1 Introduction

In the field of DSP the term ‘Transform’ is frequently used. The Transform is given this name because they transform one form of signal to another form of signal, so that it becomes easy to analyse or process than the original signal. Various types of transforms [1,2,3,4,5,6,7] and then different applications are demonstrated by different researchers like Fourier transform (FT) [8,9,10,11,12,13,14,15], fractional FT (FrFT) [16,17,18,19,20,21], short time FT (STFT) [22,23,24,25], Wavelet transform (WT) [26,27,28,29,30,31,32,33], fractional WT (FrWT) [27, 34, 35], S-transform (ST) [36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114] and fractional ST (FrST) [115,116,117,118,119,120,121]. In the FT technique, the analysed signal is either time or frequency. The time-domain signal exhibits information about the signal intensities and temporal evolution. For deterministic signals, analysis is usually based on instantaneous power spectrum or energy density spectrum. For random signals, the analysis tool depends on the auto-correlation functions and the power spectrum.

FT explores signal at various frequencies and their relative magnitudes. However, the main drawback of FT is that here no time resolution occurs, but frequency resolution occurs. That means FT can analyse the signal frequency response but cannot predict their location. To compensate this drawback, in the few decades several transforms techniques such as WT [122], Wigner transform [123], STFT [22, 23] and ST [38], and its fractional form came into existence. These techniques can represent a signal in time domain and frequency domain at the same time. However, WT cannot be used appropriately because infinite storage is required. Here no need to calculate dot-product between wavelet basis function and input signal. In WT only scale information can be expected with modulated phase information [38].

The drawback of Wigner transform is cross term, which is due to autocorrelation function. The cross terms produce noise or distortion in the signal analysis. The window size of fixed length is main drawback of STFT, hence it needs to be redefined. Therefore, STFT is not suitable for detecting the non-stationary signal where frequency is constantly changing according to instantaneous time. The STFT is usually not invertible in contrast of ST. Similarly, in the Gabor transform same problem occur due to fixed size Gaussian window. One case of non-stationary signal occur when the rotating machinery bearing breakdown in mechanical system then the defect signal received by the sensor is non-stationary in nature. Such signal contains time-bounded events. For the analysis of multicomponent and non-stationary signals, a time–frequency based method is traditional method based on either frequency or time domain. The time–frequency distributions give a 2-D representation that reflects the time-varying signals and shows the energy of distributed signal over 2-D time–frequency space. The ST gives a time–frequency representation in contrast with frequency representation by FT. At low frequency, multi resolution analysis is designed to obtain decent frequency and deficient resolution in time and vice versa at high frequency. ST and FrST is the most popular multi resolution tool that has been reviewed here. The concept behind the time–frequency joint representation is to distribute the signals into small parts followed by analysis of separate parts. In this way, the analysed signal gives more information about different frequencies [124,125,126,127,128,129].

The different time–frequency representation that exists in the literature is shown in Fig. 1. The first plot, in Fig. 1a, shows the way of representing the samples of a signal in time–frequency plane. It is clearly visible that the frequency information is absolutely lost in this method. Similarly, Fig. 1b shows the signal representation using Fourier transform. In this method frequency axis is divided uniformly but the time axis information is entirely lost. Figure 1c depicts the time–frequency representation for wavelet transform. In this method, the scaling parameter of WT is inversely proportional to frequency. Therefore, the frequency resolution is decreasing for increasing frequency. S-transform representation is shown in Fig. 1d where the Gaussian window width is frequency dependent.

Fig. 1
figure 1

Time–frequency representation method

Full size image

The width of the chosen window controls time–frequency representation in the transformed domain. ST mainly demonstrates some particular frequency segments in signal processing. It is a time–frequency restriction procedure with subordinate frequency resolution and compensates the drawback of STFT. To control the complex Fourier, signal ST is utilized as a window. However the window height and width are scaled by frequency. The response of ST shows frequency invariant amplitude in contrast with WT. It attenuates the high-frequency signals in contrast to low-frequency signals, and its generalized form is called FrST. The FrST improves FrFT and ST adaptability of signal analysis and generalized the time–frequency representation to time fractional frequency. Hence, FrST can upgrade the settling capacity and adaptability of signal analysis. Next section will demonstrate various transform with mathematical expressions.

2 Types of Transforms

Nowadays several transforms are used for various applications. Some of them are discussed in this section.

2.1 Fourier Transform (FT)

The FT is defined as a linear operator that maps a signal from time domain to an equivalent signal in frequency domain. It is also known as Fourier integral and mathematically defined as [1,2,3,4]

$${\text{X}}({\text{f}}) = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}})\exp ( - {\text{j}}2\uppi{\text{ft}})} \,{\text{dt}}$$
(1)

where \({\text{X}}({\text{f}})\) is the FT of the signal \({\text{x(t)}}\) and \(\exp ( - {\text{j}}2\uppi{\text{ft}})\) is the basis function of FT. Similarly, the inverse-FT (IFT) is defined as [1,2,3]

$${\text{x}}({\text{t}}) = \int_{ - \infty }^{\infty } {{\text{X}}({\text{f}})} \exp ({\kern 1pt} {\text{j}}2\uppi{\text{ft}}){\text{df}}$$
(2)

