# Application of S-transform to signal analysis

Application of S-transform to signal analysis This paper presents opportunities of signal analysis using S-transform. Images of the module and the angle of S-transform show the results. These images may be used as models to compare with unknown signals and to detect patterns and anomalies, even in very sophisticated signals. Keywords: analysis of signal; S-transform. where and t are time and f is . Inverse S-transform is defined as follows: h( t ) = S1 D ( , f ) d e i 2 ft df - - (2) Discrete S-transform is defined as follows: N -1 i 2 ( n+m ) k - 2 m i 2 mj 1 N -1 - n 2 N S1 D jT , e n e N = h[ kT ] e NT m-0 N k =0 Introduction S-transform was defined by Stockwell [1, 2]. It is used by many others to investigate the quality of energy wire [3], data from GPR [4, 5], etc. Its characteristics may be also used to examine other signals, including medical ones, such as electrocardiogram (ECG) [6, 7]. Utilizing S-transform, we can determine some of the signals' properties. It is especially well suited because it allows to pinpoint the property in terms of time and . Other interesting and useful characteristics of S-transform are independence of response's amplitude to of signals, progressive resolution, and information about the signal phase. This paper presents definition and various examples of computing S-transform. The paper also covers a concept of using S-transform to investigate anomalies in signals. (3) where h[kT](k=0, 1, 2, ..., N­1) is a series of discrete s of continues function h(t), T is the sampling duration, and N is the number of samples. Inverse discrete S-transform is defined as follows: h[ kT ] = N -1 1 n i 2 Nnk S1 D jT , e NT N2 n-0 (4) Results In this chapter, we present images that were created as a result of S-transform computing of following signals: ­ Sin (Figures 1­5) ­ Sin with modulation (Figures 6­10) ­ Saw signal (Figures 11­15) ­ Square signal (Figures 16­20) ­ ECG (Figures 21­25) ­ ECG with distortion (Figures 26­30) S-transform has both imaginary and real parts; therefore, the plots show module and angle as a function of time and . Hue saturation (HSV) color pallet was used to describe angles, which allows angles 0° and 360° to have the same color. Plots of the above-mentioned S-transforms show that an analysis of signal in terms of and time is possible. For various signals, we got completely different S-transforms. Anomalies as distortion or weak signal may be detected and interpreted. Methodology To analyze the signal, we shall use S-transform [6]. *Corresponding author: Piotr Szymczyk, The Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, Kraków 30-059, Poland, E-mail: piotr.szymczyk@agh.edu.pl. http://orcid.org/0000-0001-6675-0101 Magdalena Szymczyk: The Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, Kraków, Poland 224Szymczyk and Szymczyk: Application of S-transform to signal analysis 1.0 0.8 0.6 0.4 Amplitude 0.2 0 ­0.2 ­0.4 ­0.6 ­0.8 ­1.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 1:Sin. Figure 4:Flat plot of S-transform's angle for sin. 4 2 Angle 0 ­2 ­4 600 1000 500 400 300 500 200 100 0 0 Time Figure 2:Flat plot of S-transform's module for sin. Figure 5:Spatial plot of S-transform's angle for sin. 1 0.8 0.6 0.4 0.2 0 600 1000 500 400 300 500 200 100 0 0 Time 1500 Amplitude 0.2 0 ­0.2 ­0.4 ­0.6 ­0.8 ­1.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 3:Spatial plot of S-transform's module for sin. Figure 6:Sin with modulation. Szymczyk and Szymczyk: Application of S-transform to signal analysis225 4 2 Angle 0 ­2 1500 ­4 600 1000 500 300 400 500 200 100 0 0 Time Figure 7:Flat plot of S-transform's module for sin with modulation. Figure 10:Spatial plot of S-transform's angle for sin with modulation. 1 0.8 0.6 0.4 0.2 0 600 1000 500 300 400 200 500 100 0 0 Time 1500 0.8 0.7 Amplitude 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 8:Spatial plot of S-transform's module for sin with modulation. Figure 11:Saw signal. Figure 9:Flat plot of S-transform's angle for sin with modulation. Figure 12:Flat plot of S-transform's module for saw signal. 226Szymczyk and Szymczyk: Application of S-transform to signal analysis 0.5 0.4 0.3 0.2 0.1 0 600 1000 500 300 400 500 200 100 0 0 Time 1500 0.8 0.7 Amplitude 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 13:Spatial plot of S-transform's module for saw signal. Figure 16:Square signal. Figure 14:Flat plot of S-transform's angle for saw signal. Figure 17:Flat plot of S-transform's module for square signal. 4 2 Angle 0 ­2 ­4 600 1000 500 400 300 500 200 100 0 0 Time 0.8 0.6 0.4 0.2 0 600 1000 500 400 300 500 200 100 0 0 Time Figure 15:Spatial plot of S-transform's angle for saw signal. Figure 18:Spatial plot of S-transform's module for square signal. Szymczyk and Szymczyk: Application of S-transform to signal analysis227 Figure 19:Flat plot of S-transform's angle for square signal. Figure 22:Flat plot of S-transform's module for ECG. 4 2 Angle 0 ­2 ­4 600 1000 500 300 400 500 200 100 0 0 Time 0.4 0.3 0.2 0.1 0 600 800 600 500 300 400 200 400 200 100 0 0 Time Figure 20:Spatial plot of S-transform's angle for square signal. Figure 23:Spatial plot of S-transform's module for ECG. 2.6 2.4 2.2 2.0 Amplitude 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 100 200 300 400 500 600 Time 700 800 900 1000 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 21:ECG. Figure 24:Flat plot of S-transform's angle for ECG. 228Szymczyk and Szymczyk: Application of S-transform to signal analysis 4 2 Angle 0.4 0.3 0.2 0.1 0 600 800 500 300 400 400 200 100 200 0 0 600 Time 0 ­2 ­4 600 800 600 500 400 400 300 200 100 0 0 200 Time Figure 25:Spatial plot of S-transform's angle for ECG. Figure 28:Spatial plot of S-transform's module for ECG with distortion. 2.6 2.4 2.2 2.0 Amplitude 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 100 200 300 400 500 600 Time 700 800 900 1000 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 26:ECG with distortion. Figure 29:Flat plot of S-transform's angle for ECG with distortion. 4 2 Angle 0 ­2 800 ­4 600 600 500 400 400 300 200 200 100 0 0 Time Figure 27:Flat plot of S-transform's module for ECG with distortion. Figure 30:Spatial plot of S-transform's angle for ECG with distortion. Szymczyk and Szymczyk: Application of S-transform to signal analysis229 Conclusions The paper presents S-transform, which may be used for the analysis of various signals, including nonstationary ones. The analysis may be conducted on images showing module and angle (graphical representation of complex ). A database of known and anticipated signals may be created and used for image recognition, comparison (e.g. using artificial neural networks), and detection of unknown signals or anomalies. The possibility of using S-transform to examine ECG signal was also presented, although full implementation of this method requires further studies. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: None declared. Employment or leadership: None declared. Honorarium: None declared. Competing interests: The funding organization(s) played no role in the study design; in the collection, analysis, and interpretation of data; in the writing of the report; or in the decision to submit the report for publication. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bio-Algorithms and Med-Systems de Gruyter

