Preface

Joseph Fourier (1770–1830) first introduced the remarkable idea of expansion of a function in terms of trigonometric series without giving any attention to rigorous mathematical analysis. The integral formulas for the coefficients of the Fourier expansion were already known to Leonardo Euler (1707–1783) and others. In fact, Fourier developed his new idea for finding the solution of heat (or Fourier) equation in terms of Fourier series so that the Fourier series can be used as a practical tool for determining the Fourier series solution of partial differential equations under prescribed boundary conditions.

The Fourier transform originated from the Fourier integral theorem that was stated in the Fourier treatise titled *La Théore Analytique de la Chaleur,* and its deep significance has subsequently been recognized by mathematicians and physicists. It is generally believed that the theory of Fourier series and Fourier transforms is one of the most remarkable discoveries in mathematical sciences and it has widespread applications in mathematics, physics, and engineering. Both the Fourier series and Fourier transforms are related in many important ways. Many applications, including the analysis of stationary signals and real-time signal processing, make effective use of the Fourier transform in time and frequency domains.

In time-frequency analysis, the Fourier transform is one of the oldest tools to dominate signal processing. However, due to its drawbacks in the analysis of non-stationary signals, different alternative transforms have gained much popularity in recent years, including windowed Fourier transform, fractional Fourier transform, linear canonical transform, quadratic-phase Fourier transform, and so on. These transforms are known as generalizations of the classic Fourier transform.

The main reason for writing this book is to stimulate interactions among mathematicians, computer scientists, engineers, and economists, as well as biological and physical scientists. The text is suitable for advanced graduate students but is primarily intended for post-graduate students and researchers in wavelets and their applications.

The book begins with an elementary chapter that introduces general Fourier transforms like windowed Fourier transform, fractional Fourier transform, linear canonical transform, and quadratic-phase Fourier transform.

Hybrid transforms are constructed by associating the Wigner-Ville distribution (WVD) with widely known signal processing tools, such as fractional Fourier transform, linear canonical transform, offset linear canonical transform (OLCT), and their quaternion-valued versions. Chapter 2 summarizes research on hybrid transforms by reviewing a computationally efficient type of WVD-OLCT, which has simplicity in marginal properties compared to classic WVD-OLCT and WVD.

Quadratic-phase Fourier transform (QPFT) as a general integral transform has been generalized into Wigner distribution (WD) and ambiguity function (AF) to show a more powerful ability for non-stationary signal processing. Chapter 3 proposes a new version of AF associated with QPFT referred to as scaled AF. This new version of AF is defined based on the QPFT and the fractional instantaneous autocorrelation.

Chapter 4 presents analytical expressions of infinite Fourier sine and cosine transform-based Ramanujan integrals in an infinite series of hypergeometric functions using the hypergeometric technique. Moreover, as applications of Ramanujan's integrals, some closed form of infinite summation formulae involving hypergeometric functions are derived.

Chapter 5 is devoted to the recursive algorithms for harmonic analysis, one of which is the resonator-based algorithm. The approach of the parallel cascades of multipleresonators (MRs) with the common feedback is generalized as the cascaded-resonator (CR)-based structure for recursive harmonic analysis.

> **Dr. Mohammad Younus Bhat** Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, India

Section 1

Transforms

Section 1 Transforms

#### **Chapter 1**

## Introductory Chapter: The Generalizations of the Fourier Transform

*Mohammad Younus Bhat*

#### **1. Introduction**

In the world of physical science, important physical quantities such as sound, pressure, electric current, voltage, and electromagnetic fields vary with time t. Such quantities are labeled as signals/waveforms. Exemplified by signals with examples such as oral signals, optical signals, acoustic signals, biomedical signals, radar, and sonar. Indeed, signals are very common in the real world. Time-frequency analysis is a vital aid in signal analysis, which is concerned with how the frequency of a function (or signal) behaves in time, and it has evolved into a widely recognized applied discipline of signal processing. The signals can be classified under various categories. It could be done in terms of continuity (continuous v/s discrete), periodicity(periodic v/s aperiodic), stationarity(stationary v/s non-stationary), and so on. Most of the signals in nature are non-stationary (i.e., whose spectral components change with time) and apt presentation of such non-stationary signals need frequency analysis, which is local in time, resulting in the time-frequency analysis of signals. Although time frequency analysis of signals had its origin almost 70 years ago, there has been major development of the time-frequency distribution approach in the last three decades. The basic idea of these methods is to develop a joint function of time and frequency, known as a time-frequency distribution, that can describe the energy density of a signal simultaneously in both time and frequency domains. In signal processing, time-frequency analysis comprises those techniques that study signal in both the time and frequency domains simultaneously, using various time-frequency representations/tools known as integral transformations. An integral transform maps a function/signal from one function space into another function space *via* integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The integral transforms are essentially considered from the functional analysis viewpoint and as a useful technique of mathematical physics.

The classical Fourier transform (FT) is an integral transform introduced by Joseph Fourier in 1807 [1], is one of the most valuable and widely-used integral transforms that converts a signal from time versus amplitude to frequency versus amplitude. Thus FT can be considered as the time-frequency representation tool in signal processing and analysis. A fundamental limitation of the Fourier transform is that the all properties of a signal are global in scope. Information about local features of the signal, such as changes in frequency, becomes a global property of the signal in the frequency domain. In order to circumvent these drawbacks of FT, authors in Ref. [2] introduced the generalizations of FT that includes short-time Fourier transform (STFT) by performing the FT on a block-by-block basis rather than to process the entire signal at once. In spite of the fact that STFT did much to ameliorate the limitations of FT, still in some cases the STFT cannot track the signal dynamics properly for a signal with both very high frequencies of short duration and very low frequencies of long duration. To overcome these drawbacks of FT and STFT different novel generalizations of the classical Fourier transform came into existence *viz.*: the fractional Fourier transform (FRFT), the Fresnal transform, the linear canonical transform (LCT), the quadratic-phase Fourier transform (QPFT), and so on. As a generalization of classical Fourier transform, the FRFT, the LCT, the QPFT gained its ground intermittently and profoundly influenced several branches of science and engineering including signal and image processing, quantum mechanics, neural networks, differential equations, optics, pattern recognition, radar, sonar, and communication systems.

#### **2. Fourier transform and its generalizations**

#### **2.1 Fourier transform**

Joseph Fourier [1] in 1822 published first work about Fourier transform, which is an integral transform that converts a mathematical function from the time domain to the frequency domain. Fourier transform measures the frequency component of a given function. The Fourier transform has evolved into a widely recognized discipline of harmonic analysis and has been successfully applied in diverse scientific and engineering pursuits [3–6].

Let us begin with definition of the classical Fourier transform.

**Definition 1.** *The FT of any signal x t*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ð Þ ℝ *is defined and denoted as*

$$
\mathcal{F}[\varkappa(t)](\xi) = \hat{\varkappa}(\xi) = \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi t} \varkappa(t) dt,\tag{1}
$$

and corresponding inversion formula is given by

$$\mathcal{F}^{-1}(\mathcal{F}[\varkappa(t)](\xi))(t) = \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{i\xi t} \mathcal{F}[\varkappa(t)](\xi) d\xi. \tag{2}$$

**Example 1.** *Consider a function x t*ðÞ¼ *<sup>e</sup>*�*α<sup>t</sup> for t*≥0, *<sup>α</sup>* <sup>&</sup>gt;0, 0 *otherwise*; � *, then the Fourier transform of x t*ð Þ *is obtained as*

$$\begin{split} \mathcal{F}[x(t)](\xi) &= \frac{1}{\sqrt{2\pi}} \int\_{0}^{\infty} e^{-i\xi t} e^{-\alpha t} dt \\ &= \frac{1}{\sqrt{2\pi}} \int\_{0}^{\infty} (\cos \xi t - i \sin \xi t) e^{-\alpha t} dt \\ &= \frac{1}{\sqrt{2\pi}} \left\{ \int\_{0}^{\infty} \cos \xi t e^{-\alpha t} dt - i \int\_{0}^{\infty} \sin \xi t e^{-\alpha t} dt \right\} \\ &= \frac{1}{\sqrt{2\pi}} \left\{ \frac{\alpha}{\alpha^{2} + \xi^{2}} - \frac{i\xi}{\alpha^{2} + \xi^{2}} \right\}. \end{split}$$

*Introductory Chapter: The Generalizations of the Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.112175*

**Example 2.** *Consider the function*

$$\mathfrak{x}(t) = \begin{cases} \sin 3t & \text{for } -\pi \le t \le \pi, \\ 0 & \text{otherwise.} \end{cases}$$

*Then the Fourier transform of x t*ð Þ *is obtained as*

$$\begin{split} \mathcal{F}[\mathbf{x}(t)](\xi) &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} (\cos \xi t - i \sin \xi t) \sin 3t dt \\ &= \frac{-i}{\sqrt{2\pi}} \int\_{-\pi}^{\pi} \sin \xi t \sin 3t dt \\ &= \frac{i 3 \sqrt{2} \sin \xi \pi}{\sqrt{\pi} (\xi^2 - 9)}. \end{split}$$

Next, we shall study some properties of FT. **Theorem 1** (Translation)**.** *The Fourier transform of any function x t*ð Þ � *k is given by*

$$
\mathcal{F}[\varkappa(t-k)](\xi) = e^{-i\xi k} \mathcal{F}[\varkappa(t)](\xi). \tag{3}
$$

*Proof.* From Definition 1, we have

$$\begin{split} \mathcal{F}[\mathbf{x}(t-k)](\xi) &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi t} \mathbf{x}(t-k) dt \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi(u+k)} \mathbf{x}(u) du \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi k} e^{-i\xi u} \mathbf{x}(u) du \\ &= \frac{1}{\sqrt{2\pi}} e^{-i\xi k} \int\_{\mathbb{R}} e^{-i\xi u} \mathbf{x}(u) du \\ &= e^{-i\xi k} \mathcal{F}[\mathbf{x}(t)](\xi). \end{split}$$

This completes the proof. □ **Theorem 2** (Modulation)**.** *The Fourier transform of any function e<sup>i</sup>ξ*0*<sup>t</sup> x t*ð Þ *is given by*

$$
\mathcal{F}\left[e^{i\xi\_0 t}\mathbf{x}(t)\right](\xi) = \mathcal{F}[\mathbf{x}(t)](\xi-\xi\_0). \tag{4}
$$

*Proof.* From Definition 1, we have

$$\begin{split} \mathcal{F}\left[e^{i\xi\_{0}t}\mathfrak{x}(t)\right](\xi) &= \frac{1}{\sqrt{2\pi}}\int\_{\mathbb{R}} e^{-i\xi t}e^{i\xi\_{0}t}\mathfrak{x}(t)dt\\ &= \frac{1}{\sqrt{2\pi}}\int\_{\mathbb{R}} e^{-i(\xi-\xi\_{0})t}\mathfrak{x}(t)dt\\ &= \mathcal{F}[\mathfrak{x}(t)](\xi-\xi\_{0}). \end{split}$$

This completes the proof. □ **Theorem 3** (Orthogonality relation)**.** *The Fourier transform of the functions x t*ð Þ *and y t*ð Þ *in L*<sup>2</sup> ð Þ ℝ *satisfies the following orthogonality relation*

$$
\langle \mathcal{F}[\mathfrak{x}(t)], \mathcal{F}[\mathfrak{y}(u)] \rangle = \langle \mathfrak{x}(t), \mathfrak{y}(u) \rangle. \tag{5}
$$

*Proof.* We have

$$
\begin{split}
\langle\mathcal{F}[\mathbf{x}(t)],\mathcal{F}[\mathbf{y}(u)]\rangle &= \int\_{\mathbb{R}} \mathcal{F}[\mathbf{x}(t)](\xi) \overline{\mathcal{F}[\mathbf{y}(u)](\xi)} d\xi \\ &= \int\_{\mathbb{R}} \mathcal{F}[\mathbf{x}(t)](\xi) \overline{\left(\frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi u} \mathbf{y}(u) du\right)} d\xi \\ &= \int\_{\mathbb{R}} \left(\frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi u} \mathbf{x}(t) dt\right) \left(\frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{i\xi u} \overline{\mathbf{y}(u)} du\right) d\xi \\ &= \int\_{\mathbb{R}^2} \mathbf{x}(t) \overline{\mathbf{y}(u)} \left(\frac{1}{2\pi} \int\_{\mathbb{R}} e^{i\xi (u-t)} d\xi\right) dt du \\ &= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\mathbf{y}(u)} \delta(u-t) dt du \\ &= \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\mathbf{y}(u)} dt \\ &= \langle \mathbf{x}(t), \mathbf{y}(u) \rangle.
\end{split}
$$

This completes the proof. □ Note: If we take *x t*ðÞ¼ *y t*ð Þ, the orthogonality relation yields Plancherel's Theorem for the Fourier transforms that states the energy of a signal ln the time domain, is the same as the energy in the frequency domain given as

$$\|\|\mathcal{F}(\mathfrak{x}(t))\|\| = \|\mathfrak{x}(t)\|\|.\tag{6}$$

Next, we show that the inverse Fourier operator is the adjoint of the Fourier operator.