The discrete FT (DFT) can be obtained by discretizing the both time and frequency axes into N samples and is defined [5,6,7] as

$${\text{X}}[{\text{k}}] \, = \, \sum\limits_{{{\text{n}} = 0}}^{{{\text{N}} - 1}} {{\text{x}}[{\text{n}}]} \exp \left( { - {\text{j}}\frac{{2\uppi{\text{kn}}}}{\text{N}}} \right);\quad {\text{k}} = 0,1,2, \ldots ,{\text{N}} - 1$$
(3)

and inverse DFT (IDFT) is defined as [5, 6]

$${\text{x}}[{\text{n}}] = \frac{1}{\text{N}}\sum\limits_{{{\text{k}} = 0}}^{{{\text{N}} - 1}} {{\text{X}}[{\text{k}}]} \exp \left( {{\text{j}}\frac{{2\uppi{\kern 1pt} {\text{nk}}}}{\text{N}}} \right);\quad {\text{n}} = 0,1,2, \ldots ,{\text{N}} - 1$$
(4)

2.2 Fractional Fourier Transform (FrFT)

The generalized form of FT with an angle \(\upalpha\) is called fractional FT (FrFT). It is defined for entire time–frequency plane where \(\upalpha\) is the counter clockwise angle from time-axis. Therefore, FrFT with \(\upalpha = \frac{\uppi}{2}\) is FT. The FrFT with the angle \(\upalpha\) of a function \(x(t)\) in transform-domain \({\text{v}}\) in time–frequency plane is written as [16,17,18,19,20]

$${\text{X(v) = }}\int_{ - \infty }^{\infty } {{\text{x(t)K}}_{\upalpha} \left( {\text{t, v}} \right){\text{dt}}}$$
(5)

where

$${\text{K}}_{\upalpha} \left( {{\text{t,}}\,{\text{v}}} \right) = \left\{ {\begin{array}{*{20}l} {B_{\upalpha} \exp \left( {j\uppi(\text{v}^{2} + \text{t}^{2} )\cot\upalpha - j{\kern 1pt} 2{\pi vt}\csc\upalpha} \right);} \hfill & {\upalpha \ne {n\pi }} \hfill \\ {\updelta(\text{t} - \text{v}),} \hfill & {\upalpha = 2{n\pi }} \hfill \\ {\updelta(\text{t} + \text{v}),} \hfill & {\upalpha = (2\text{n} - 1)\uppi} \hfill \\ \end{array} } \right.$$
(6)

and \(B_{\upalpha} = \sqrt {1 - \text{j}\cot\upalpha}\), \(n \in {\mathbf{\mathbb{Z}}}\) and \(\updelta(\text{t})\) is the unit impulse function. Here, \({\text{K}}_{\upalpha} \left( {\text{t, v}} \right)\) is the basis function of FrFT. Similarly FrFT with respect to angle \(-\upalpha\) is called inverse fraction FT (IFrFT) and defined as [16, 17]

$$\text{x}(\text{t}) = \int_{ - \infty }^{\infty } {{\text{X}}({\text{v}})} {\rm K}^{ * }_{\alpha } \left( {{\text{t}},{\text{v}}} \right){\text{dv}}$$
(7)

where \(*\) represents the complex conjugate operation. The FrFT represents fractional spectrum; however, it has one major disadvantage of utilizing the global kernel. FrFT only provides spectral contents with no information around time confinement of FrFT spectrum parts. In this way, the exploration of a non-stationary signals that FrFT spectrum quality variation with time is still required.

2.3 Short Time-FT (STFT)

STFT is a classical transform for time–frequency analysis. In this transform, first the original signal \({\text{x(t)}}\) multiplies with the window \({\text{w}}({\text{t}} -\uptau)\) and then computation of FT of the windowed signal is performed. FT change signal domain from time to frequency by coordinating over time axis. But for non-stationary signal, i.e., the frequency components are a function of time, at that point they can’t point-out, when a specific frequency rises. The STFT overcome this drawback of FT by presenting a window \({\text{w}}({\text{t}} -\uptau)\). The window is intended to remove a little segment of continuous time signal \({\text{x(t)}}\) and after that taking FT. The changed coefficient has two free parameters are frequency ‘\({\text{f}}\)’ and time ‘\(\uptau\)’. The STFT is written as [22,23,24,25]

$${\text{X}}(\uptau,{\text{ f}}) \, = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}}){\text{w}}({\text{t}} -\uptau) \ldots \exp ( - {\text{j}}2\uppi{\text{ft}})} {\text{dt}}$$
(8)

and inverse STFT (ISTFT) is written as [22,23,24,25]

$${\text{x}}({\text{t}}) \, = \, \frac{1}{{2\uppi}}\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {{\text{X}}(\uptau,{\text{ f}}){\text{w}}({\text{t}} -\uptau)} } \exp ({\text{j}}2\uppi{\text{ft}}){\text{d}}\uptau{\text{df}}$$
(9)

2.4 Short-Time Fractional FT (STFrFT)

The short time FrFT (STFrFT) is improved form of FrFT for analysis of a non-stationary signal. The thought behind STFrFT was dividing the signals by utilizing a period over narrow window and acting the FrFT spectrum for every section. Since, the FrFT is calculated for every window section of the signal. STFrFT gives an accurate joint signal representation in both FrFT and time domain. It was introduced to improve their concentration and modify the STFT by using a window. The STFrFT was given in [18, 19], and versatile STFrFT is given in [20], here rotate a signal in time–frequency space by FrFT before executing STFT. A signal is multiplying with a window function called STFrFT. It is distinct as

$${\text{STFrFT}}({\text{t}},{\text{v}}) = \int_{ - \infty }^{\infty } {{\text{x}}(\uptau)} {\text{g}}({\text{t}} -\uptau){\text{k}}(\uptau,{\text{v}}){\text{d}}\uptau$$
(10)

In a few cases, the window function, with short time support is symmetry and real. However, fixed window width is a major disadvantage of STFrFT. Therefore, it did not give high resolution in FrFT and time domain. However, the efficacy of STFrFT is bounded using the uncertainty principle.

