dB in comparison withthe maximum coefficient. The pseudo Wigner-Ville distribu-

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Fig. 3 a Sinusoidal signal with three frequency components, b S-transform with Gaussian window, c S-transform with proposed window,d the S-transform with Gaussian window of the same Sinusoidal signal

which is corrupted by additive white Gaussian noise with zero meanand a SNR of 10 dB, e the S-transform with the proposed window ofthe same noisy signal

tion is omitted in this figure, because of high amount of crossterms. As shown, by increasing the frequency and changerate of the chirp signal by time, the instantaneous bandwidthis increased. However, the proposed method provides lessinstantaneous bandwidth compared with the other appliedtime–frequency methods.

5 Conclusion

In this paper, we have proposed a new window for the S-transform to make the time and frequency resolutions more

uniform than the former windows applied in the S-transform.The amplitude of the previously applied widows in the S-transform scales linearly with the frequency. Therefore, thewidth of these windows in frequency domain increases lin-early with frequency. This increase in the window widthdeclines the frequency resolution of high-frequency compo-nents.

The width of the proposed window in the frequencydomain is increased with the square root of frequency. Thisrelatively gradual variation of the proposed window improvesthe time resolution of low-frequency components as wellas the frequency resolution of high frequencies. Simulation

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Fig. 4 a Time series of two logarithmic chirps, b reassigned spectrogram, c S-method distribution d pseudo Wigner-Ville distribution, e smoothedpseudo Wigner-Ville distribution, f S-transform with Gaussian window, g S-transform with the proposed window

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Fig. 5 a Time series of two convex quadratic chirps, b S-transform with Gaussian window (γGS = 10), c S-transform with the proposed window(γPr = 3)

Fig. 6 Comparison of different time–frequency methods in estimatingthe instantaneous frequency and bandwidth of the rising logarithmicchirp signal

results show that the proposed window makes the resolutionsmore uniform than applying other windows in the time andfrequency domains. The result is suitable for some applica-tions such as spectrum sensing according to more uniformresolutions in the time–frequency plane.

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