**Theorem 4.** *Let x t*ð Þ *and y t*ð Þ *in L*<sup>2</sup> ð Þ ℝ *, then*

$$
\langle \mathcal{F}[\mathbf{x}(t)](\xi), \mathbf{y}(\xi) \rangle = \langle \mathbf{x}(t), \mathcal{F}^{-1}[\mathbf{y}](t) \rangle. \tag{7}
$$

*Proof.* We have

$$
\begin{split}
\langle\mathcal{F}[\mathbf{x}(t)],\mathbf{y}(t)\rangle &= \int\_{\mathbb{R}} \mathcal{F}[\mathbf{x}(t)](\xi)\overline{\mathbf{y}(\xi)}d\xi \\&= \int\_{\mathbb{R}} \left(\frac{1}{\sqrt{2\pi}}\int\_{\mathbb{R}} e^{-i\xi t}\mathbf{x}(t)dt\right)\overline{\mathbf{y}(\xi)}d\xi \\&= \int\_{\mathbb{R}} \mathbf{x}(t) \left(\frac{1}{\sqrt{2\pi}}\int\_{\mathbb{R}} e^{-i\xi t}\overline{\mathbf{y}(\xi)}d\xi\right)dt \\&= \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\left(\frac{1}{\sqrt{2\pi}}\int\_{\mathbb{R}} e^{i\xi t}\mathbf{y}(\xi)d\xi\right)}dt \\&= \int\_{\mathbb{R}} \mathbf{x}(t)\overline{\mathcal{F}^{-1}[\mathbf{y}](t)}dt \\&= \left\langle \mathbf{x}(t), \mathcal{F}^{-1}[\mathbf{y}](t) \right\rangle.
\end{split}
$$

This completes the proof. □

**Theorem 5.** *Let x t*ð Þ *and y t*ð Þ *in L*<sup>2</sup> ð Þ ℝ *, then*

$$
\mathcal{F}[(\mathfrak{x}\ast\mathfrak{y})](\xi) = \sqrt{2\pi}\,\,\mathcal{F}[\mathfrak{x}(t)](\xi)\,\mathcal{F}[\mathfrak{y}(t)](\xi),\tag{8}
$$

where *x* ∗ *y* denotes the convolution of the functions *x t*ð Þ and *y t*ð Þ and is given by

$$(\mathfrak{x} \* \mathfrak{y})(t) = \int\_{\mathbb{R}} \mathfrak{x}(t)\mathfrak{y}(\mathfrak{u} - t)dt.$$

*Proof.* By applying definition of Fourier transform to the convolution of the functions *x t*ð Þ and *y t*ð Þ, we obtain

$$\begin{split} \mathcal{F}[(\mathbf{x}\*\mathbf{y})](\xi) &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} (\mathbf{x}\*\mathbf{y})(u) e^{-i\xi u} du \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} \left( \int\_{\mathbb{R}} \mathbf{x}(t) \mathbf{y}(u-t) dt \right) e^{-i\xi u} du \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbf{x}(t) \mathbf{y}(v) e^{-i\xi (t+v)} dv dt \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} \int\_{\mathbb{R}} e^{-i\xi t} \mathbf{x}(t) \mathbf{y}(v) e^{-i\xi v} dv dt \\ &= \sqrt{2\pi} \left\{ \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi t} \mathbf{x}(t) dt \right\} \left\{ \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi v} \mathbf{y}(v) dv \right\}. \end{split}$$
 
$$= \sqrt{2\pi} \mathcal{F}[\mathbf{x}(t)](\xi) \mathcal{F}[\mathbf{y}(t)](\xi).$$

This completes the proof. □

#### **2.2 Windowed Fourier transform**

**Definition 2.** *Let* Ψ *be a given window function in L*<sup>2</sup> ð Þ ℝ , *then the window Fourier transform (WFT) of any function x t*ð Þ∈*L*<sup>2</sup> ð Þ ℝ *is defined and denoted as*

$$\mathcal{V}\Psi[\mathbf{x}(t)](b,\xi) = \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi t} \mathbf{x}(t) \overline{\Psi(t-b)} dt, \quad b,\xi \in \mathbb{R}. \tag{9}$$

Further, the WFT (9) can be rewritten as

$$\mathcal{W}\_{\Psi}[\mathfrak{x}(t)](b,\xi) = \mathcal{F}\left[\mathfrak{x}(t)\overline{\Psi(t-b)}\right].\tag{10}$$

Applying inverse FT (2), (10) yields

$$\begin{split} \overline{\varkappa(t)\Psi(t-b)} &= \mathcal{F}^{-1}[\mathcal{V}\_{\Psi}[\varkappa(t)](b,\xi)] \\ &= \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} \varepsilon^{i\xi t} \mathcal{V}\_{\Psi}[\varkappa(t)](b,\xi) d\xi \end{split} \tag{11}$$

Multiplying (11) both sides by Ψð Þ *t* � *b* and then integrating with respect to *db*, we get

*Time Frequency Analysis of Some Generalized Fourier Transforms*

$$\left\|\mathbf{x}(t)\right\|\Psi\|^2 = \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} \int\_{\mathbb{R}} e^{i\xi t} \mathcal{V}\_{\Psi}[\mathbf{x}(t)](b,\xi)\Psi(t-b)d\xi db\,.$$

Equivalently, we have

$$\mathbf{x}(t) = \frac{1}{\sqrt{2\pi} \|\Psi\|^2} \int\_{\mathbb{R}} \int\_{\mathbb{R}} e^{i\xi t} \mathcal{V}\_{\Psi}[\mathbf{x}(t)](b,\xi)\Psi(t-b)d\xi db. \tag{12}$$

Eq. (12) gives the inversion formula corresponding to WFT (9).

**Theorem 6** (Orthogonality relation)**.** *For any two functions x t*ð Þ, *y t*ð Þ *in L*<sup>2</sup> ð Þ ℝ , *we have following relation*

$$
\langle \mathcal{V}\Psi[\mathbf{x}(t)](\mathbf{b},\xi), \mathcal{V}\Psi[\mathbf{y}(t)](\mathbf{b},\xi) \rangle = \|\Psi\|^2 \langle \mathbf{x}(t), \mathbf{y}(t) \rangle. \tag{13}
$$

*Proof.* By Definition (2), we have

$$\begin{split} \langle \mathcal{V}\Psi[\mathbf{x}(t)](b,\xi), \mathcal{V}\_{\Psi}[\mathbf{y}(t)](b,\xi) \rangle \\ &= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathcal{V}\_{\Psi}[\mathbf{x}(t)](b,\xi) \overline{\mathcal{V}\_{\Psi}[\mathbf{y}(t)](b,\xi)} d\xi db \\ &= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathcal{V}\_{\Psi}[\mathbf{x}(t)](b,\xi) \overline{\left(\frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi t} \mathcal{Y}(t) \overline{\Psi(t-b)} dt\right)} d\xi db \\ &= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \left(\frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{i\xi t} \mathcal{V}\_{\Psi}[\mathbf{x}(t)](b,\xi) d\xi\right) \overline{\mathcal{V}(t)} \Psi(t-b) dt db. \end{split} \tag{14}$$

By virtue of Eq. (11), (14) yields

$$\begin{split} \langle \mathcal{V}\Psi[\mathbf{x}(t)](b,\xi), \mathcal{V}\_{\Psi}[\mathbf{y}(t)](b,\xi) \rangle \\ &= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\Psi(t-b)} \Psi(t-b) \overline{\boldsymbol{y}(t)} dt db \\ &= \int\_{\mathbb{R}} \boldsymbol{\varkappa}(t) \overline{\boldsymbol{y}(t)} dt \int\_{\mathbb{R}} \overline{\Psi(t-b)} \Psi(t-b) db \\ &= ||\Psi||^{2} \langle \boldsymbol{\varkappa}(t), \boldsymbol{y}(t) \rangle. \end{split} \tag{15}$$

This completes the proof. □ Next, we introduce the fractional Fourier transform as a generalization of the classical Fourier transform.

#### **2.3 Fractional Fourier transform**

It is well known that when one performs the FT two times, the time-reverse operation is obtained. When one performs the FT three times, the inverse FT is obtained. Furthermore, performing the FT four times is equivalent to performing an identity operation. Now, one may think what will be obtained when the FT is performed a noninteger number of times The fractional Fourier transform (FRFT) can be viewed as performing the FT 2f g *α=π* times, where 2f g *α=π* can be a non-integer value. The fractional Fourier transform (FRFT) has played an important role in signal processing [7] optics [8, 9], image processing [10], and quantum mechanics [11]. As a generalization of the conventional Fourier transform (FT), the FRFT implements an order parameter

*Introductory Chapter: The Generalizations of the Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.112175*

which acts on the conventional Fourier transform operator and can process timevarying signals and non-stationary signals. With variation of the fractional parameter, the FRFT transforms the signal into the fractional Fourier domain representation, which is oriented by corresponding rotation angle with respect to the time axis in the counterclockwise direction. Using a global kernel, the FRFT shows the overall fractional Fourier domain contents. Hence, the time-frequency representation should be extended to the time-fractional Fourier frequency domain. Let us define fractional Fourier transform.

**Definition 3.** *Let x t*ð Þ *be a signal in L*<sup>2</sup> ð Þ ℝ *, then the fractional Fourier transform of x t*ð Þ *is defined as*

$$\mathcal{F}\_a[\mathbf{x}(t)](\xi) = \int\_{\mathbb{R}} K\_a(t,\xi)\mathbf{x}(t)dt,\tag{16}$$

where *α* is a angular parameter and *Kα*ð Þ *t*, *ξ* is the kernel of the FRFT and is given by

$$K\_{a}(t,\xi) = \begin{cases} \sqrt{\frac{1-i\cot a}{2\pi}} e^{\frac{i}{2}(t^{2}+\xi^{2})\cot a - i\xi\csc a} & \text{for} \quad a \neq n\pi, \\\delta(t-\xi) & \text{for} \quad a = 2n\pi, \\\delta(t+\xi) & \text{for} \quad a = (2n+1)\pi, \quad n \in \mathbb{Z}. \end{cases} \tag{17}$$

and the corresponding inversion formula is also a FRFT with angle �*α* and is given by

$$\mathbf{x}(t) = \mathcal{F}\_{-a}\{\mathcal{F}\_a[\mathbf{x}(t)](\xi)\}(t) = \int\_{\mathbb{R}} \mathcal{F}\_a[\mathbf{x}(t)](\xi) K\_{-a}(t,\xi) d\xi. \tag{18}$$

It is easy to see that, when *α* ¼ 0, *π=*2, *π* and 3*π=*2, the FRFT is reduced to the identity operation, the FT, time-reverse operation, and the IFT, respectively.

Assuming that *u t*ðÞ¼ *eit*<sup>2</sup> cot *<sup>α</sup>=*<sup>2</sup>*x t*ð Þ, then for *<sup>α</sup>* ¼6 *<sup>n</sup><sup>π</sup>* the FRFT (16) can be rewritten as

$$\begin{split} \mathcal{F}\_a[\mathbf{x}(t)](\xi) \\\\ = \sqrt{\frac{1 - i \cot a}{2\pi}} e^{i\xi^2 \cot a/2} \left( \frac{1}{\sqrt{2\pi}} \int\_{\mathbb{R}} e^{-i\xi \text{tr} \mathbf{c} \mathbf{a}} u(t) \right) \\\\ = \sqrt{\frac{1 - i \cot a}{2\pi}} e^{i\xi^2 \cot a/2} \mathcal{F}[u](\xi \text{csc} \mathbf{a}). \end{split} \tag{20}$$

It is clear from (20) that the FRFT can be viewed as a chirp-Fourier-chirp transformation.

Next, we highlight some properties of FRFT.

**Theorem 7.** *Let x t*ð Þ, *y t*ð Þ∈*L*<sup>2</sup> ð Þ ℝ *and k*, *ξ*<sup>0</sup> ∈ ℝ*, then the FRFT satisfies following properties:*

$$\text{1.Translation: } \mathbb{F}\_a[\mathbf{x}(t-k)](\xi) = e^{\frac{1}{2}ik^2 \cos a \sin a - ik \xi \sin a} \mathbb{F}\_a[\mathbf{x}(t)](\xi)(\xi - k \cos a).$$

$$\text{2.Modulation: } \mathcal{F}\_a \left[ e^{i\xi\_0 t} \mathbf{x}(t) \right](\xi) = e^{i\xi\_0 \xi \cos a - \frac{i}{2} \theta^2 \sin a \cos a} \mathcal{F}\_a[\mathbf{x}(t)](\xi - \xi\_0 \sin a).$$

3.*Orthogonality Relation:* h i ℱ*α*½ � *x t*ð Þ , ℱ*α*½ � *y t*ð Þ ¼ h i *x t*ð Þ, *y t*ð Þ *:*

*Proof.* For the sake of brevity, we omit proof of translation and modulation properties and prove only orthogonality relation.

We have

$$
\langle \overline{\mathcal{F}\_a}[\mathbf{x}(t)], \overline{\mathcal{F}\_a}[\mathbf{y}(t)] \rangle = \int\_{\mathbb{R}} \overline{\mathcal{F}\_a}[\mathbf{x}(t)](\xi) \overline{\mathcal{F}\_a}[\mathbf{y}(t)](\xi) d\xi
$$

$$
= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \int\_{\mathbb{R}} K\_a(t, \xi) \mathbf{x}(t) \overline{K\_a(s, \xi)} \mathbf{y}(s) d\xi dt d\xi
$$

$$
= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\mathbf{y}(s)} \left( \int\_{\mathbb{R}} K\_a(t, \xi) \overline{K\_a(s, \xi)} d\xi \right) ds dt
$$

$$
= \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\mathbf{y}(s)} \delta(t - s) ds dt
$$

$$
= \int\_{\mathbb{R}} \mathbf{x}(t) \overline{\mathbf{y}(s)} dt
$$

$$
= \langle \mathbf{x}(t), \mathbf{y}(t) \rangle.
$$

This completes the proof. □ Since the FRFT is a generalization of the FT, many properties, applications, and operations associated with FT can be generalized by using the FRFT. The FRFT is more flexible than the FT and performs even better in many signal processing and optical system analysis applications.

In the sequel, we introduce linear canonical transform, which is a generalized version of the classical Fourier transform with four parameters.

#### **2.4 Linear canonical transform**

The linear canonical transform (LCT) introduced by Moshinsky and Quesne [12] has a total of four parameters. It is not only a generalization of the FT, but also a generation of the FRFT, the scaling operation. As the FRFT, the LCT was first used for solving differential equations and analyzing optical systems. Recently, after the applications of FRFT were developed, the roles of the LCT for signal processing have also been examined. Due to the extra degrees of freedom and simple geometrical manifestation, the LCT is more flexible than other transforms and is as such suitable as well as powerful tool for investigating deep problems in science and engineering [13–16]. Now, we shall define linear canonical transform (LCT).

$$\textbf{Definition 4.}\text{ Consider the second order matrix }\textbf{M}\_{2 \times 2} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\text{. Then the linear}$$

*canonical transform of any x t*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ð Þ ℝ *with respect to the uni-modular matrix M*<sup>2</sup>�<sup>2</sup> ¼

$$
\begin{bmatrix} a & b \\ c & d \end{bmatrix} \text{ is defined by}
$$

$$
\mathcal{R}\_M[\mathbf{x}(t)](\xi) = \begin{cases}
\int\_{\mathbb{R}} \mathsf{K}\_M(t, \xi) \mathbf{x}(t) dt & b \neq \mathbf{0} \\\\ \sqrt{d} \exp\left\{\frac{c d \xi^2}{2}\right\} \mathbf{f}(d\xi) & b = \mathbf{0}.
\end{cases} \tag{21}
$$

**10**

where K*M*ð Þ *t*, *ξ* is the kernel of linear canonical transform and is given by

$$\mathcal{K}\_{\mathcal{M}}(t,\xi) = \frac{1}{\sqrt{2\pi i b}} e^{\frac{i}{\mathfrak{D}} \left(at^2 - 2t\xi + d\xi^2\right)}, \quad b \neq 0. \tag{22}$$

*When b* 6¼ 0, *the inverse LCT is given by*

$$f(t) = \mathcal{E}\_{M^{-1}}\left\{\mathcal{E}\_M[\mathbf{x}(\left(t\right)\right](\xi)](t) = \int\_{\mathbb{R}} \mathcal{E}^{M}[\mathbf{x}(t)](\xi) \overline{\mathcal{K}\_M(t,\xi)} d\xi \tag{23}$$

where the kernel <sup>K</sup>*M*ð Þ¼ *<sup>t</sup>*, *<sup>ξ</sup>* <sup>K</sup>*<sup>M</sup>*�<sup>1</sup> ð Þ *<sup>t</sup>*, *<sup>ξ</sup>* and *<sup>M</sup>*�<sup>1</sup> denotes the inverse of matrix *<sup>M</sup>*. For typographical convenience we write the matrix *M* ¼ ð Þ *a*; *b*;*c*; *d* .