2.5 Wavelet Transform (WT)

Wavelet is used for the analysis of transient, time-varying or non-stationary phenomenon. It has energy concentrated in time and oscillating wave like characteristics. It can also allow instantaneous TF analysis with a mathematical definition. The main objective of WT is to define the basis function and to provide an efficient method for computations. In WT, a function \({\text{x(t)}}\) is often be better defined or analyzed, if articulated as a linear decomposition using \({\text{x(t) = }}\sum\limits_{l} {{\text{a}}_{l} } {\varphi }_{l} ( {\text{t)}}\), where \(l\) represents integer index, \({\varphi }_{l} ( {\text{t)}}\) is extension set and \(\text{a}_{l}\) is real-valued expansion coefficients. Similar to FT, WT defines with the extension of a set of basis function. Unlike to FT, WT does not expand in the trigonometric polynomial forms but expand in wavelet forms. The mother wavelet is observed that every application using fast FT (FFT) is formulated by wavelet to deliver more confined temporal and frequency information. In STFT, very small-frequency components cannot be detected in the spectral because fixed size window; however, WT overcomes this STFT problem. The wavelet analysis is considered as a complex function that satisfy the following circumstances [27,28,29,30,31,32,33]

$$\int_{ - \infty }^{\infty } { |\uppsi ( {\text{t)}}} |^{ 2} < \infty$$
(11)
$${\text{C}}_{\uppsi} { = 2}\uppi\int_{ - \infty }^{\infty } {\frac{{ |\Psi (\upomega ) |^{ 2} }}{{ |\upomega |}}} {\text{d}}\upomega \, < \, \infty$$
(12)

where \(\Psi\) is FT of mother wavelet function \(\uppsi\). Here, condition (11) gives finite energy of wavelet function \(\uppsi\) and (12) is admissibility condition, indicates that if \(\Psi (\upomega)\) it is smooth, then \(\Psi (\upomega) = 0\).

2.5.1 Continuous Wavelet Transform (CWT)

If wavelet function \(\uppsi\) fulfils the above conditions, then WT of a function \({\text{x(t)}}\) and wavelet function \(\uppsi ( {\text{t)}}\) is written as [121, 130]

$${\text{X}}({\text{b}},{\text{a}}) = \frac{1}{{\sqrt {\text{a}} }}\int_{ - \infty }^{\infty } {{\text{x}}({\text{t}})\uppsi^{*} \left( {\frac{{{\text{t}} - {\text{b}}}}{\text{a}}} \right)} \;{\text{dt}}$$
(13)
$${\text{X}}({\text{b}},{\text{a}}) = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}})\uppsi_{{{\text{a}},{\text{b}}}}^{ * } } {\text{dt}}$$
(14)

where \(\uppsi_{{{\text{a}},{\text{b}}}} = \frac{1}{{\sqrt {\text{a}} }}\uppsi\left( {\frac{{{\text{t}} - {\text{b}}}}{\text{a}}} \right)\) and \(\uppsi^{ *}\) represent the complex conjugate of wavelet function \(\uppsi\) and defined on open interval \(( {\text{b, a)}}\) half plane (\({\text{b}} \in {\mathbb{R}},\quad {\text{a}} > 0\)), where the scale of analysing and time shifting wavelet represented by \({\text{a}}\) and \({\text{b}}\) respectively. The signal \({\text{x(t)}}\) is achieved from WT and \({\text{X(b, a)}}\) by the inversion formula. The inverse CWT (ICWT) is written as

$${\text{x}}({\text{t}}) = \frac{1}{{{\text{C}}_{\uppsi} }}\int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {\frac{{{\text{X}}({\text{b}},{\text{a}})\,\uppsi_{{{\text{a}},{\text{b}}}} ({\text{t}})}}{{{\text{a}}^{2} }}} } {\text{ da}}\,{\text{db}}$$
(15)

2.5.2 Discrete Wavelet Transform (DWT)

In discrete form of WT, scale and shift factors are discretized as \({\text{a = a}}_{ 0}^{\text{m}}\) and \({\text{b = nb}}_{0}\). Therefore, wavelet is also discretized as

$$\uppsi_{{{\text{m}},{\text{n}}}} ({\text{t}}) = {\text{a}}_{0}^{{ - \frac{\text{m}}{2}}} \,\uppsi\left( {\frac{{{\text{t}} - {\text{nb}}_{0} }}{{{\text{a}}_{0}^{\text{m}} }}} \right)$$
(16)

where \({\text{m}}\) and \({\text{n}}\) are an integers. The DWT is defined as [27, 28]

$${\text{X}}_{{{\text{m,}}{\kern 1pt} {\text{n}}}} { = }\int_{{{ - }\infty }}^{\infty } {{\text{x(t)}}\,\uppsi_{{{\text{m,}}{\kern 1pt} {\text{n}}}}^{ *} ( {\text{t)dt}}}$$
(17)

and inverse DWT (IDWT) is defined in [27, 28] as

$${\text{x(t) = k}}_{\uppsi} \, \sum\limits_{\text{m}} {\sum\limits_{\text{n}} {{\text{X}}_{{{\text{m,}}{\kern 1pt} {\text{n}}}}\uppsi_{{{\text{m,}}{\kern 1pt} {\text{n}}}} ( {\text{t)}}} }$$
(18)

where normalization constant is \({\text{k}}_{\uppsi}\). In the scale time plane function \(\uppsi_{{{\text{m,}}{\kern 1pt} {\text{n}}}} ( {\text{t)}}\) gives the sampling points, linear sampling in time direction but logarithmic in scale direction. In general, \({\text{a}}_{0}\) is chosen as \({\text{a}}_{ 0} { = 2}^{{\frac{ 1}{\text{J}}}}\), where, \({\text{J}}\) is an integer value [27]. The generalized form of wavelet called fractional WT is given in the next part.