# Application of S-transform to signal analysis

, Volume 11 (4) – Dec 1, 2015
7 pages

/lp/de-gruyter/application-of-s-transform-to-signal-analysis-bmAQhGPvPi
Publisher
de Gruyter
ISSN
1895-9091
eISSN
1896-530X
DOI
10.1515/bams-2015-0028
Publisher site
See Article on Publisher Site

### Abstract

This paper presents opportunities of signal analysis using S-transform. Images of the module and the angle of S-transform show the results. These images may be used as models to compare with unknown signals and to detect patterns and anomalies, even in very sophisticated signals. Keywords: analysis of signal; S-transform. where and t are time and f is . Inverse S-transform is defined as follows: h( t ) = S1 D ( , f ) d e i 2 ft df - - (2) Discrete S-transform is defined as follows: N -1 i 2 ( n+m ) k - 2 m i 2 mj 1 N -1 - n 2 N S1 D jT , e n e N = h[ kT ] e NT m-0 N k =0 Introduction S-transform was defined by Stockwell [1, 2]. It is used by many others to investigate the quality of energy wire [3], data from GPR [4, 5], etc. Its characteristics may be also used to examine other signals, including medical ones, such as electrocardiogram (ECG) [6, 7]. Utilizing S-transform, we can determine some of the signals' properties. It is especially well suited because it allows to pinpoint the property in terms of time and . Other interesting and useful characteristics of S-transform are independence of response's amplitude to of signals, progressive resolution, and information about the signal phase. This paper presents definition and various examples of computing S-transform. The paper also covers a concept of using S-transform to investigate anomalies in signals. (3) where h[kT](k=0, 1, 2, ..., N­1) is a series of discrete s of continues function h(t), T is the sampling duration, and N is the number of samples. Inverse discrete S-transform is defined as follows: h[ kT ] = N -1 1 n i 2 Nnk S1 D jT , e NT N2 n-0 (4) Results In this chapter, we present images that were created as a result of S-transform computing of following signals: ­ Sin (Figures 1­5) ­ Sin with modulation (Figures 6­10) ­ Saw signal (Figures 11­15) ­ Square signal (Figures 16­20) ­ ECG (Figures 21­25) ­ ECG with distortion (Figures 26­30) S-transform has both imaginary and real parts; therefore, the plots show module and angle as a function of time and . Hue saturation (HSV) color pallet was used to describe angles, which allows angles 0° and 360° to have the same color. Plots of the above-mentioned S-transforms show that an analysis of signal in terms of and time is possible. For various signals, we got completely different S-transforms. Anomalies as distortion or weak signal may be detected and interpreted. Methodology To analyze the signal, we shall use S-transform [6]. *Corresponding author: Piotr Szymczyk, The Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, al. A. Mickiewicza 30, Kraków 30-059, Poland, E-mail: piotr.szymczyk@agh.edu.pl. http://orcid.org/0000-0001-6675-0101 Magdalena Szymczyk: The Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Automatics and Biomedical Engineering, AGH University of Science and Technology, Kraków, Poland 224Szymczyk and Szymczyk: Application of S-transform to signal analysis 1.0 0.8 0.6 0.4 Amplitude 0.2 0 ­0.2 ­0.4 ­0.6 ­0.8 ­1.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 1:Sin. Figure 4:Flat plot of S-transform's angle for sin. 4 2 Angle 0 ­2 ­4 600 1000 500 400 300 500 200 100 0 0 Time Figure 2:Flat plot of S-transform's module for sin. Figure 5:Spatial plot of S-transform's angle for sin. 1 0.8 0.6 0.4 0.2 0 600 1000 500 400 300 500 200 100 0 0 Time 1500 Amplitude 0.2 0 ­0.2 ­0.4 ­0.6 ­0.8 ­1.0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 3:Spatial plot of S-transform's module for sin. Figure 6:Sin with modulation. Szymczyk and Szymczyk: Application of S-transform to signal analysis225 4 2 Angle 0 ­2 1500 ­4 600 1000 500 300 400 500 200 100 0 0 Time Figure 7:Flat plot of S-transform's module for sin with modulation. Figure 10:Spatial plot of S-transform's angle for sin with modulation. 