By changing the matrix parameter *M* ¼ ð Þ *a*; *b*;*c*; *d* , the LCT boils down to various integral transforms such as:

• When *M* ¼ ð Þ 0, 1, �1, 0 , the LCT turns out to be Fourier transform(FT):

$$
\mathcal{E}\_M[\mathfrak{x}(t)] = \sqrt{-i}\mathcal{F}[\mathfrak{x}(t)].
$$

• When *M* ¼ ð Þ 0, �1,1,0 , the LCT turns out to be inverse Fourier transform(IFT):

$$
\mathcal{E}\_M[\mathbf{x}(t)] = \sqrt{i} \mathcal{F}^{-1}[\mathbf{x}(t)].
$$

• When *M* ¼ ð Þ cos *α*, sin *α*, � sin *α*, cos *α* , the LCT becomes the FRFT:

$$
\mathcal{E}\_M[\varkappa(t)] = \sqrt{e^{-ia}} \mathcal{F}\_a[\varkappa(t)].
$$

• When *<sup>M</sup>* <sup>¼</sup> *<sup>λ</sup>*, 0, 0, <sup>1</sup> *λ* � �, the LCT becomes a scaling operation:

$$
\mathcal{R}\_M[\mathbf{x}(t)] = \sqrt{\frac{1}{\lambda}} \mathbf{x}\left(\frac{\xi}{\lambda}\right).
$$

• When *M* ¼ ð Þ 1, 0, *β*, 1 , the LCT becomes a chirp multiplication operation:

$$
\mathcal{H}\_M[\mathfrak{x}(t)] = e^{\frac{i}{\hbar}\theta\xi^2}\mathfrak{x}(\xi).
$$

Moreover Fresnel transform can be viewed with matrix 1, ð Þ *b*,0,1 and the Laplace transform can be obtained with 0, ð Þ *i*, *i*, 0 .

From (21), we have for *b* 6¼ 0

$$\begin{split} \mathcal{E}\_{\mathcal{M}}[\mathbf{x}(t)](\xi) &= \int\_{\mathbb{R}} \mathcal{K}\_{\mathcal{M}}(t,\xi)\mathbf{x}(t)dt \\ &= \frac{1}{\sqrt{2\pi i b}} \int\_{\mathbb{R}} e^{\frac{i}{\hbar}(at^2 - 2t\xi + d\xi^2)} \mathbf{x}(t)dt \\ &= \frac{1}{\sqrt{2\pi i b}} e^{\frac{i}{\hbar}d\xi^2} \int\_{\mathbb{R}} e^{-\frac{i}{\hbar}t} \left(\mathbf{x}(t)e^{\frac{i}{\hbar}at^2}\right) dt \\ &= \frac{1}{\sqrt{2\pi i b}} e^{\frac{i}{\hbar}d\xi^2} \mathcal{F}[\mathbf{g}(t)](\xi/b), \end{split} \tag{24}$$

where *g t*ðÞ¼ *x t*ð Þ*e i* 2*bat*<sup>2</sup> .

Thus, it is clear from (24), that LCT can be regarded as a chirp-Fourier-chirp transformation.

Next, we investigate some basic properties associated with LCT.

**Theorem 8.** *Let x t*ð Þ, *y t*ð Þ∈*L*<sup>2</sup> ð Þ ℝ *and k*, *ξ*<sup>0</sup> ∈ ℝ*, then the LCT satisfies following properties:*

$$\text{1.Translation: } \mathcal{E}\_M[\varkappa(t-k)](\xi) = e^{ikc\xi - \frac{i}{2}\hbar^2 ac} \mathcal{E}\_M[\varkappa(t)](\xi - ak) \dots$$


*Proof.* To be specific, we shall only prove the translation property, the rest of the properties follows similarly.

For any real *k*, we have

$$\begin{split} \mathcal{H}\_{M}[\mathbf{x}(t-k)](\xi) &= \int\_{\mathbb{R}} \mathcal{K}\_{M}(t,\xi) \mathbf{x}(t-k) dt \\ &= \frac{1}{\sqrt{2\pi i b}} \int\_{\mathbb{R}} e^{\frac{i}{\mathsf{B}} \left(at^{2} - 2t\xi + d\xi^{2}\right)} \mathbf{x}(t) dt \\ &= \frac{1}{\sqrt{2\pi i b}} \int\_{\mathbb{R}} e^{\frac{i}{\mathsf{B}} \left(a(s+k)^{2} - 2(s+k)\xi + d\xi^{2}\right)} \mathbf{x}(s) ds \\ &= e^{ikc\xi - \frac{i}{\mathsf{B}}\hat{k}^{2}ac} \frac{1}{\sqrt{2\pi i b}} \int\_{\mathbb{R}} e^{\frac{i}{\mathsf{B}} \left(at^{2} - 2(\xi - ak) + d\left(\xi - ak^{2}\right)\right)} \mathbf{x}(s) ds \\ &= e^{ikc\xi - \frac{i}{\mathsf{B}}\hat{k}^{2}ac} \mathcal{Q}\_{\mathsf{R}}[\mathbf{x}(t)](\xi - ak). \end{split}$$

This completes the proof. □ Finally, we will define quadratic-phase Fourier transform.

#### **2.5 Quadratic-phase Fourier transform**

The most neoteric generalization of the classical Fourier transform (FT) with five real parameters appeared *via* the theory of reproducing kernels is known as the quadratic-phase Fourier transform (QPFT) [17]. It treats both the stationary and nonstationary signals in a simple and insightful way that are involved in radar, signal processing, and other communication systems [18–25]. Here, we gave the notation and definition of the quadratic-phase Fourier transform and study some of its properties.

**Definition 5.** *For a real parameter set* Λ ¼ ð Þ *a*, *b*,*c*, *d*,*e with b* 6¼ 0, *the quadraticphase Fourier transform of any signal f* ∈*L*<sup>2</sup> ð Þ ℝ *is defined as*

$$\mathcal{Q}\_{\Lambda}[\mathfrak{x}(t)](\xi) = \int\_{\mathbb{R}} K\_{\Lambda}(t,\xi)\mathfrak{x}(t)dt,\tag{25}$$

where *k*Λð Þ *t*, *ξ* is the kernel signal of the QPFT and is given by

$$K\_{\Lambda}(t,\xi) = \frac{1}{\sqrt{2\pi}}e^{-i\left(at^2 + b\xi t + c\xi^2 + dt + e\xi\right)},\tag{26}$$

and corresponding inversion formula is given by

$$\mathbf{x}(t) = \mathcal{Q}\_{\Lambda}^{-1}(\mathcal{Q}\_{\Lambda}[\mathbf{x}(t)](\xi))(t) = \int\_{\mathbb{R}} \overline{K\_{\Lambda}(t,\xi)} \mathcal{Q}\_{\Lambda}[\mathbf{x}(t)](\xi) d\xi. \tag{27}$$

The novel QPFT (5) can be considered as a cluster of several existing integral transforms ranging from the classical Fourier to the much recent special affine Fourier transform. Nevertheless, many signal processing operations, such as scaling,shifting and time reversal, can also be performed *via* the QPFT (5).

Now, we will establish some properties of the quadratic-phase Fourier transform. **Theorem 9.** *Let x t*ð Þ, *y t*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ð Þ ℝ *and k*, *ξ*<sup>0</sup> ∈ ℝ*, then the QPFT satisfies following properties:*

$$\begin{aligned} \text{1.Modulation: } & \mathcal{Q}\_{\Lambda} \left[ \boldsymbol{\varepsilon}^{\dagger \xi\_{0} t} \boldsymbol{\kappa}(t) \right] (\boldsymbol{\xi}) = \boldsymbol{\varepsilon}^{\dagger} (\boldsymbol{\varepsilon}^{\dagger} (\boldsymbol{b}^{-2} \boldsymbol{\varepsilon}\_{0}^{\dagger} - \boldsymbol{b}^{-1} \boldsymbol{\xi} \boldsymbol{\varepsilon}\_{0}) \cdots \boldsymbol{\kappa}^{-1} \boldsymbol{\xi}\_{0}) \boldsymbol{\mathcal{Q}}\_{\Lambda} [\boldsymbol{\kappa}(t)] (\boldsymbol{\xi} - \boldsymbol{b}^{-1} \boldsymbol{\xi}\_{0}) \boldsymbol{\lambda}, \\\\ \text{2.Parity: } & \mathcal{Q}\_{\Lambda} [\boldsymbol{\kappa}(-t)] (\boldsymbol{\xi}) = \mathcal{Q}\_{\Lambda} [\boldsymbol{\kappa}(t)] (-\boldsymbol{\xi}), \text{ where } \boldsymbol{\Lambda}^{\prime} = (\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}, -\boldsymbol{d}, -\boldsymbol{e}). \\\\ \text{3.conjunction: } & \mathcal{Q}\_{\Lambda} \left[ \overline{\boldsymbol{\kappa}(t)} \right] (\boldsymbol{\xi}) = \overline{\mathcal{Q}\_{-\Lambda} [\boldsymbol{\kappa}(t)] (\boldsymbol{\xi})}, \text{ where } -\Lambda = (-a, -b, -c, -d, -e). \\\\ \text{4.Orthogonality Relation: } & \{ \mathcal{Q}\_{\Lambda} [\boldsymbol{\kappa}(t)], \mathcal{Q}\_{\Lambda} [\boldsymbol{\eta}(t)] \} = \frac{1}{b} \langle \boldsymbol{\kappa}(t), \boldsymbol{y}(t) \rangle. \end{aligned}$$

*Proof.* For the sake of brevity, we avoid proof. □

$$\beth$$

### **Author details**

Mohammad Younus Bhat Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, India

\*Address all correspondence to: gyounusug@gmail.com

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

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[3] Boggess A, Narcowich FJ. A First Course in Wavelets with Fourier Analysis. Upper Saddle River, New Jersey: Prentice Hall; 2001

[4] Bracewell RN. The Fourier Transform and its Applications. Third ed. Boston: McGraw-Hill; 2000

[5] Howell KB. Principles of Fourier Analysis. Boca Raton: Chapman & Hall/ CRC; 2001

[6] Olson T. Applied Fourier Analysis. NewYork: Birkhauser; 2017

[7] Almeida L. The fractional Fourier transform and time-frequency representations. IEEE Transactions on Signal Processing. 2022;**42**(11): 3084-3091

[8] Ozaktas HM, Mendlovic D. Fourier transforms of fractional order and their optical interpretation. Optics Communication. 1993;**101**(3–4):163-169

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[10] Djurovic I, Stankovic S, Pitas I. Digital watermarking in the fractional Fourier transformation domain. Journal of Network and Computer Applications. 2001;**24**(2):167-173

[11] Namias V. The fractional order Fourier transform and its application to quantum mechanics. IMA Journal of Applied Mathematics. 1980;**25**(3): 241-265

[12] Moshinsky M, Quesne C. Linear canonical transformations and their unitary representations. Journal of Mathematical Physics. 1971;**12**(8): 1772-1780

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[17] Saitoh S. Theory of reproducing kernels: Applications to approximate solutions of bounded linear operator functions on Hilbert spaces. American Mathematical Society Transformation Series. 2010;**230**(2):107-134

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Quadratic Fourier transforms. Annals of Functional Analysis AFA. 2014;**5**(1): 10-23

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[23] Prasad A, Sharma PB. The quadraticphase Fourier wavelet transform. Mathematicsl Methods in the Applied Sciences. DOI: 10.1002/mma.6018

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## **Chapter 2** Hybrid Transforms

*Didar Urynbassarova and Altyn Urynbassarova*

#### **Abstract**

Hybrid transforms are constructed by associating the Wigner-Ville distribution (WVD) with widely-known signal processing tools, such as fractional Fourier transform, linear canonical transform, offset linear canonical transform (OLCT), and their quaternion-valued versions. We call them hybrid transforms because they combine the advantages of both transforms. Compared to classical transforms, they show better results in applications. The WVD associated with the OLCT (WVD-OLCT) is a class of hybrid transform that generalizes most hybrid transforms. This chapter summarizes research on hybrid transforms by reviewing a computationally efficient type of the WVD-OLCT, which has simplicity in marginal properties compared to WVD-OLCT and WVD.

**Keywords:** time-frequency analysis, Wigner-Ville distribution, offset linear canonical transform, hybrid transform, linear frequency modulated (LFM) signal

#### **1. Introduction**

The linear canonical transform (LCT) [1–4] and its generalization, the offset linear canonical transform (OLCT) [5, 6] are introduced to study non-stationary signals (audio, image, biomedical, linear frequency modulated (LFM) signals). OLCT has five degrees of freedom, and LCT has three degrees of freedom, which makes them more flexible than the well-known fractional Fourier transform (FrFT) [7] with one degree of freedom and the Fourier transform (FT) with no freedom. Various applications of LCT have been found in the different fields of optics and signal processing. In fact, the properties and applications of the OLCT are similar to the LCT, but they are more general than the LCT, thanks to its two extra parameters, which correspond to timeshift and frequency modulation. It is proven that the Wigner-Ville distribution (WVD) plays a major role in time-frequency signal analysis and processing.

The LFM signal is used in communications, radar and sonar systems. Consequently, LFM signal detection and estimation is one of the most important topics in engineering. The WVD and LCT/OLCT are used in LFM signal processing, but they have their disadvantages:


This results in poor performance under a low signal-to-noise ratio for detection and estimation. Recently, for the purpose to improve the performance of LFM signal detection and estimation, several researchers have associated WVD with the FrFT, LCT, and OLCT, respectively [8–27]. Results show that such transforms exploit the advantages of both transforms, which is why we call them hybrid transforms. The aim of this chapter is to review and summarize research on hybrid transforms by studying WVD association with the OLCT (WVD-OLCT) definitions and properties.

#### **2. Preliminaries**

#### **2.1 Wigner-Ville distribution**

FT analysis originated long ago and is used in many areas of mathematics and engineering, including quantum mechanics, wave propagation, turbulence, signal analysis and processing. In spite of remarkable success, the FT analysis seems to be inadequate for studying some problems for the following reasons:


Therefore, we see that FT is sufficient to study signals that are statistically invariant over time, e.g. stationary signals. Naturally, we are surrounded by many signals: audio, video, radar, biomedical signals, etc., all those signals are non-stationary. FT is insufficient to do a complete analysis for such signals because it requires both timefrequency representations of the signal. So it was necessary to define a single transformation of time and frequency domains.