2.6 Fractional Wavelet Transform (FrWT)

Fractional WT (FrWT) is a generalized form of WT with an angle parameter \(\upalpha\), but when the angle \(\upalpha\) is \(\frac{\uppi}{2}\), FrWT is similar to WT. FrWT rectifies the limitation of WT and FrFT. FrWT with an angle \(\upalpha\) of a continuous function \(\text{x}(\text{t})\) defined as [27, 34, 35]

$$W_{\text{x}}^{\upalpha} (\text{a},\text{b}) = \int\limits_{\mathbb{R}} {\text{x}(\text{t}){\varphi }_{{\upalpha,\text{a},\text{b}}}^{ * } (\text{t})\text{dt}}$$
(19)

and transform kernel defined as [27, 34, 35]

$${\varphi }_{{\upalpha,{\text{a}},{\text{b}}}} ({\text{t}}) = \frac{1}{{\sqrt {\text{a}} }}{\varphi }\left( {\frac{{{\text{t}} - {\text{b}}}}{\text{a}}} \right)\exp \left( { - {\text{j}}\frac{{{\text{t}}^{2} - {\text{b}}^{2} }}{2}\cot\upalpha} \right)$$
(20)

where \(\text{a} \in {\mathbb{R}}^{ + }\),\(\text{b} \in {\mathbb{R}}\) and \(\text{x}(\text{t}) \in \text{L}^{2} ({\mathbb{R}})\). Similar to FrFT, FrWT gets compensations of multiresolution analysis (MRA) of WT, which has fractional domain signal representation capability.

2.7 S-Transform (ST)

S-transform (ST) is hybrid of WT and STFT in time–frequency domain. It overcomes the drawbacks of STFT and deficiency of phase in WT. ST utilizes a Gaussian function, that width and height is constrained by frequency. The ST gives signal clarity in contrast to different transforms since it doesn’t have cross terms issues. ST displays invariant frequency amplitude in compare with WT and also analysed phase and power spectrums. The ST diminishes high-frequency as compare to low-frequency signal [2, 3]. ST of a signal ‘\({\text{x(t)}}\)’ is represented by \({\text{X}}(\uptau,{\text{f}})\) and defined as [1,2,3]

$${\text{X}}(\uptau,{\text{ f}}) = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}}){\text{g}}({\text{t}} -\uptau,{\text{f}})\exp ( - {\text{j}}2\uppi{\text{f}}{\kern 1pt} {\text{t}})\,{\text{dt}}}$$
(21)
$${\text{X}}(\uptau,{\text{ f}}) = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}})\frac{{|{\text{f}}|}}{{\sqrt {2\uppi} }}\exp \left( {\frac{{ - (t -\uptau)^{2} {\text{f}}^{2} }}{2}} \right)} \;\exp ( - {\text{j}}2\uppi{\text{ft}})\,{\text{dt}}$$
(22)

where Gaussian function, \({\text{g}}({\text{t}} -\uptau,{\text{ f}})\) controlled by frequency ‘\({\text{f}}\)’ and time shift ‘\(\uptau\)’. If \({\text{X(}}\uptau , {\text{ f)}}\) is integrated with respect to ‘\(\uptau\)’ then, it give FT of \({\text{x(t)}}\), written as

$$\int_{ - \infty }^{\infty } {{\text{X}}\left( {\uptau,{\text{ f}}} \right)\,} {\text{d}}\uptau = \int_{ - \infty }^{\infty } {\left( {\int_{ - \infty }^{\infty } {{\text{x}}\left( {\text{t}} \right)} .{\text{g}}\left( {{\text{t}} -\uptau,{\text{ f}}} \right)\exp ( - {\text{j}}2\uppi{\text{ft}}){\text{dt}}} \right)} {\text{d}}\uptau$$
(23)

Using the normalized condition of the Gaussian function is written as

$$\int_{ - \infty }^{\infty } {{\text{g}}\left( {{\text{t}} -\uptau,{\text{ f}}} \right){\text{d}}\uptau} = \int_{ - \infty }^{\infty } {\frac{{|{\text{f}}{\kern 1pt} |}}{{\sqrt {2\uppi} }}\exp \left( {\frac{{ - \left( {{\text{t}} -\uptau} \right)^{2} {\text{f}}^{2} }}{2}} \right)} \,{\text{d}}\uptau = 1$$
(24)

and (23) can be written as

$$\int_{ - \infty }^{\infty } {{\text{X}}\left( {\uptau,{\text{ f}}} \right)} \,{\text{d}}\uptau = \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}}) \, {\kern 1pt} \exp ( - {\text{j}}2\uppi{\text{ft}})\,{\text{dt}} = {\text{X}}({\text{f}})}$$
(25)

where FT of \({\text{x(t)}}\) is represented by \({\text{X(f}}{\kern 1pt} )\).