1 0.8 0.6 0.4 0.2 0 600 1000 500 300 400 200 500 100 0 0 Time 1500 0.8 0.7 Amplitude 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 8:Spatial plot of S-transform's module for sin with modulation. Figure 11:Saw signal. Figure 9:Flat plot of S-transform's angle for sin with modulation. Figure 12:Flat plot of S-transform's module for saw signal. 226Szymczyk and Szymczyk: Application of S-transform to signal analysis 0.5 0.4 0.3 0.2 0.1 0 600 1000 500 300 400 500 200 100 0 0 Time 1500 0.8 0.7 Amplitude 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time 0.1 Figure 13:Spatial plot of S-transform's module for saw signal. Figure 16:Square signal. Figure 14:Flat plot of S-transform's angle for saw signal. Figure 17:Flat plot of S-transform's module for square signal. 4 2 Angle 0 ­2 ­4 600 1000 500 400 300 500 200 100 0 0 Time 0.8 0.6 0.4 0.2 0 600 1000 500 400 300 500 200 100 0 0 Time Figure 15:Spatial plot of S-transform's angle for saw signal. Figure 18:Spatial plot of S-transform's module for square signal. Szymczyk and Szymczyk: Application of S-transform to signal analysis227 Figure 19:Flat plot of S-transform's angle for square signal. Figure 22:Flat plot of S-transform's module for ECG. 4 2 Angle 0 ­2 ­4 600 1000 500 300 400 500 200 100 0 0 Time 0.4 0.3 0.2 0.1 0 600 800 600 500 300 400 200 400 200 100 0 0 Time Figure 20:Spatial plot of S-transform's angle for square signal. Figure 23:Spatial plot of S-transform's module for ECG. 2.6 2.4 2.2 2.0 Amplitude 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 100 200 300 400 500 600 Time 700 800 900 1000 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 21:ECG. Figure 24:Flat plot of S-transform's angle for ECG. 228Szymczyk and Szymczyk: Application of S-transform to signal analysis 4 2 Angle 0.4 0.3 0.2 0.1 0 600 800 500 300 400 400 200 100 200 0 0 600 Time 0 ­2 ­4 600 800 600 500 400 400 300 200 100 0 0 200 Time Figure 25:Spatial plot of S-transform's angle for ECG. Figure 28:Spatial plot of S-transform's module for ECG with distortion. 2.6 2.4 2.2 2.0 Amplitude 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0 100 200 300 400 500 600 Time 700 800 900 1000 50 100 150 200 250 300 350 400 450 500 100 200 300 400 500 600 Time 700 800 900 1000 Figure 26:ECG with distortion. Figure 29:Flat plot of S-transform's angle for ECG with distortion. 4 2 Angle 0 ­2 800 ­4 600 600 500 400 400 300 200 200 100 0 0 Time Figure 27:Flat plot of S-transform's module for ECG with distortion. Figure 30:Spatial plot of S-transform's angle for ECG with distortion. Szymczyk and Szymczyk: Application of S-transform to signal analysis229 Conclusions The paper presents S-transform, which may be used for the analysis of various signals, including nonstationary ones. The analysis may be conducted on images showing module and angle (graphical representation of complex ). A database of known and anticipated signals may be created and used for image recognition, comparison (e.g. using artificial neural networks), and detection of unknown signals or anomalies. The possibility of using S-transform to examine ECG signal was also presented, although full implementation of this method requires further studies. Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission. Research funding: None declared. Employment or leadership: None declared. Honorarium: None declared. Competing interests: The funding organization(s) played no role in the study design; in the collection, analysis, and interpretation of data; in the writing of the report; or in the decision to submit the report for publication.

### Journal

Bio-Algorithms and Med-Systemsde Gruyter

Published: Dec 1, 2015

### References

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