Historically, Eugene Paul Wigner, the 1963 Nobel Prize winner in physics, in 1932 first introduced a fundamental nonlinear transformation to study quantum corrections for classical statistical mechanics in the form [28].

$$\mathcal{W}\_{\Psi}(\mathbf{x}, p) = \frac{1}{h} \int\_{\mathbb{R}} \Psi \left( \mathbf{x} - \frac{\tau}{2} \right) \overline{\Psi} \left( \mathbf{x} + \frac{\tau}{2} \right) \exp \left( \frac{ip\tau}{\hbar} \right) d\tau,\tag{1}$$

where the wave function *ψ*ð Þ *x* satisfies the one-dimensional Schrödinger equation, the quantum mechanical position *x* and momentum *p* are independent variables, and *h* ¼ 2*π*ℏ is the Planck constant. The Wigner distribution W*<sup>ψ</sup>* ð Þ *x*, *p* has many important properties and is found to behave as a distribution function defined on a phase space consisting of points ð Þ *x*, *p :* The most remarkable properties of the Wigner distribution include the marginal integrals in the position and momentum domains as follows [29, 30].

$$\begin{aligned} \int\_{\mathbb{R}} \mathcal{W}\_{\Psi}(\mathbf{x}, p) d\mathbf{x} &= |\varrho(p)|^2, \\ \int\_{\mathbb{R}} \mathcal{W}\_{\Psi}(\mathbf{x}, p) dp &= |\wp(\mathbf{x})|^2, \end{aligned} \tag{2}$$

and the total energy of the wave function *ψ* in the ð Þ *x*, *p* space

$$\int\_{\mathbb{R}^2} \mathcal{W}\_{\Psi}(\mathbf{x}, p) d\mathbf{x} dp = \int\_{\mathbb{R}} |\varphi(\mathbf{x})|^2 d\mathbf{x} = ||\varphi||. \tag{3}$$

In the context of non-stationary signal analysis, in 1948 Jean-Andre Ville independently re-derived the Wigner distribution given in Eq. (1) as a quadratic representation of the local time-frequency energy of a signal [31]. Besides linear time-frequency representations of a signal like the Gabor transform, the Zak transform, and the shorttime Fourier transform, the WVD (or Wigner-Ville transform (WVT)) occupies a central position in the field of quadratic time-frequency representations and it is recognized as a valuable method/tool for time-frequency of time-varying signals and non-stationary random processes.

With its remarkable structure and properties, the WVD has been regarded as the main distribution of all the time-frequency distributions and used as the classical and fundamental time-frequency analysis tool in different areas of physics and engineering. Particularly, it has been used for instantaneous frequency estimation, spectral analysis of random signals, detection and classification, algorithms for computer implementation, and has a wide range of applications in vision, X-ray diffraction of crystals, pattern recognition, radar, and sonar. Additionally, it has been applied to the analysis of seismic data, speech, and phase distortions in audio engineering problems.

Definition 1 (WVD). If *f* belong to the Hilbert space *L*<sup>2</sup> ð Þ , the WVD W*<sup>f</sup>* of signal *f* is defined as [3, 29, 30].

$$\mathcal{W}\_f(t,u) = \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{-iu\tau} d\tau. \tag{4}$$

It is easy to see that the WVD is the FT of the instantaneous autocorrelation function

$$R\_f(t, \tau) = f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)}\tag{5}$$

with respect to *τ:*

Some main properties of WVD are summarized in **Table 1**. For some recent works and surveys on the WVD, we refer readers to [3, 29, 30] and the references therein.

#### **2.2 Linear canonical transform**

The LCT is a four-parameter ð Þ *a*, *b*, *c*, *d* integral transform that was introduced in the 1970s by Collins, and Moshinsky and Quesne to analyze optical systems and solve differential equations [1, 2]. After the fast algorithm for calculating the discrete LCT was proposed in [32], the LCT was widely used to process non-stationary signals. It has been applied in radar system analysis, filter design, watermarking, phase retrieval, pattern recognition, signal synthesis, and in other areas of engineering sciences. With intensive research, many properties of the LCT are well studied. Transforms and operations, such as the FT, FrFT, Fresnel transform (FRST), Laplace transform, fractional Laplace transform, time scaling, and chirp operations are the special cases of the LCT.


**Table 1.** *Properties of the WVD.*

In some works, the LCT is known under different names as the Collins formula, Moshinsky and Quesne integrals, extended fractional Fourier transform, quadraticphase integral or quadratic-phase system, generalized Fresnel transform, generalized Huygens integral [33], ABCD transform [34], and affine Fourier transform [35], etc.

Definition 2 (LCT). The LCT L**<sup>A</sup>** of a signal *f t*ð Þ with matrix **A** ¼ ð Þ *a*, *b*, *c*, *d* , where *a*,*b*,*c*,*d*∈ are real parameters and detð Þ¼ **A** *ad* � *bc* ¼ 1, is defined as [2–4]

$$\mathcal{L}\_{\mathbf{A}}\{f(t)\}(u) = \begin{cases} \int\_{\mathbb{R}} f(t) \frac{\mathbf{1}}{\sqrt{i2\pi b}} e^{i\left(\frac{d^2}{2b} - \frac{1}{b}tu + \frac{d}{2b}u^2\right)} dt, b \neq 0, \\\ \sqrt{d} e^{\sharp du^2} f(du), b = 0. \end{cases} \tag{6}$$

From the definition of LCT, we can see that, when the parameter *b* ¼ 0, the LCT is a scaling transformation coupled with amplitude and quadratic phase modulation and it is of no particular interest to our object. Therefore, without loss of generality, in this chapter we always assume *b* 6¼ 0.

A detailed and comprehensive view of LCT can be found in [2, 3] and the references therein.

#### **2.3 Offset linear canonical transform**

The OLCT is a six-parameter ð Þ *a*, *b*, *c*, *d*, *u*0, *ω*<sup>0</sup> integral transform, which has been shown as a powerful tool and received much attention in signal processing and optics. It is a time-shifted and frequency-modulated version of the LCT. In some works OLCT called the special affine Fourier transform [35–37] and the inhomogeneous canonical transform [38].

Definition 3 (OLCT). The OLCT O**<sup>A</sup>** of a signal *f t*ð Þ with real parameters of matrix **A** ¼ ð Þ *a*, *b*, *c*, *d*, *u*0, *ω*<sup>0</sup> , where *a*,*b*,*c*,*d*,*u*0,*ω*<sup>0</sup> ∈ are real parameters and detð Þ¼ **A** 1, is defined as [6, 19]

$$\mathcal{O}\_{\mathbf{A}}\{f(t)\}(u) = \begin{cases} \int\_{\mathbb{R}} f(t) K\_{\mathbf{A}}(t, \ u) dt, \ b \neq \mathbf{0}, \\\sqrt{d} \epsilon^{\frac{sd}{2}(u - u\_0)^2 + jou} f(d(u - u\_0)), \ b = \mathbf{0}. \end{cases} \tag{7}$$

where *K***A**ð Þ *t*, *u* is the OLCT kernel and expressed as

$$K\_{\mathbf{A}}(t,u) = \frac{1}{\sqrt{i2\pi b}} e^{i\left(\frac{a}{2b}t^2 - \frac{1}{b}(u-u\_0) + \frac{d}{2b}\left(u^2 + u\_0^2\right) - \frac{u}{b}(du\_0 - bu\_0)\right)}.\tag{8}$$

From Eq. (7) it can be seen that for case *b* ¼ 0 the OLCT is simply a time scaled version of *f* multiplied by a linear chirp. Therefore, from now we restrict our attention to OLCT for the case *b* 6¼ 0*:* And without loss of generality, we assume *b*>0 in the following sections of this chapter.

A number of widely known classical transforms and mathematical operations related to signal processing and optics are special cases of the OLCT. The OLCT converts to its special cases when taking different parameters of matrix **A***:* For example, the OLCT with parameters ð*a*, *b*, *c*, *d*, *u*0, *ω*0Þ ¼ ð Þ *a*, *b*, *c*, *d*, 0, 0 reduces to LCT; when **A** ¼ ð Þ cos *θ*, sin *θ*, � sin *θ*, cos *θ*, 0, 0 , it becomes the FrFT; when **A** ¼ ð Þ 0, 1, �1, 0, 0, 0 , the OLCT becomes FT; when **A** ¼ ð Þ 1, *b*, 0, 1, 0, 0 , it becomes FRST; and when **<sup>A</sup>** <sup>¼</sup> *<sup>d</sup>*�<sup>1</sup> , 0, 0, *d*, 0, 0 � �, it becomes time scaling operation. Multiplication by Gaussian or chirp function is obtained with an **A** ¼ ð Þ 1, 0, *τ*, 1, 0, 0 [1]. The offset Fourier transform **A** ¼ ð Þ 0, 1, �1, 0, *u*0, *ω*<sup>0</sup> , offset fractional Fourier transform **A** ¼ ð Þ cos *θ*, sin *θ*, � sin *θ*, cos *θ*, *u*0, *ω*<sup>0</sup> , frequency modulation **A** ¼ ð Þ 1, 0, 0, 1, 0, *ω*<sup>0</sup> , and time shifting **A** ¼ ð Þ 1, 0, 0, 1, *u*0, 0 are also special cases of the OLCT. The OLCT is able to extend their properties and applications and can solve some problems that cannot be solved well by these operations. In fact, offset versions of FT, FrFT, and LCT are similar to the classical FT, FrFT, and LCT, but they are more flexible than the classical ones, and mainly useful for analyzing optical systems with prisms or shifted lenses. The OLCT has a close relationship with its special cases. So it is practically useful to develop relevant theorems for OLCT. By developing theories for OLCT, we can gain a deeper understanding of its special cases and transfer knowledge from one subject to another. As a generalization of many other linear transforms, the OLCT has found wide applications in applied mathematics, signal processing, and optical system modeling [5, 6, 19, 34, 35, 37].

#### **2.4 Previous results**

With the development of the FrFT, Lohmann in [8] and Almeida in [9] investigated the relationship between the WVD and the FrFT. They show that the WVD of the FrFTed signal can be seen as a rotation of the WVD in the time-frequency plane. In this direction, based on the properties of the FrFT, the LCT, and the WVD, Pei and Ding [10] investigated and discussed the relations between the common fractional and canonical operators. The WVD associated with the LCT, named LCWD, denoted as WD**A**, given in [10] is useful for the separation of multi-component signals. It is defined as [10, 18].

$$\mathcal{W}\mathcal{D}\_{\mathbf{A}}(\boldsymbol{u},\boldsymbol{v}) = \int\_{\mathbb{R}} \mathcal{L}\_{\mathbf{A}}\left(\boldsymbol{u} + \frac{\boldsymbol{\pi}}{2}\right) \overline{\mathcal{L}\_{\mathbf{A}}\left(\boldsymbol{u} - \frac{\boldsymbol{\pi}}{2}\right)} e^{-i\boldsymbol{v}\boldsymbol{\pi}} d\boldsymbol{\pi},\tag{9}$$

where L**A**ð Þ *u* is the LCT of signal *f t*ð Þ with parameter matrix **A** ¼ ð Þ *a*, *b*, *c*, *d :*

Unlike the definition of LCWD, Bai et al. obtained generalized type of WVD in the LCT domain, named WVD-LCT (or WDL), denoted as WDL*<sup>f</sup>* , by substituting FT kernel *<sup>e</sup>*�*iu<sup>τ</sup>* with LCT kernel <sup>1</sup>ffiffiffiffiffiffi *<sup>i</sup>*2*π<sup>b</sup>* <sup>p</sup> *e i <sup>a</sup>* 2*bτ*2�<sup>1</sup> *<sup>b</sup>τu*<sup>þ</sup> *<sup>d</sup>* <sup>2</sup>*bu*<sup>2</sup> ð Þ [11].

$$\mathcal{WDL}\_f(t,u) = \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} \frac{1}{\sqrt{i2\pi b}} e^{i\left(\frac{d}{\mathfrak{B}}\tau^2 - \frac{1}{\mathfrak{F}}\tau u + \frac{d}{\mathfrak{B}}a^2\right)} d\tau. \tag{10}$$

The WVD-LCT generalizes the LCWD and WVD. It is easy to see that the WVD-LCT is the LCT of the instantaneous autocorrelation function *Rf*ð Þ *t*, *τ* with respect to *τ*

$$\mathcal{WDL}\_f(t,\mathfrak{u}) = \mathcal{L}\_\mathbf{A}\{R\_f(t,\ \mathfrak{r})\}.\tag{11}$$

Also, in [11] authors derived the main properties and applications of the WVD-LCT in the LFM signal detection. Uncertainty principles for the WVD-LCT were studied in [13, 25]. Song et al. presented WVD-LCT applications for quadratic frequency modulated signal parameter estimation in [14]. Convolution and correlation theorems for WVD-LCT are obtained in [16]. In [26] authors proposed a new method of instantaneous frequency estimation by associating the WVD with the LCT, which has a higher capacity for anti-noise and a higher estimation accuracy than WVD. Zhang unified LCWD and WVD-LCT [20], and then presented its special cases with less parameters [21, 22]. Urynbassarova et al. presented the WVD associated with the instantaneous autocorrelation function in the LCT domain, named WL, which has elegance and simplicity in marginal properties and affine transformation relationships compared to the WVD [17]. Similar to this in [27] Xin and Li proposed a new definition of WVD associated with LCT, and its integration form, which estimates two phase coefficients of LFM signal simultaneously and effectively suppresses cross terms for multi-component LFM signal. In [19] introduced the WVD association with the OLCT (WVD-OLCT), which is a generalization of the WVD-LCT and its special cases. Recently, in order to study higher dimensions, WVD associations with the quaternion LCT/OLCT were studied in [39–42], and WVD in the framework of octonion LCT was proposed by Dar and Bhat [43].

#### **3. Definition**

The WVD given in Eq. (4) can be re-written as

$$\mathcal{W}\_f(t,u) = \int\_{\mathbb{R}} f\_{\mathcal{F}}\left(t + \frac{\tau}{2}\right) \overline{f\_{\mathcal{F}}\left(t - \frac{\tau}{2}\right)} d\tau,\tag{12}$$

where *<sup>f</sup>* <sup>ℱ</sup> equals to *f t*ð Þ multiplied with FT kernel *<sup>e</sup>*�*iut:* By substituting FT kernel *e*�*iut* with OLCT kernel (Eq. (8)), we will get the following definition of the WVD in the OLCT domain, named WOL, denoted as WOL*<sup>f</sup>* , which is the type of the WVD-OLCT.