The two-dimensional ST (2-D ST) is written as

$${\text{X(}}\uptau_{ 1} ,\uptau_{ 2} , {\text{f}}_{1} , {\text{f}}_{2} )= \int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {{\text{x}}\left( {{\text{t}}_{ 1} , {\text{t}}_{ 2} } \right)\frac{{\left| {{\text{f}}_{1} } \right|\left| {{\text{f}}_{2} } \right|}}{{ 2\uppi}}{ \exp }\left( { - \frac{{\left( {{\text{t}}_{ 1} -\uptau_{ 1} } \right)^{ 2} {\text{f}}_{1}^{ 2} }}{ 2} - \frac{{\left( {{\text{t}}_{ 2} -\uptau_{ 2} } \right)^{ 2} {\text{f}}_{2}^{ 2} }}{ 2}} \right){ \exp }\left( { - {\text{j2}}\uppi\left( {{\text{f}}_{1} {\text{x + f}}_{2} {\text{y}}} \right)} \right){\text{dt}}_{ 1} {\text{dt}}_{ 2} } }$$
(26)

The two-dimensional inverse S-transform (2-DIST) is defined as

$${\text{x}}\left( {{\text{t}}_{ 1} , {\text{t}}_{ 2} } \right) = \int_{ - \infty }^{\infty } {\int_{ - \infty }^{\infty } {\left( {\int_{{{\text{f}}_{1l} }}^{{{\text{f}}_{{1_{\text{h}} }} }} {\int_{{{\text{f}}_{{2_{l} }} }}^{{{\text{f}}_{{2_{h} }} }} {{\text{X(}}\uptau_{ 1} ,\uptau_{ 2} , {\text{f}}_{1} , {\text{f}}_{2} ) {\text{d}}\uptau_{ 1} {\text{d}}\uptau_{ 2} } } } \right)} } { \exp }\left( {{\text{j2}}\uppi\left( {{\text{f}}_{1} {\text{x + f}}_{2} {\text{y}}} \right)} \right){\text{df}}_{1} {\text{df}}_{2}$$
(27)

2.7.1 Discrete S-Transform (DST)

Due to the advent of discrete systems, the discretization of every mathematical tool becomes necessary to increase the span of its applications. Considering the discrete version of the signal \({\text{x(t)}}\) as \({\text{x[kT]}}\) where, \({\text{k = 0,1,2,}} \ldots ,{\text{N}} - 1\) and sampling interval \({\text{T}}\), its discrete FT (DFT) is given by [1]

$${\text{X}}\left[ {\frac{{\text{n}}}{{\text{NT}}}} \right] = \frac{{1}}{{\text{N}}}\sum\limits_{{{\text{k}} = 0}}^{{{\text{N}} - 1}} {{\text{x}}[{\text{kT}}]} \;\exp \left( { - \frac{{{\text{j2}}\uppi{\text{nk}}}}{{\text{N}}}} \right)$$
(28)

where,\({\text{n = 0,1,2,}} \ldots ,{\text{N}} - 1\). For discrete, ST is the projection of vectors define using time sequences \({\text{x[kT]}}\) over vector a set of vectors. Using (22), in Discrete case of \({\text{x[kT]}}\) can be defined [1,2,3,4] as

$${\text{X}}\left[ {{\text{iT,}}\frac{{\text{n}}}{{\text{NT}}}} \right] = \sum\limits_{{{\text{m}} = 0}}^{{{\text{N}} - 1}} {{\text{x}}\left[ {\frac{{\text{m + n}}}{{\text{NT}}}} \right]\;{\exp }\left( { - \frac{{2\uppi^{ 2} {\text{m}}^{2}}}{{{\text{n}}^{2} }}} \right)\,{\exp}\left({\frac{{{\text{j2}}\uppi{\text{mi}}}}{{\text{N}}}}\right);} \quad {\text{n}} \ne 0$$
(29)

For \({\text{n = 0}}\), it is written as

$${\text{X}}\left[ {\text{iT}} \right]{ = }\frac{{1}}{{\text{N}}}\sum\limits_{{{\text{m}} = 0}}^{{{\text{N}} - 1}} {{\text{x}}\left[ {\frac{{\text{m}}}{{\text{NT}}}} \right]}$$
(30)

where,\({\text{i,}}\,{\text{m}}\;{\text{and}}\;{\text{n}}\,{ = }\, 0 , 1 , 2 ,\ldots , {\text{N}} - 1\). The inverse discrete ST (IDST) is defined as [1, 2]

$${\text{x}}[{\text{kT}}]{ = }\sum\limits_{{{\text{n}} = 0}}^{{{\text{N}} - 1}} {\left( {\frac{{1}}{{\text{N}}}\sum\limits_{{{\text{i}} = 0}}^{{{\text{N}} - 1}} {{\text{X}}\left[ {{\text{iT}},\frac{{\text{n}}}{{\text{NT}}}} \right]} } \right)}\, { \exp }\left( {\frac{{{{\text{j2}}\uppi{\text{nk}}}}}{{\text{N}}}} \right)$$
(31)

Subsequently, in literature, other definitions of discrete S-transform were also reported [9]. Here one such definition is presented, which was called as Discrete Orthonormal ST (DOST) [9,10,11], which is defined in terms of ‘N’ unit length basis vector, which as:

$${\text{X[kT]}}_{{ [ {\text{v,}}\upbeta ,\uptau ]}} = \frac{{1}}{{{\sqrt\upbeta}}}\sum\limits_{{{\text{f}}{\kern 1pt} = {\text{v}} - {\raise0.7ex\hbox{$\upbeta$} \!\mathord{\left/ {\vphantom {\upbeta 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$ 2$}}}}^{{{\text{v}}{\kern 1pt} { + }\;{\raise0.7ex\hbox{$\upbeta$} \!\mathord{\left/ {\vphantom {\upbeta 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$ 2$}}{ + 1}}} {{ \exp }\left( {{\text{j2}}\uppi\frac{\uptau}{\upbeta}{\text{f}}} \right) . {\text{exp}}\left( { - {\text{j2}}\uppi\frac{{\text{k}}}{{\text{N}}}{\text{f}}} \right) . {\text{exp(}} - {\text{j2}}\uppi \uptau )} ;\quad {\text{k = 0,1,2,}} \ldots , {\text{N}} - 1$$
(32)
$${\text{X[kT]}}_{{ [ {\text{v,}}\upbeta ,\uptau ]}} { = }\frac{{{ \exp }\left( { - {\text{j2}}\upalpha\left( {\frac{{ 2 {\text{v}} -\upbeta - 1}}{ 2}} \right)} \right) - { \exp }\left( { - {\text{j2}}\upalpha\left( {\frac{{ 2 {\text{v + }}\upbeta{ - 1}}}{ 2}} \right)} \right)}}{{ 2\sqrt\upbeta \;{ \sin }\upalpha}}{\text{j}}\,{\text{exp(}} - {\text{j2}}\uppi \uptau )$$
(33)

where \(\upalpha = \uppi\left( {\frac{{\text{k}}}{{\text{N}}} - \frac{{\uptau}}{{\upbeta}}} \right)\) is a center of the temporal window for \({\text{kth}}\) basis vector. Mathematically, these basis vectors are orthonormal

$$\frac{ 1}{\text{N}}\int_{ 0}^{\text{N}} {{\text{X[kT]}}_{{ [{v^{\prime}} ,{\beta^{\prime}} ,{\tau^{\prime}} ]}} } {\text{X}}^{ *} [ {\text{kT]}}_{{ [ {\text{v,}}\upbeta ,\uptau ]}} {\text{dk = }}\updelta_{{{v^{\prime}}{\text{v}}}}\updelta_{{{\beta^{\prime}\beta }}}\updelta_{{{\tau^{\prime}\tau }}}$$
(34)

where \(\updelta_{{{v^{\prime}}{\text{v}}}} { = }\left\{ {\begin{array}{*{20}l} {1;} \hfill & {{v^{\prime}} = {\text{v}}} \hfill \\ {0;} \hfill & {\text{else}} \hfill \\ \end{array} } \right.\) is a delta function. The discrete inverse ST (IDST) is defined as

$${\text{x[kT] = }}\sum\limits_{{{\text{n}} = 0}}^{{{\text{N}} - 1}} {\left( {\frac{{1}}{{\text{N}}}\sum\limits_{{{\text{i}} = 0}}^{{{\text{N}} - 1}} {{\text{X}}\left[ {{\text{iT,}}\frac{{\text{n}}}{{\text{NT}}}} \right]} } \right)} { \exp }\left( {\frac{{{{\text{j2}}\uppi{\text{nk}}}}}{{\text{N}}}} \right)$$
(35)

In Table 1 described the basis function of different transforms.

Table 1 Basis of the transforms
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2.7.2 Relation Between ST and Others Types of Transforms

The relation between ST and some other types of transform is given as

  1. (i)

    Relation between ST and STFT

    STFT of a function \({\text{x(t)}}\) is defined in (8), and the ST is defined in (22). ST is considered as a specific type of STFT with a Gaussian window function.

    The STFT of a signal \({\text{x(t)}}\) is defined as [6]

    $$STFT (\uptau ,\,{\text{f ) = }}\int_{ - \infty }^{\infty } {{\text{x(t) w(t}} -\uptau ) {\text{ exp(}} - {\text{j2}}\uppi{\text{f t)}}} {\text{ dt}}$$
    (36)

    where,\({\text{w(t}} -\uptau )\) is represent window function.

    $${\text{ X(}}\uptau ,\,{\text{f )}} = \int_{ - \infty }^{\infty } {{\text{x(t)}}\,{\text{g(t}} -\uptau ,\,{\text{f ) exp(}} - {\text{j2}}\uppi{\text{f}}{\kern 1pt} {\text{t)dt}}}$$
    (37)

    where Gaussian function, \({\text{g(t}} -\uptau , {\text{ f )}}\) controlled by frequency ‘\({\text{f}}\)’ and time shift ‘\(\uptau\)’. Hence, after the comparative analysis, concluded that ST is a specific instance of the STFT with a Gaussian window function.

  2. (ii)

    Relation between ST and CWT

    The WT of a signal \({\text{x(t)}}\) defined as [8]

    $${\text{W(}}\uptau , {\text{d) = }}\int_{ - \infty }^{\infty } {\text{x(t) }} {\text{m(t}} -\uptau ,\,{\text{d)dt}}$$
    (38)

    where \(\uptau\) and \({\text{d}}\) represents, the time of spectral localization and dilation factor respectively. The dilation factor decides the ‘width’ of the wavelet and thus, it controls the resolution. \({\text{W(}}\uptau ,\,{\text{d)}}\) denoted by a scaled copy of the fundamental mother wavelet and must have zero mean [2]. Hence, after the comparative analysis, the ST is derived, when WT multiplied by the phase factor, thus the ST can be defined as

    $${\text{X(}}\uptau ,\,{\text{f ) = W(}}\uptau ,\,{\text{d)}}{\kern 1pt} \,{\text{exp(}} - {\text{j2}}\uppi{\text{f}}\uptau )$$
    (39)

    where the mother wavelet is defined as

    $${\text{m(t,f}}{\kern 1pt} ) { = }\frac{{ | {\text{f}}{\kern 1pt} |}}{{\sqrt { 2\uppi} }}{ \exp }\left( {\frac{{ - {\text{t}}^{ 2} {\text{f}}^{ 2} }}{ 2}} \right){ \exp }\left( { - {\text{j2}}\uppi{\text{ft}}} \right)$$
    (40)

    where the dilation factor ‘\({\text{d}}\)’is inverse of the frequency ‘\({\text{f}}\)’. The ST gives better, dependent frequency resolution in time frequency analysis with least noise.