Definition 4 (WOL). The WOL WOL*<sup>f</sup>* of signal *f* for the parameter matrix **A** ¼ ð Þ *a*, *b*, *c*, *d*, *u*0, *ω*<sup>0</sup> is defined as follows [18]

$$\mathcal{WOL}\_f(t,u) = \frac{1}{2\pi|b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\frac{i}{\hbar}\tau t} e^{\frac{i}{\hbar}\tau(u\_0 - u)} d\tau.$$

The WOL is reduced to the WL, when **A** ¼ ð Þ *a*, *b*, *c*, *d*, 0, 0 ,

$$\mathcal{WOL}\_f^{(a,\ b,\ c,\ d,\ 0,\ 0)}(t,u) = \mathcal{Wcl}\_f(t,u). \tag{13}$$

Obviously, when the parameter matrix has the special form **A** ¼ ð Þ 0, 1, �1, 0, 0, 0 , the WOL is reduced to the WVD

$$\mathcal{WOL}\_f^{(0,\ 1,\ -1,\ 0,\ 0,\ 0)}(t,u) = \mathcal{W}\_f(t,u). \tag{14}$$

It is clear from Eq. (13) and Eq. (14) that the WOL is a generalization of the WL and the WVD.

#### **4. Properties**

Bellow we list some basic properties of the WOL. *Conjugation symmetry property.* The conjugation symmetry property of the WOL is expressed as

$$\mathcal{WOL}\_f(\mathfrak{t}, \mathfrak{u}) = \overline{\mathcal{WOL}\_f(\mathfrak{t}, \mathfrak{u})}.\tag{15}$$

Proof. From the Definition 4, we have

$$\begin{split} \overline{\mathcal{WOL}\_f(t,u)} &= \overline{\int\_{\mathbb{R}} f\left(t+\frac{\tau}{2}\right) \overline{f\left(t-\frac{\tau}{2}\right)} e^{\stackrel{i\omega}{\tau}t} e^{\stackrel{i}{\tau}(u\_0-u)} d\tau} \\ &= \int\_{\mathbb{R}} \overline{f\left(t+\frac{\tau}{2}\right)} f\left(t-\frac{\tau}{2}\right) e^{\stackrel{i\omega}{\tau}(-\tau)t} e^{\stackrel{i}{\tau}(-\tau)(u\_0-u)} d\tau, \end{split} \tag{16}$$

let �*τ* ¼ *τ*<sup>0</sup> , then we will arrive at

$$\begin{split} \overline{\mathcal{WOL}\_f(t, u)} &= \int\_{\mathbb{R}} f\left(t + \frac{\tau'}{2}\right) \overline{f\left(t - \frac{\tau'}{2}\right)} e^{\frac{i\omega}{\hbar}\tau t} e^{\frac{i}{\hbar}\tau t \left(u\_0 - u\right)} d\tau' \\ &= \mathcal{WOL}\_f(t, u) . \blacksquare} \end{split} \tag{17}$$

This property shows that the WOL is always a real number. *Time marginal property.*

The time marginal property of the WOL is given as

$$\int\_{\mathbb{R}} \mathcal{W} \mathcal{OL}\_f(t, u) du = \left| f(t) \right|^2. \tag{18}$$

Proof.

$$\begin{split} \int\_{\mathbb{R}} \mathcal{WOL}\_{f}(t,u) du &= \frac{1}{2\pi |b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\dot{\mu}\tau} e^{\dot{\mu}\tau(u u - u)} d\tau du \\ &= \frac{1}{2\pi |b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\dot{\overline{\nu}}\tau t} e^{\dot{\overline{\nu}}u \tau} \left(\int\_{\mathbb{R}} e^{-\dot{\overline{\nu}}\tau t} du\right) d\tau \\ &= \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\dot{\overline{\nu}}\tau^2} e^{\dot{\overline{\nu}}u \circ \tau} \delta(\tau) d\tau \\ &= |f(t)|^2 \cdot \blacksquare \end{split} \tag{19}$$

*Frequency marginal property.*

The frequency marginal property of the WOL is given by

$$\int\_{\mathbb{R}} \mathcal{WOL}\_f(t, u) dt = \left| \hat{f}(u) \right|^2. \tag{20}$$

Proof.

$$\int\_{\mathbb{R}} \mathcal{WOL}\_{\tilde{f}}(t, u) dt = \frac{1}{2\pi |b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\frac{i\omega t}{\tilde{\tau}}} e^{-\frac{i}{\tilde{\tau}} (u - u\_0)\tau} d\tau dt$$

$$= \frac{1}{2\pi |b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\frac{i\omega}{\tilde{\tau}} \left(t^2 + t\tau + \frac{\tau^2}{4} - t^2 + t\tau - \frac{\tau^2}{2}\right)} e^{-\frac{i}{\tilde{\tau}} (u - u\_0)\left(t + \frac{\tau}{2} + \frac{\tau}{2} - t\right)} d\tau dt$$

$$= \frac{1}{2\pi |b|} \int\_{\mathbb{R}} f\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{\frac{i\omega}{\tilde{\tau}} \left(t + \frac{\tau}{2}\right)^2} e^{-\frac{i}{\tilde{\tau}} \left(t - \frac{\tau}{2}\right)^2} e^{-\frac{i}{\tilde{\tau}} (u - u\_0)\left(t + \frac{\tau}{2} + \frac{\tau}{2} - t\right)} d\tau dt. \tag{21}$$

Let *<sup>ω</sup>* <sup>¼</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>τ</sup>* <sup>2</sup> and let *<sup>υ</sup>* <sup>¼</sup> *<sup>t</sup>* � *<sup>τ</sup>* <sup>2</sup> , then above equation reduces to the final result

$$\begin{split} \int\_{\mathbb{R}} \mathcal{WOL}\_f(t, u) dt &= \frac{1}{2\pi |b|} \int\_{\mathbb{R}^2} f(\omega) \overline{f(\nu)} e^{\frac{i}{\mathfrak{W}^\mu} \nu^2} e^{-\frac{i}{\mathfrak{W}^\mu} \nu^2} e^{\frac{i}{\mathfrak{W}^\mu(\nu-\nu)}} e^{-\frac{i}{\mathfrak{W}^\mu(\nu-\nu)} d\alpha d\nu} \\ &= \left| \widehat{f}(u) \right|^2. \end{split} \tag{22}$$

*Energy distribution property.*

The energy distribuition property of the WOL is given as

$$\int\_{\mathbb{R}^2} \mathcal{W} \mathcal{O} \mathcal{L}\_f(t, u) dt du = \int\_{\mathbb{R}} |f(t)|^2 dt. \tag{23}$$

Proof.

$$\int\_{\mathbb{R}^2} \mathcal{WCC}\_{\mathcal{T}}(t, u) dt du = \frac{1}{2\pi |b|} \int\_{\mathbb{R}^{\tilde{\mathbb{R}}}} \overline{f}\left(t + \frac{\tau}{2}\right) \overline{f\left(t - \frac{\tau}{2}\right)} e^{j\omega\_{\mathsf{f}}t} e^{-\frac{i}{\mathsf{p}}(u - u\_0)\tau} d\tau dt du$$

$$= \frac{1}{2\pi |b|} \int\_{\mathbb{R}^{\tilde{\mathbb{R}}}} \overline{f\left(t + \frac{\tau}{2}\right)} \overline{f\left(t - \frac{\tau}{2}\right)} e^{j\omega\_{\mathsf{f}}\tau} e^{j\omega\_{\mathsf{f}}\tau} \left(\int\_{\mathbb{R}} e^{-j\omega\_{\mathsf{f}}\tau} du\right) d\tau dt \qquad (24)$$

$$= \int\_{\mathbb{R}} |\![f(t)]\!]^2 dt. \blacksquare$$

*Moyal's formula.*

The Moyal's formula of the WOL is presented as

$$\int\_{\mathbb{R}^2} \mathcal{WOL}\_{\mathcal{f}}(\mathfrak{t}, u) \overline{\mathcal{WOL}\_{\mathfrak{g}}(\mathfrak{t}, u)} dt du = \left| \langle \mathfrak{f}, \,\, \mathfrak{g} \rangle \right|^2. \tag{25}$$

Proof.

ð 2 WOL*f*ð Þ *t*, *u* WOL*g*ð Þ *t*, *u dtdu* ¼ <sup>¼</sup> <sup>1</sup> 2*π*j j *b* ð 4 *f t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*f t* � *<sup>τ</sup>* 2 � �*<sup>e</sup> ia b τt e i <sup>b</sup>u*0*τ e* �*i buτ* � 1 <sup>2</sup>*π*j j *<sup>b</sup> g t* <sup>þ</sup> *<sup>τ</sup>*<sup>0</sup> 2 � �*g t* � *<sup>τ</sup>*<sup>0</sup> 2 � �*<sup>e</sup>* �*ia <sup>b</sup> τ*0*t e* �*i bu*0*τ e i <sup>b</sup>uτ*0 *dτdτ* 0 *dtdu* <sup>¼</sup> <sup>1</sup> 2*π*j j *b* ð 3 *f t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*f t* � *<sup>τ</sup>* 2 � �*<sup>e</sup> ia b τt e i <sup>b</sup>u*0*τ e* �*i buτ dτ* � 1 2*π*j j *b* ð *g t* <sup>þ</sup> *<sup>τ</sup>*<sup>0</sup> 2 � �*g t* � *<sup>τ</sup>*<sup>0</sup> 2 � �*<sup>e</sup>* �*ia <sup>b</sup> τ*0*t e* �*i <sup>b</sup>u*0*τ e i <sup>b</sup>uτ*0 *dτ* 0 *dtdu* <sup>¼</sup> <sup>1</sup> 2*π*j j *b* ð 2 *f t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*f t* � *<sup>τ</sup>* 2 � �*<sup>e</sup> ia b τt e i <sup>b</sup>u*0*τ dτ* � ð *g t* <sup>þ</sup> *<sup>τ</sup>*<sup>0</sup> 2 � �*g t* � *<sup>τ</sup>*<sup>0</sup> 2 � �*<sup>e</sup>* �*ia <sup>b</sup> τ*0*t e* �*i <sup>b</sup>u*0*τ dτ* 0 *dt* <sup>1</sup> 2*π*j j *b* ð *e i <sup>b</sup>u τ*ð Þ <sup>0</sup> �*<sup>τ</sup> du* <sup>¼</sup> <sup>1</sup> 2*π*j j *b* ð 2 *f t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*f t* � *<sup>τ</sup>* 2 � �*<sup>e</sup> ia b τt e i <sup>b</sup>u*0*τ dτ* ð *g t* <sup>þ</sup> *<sup>τ</sup>*<sup>0</sup> 2 � �*g t* � *<sup>τ</sup>*<sup>0</sup> 2 � �*<sup>e</sup>* �*ia <sup>b</sup> τ*0*t e* �*i <sup>b</sup>u*0*τ δ τ* � *τ*<sup>0</sup> ð Þ*dτ* 0 *dt* <sup>¼</sup> <sup>1</sup> 2*π*j j *b* ð ð *f t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*f t* � *<sup>τ</sup>* 2 � �*g t* <sup>þ</sup> *<sup>τ</sup>* 2 � �*g t* � *<sup>τ</sup>* 2 � �*dt* � �*dτ:* (26)

Now, we make the change of variable *<sup>μ</sup>* <sup>¼</sup> *<sup>t</sup>* � *<sup>τ</sup>* <sup>2</sup> , and come to

$$\begin{split} \int\_{\mathbb{R}^2} \mathcal{WOL}\_f(t, u) \overline{\mathcal{WOL}\_\mathbf{g}(t, u)} dt du &= \frac{1}{2\pi |b|} \Big[ \int\_{\mathbb{R}} f(\mu + \tau) \overline{\mathcal{g}(\mu + \tau)} d\tau \overline{\Big[ \int\_{\mathbb{R}} f(\mu) \overline{\mathcal{g}(\mu)} d\mu \Big]} \\ &= \frac{1}{2\pi |b|} |\langle f, \cdot \mathbf{g} \rangle|^2. \ \blacksquare \end{split} \tag{27}$$


**Table 2.** *Properties of the WOL.*

Some main properties of WOL are summarized in **Table 2**. The comprehensive view on the WOL can be seen in [17, 18].

#### **5. Conclusion**

In this chapter, we thoroughly revised research on hybrid transforms, which are constructed by associating WVD with well-known signal processing tools, such as FrFT, LCT, and OLCT. The WVD-OLCT generalizes most hybrid transforms, and the WOL is its special type. It is proven that hybrid transforms have better output in detection and estimation applications. Since the idea of associating two transforms is novel, it needs deep theoretical analysis and lacks diverse applications. Interested readers can develop hybrid transforms into quaternion and octonion algebra. These studies may be helpful in color image processing.

#### **Acknowledgements**

This research was funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP14871252).

#### **Author details**

Didar Urynbassarova<sup>1</sup> \* and Altyn Urynbassarova2,3

1 National Engineering Academy of the Republic of Kazakhstan, Almaty, Kazakhstan

2 Department of Technology and Ecology, School of Society, Technology and Ecology, Narxoz University, Almaty, Kazakhstan

3 Faculty of Information Technology, Department of Information Security, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan

\*Address all correspondence to: didaruzh@mail.ru

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 3**

## Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform

*Mohammad Younus Bhat, Aamir Hamid Dar, Altaf Ahmad Bhat and Deepak Kumar Jain*

#### **Abstract**

Quadratic-phase Fourier transform (QPFT) as a general integral transform has been considered into Wigner distribution (WD) and Ambiguity function (AF) to show more powerful ability for non-stationary signal processing. In this article, a new version of ambiguity function (AF) coined as scaled ambiguity function associated with the Quadratic-phase Fourier transform (QPFT) is proposed. This new version of AF is defined based on the QPFT and the fractional instantaneous autocorrelation. Firstly, we define the scaled ambiguity function associated with the QPFT (SAFQ). Then, the main properties including the conjugate-symmetry, shifting, scaling, marginal and Moyal's formulae of SAFQ are investigated in detail, the results show that SAFQ can be viewed as the generalization of the classical AF. Finally, the newly defined SAFQ is used for the detection of linear-frequency-modulated (LFM) signals.

**Keywords:** ambiguity function, quadratic-phase Fourier transform, Moyal's formula, modulation, linear frequency-modulated signal

#### **1. Introduction**

The Fourier transform is indeed an indispensable tool for the time-frequency analysis of the stationary signals. Due to its success stories FT has profoundly influenced the mathematical, biological, chemical and engineering communities over decades, but FT can not analyze non-stationary signals as it can not provide any valid information despite the localization properties of the spectral contents. FT only allows us to visualize the signals either in time or frequency domain, but not in both domains simultaneously. In Refs. [1–3], Castro et al. introduced a superlative generalized version of the Fourier transform(FT) called quadratic-phase Fourier transform(QPFT), which not only treats uniquely both the transient and non-transient signals in a nice fashion but also with non-orthogonal directions. The QPFT is actually a generalization of several well known transforms like Fourier, fractional Fourier and linear canonical transforms, offset linear canonical transform whose kernel is in the exponential form.