  3. (iii)

    Relation between FT and ST

    The above definition of FT and ST of function \({\text{x(t)}}\) give a relation between FT and ST is written as

    $${\text{X(f}}{\kern 1pt} )= \int_{ - \infty }^{\infty } {{\text{X(}}\uptau , {\text{ f}}{\kern 1pt} )} \,{\text{d}}\uptau$$
    (41)

    Hence, using FT, inverse ST (IST) is defined as [36]

    $${\text{x(t)}} = \int_{ - \infty }^{\infty } {\left\{ {\int_{ - \infty }^{\infty } {{\text{X(}}\uptau , {\text{ f}}{\kern 1pt} )\,{\text{d}}\uptau} } \right\}} {\text{ exp(j2}}\uppi{\text{ft)}}\,{\text{df}}$$
    (42)

2.7.3 Properties of S-Transform

Based on the survey [39], there are many existing properties of ST as defined in Table 2.

Table 2 Existing properties of S-transform
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Figure 2 shows that the application of ST is increases yearly, where the maximum publications are in the year of 2015 to 2019.

Fig. 2
figure 2

Number of publications in S-transform

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Table 3 described the different applications of ST in different area of Geo-informatics, Image processing, Signal processing, Bio-medical, Power system and Communication by several authors.

Table 3 Applications of S-transform
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2.8 Fractional S-Transform (FrST)

The FrST was first introduced in 2012 [115], as a way to deal with synthetic Ricker wavelet and seismic data. It is defined as a cascade of FrST and ST. The Fig. 3 show, time-fractional frequency plane in FrST domain.

Fig. 3
figure 3

The time-fractional frequency plane

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In 2012, [115] expressed ST of a signal in terms of FrFT and Gaussian window, they are also known as FrST. Actually authors originate an equivalent expression of ST in FrFT domain. FrST of a function \({\text{x}}({\text{t}})\) with an angle \(\upalpha\) is written in [115,116,117] as

$$X^{\upalpha} \left( {\uptau,{\text{v}}} \right) \, = \, \int_{ - \infty }^{\infty } {{\text{x}}({\text{t}}){\text{g}}\left( {\uptau - {\text{t}},{\text{v}}} \right)\upkappa_{\upalpha} \left( {{\text{t}},{\text{v}}} \right){\text{dt}}}$$
(43)

where ‘\({\text{v}}\)’ is fractional frequency. The FrST is function of fractional frequency (\({\text{v}}\)) and time (\({\text{t}}\)) and Gaussian function \({\text{g}}\left( {\uptau - {\text{t, v}}} \right)\) is scalable with ‘\({\text{t}}\)’ and FrFT frequency ‘\({\text{v}}\)’.

$${\text{g}}\left( {\text{t, v}} \right){ = }\frac{{ | {\text{v(csc}}\upalpha ) |^{\text{p}} }}{{\sqrt { 2\uppi} {\text{q}}}}{ \exp }\left( {\frac{{ - {\text{t}}^{ 2} \left\{ {{\text{v(csc}}\upalpha )} \right\}^{{ 2 {\text{p}}}} }}{{ 2 {\text{q}}^{ 2} }}} \right)$$
(44)

where transformation kernel \(\upkappa_{\upalpha} \left( {\text{t, v}} \right)\) is defined in (2) and \(\upalpha{ = }\frac{{{\text{a}}\uppi}}{ 2}\). In (7) the Gaussian function varies with frequency. The fractional factor \({\text{a}} \in \left[ {0, \, 4} \right]\) and rotation angle \(\upalpha{ = }\frac{{{\text{a}}\uppi}}{ 2}\), with \({\text{p}} = {\text{q}} = 1\) and \({\text{a}} = 1\), then FrST coincides with ST as

$${\text{X}}\left( {\uptau,{\text{ v}}} \right) = \int_{ - \infty }^{\infty } {x({\text{t}})} \frac{{|{\text{v}}|}}{{\sqrt {2\uppi} }}\,\exp \left( { - \frac{{({\text{t}} -\uptau)^{2} {\text{v}}^{2} }}{2}} \right)\exp ( - {\text{j}}2\uppi{\text{vt}})$$
(45)

The generalized form of ST is called FrST, it is better than ST and FrFT. The FrST is improve the flexibility of time–frequency analysis and energy of signal spectra. FrST can efficiently improve the time–frequency resolution capacity as compared to ST.

The inverse FrST (IFrST) is given as [115]

$${\text{x}}({\text{t}}) = \int_{ - \infty }^{\infty } {\left\{ {\int_{ - \infty }^{\infty } {{\text{X}}^{ * } \left( {\uptau,{\text{v}}} \right){\text{d}}\uptau} } \right\}\upkappa_{\upalpha}^{ * } ({\text{t}},{\text{v}}){\text{d}}v}$$
(46)

where ‘\(*\)’ represents complex conjugate.