Many researches have been carried on quadratic-phase Fourier transform(see [4, 5]). With the fact that the QPFT is monitored by a bunch of free parameters, it has evolved as an effective tool for the representation of signals. A notable consideration has been given in the extension of the Wigner distributions to the classical QPFT and its generalizations. More can be found in Refs. [6–9].

On the other hand, the classical ambiguity function (AF) and Wigner distribution (WD) are the basic parametric time-frequency analysis tools, evolved for the analysis of time-frequency characteristics of non-stationary signals [10–14]. At the same time, the linear frequency-modulated (LFM) signal, a typical non-stationary signal, is widely used in communications, radar and sonar system. Many algorithms and methods have been proposed in view of LFM. The most important among them are the AF and WD [10, 13, 15–19], defined as the Fourier transform of the classical instantaneous autocorrelation function *<sup>ω</sup> <sup>t</sup>* <sup>þ</sup> *<sup>τ</sup>* 2 *<sup>ω</sup>*<sup>∗</sup> *<sup>t</sup>* � *<sup>τ</sup>* 2 for *t* and *τ*, (superscript ∗ denotes complex conjugate) respectively. It is well known that the AF offers perfect localization (localized on a straight line) to the mono-component LFM signals but cross terms appear while dealing with multi-component LFM signals as they are quadratic in nature. However these cross terms become troublesome if the frequency rate of one component approaches other. This drawback of AF gave rise to a series of different classes of time- frequency representation tools (see [20–27]). In Ref. [28], authors used fractional instantaneous auto-correlation *<sup>ω</sup> <sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 *<sup>ω</sup>*<sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 found in the definition of fractional bi-spectrum [29], which is parameterized by a constant *k*∈ <sup>þ</sup> to introduced a scaled version of the conventional WD. Later Dar and Bhat [30] introduced the scaled version of Ambiguity function and Wigner distribution in the linear canonical transform domain. They also introduced scaled version of Wigner distribution in the offset linear canonical transform [31–35], hence provides a novel way for the improvement of the cross-term reduction time–frequency resolution and angle resolution.

Keeping in mind the degree of freedom corresponding to the choice of a factor *k* in the fractional instantaneous auto-correlation and the extra degree of freedom present in QPFT, we introduce a novel scaled ambiguity function in the quadratic-phase Fourier transform domain (SAFQ), which gives a unique treatment for all classical classes of AF's. Hence, it is good to study rigorously the SAFQ which will be effective for signal processing theory and applications especially for detection and estimation of LFM signals.

#### **1.1 Paper contributions**

The contributions of this paper are summarized below:


*Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.108668*

#### **1.2 Paper outlines**

The paper is organized as follows: In Section 2, we gave a brief review of QPFT and introduce AF associated with it. The definition and the properties of the SAFQ are studied in Section 3. In Section 4, the applications of the proposed distribution for the detection of single-component and bi-component LMF signals is provided. Finally, a conclusion is drawn in Section 5.

#### **2. Preliminary**

In this section, we gave the definitions of the Quadratic-phase Fourier transform (QPFT), the ambiguity function associated with QPFT and the scaled ambiguity function which will be needed throughout the paper.

#### **2.1 Quadratic-phase Fourier transform (QPFT)**

For a given set of parameters of Ω ¼ ð Þ *A*, *B*, *C*, *D*, *E* ,*B* 6¼ 0 the quadratic-phase Fourier transform any signal *ω*ð Þ*t* is defined by [1–3]

$$\mathcal{Q}^{\Omega}[o](u) = \int\_{\mathbb{R}} o(t) K\_{\Omega}(t, u) dt,\tag{1}$$

where the quadratic-phase Fourier kernel *K*Ωð Þ *t*, *w* is given by

$$K\_{\Omega}(t,u) = \sqrt{\frac{B}{2\pi i}} e^{\left(At^2 + Btu + Cu^2 + Dt + Eu\right)}, \quad A, B, C, D.E \in \mathbb{R}.\tag{2}$$

#### **2.2 Ambiguity function in the quadratic-phase fourier domain (AFQ)**

Authors in Refs. [7, 8] defined the AF associated with the LCT, using the same procedure we can define the AF associated with QPFT (AFQ) as

$$AFQ\_{\alpha(t)}^{\Omega}(\tau, u) = \int\_{\mathbb{R}} w\left(t + \frac{\tau}{2}\right) \alpha^\*\left(t - \frac{\tau}{2}\right) K\_{\Omega}(\tau, u) dt,\tag{3}$$

#### **2.3 Scaled ambiguity function**

For a finite energy signal the scaled Ambiguity function (SAF) is defined as Ref. [30].

$$\text{SAF}\_{o(t)}(\tau,\mathfrak{u}) = \int\_{\mathbb{R}} \alpha \left(t + k\frac{\tau}{2}\right) \alpha^\* \left(t - \frac{\tau}{2}\right) e^{-i\omega t} dt,\tag{4}$$

where *k*∈ <sup>þ</sup> the set of positive rational numbers.

#### **3. Scaled ambiguity function associated with quadratic-phase fourier transform (SAFQ)**

In this section, we shall introduce the notion of the scaled Ambiguity function associated with QPFT followed by some of its basic properties.

#### **3.1 Definition of the scaled AFQ**

Thanks to the scaled AF, we obtain obtain different expressions for the SAFQ as follows:

$$SAF\_{o(t)}(\tau,\mu) = \int\_{\mathbb{R}} a\left(t + k\frac{\tau}{2}\right) \boldsymbol{\alpha}^\*\left(t - k\frac{\tau}{2}\right) e^{-i\boldsymbol{t}\boldsymbol{t}} d\boldsymbol{t}$$

$$= \int\_{\mathbb{R}} a\left(t + k\frac{\tau}{2}\right) e^{-i\frac{\pi}{2}\left(t + k\frac{\tau}{2}\right)} \boldsymbol{\alpha}^\*\left(t - \frac{\tau}{2}\right) e^{-i\frac{\pi}{2}\left(t - k\frac{\tau}{2}\right)} d\boldsymbol{t} \tag{5}$$

$$= \int\_{\mathbb{R}} \overline{\boldsymbol{\alpha}\_{\boldsymbol{u}}\left(t + k\frac{\tau}{2}\right)} \hat{\boldsymbol{\alpha}}\_{\boldsymbol{u}}^\*\left(t - k\frac{\tau}{2}\right) d\boldsymbol{t},$$

where

$$
\overline{\boldsymbol{\alpha}}\_{\mathfrak{u}}(t) = \boldsymbol{\alpha}(t)\boldsymbol{e}^{-i\frac{\mathfrak{v}}{2}} \quad \text{and} \quad \boldsymbol{\hat{\boldsymbol{\alpha}}}\_{\mathfrak{u}}(t) = \boldsymbol{\alpha}(t)\boldsymbol{e}^{i\frac{\mathfrak{v}}{2}}.\tag{6}
$$

On replacing the Fourier kernel in (6) with the QPFT kernel, we obtain

$$
\overline{\boldsymbol{\alpha}}\_{\boldsymbol{u}}^{\Omega}(t) = \boldsymbol{\alpha}(t)K\_{\Omega}\left(t, \frac{\boldsymbol{u}}{2}\right) \quad \text{and} \quad \widehat{\boldsymbol{\alpha}}\_{\boldsymbol{u}}^{\Omega}(t) = \boldsymbol{\alpha}(t)K\_{\Omega}\left(t, -\frac{\boldsymbol{u}}{2}\right). \tag{7}
$$

Thus, we obtain a new version of scaled AF associated with the QPFT by replacing *ωu*ð Þ*t* with *ω*<sup>Ω</sup> *<sup>u</sup>* ð Þ*<sup>t</sup>* and *<sup>ω</sup>*^*u*ð Þ*<sup>t</sup>* with *<sup>ω</sup>*^<sup>Ω</sup> *<sup>u</sup>* ð Þ*t* in (5), i.e.,

$$\begin{split} \mathcal{S}AF^{\Delta}\_{w(t)}(\tau,u) &= \int\_{\mathbb{R}} \overline{\boldsymbol{w}}\_{u}^{\Delta} \left(t+k\frac{\tau}{2}\right) \dot{\boldsymbol{x}}\_{u}^{\Delta\*} \left(t-k\frac{\tau}{2}\right) dt \\ &= \int\_{\mathbb{R}} \boldsymbol{w} \left(t+k\frac{\tau}{2}\right) \mathcal{K}\_{\Omega} \left(t+k\frac{\tau}{2}, \frac{u}{2}\right) \boldsymbol{\alpha}^\* \left(t-k\frac{\tau}{2}\right) \mathcal{K}\_{\Omega}^\* \left(t-k\frac{\tau}{2}, \frac{-u}{2}\right) dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{w} \left(t+k\frac{\tau}{2}\right) \boldsymbol{\alpha}^\* \left(t-k\frac{\tau}{2}\right) \boldsymbol{e}^{i[(2\lambda k\tau + Bu)\tau + Dk\tau + Eu]} dt. \end{split} \tag{8}$$

With the virtue of above equation we have following definition.

Definition 3.1. *The scaled Ambiguity function associated with quadratic-phase Fourier transform of a signal* <sup>0</sup> *<sup>ω</sup>*ð Þ*<sup>t</sup>* <sup>0</sup> *in L*<sup>2</sup> ð Þ *with respect the real parameter set* Ω ¼ ð Þ *A*, *B*, *C*, *D*, *E* ,*B* 6¼ 0 *is defined as*

$$\text{SAF}\_{o(t)}^{\Omega}(\tau, u) = \frac{B}{2\pi} \int\_{\mathbb{R}} \alpha \left(t + k\frac{\tau}{2}\right) \alpha^\* \left(t - k\frac{\tau}{2}\right) e^{i[(2Ak\tau + Bu)t + Dk\tau + Eu]} dt,\tag{9}$$

where *k*∈ þ.

It is worth to mention that if we change the parameter Ω ¼ ð Þ *A*, *B*, *C*, *D*, *E* in the Definition 3.1, we have the following important deductions:

i. When the parameter Ω ¼ ð Þ *A=*2*B*, �1*=B*, *C=*2*B*,0,0 is chosen and multiplying the right side of (9) by �1, the SAFQ (9) yields the scaled ambiguity function associated with linear canonical transform [30]:

*Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.108668*

$$\text{SAF}\_{o(t)}^{\Omega}(\tau, u) = \frac{1}{2\pi B} \int\_{\mathbb{R}} \alpha \left(t + k\frac{\tau}{2}\right) \alpha^\* \left(t - k\frac{\tau}{2}\right) e^{i\frac{\mathbf{j}}{\hbar}(Ak\tau - u)t} dt. \tag{10}$$

ii. For the set Ω ¼ ð Þ cot *ζ=*2, �csc*ζ*, cot *ζ=*2,0,0 ,*ζ* 6¼ 2*π* and multiplying the right side of (9) by �1 the SAFQ (9) yields the novel scaled AF associated with fractional Fourier transform:

$$\text{SAF}^{\zeta}\_{o(t)}(t,u) = \frac{1}{2\pi\sin\zeta} \int\_{\mathbb{R}} a\left(t+k\frac{\theta}{2}\right) \omega^\*\left(t-k\frac{\tau}{2}\right) e^{i(\left(k\cot\zeta\tau-u\csc\zeta\right)t}dt.\tag{11}$$

iii. When the parameter is choosen as Ω ¼ ð Þ 0,1,0,0,0 is chosen, the scaled AFQ (4) boils down to the classical scaled AF given in Ref. [30]. In addition of above if we take *k* ¼ 1, it reduce to classical Amniguity function.

#### **3.2 Properties of the scaled AFOL**

In this subsection, we investigate some general properties of the scaled AFQ with their detailed proofs. These properties play vital role in signal representation. We shall see the differences between the scaled versions and conventional ones.

**Property 3.1 (symmetry property)** *For <sup>ω</sup>*ð Þ*<sup>t</sup>* <sup>∈</sup>*L*<sup>2</sup> ð Þ , *then scaled AFOL of the signals <sup>ω</sup>*<sup>∗</sup> ð Þ*<sup>t</sup> and P*½ � *<sup>ω</sup>*ð Þ*<sup>t</sup> have the following forms*

$$\text{SAF}\_{o(t)^\*}^{\Omega}(\tau, u) = \text{SAF}\_{o(t)}^{\Omega'}(-\tau, -u) \tag{12}$$

where Ω<sup>0</sup> ¼ �ð Þ *A:* � *B*, *C*, �*D*, �*E : and*

$$\text{SAF}^{\Omega}\_{P[o(t)]}(\tau, u) = -\text{SAF}^{\overline{\Omega}}\_{o(t)}(-\tau, -u), \tag{13}$$

where *P*½ �¼ *ω*ð Þ*t ω*ð Þ �*t* and Ω ¼ ð Þ *A*, *B*, *C*, �*D*, �*E* . *Proof.* From Definition 3.1, we have

$$\begin{split} & \mathcal{S}AF\_{\boldsymbol{w}(\cdot)}^{\Omega}(\tau,\boldsymbol{u}) \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{w}^\* \left(t + k\frac{\tau}{2}\right) \boldsymbol{w} \Big(t - k\frac{\tau}{2}\Big) \boldsymbol{e}^{i[(2Ak\tau + Bu) + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{w} \Big(t + k\frac{(-\tau)}{2}\Big) \boldsymbol{w}^\* \left(t - k\frac{(-\tau)}{2}\right) \boldsymbol{e}^{i[(2Ak\tau + Bu) + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{w} \Big(t + k\frac{(-\tau)}{2}\Big) \boldsymbol{w}^\* \left(t - k\frac{(-\tau)}{2}\right) \times \boldsymbol{e}^{i[(2(-A)k(-\tau) + (-B)(-u))t + (-D)k(-\tau) + (-E)(-u)]} dt \\ &= \mathcal{S}AF\_{\boldsymbol{w}(\cdot)}^{\Omega'}(-\tau, -u), \quad \text{where} \quad \Omega' = (-A, -B, C, -D, -E). \end{split}$$

which prove (12). Now, we move forward to prove (13) From (9), we have

*SAF*<sup>Ω</sup> *<sup>P</sup>*½ � *<sup>ω</sup>*ð Þ*<sup>t</sup>* ð Þ *<sup>τ</sup>*, *<sup>u</sup>* ¼ *B* 2*π* ð *<sup>P</sup><sup>ω</sup> <sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*Pω*<sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*<sup>e</sup> <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu dt* ¼ *B* 2*π* ð *<sup>ω</sup>* �*<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> �*<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*<sup>e</sup> <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu dt* ¼ *B* 2*π* ð *<sup>ω</sup>* �*<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> �*<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*<sup>e</sup> <sup>i</sup>*½ � f g <sup>2</sup>*Ak*ð Þþ �*<sup>τ</sup> <sup>B</sup>*ð Þ �*<sup>u</sup>* ð Þþ � �*<sup>t</sup>* ð Þ *<sup>D</sup> <sup>k</sup>*ð Þþ � �*<sup>τ</sup>* ð Þ� *<sup>E</sup>* ð Þ *<sup>u</sup> dt* ¼ � *<sup>B</sup>* 2*π* ð *ω υ* <sup>þ</sup> *<sup>k</sup>* �*<sup>τ</sup>* 2 � �*ω*<sup>∗</sup> *<sup>υ</sup>* � *<sup>k</sup>* �*<sup>τ</sup>* 2 � �*<sup>e</sup> <sup>i</sup>*½ � f g <sup>2</sup>*Ak*ð Þþ �*<sup>τ</sup> <sup>B</sup>*ð Þ �*<sup>u</sup> <sup>υ</sup>*þ �ð Þ *<sup>D</sup> <sup>k</sup>*ð Þþ � �*<sup>τ</sup>* ð Þ� *<sup>E</sup>* ð Þ *<sup>u</sup> dυ* ¼ �*SAF*<sup>Ω</sup> *<sup>ω</sup>*ð Þ*<sup>t</sup>* ð Þ �*τ*, �*<sup>u</sup>* , <sup>Ω</sup> <sup>¼</sup> ð Þ *<sup>A</sup>*, *<sup>B</sup>*, *<sup>C</sup>*, �*D*, �*<sup>E</sup> :*

which completes the proof. □ **Property 3.2 (Time shift).** *The SAFQ of a signal ω*ð Þ *t* � *λ can be expressed as:*

$$
\delta AF^{\Omega}\_{\alpha(t-\lambda)}(\tau,\mu) = e^{i\lambda(2Ak\tau + Bu)} \text{SAF}^{\Omega}\_{\alpha(t)}(\tau,\mu). \tag{14}
$$

*Proof.* From (9), we obtain

$$\text{SAF}\_{o(t-\lambda)}^{\Omega}(\tau, u) = \frac{B}{2\pi} \int\_{\mathbb{R}} \left. o\left(t - \lambda + k\frac{\tau}{2}\right) \alpha^\* \left(t - \lambda - k\frac{\tau}{2}\right) e^{i[(2A\mathbb{k}\tau + Bu)t + Dk\tau + Eu]} dt \dots$$

Setting *t* � *λ* ¼ *s*, we have from last equation

$$\begin{split} \mathbb{S} \mathbf{A} F^{\Omega}\_{\boldsymbol{\alpha}(t-\boldsymbol{k})} (\boldsymbol{\tau}, \boldsymbol{u}) &= \frac{\mathrm{B}}{2\pi} \int\_{\mathbb{R}} \boldsymbol{\alpha} \left( \boldsymbol{s} + \boldsymbol{k} \frac{\boldsymbol{\tau}}{2} \right) \boldsymbol{\alpha}^{\*} \left( \boldsymbol{s} - \boldsymbol{k} \frac{\boldsymbol{\tau}}{2} \right) \boldsymbol{e}^{i[(2A\boldsymbol{k}\boldsymbol{\tau} + \boldsymbol{B}\boldsymbol{u})\boldsymbol{\gamma} + D\boldsymbol{k}\boldsymbol{\tau} + \boldsymbol{E}\boldsymbol{u}]} d\boldsymbol{s} \\ &= \mathrm{e}^{i\boldsymbol{k}(2A\boldsymbol{k}\boldsymbol{\tau} + \boldsymbol{B}\boldsymbol{u})} \frac{\mathrm{B}}{2\pi} \int\_{\mathbb{R}} \boldsymbol{\alpha} \left( \boldsymbol{s} + \boldsymbol{k} \frac{\boldsymbol{\tau}}{2} \right) \boldsymbol{\alpha}^{\*} \left( \boldsymbol{s} - \boldsymbol{k} \frac{\boldsymbol{\tau}}{2} \right) \boldsymbol{e}^{i[(ak\boldsymbol{\tau} - \boldsymbol{u})\boldsymbol{v} + k\boldsymbol{u}\boldsymbol{q}\boldsymbol{\tau} - \boldsymbol{u}(d\boldsymbol{u}\boldsymbol{q} - \boldsymbol{k}\boldsymbol{w}\_{0})]} \mathrm{d}\boldsymbol{s} \\ &= \mathrm{e}^{i\boldsymbol{k}(2A\boldsymbol{k}\boldsymbol{\tau} + \boldsymbol{B}\boldsymbol{u})} \mathrm{S} \boldsymbol{A} \boldsymbol{F}^{\Omega}\_{\boldsymbol{\alpha}(t)}(\boldsymbol{\tau}, \boldsymbol{u}). \end{split}$$

Which completes the proof of (14). □

**Property 3.3 (Frequency shift).** *The SAFQ of a signal <sup>ω</sup>*ð Þ*<sup>t</sup> <sup>e</sup>ivt can be expressed as:*

$$\text{SAF}^{\Omega}\_{\boldsymbol{w}(t)\varepsilon^{\text{int}}}(\boldsymbol{\tau},\boldsymbol{u}) = e^{i\boldsymbol{u}\boldsymbol{\tau}}\text{SAF}^{\Omega}\_{\boldsymbol{w}(t)}(\boldsymbol{\tau},\boldsymbol{u})\tag{15}$$

*Proof.* From (9), we have

$$\begin{split} \mathcal{S}AF^{\Omega}\_{o(t)e^{i\omega}}(\tau,u) &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{\alpha}\left(t+k\frac{\tau}{2}\right) e^{i\nu\left(t+k\frac{\tau}{2}\right)} \boldsymbol{\alpha}^\*\left(t-\frac{\tau}{2}\right) e^{-i\nu\left(t-k\frac{\tau}{2}\right)} \\ &\quad \times e^{i\left[\left(2Ak\tau+Bu\right)t+Dk\tau+Eu\right]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{\alpha}\left(t+k\frac{\tau}{2}\right) \boldsymbol{\alpha}^\*\left(t-\frac{\tau}{2}\right) e^{i\nu k\tau} \\ &\quad \times e^{i\left[\left(2Ak\tau+Bu\right)t+Dk\tau+Eu\right]} dt \\ &= e^{i\nu k\tau} \frac{B}{2\pi} \int\_{\mathbb{R}} \boldsymbol{\alpha}\left(t+k\frac{\tau}{2}\right) \boldsymbol{\alpha}^\*\left(t-\frac{\tau}{2}\right) \\ &\quad \times e^{i\left[\left(2Ak\tau+Bu\right)t+Dk\tau+Eu\right]} dt \\ &= e^{i\nu k\tau} \mathcal{S}AF^{\Omega}\_{o(t)}(\tau,u). \end{split}$$

Which completes the proof □ **Property 3.4 (Non-linearity).** *Let <sup>ω</sup>*ðÞ¼ *<sup>t</sup> <sup>ω</sup>*1ðÞþ*<sup>t</sup> <sup>ω</sup>*2ð Þ*<sup>t</sup> be in L*<sup>2</sup> ð Þ , *then we have*

$$\text{SAF}\_{a(t)}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{\mu}) = \text{SAF}\_{a\mathbf{u}(t)}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{\mu}) + \text{SAF}\_{a\mathbf{u}(t)}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{\mu}) + \text{SAF}\_{a\mathbf{u}, a\mathbf{u}}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{\mu}) + \text{SAF}\_{a\mathbf{u}, a\mathbf{u}}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{\mu}) \tag{16}$$

Proof. From Definition 3.1, we have

*SAF*<sup>Ω</sup> *<sup>ω</sup>*ð Þ*<sup>t</sup>* ð Þ *<sup>τ</sup>*, *<sup>u</sup>* ¼ *B* 2*π* ð ð Þ *<sup>ω</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �ð Þ *<sup>ω</sup>*<sup>1</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*<sup>e</sup> <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu dt* ¼ *B* 2*π* ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � � <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 h� � � � *ω*1 <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � ��i*<sup>e</sup> <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu dt* ¼ *B* 2*π* ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>1</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 h � � <sup>þ</sup>*ω*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>1</sup> <sup>∗</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �i �*e <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu dt*

$$=\mathsf{SAF}\_{\boldsymbol{\alpha}\boldsymbol{\upgamma}}^{\boldsymbol{\Omega}}(\boldsymbol{\tau},\boldsymbol{u}) + \mathsf{SAF}\_{\boldsymbol{\alpha}\boldsymbol{\upgamma}}^{\boldsymbol{\Omega}}(\boldsymbol{\tau},\boldsymbol{u}) + \mathsf{SAF}\_{\boldsymbol{\alpha}\boldsymbol{\upgamma},\boldsymbol{\alpha}\boldsymbol{\upgamma}}^{\boldsymbol{\Omega}}(\boldsymbol{\tau},\boldsymbol{u}) + \mathsf{SAF}\_{\boldsymbol{\alpha}\boldsymbol{\upgamma},\boldsymbol{\alpha}\boldsymbol{\upgamma}}^{\boldsymbol{\Omega}}(\boldsymbol{\tau},\boldsymbol{u}).$$

Thus completes the proof. □

**Property 3.5 (Frequency marginal property).** *The frequency marginal property of SAFQ is given by*

$$\int\_{\mathbb{R}} \mathcal{S} A F\_{o(t)}^{\Omega}(\tau, u) d\tau = \frac{1}{k} \mathcal{Q}^{\Omega}[o(t)] \left(\frac{u}{2}\right) \mathcal{Q}^{\*\Omega}[o(t)] \left(\frac{-u}{2}\right) \tag{17}$$

*Proof.* From Definition 3.1, we have

$$\int\_{\mathbb{R}} \mathcal{S}AF\_{\alpha(t)}^{\Omega}(\tau, u)d\tau = \frac{B}{2\pi} \int\_{\mathbb{R}^2} \omega\left(t + k\frac{\tau}{2}\right) \omega^\*\left(t - \frac{\tau}{2}\right) e^{i[(2Ak\tau + Bu)t + Dk\tau + Eu]} dt d\tau.$$

Making change of variable *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* <sup>2</sup> ¼ *s*, above equation yields

$$\int\_{\mathbb{R}} \mathcal{S}AF\_{\alpha(t)}^{\Omega}(\tau, u)d\tau = \frac{B}{\pi \mathsf{k}} \int\_{\mathbb{R}^2} \alpha(s)\alpha^\*(2t - s)e^{i[\{4A(s - t) + Bu\}t + 2D(s - t) + Eu]} dsdt.$$

Now setting 2*t* ¼ *s* þ *v*, we get

$$\begin{split} &\int\_{\mathbb{R}} \left[\underline{\mathcal{S}} \boldsymbol{A} \underline{\mathcal{S}}\_{\alpha(\cdot)}^{\mathcal{D}}(\tau, u) d\tau \right] \\ &= \frac{B}{2\pi k} \int\_{\mathbb{R}^2} \alpha(s) \alpha^\*(p) e^{i\left[\left(\frac{1}{24} \left(s - \frac{i\pi}{2}\right) - \mathrm{Bu}\right) \left(\frac{i\pi}{2}\right) + \mathcal{D} \left(s - \frac{i\pi}{2}\right) + \mathrm{Eu}\right]} d\mathrm{s} d\nu \\ &= \frac{B}{2\pi k} \int\_{\mathbb{R}^2} \alpha(s) \alpha^\* \left(\nu\right) e^{i\left[\left(2A(s-v) + \mathrm{Bu}\right) \left(\frac{i\pi}{2}\right) + \mathcal{D} \left(s - v\right) + \mathrm{Eu}\right]} d\mathrm{s} d\nu \\ &= \frac{B}{2\pi k} \int\_{\mathbb{R}^2} \alpha(s) \alpha^\* \left(\nu\right) e^{i\left[A\left(\frac{s^2}{2} - v^2\right) + \mathrm{Bu}\left(\frac{i\pi}{2}\right) + D \left(s - v\right) + \mathrm{Eu}\right]} d\mathrm{s} d\nu \\ &= \frac{B}{2\pi k} \int\_{\mathbb{R}^2} \alpha(s) e^{i\left[A\nu^2 + B\nu \left(\frac{-s}{2}\right) + D \nu \left(\frac{-s}{2}\right)\right]} d\mathrm{s} \\ & \times \Big[\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\/$$

Which completes the proof. □

**Property 3.6 (Scaling property).** *For a signal <sup>ω</sup>*~ðÞ¼ *<sup>t</sup>* ffiffiffi *<sup>σ</sup>* <sup>p</sup> *ω σ*ð Þ*<sup>t</sup> the SAFQ has the following form:*

$$
\Delta AF\_{\tilde{w}(t)}^{\Omega}(\tau, u) = \text{SAF}\_{w(t)}^{\Omega'} \left( \sigma \tau, \frac{u}{\sigma} \right), \tag{18}
$$

where <sup>Ω</sup><sup>0</sup> <sup>¼</sup> *<sup>A</sup> <sup>σ</sup>*<sup>2</sup> , *B*, *C*, *<sup>D</sup> <sup>σ</sup>* , *<sup>σ</sup><sup>E</sup>* � �. *Proof.* From (9), we have

$$\mathcal{L}AF\_{\check{w}(t)}^{\Omega}(\tau, u) = \frac{\sigma B}{2\pi} \int\_{\mathbb{R}} \omega \left(\sigma t + \sigma k \frac{\tau}{2}\right) \omega^\* \left(\sigma t - \sigma k \frac{\tau}{2}\right) e^{i[(2Ak\tau + Bu)t + Dk\tau + Eu]} dt.$$