According to [115] marginal condition of FrST yield,

$$\int_{ - \infty }^{\infty } {{\text{X}}^{\upalpha} \left( {\uptau,{\text{ v}}} \right){\text{d}}\uptau} = {\text{X}}_{\upalpha} ({\text{v}})$$
(47)

So, the IFrST is define as [115]

$${\text{x}}({\text{t}}) = \int_{ - \infty }^{\infty } {{\text{X}}_{\upalpha} ({\text{v}})\upkappa_{\upalpha}^{ * } ({\text{t}},{\text{v}}){\text{dv}}}$$
(48)

where \({\text{X}}_{\upalpha} ({\text{v}})\) is fractional FT of function \({\text{x}}({\text{t}})\).

Based on the survey, properties of fractional ST are already documented in the literature [115, 130], given in Table 4.

Table 4 Existing properties of fractional ST
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2.9 Application of Fractional S-Transform

A signal is defined in the concept of fractional ST based on the ideas of ST and FrFT. Here, extend ST time–frequency domain to the time-fractional frequency of FrST and also study the mathematical properties of FrST likes- linearity, scaling, inverse fractional ST and time marginal condition [115]. Generalization of the results of ST on the spaces of type W is given in [116]. The FrST on space of type S is demonstrated in [117]. It is useful for the analysis of time–frequency behaviour of test function and distributions. In [118] extends the result of ultra-distribution for the FrST given by Singh to the Bohemian spaces. Since the diverse seismic signal has distinctive ideal fractional parameters which are not helpful for multichannel seismic data analysis. Therefore utilizing FrST first decay the basic frequency after that, they analyse the minimum frequency [119]. The FrST depends on the concept of time-bandwidth product criterion and time–frequency rotation property of FrFT, and furthermore, present the standardized second-order pivotal moment calculation technique for finding the optimal order; in contrast of time-bandwidth product [120]. The theory of ideal FrST is derived from the properties of FrFT and GTBP criterion. It is a novel fractional lower order ST (FLOST) time–frequency representation technique. The FLOST time–frequency sifting algorithm depends on FLOST time–frequency representation systems [121]. The properties of FrST are described in [130]. Figure 4 shows the various applications of different transforms in different areas of Geo-informatics, Image processing, Signal processing, Bio-medical, Power system and Radar signal and Communication.

Fig. 4
figure 4

Applications of transforms

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Figure 5 shows that the maximum applications of ST used in the areas of image processing and digital signal processing whereas minimum applications is used in the area of radar signals and communication system.

Fig. 5
figure 5

Graphical presentation of S-transform application

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2.10 Detection of LFM Signal in the Presence of Echo

A LFM chirp signal is a short burst sinusoidal signal with constant amplitude and its frequency varies linearly over the pulse time. Here, a scenario of radar signal processing is considered for the detection of desired signal in the presence of echo signal. The type of both desired and echo signal is a non-stationary chirp signal. For illustration of detection of LFM signal in the presence of echo, following expression of desired signal and echo is considered,

$${\text{x}}_{\text{in}} {\text{ = e}}^{{{\text{j2}}\uppi ( {\text{bt + at}}^{ 2} )}}$$
(49)

where \({\text{b}}\) is the centre frequency of LFM chirp signal and chirp rate \(2 {\text{a}}\). The echo signal is defined as

$${\text{n(t) = }}\frac{ 1}{{ 1 0^{{{\raise0.7ex\hbox{${\text{SNR}}$} \!\mathord{\left/ {\vphantom {{\text{SNR}} { 2 0}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${ 2 0}$}}}} }}{\text{e}}^{{{\text{j2}}\uppi ( {\text{dt + ct}}^{ 2} )}}$$
(50)

where \({\text{d}}\) is the centre frequency of LFM signal in the presence of echo and chirp rate \(2 {\text{c}}\). The simulated results is computed with \({\text{a = 8, b = 55, c = 5 and d = 35}}\).

The methodology adopted for the detection of LFM signal in the presence of echo is shown in Fig. 6. First the detection of LFM signal, FrST is used as the transform technique. The optimum angle of FrST is decided by the angle where FrST of the signal is maximally concentrated in transform domain. The filter transfer function is considered as uniform distribution in the transform-domain for which the transformed signal is concentrated and non-zero. Thereafter, the inverse transform is applied to determine the filtered signal in time-domain.

Fig. 6
figure 6

Filtering of LFM signal

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Subsequently, the mean square error (MSE) of filtered signal and original signal is determined and plotted with respect to signal-to-noise ratio. Similarly the detection is performed with ST. The complete plot of MSE for these transform is shown in Fig. 7 for a comparative analysis.

Fig. 7
figure 7

Mean square error of FrST and ST

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This plot clearly shows that detection of LFM signal using FrST is a superior technique than the ST methods.

2.11 Conclusion

The S-transform is a useful analysing tool in time–frequency localization methods with frequency dependent resolution. It is depends on moving and scalable Gaussian window function, and compensate STFT low resolution and phase information in WT. ST is a hybrid combination of the WT and STFT. This article includes the relation of several transform with FrST followed by definition and properties of ST and FrST. Subsequently, a brief survey of ST and FrST based on its application techniques between 1996 and 2019 is presented. Thereafter, a comparative analysis is performed for the filtering of chirp signal in the presence of echo signal. This analysis clearly established that FrST is superior tool than ST.