Setting *σt* ¼ *η*, above equation yields

$$\begin{split} & \quad \|SAF\_{\boldsymbol{\alpha}(t)}^{\Omega}(\tau,u) \\ &= \frac{\sigma B}{2\pi} \Bigg[ \int\_{\mathbb{R}} \boldsymbol{\alpha} \left( \boldsymbol{\sigma}t + \sigma k \frac{\tau}{2} \right) \boldsymbol{\alpha}^\* \left( \boldsymbol{\sigma}t - \sigma k \frac{\tau}{2} \right) \boldsymbol{e}^{i\left[ (2A\boldsymbol{\tau} + Bu)\frac{\tau}{\sigma} + Dk\tau + Eu \right]} \cdot \frac{d\eta}{\sigma} \\ &= \frac{\sigma B}{2\pi} \Bigg[ \boldsymbol{\alpha} \left( \boldsymbol{\sigma}t + \sigma k \frac{\tau}{2} \right) \boldsymbol{\alpha}^\* \left( \boldsymbol{\sigma}t - \sigma k \frac{\tau}{2} \right) \boldsymbol{e}^{i\left[ \left( 2\frac{A}{\sigma^2}b(\boldsymbol{\sigma}\boldsymbol{\tau}) + B\frac{\tau}{\sigma} \right) \boldsymbol{\eta} + Dk\tau + Eu \right]} \cdot \frac{d\eta}{\sigma} \\ &= \frac{B}{2\pi} \Bigg[ \boldsymbol{\alpha} \left( \boldsymbol{\sigma}t + \sigma k \frac{\tau}{2} \right) \boldsymbol{\alpha}^\* \left( \boldsymbol{\sigma}t - \sigma k \frac{\tau}{2} \right) \boldsymbol{e}^{i\left[ \left( 2\frac{A}{\sigma^2}b(\boldsymbol{\sigma}\boldsymbol{\tau}) + B\left( \frac{\tau}{\sigma} \right) \right) \boldsymbol{\eta} + \frac{D}{\sigma}b(\boldsymbol{\sigma}\boldsymbol{\tau}) + \sigma E \left( \frac{u}{\sigma} \right) \right]}{\left( \boldsymbol{\sigma} \right)} \Bigg] d\eta \\ &= S A F\_{\boldsymbol{\alpha}(t)}^{\Omega'} \left( \boldsymbol{\sigma}\tau, \frac{u}{\sigma} \right), \end{split}$$

*Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.108668*

where <sup>Ω</sup><sup>0</sup> <sup>¼</sup> *<sup>A</sup> <sup>σ</sup>*<sup>2</sup> , *B*, *C*, *<sup>D</sup> <sup>σ</sup>* , *<sup>σ</sup><sup>E</sup>* � �. This proves (18). □ **Property 3.7 (Moyal formula).** *The Moyal formula of the SAFQ has the following form:*

$$\int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathbb{S}AF\_{\boldsymbol{\alpha}\_{1}(t)}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{u}) \Big[\mathbb{S}AF\_{\boldsymbol{\alpha}\_{2}(t)}^{\Omega}(\boldsymbol{\tau}, \boldsymbol{u})\Big]^{\*} d\boldsymbol{\tau} d\boldsymbol{u} = \frac{B}{2\pi k} \left| \langle \boldsymbol{\alpha}\_{1}(t), \boldsymbol{\alpha}\_{2}(t) \rangle \right|^{2}. \tag{19}$$

*Proof.* From (9), we have

ð ð *SAF*<sup>Ω</sup> *<sup>ω</sup>*1ð Þ*<sup>t</sup>* ð Þ *<sup>t</sup>*, *<sup>u</sup> SAF*<sup>Ω</sup> *<sup>ω</sup>*2ð Þ*<sup>t</sup>* ð Þ *t*, *u* h i <sup>∗</sup> *dtdu* <sup>¼</sup> *<sup>B</sup>* 2*π* � �<sup>2</sup>ð ð ð ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>2</sup> *t* <sup>0</sup> <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> *<sup>t</sup>* <sup>0</sup> � *<sup>k</sup> <sup>τ</sup>* 2 � � �*e <sup>i</sup>*½ � ð Þ <sup>2</sup>*Akτ*þ*Bu <sup>t</sup>*þ*Dkτ*þ*Eu e* �*i* ð Þ 2*Akτ*þ*Bu t* ½ � <sup>0</sup> <sup>þ</sup>*Dkτ*þ*Eu dτdt*<sup>0</sup> *dtdu* <sup>¼</sup> *<sup>B</sup>* 2*π* � �<sup>2</sup>ð ð ð ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � �*e i*ð Þ 2*Akτ*þ*Bu t*�*t* <sup>0</sup> ð Þ*dτdudtdt*<sup>0</sup> ¼ *B* 2*π* ð ð ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � �*e i*2*Akτ t*�*t* <sup>0</sup> ð Þ *B* 2*π* ð *e iBu t*�*t* <sup>0</sup> ð Þ*du* � �*dτdtdt*<sup>0</sup> ¼ *B* 2*π* ð ð ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � � �*e i*2*Akτ t*�*t* <sup>0</sup> ð Þ*<sup>δ</sup> <sup>t</sup>* � *<sup>t</sup>* <sup>0</sup> ð Þ*dt*<sup>0</sup> *dτdt* ¼ *B* 2*π* ð ð *<sup>ω</sup>*<sup>1</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>1</sup> *<sup>t</sup>* � *<sup>τ</sup>* 2 � �*ω*<sup>∗</sup> <sup>2</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* 2 � �*ω*<sup>2</sup> *<sup>t</sup>* � *<sup>k</sup> <sup>τ</sup>* 2 � �*dτdt*

By making the change of variable *<sup>s</sup>* <sup>¼</sup> *<sup>t</sup>* <sup>þ</sup> *<sup>k</sup> <sup>τ</sup>* <sup>2</sup> , we have

$$\int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathcal{W}\_{a\_{\mathrm{2}}(t)}^{A,k}(t,u) \left[\mathcal{W}\_{a\_{\mathrm{2}}(t)}^{A,k}(t,\cdot,u)\right]^\* d\tau du = \frac{B}{k\pi} \int\_{\mathbb{R}} \int\_{\mathbb{R}} a\_{\mathrm{2}}(s) a\_{1}^{\*}(2t-s) a\_{2}^{\*}(s) a\_{2}(2t-s) ds dt$$

Now taking 2*t* � *s* ¼ *v*, we obtain

$$\begin{split} \int\_{\mathbb{R}} \int\_{\mathbb{R}} \mathcal{W}\_{\boldsymbol{\alpha}\_{1}}^{A,k}(t,u) \left[\mathcal{W}\_{\boldsymbol{\alpha}\_{2}}^{A,k}(t,u)\right]^{\*} d\boldsymbol{\pi} du &= \frac{B}{2\pi k} \int\_{\mathbb{R}} \int\_{\mathbb{R}} \boldsymbol{\alpha}\_{1}(s) \boldsymbol{\alpha}\_{1}^{\*}(v) \boldsymbol{\alpha}\_{2}^{\*}(s) \boldsymbol{\alpha}\_{2}(v) ds dv \\ &= \frac{B}{2\pi k} \left(\int\_{\mathbb{R}} \boldsymbol{\alpha}\_{1}(s) \boldsymbol{\alpha}\_{2}^{\*}(s) d\boldsymbol{x}\right) \left(\int\_{\mathbb{R}} \boldsymbol{\alpha}\_{1}^{\*}(v) \boldsymbol{\alpha}\_{2}(v) dv\right) \\ &= \frac{B}{2\pi k} \left| \langle \boldsymbol{\alpha}\_{1}(t), \boldsymbol{\alpha}\_{2}(t) \rangle \right|^{2}. \end{split}$$

Thus completes the proof. □

#### **4. Applications of the scaled AFQ**

In engineering the most important research topics is the detection of LFM signals as they are widely used in communications, information and optical systems. In this section our main goal is to use scaled AFQ in detection of one-component and bicomponent LFM signals, respectively.

• **One component LFM signal:** A one-component LFM signal is chosen as

$$\rho(t) = e^{i\left(\theta\_1 t + \theta\_2 t^2\right)}\tag{20}$$

where *ϑ*<sup>1</sup> and *ϑ*<sup>2</sup> represent the initial frequency and frequency rate of *ω*ð Þ*t* , respectively. Then, we obtain the SAFQ of a signal *ω*ð Þ*t* as shown in the following theorem.

Theorem 4.1 *The SAFQ of ω*ðÞ¼ *t e i ϑ*1*t*þ*ϑ*2*t* <sup>2</sup> ð Þ *can be presented as*

$$\text{SAF}\_{w(t)}^{\Omega}(\mathfrak{r}, \mathfrak{u}) = e^{i[k(\theta\_1 + D)\mathfrak{r} + \text{Eu}]} \delta[2k(\theta\_2 + A)\mathfrak{r} + Bu]. \tag{21}$$

*Proof*. By Definition 3.1, we have

$$\begin{split} & \quad \Delta F\_{\alpha(t)}^{B}(\tau, u) \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \alpha \left( t + k\frac{\tau}{2} \right) a^{\ast} \left( t - k\frac{\tau}{2} \right) e^{i[(2Ak\tau + Bu)\tau + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \epsilon^{i} \left[ \vartheta\_{1}(t + k\frac{\tau}{2}) + \vartheta\_{2}(t + k\frac{\tau}{2}) \right] e^{-i\left[\vartheta\_{1}(t - k\frac{\tau}{2}) + \vartheta\_{2}(t - k\frac{\tau}{2}) \right]} \\ & \qquad \times \epsilon^{j[(2Ak\tau + Bu)\tau + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \epsilon^{j} \left[ \vartheta\_{1} t + \vartheta\_{1} k\dot{\varepsilon} + \vartheta\_{2} t^{2} + \vartheta\_{2} k\dot{\varepsilon} + \vartheta\_{2} k^{2} \frac{\tau}{4} \right] - \epsilon^{j} \left[ \vartheta\_{1} t - \vartheta\_{1} k\dot{\varepsilon} + \vartheta\_{2} t^{2} - \vartheta\_{2} k\varepsilon + \vartheta\_{2} k^{2} \frac{\tau}{4} \right] \\ & \qquad \times \epsilon^{j[(2Ak\tau + Bu)\tau + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \epsilon^{j[(\vartheta\_{1} k\tau + D)\tau + Eu]} \int\_{\mathbb{R}} \epsilon^{j[2A(\vartheta\_{1} + A)\tau + Bu]} dt \\ &= \frac{B}{2\pi} \epsilon^{j[k(\vartheta\_{1} + D)\tau + Eu]} \int\_{\mathbb{R}} \epsilon^{j[2A(\vartheta\_{1} + A)\tau + Bu]} dt \\ &= \epsilon^{j[$$

□ From above Theorem, we can conclude that the that the SAFQ of a one-component signal (20) are able to generate impulses in ð Þ *τ*, *u* plane at a straight line ð*Bu* þ 2*k*ð Þ *ϑ*<sup>2</sup> þ *A τ*Þ ¼ 0 and is dependent on the scaling factor *k* and the parameter Ω ¼ ð Þ *A*, *B*, *C*, *D*, *E* . Therefore, the SAFQ can be applied to the detection of onecomponent LFM signals and is very useful and effective as there is choice of selecting the scaling factor *k* and the parameter Ω.

• **Bi-component LFM signal:** Consider the following bi-component LFM signal *ω*ð Þ*t* it is well known that the bi-component LFM signal can be expressed by the summation of two single component LFM signals, i.e.,

$$
\alpha(t) = \alpha\_1(t) + \alpha\_2(t), \tag{23}
$$

*Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.108668*

where *ω*1ðÞ¼ *t e i ξ*1*t*þ*η*1*t* <sup>2</sup> ð Þð Þ *<sup>η</sup>*<sup>1</sup> 6¼ <sup>0</sup> , *<sup>ω</sup>*2ðÞ¼ *<sup>t</sup> <sup>e</sup> i ξ*2*t*þ*η*2*t* <sup>2</sup> ð Þð Þ *<sup>η</sup>*<sup>2</sup> 6¼ <sup>0</sup> and *<sup>η</sup>*<sup>1</sup> 6¼ *<sup>η</sup>*2*:* Now using the non-linearity property (8), the SAFQ of the signal *ω*ð Þ*t* given in (23) can be computed as follows:

$$\begin{split} \mathsf{SAF}\_{\mathsf{o}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) &= \mathsf{SAF}\_{\mathsf{o}\_{1}(t) + \mathsf{o}\_{2}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) \\ &= \mathsf{SAF}\_{\mathsf{o}\_{1}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) + \mathsf{SAF}\_{\mathsf{o}\_{2}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) + \mathsf{SAF}\_{\mathsf{o}\_{1}(t),\mathsf{o}\_{2}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) + \mathsf{SAF}\_{\mathsf{o}\_{2}(t),\mathsf{o}\_{1}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) \\ &= \mathfrak{e}^{i[\mathfrak{k}(\xi\_{1} + D)\mathfrak{r} + \mathsf{E}\mathfrak{u}]} \delta[2\mathfrak{k}(\eta\_{1} + \mathsf{A})\mathfrak{r} + \mathsf{B}\mathfrak{u}] \\ &+ \mathsf{e}^{i[\mathfrak{k}(\xi\_{1} + D)\mathfrak{r} + \mathsf{E}\mathfrak{u}]} \delta[2\mathfrak{k}(\eta\_{2} + \mathsf{A})\mathfrak{r} + \mathsf{B}\mathfrak{u}] + \mathsf{SAF}\_{\mathsf{o}\_{2}(t),\mathsf{o}\_{2}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}) + \mathsf{SAF}\_{\mathsf{o}\_{2}(t),\mathsf{o}\_{1}(t)}^{\mathsf{D}}(\mathfrak{r},\mathfrak{u}). \end{split}$$

The first two terms in last equation stands for the auto-terms of one-component signals, whereas the rest represent the cross terms that are given by

$$\begin{split} & \quad \text{SAF}\_{\text{or}\_{1}(t),\text{or}\_{2}(t)}^{\text{SA}}(t,u) \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} \alpha\_{1} \left(t + k\frac{\tau}{2}\right) \alpha\_{2}^{\*} \left(t - k\frac{\tau}{2}\right) e^{i[(2Ak\tau + Bu)t + Dk\tau + Eu]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} e^{i\left[\frac{\tau}{2}(t + k\frac{\tau}{2}) + \eta\_{1}\left(t + k\frac{\tau}{2}\right)^{2}\right]} e^{-i\left[\frac{\tau}{2}(t - k\frac{\tau}{2}) + \eta\_{2}\left(t - k\frac{\tau}{2}\right)^{2}\right]} e^{i\left[(2Ak\tau + Bu)t + Dk\tau + Eu\right]} dt \\ &= \frac{B}{2\pi} \int\_{\mathbb{R}} e^{i\left[\frac{\tau}{2}t + \xi\_{1}k\frac{\tau}{2} + \eta\_{1}t^{2}\frac{\tau}{4} + \eta\_{1}k\frac{\tau}{4}\right]} e^{-i\left[\frac{\tau}{2}t - \xi\_{2}k\frac{\tau}{2} + \eta\_{2}t^{2} + \eta\_{2}k^{2}\frac{\tau}{4} - \eta\_{2}k\pi\right]} \\ & \qquad \qquad \times e^{i\left[(2Ak\tau + Bu)t + Dk\tau + Eu\right]} dt \\ &= \frac{B}{2\pi} e^{i\left[\frac{2\eta\_{1}\eta\_{2}}{4}k^{2}\tau^{2} + \frac{\zeta\_{1} + \zeta\_{2} + 2D}{2}k\tau + Eu\right]} \int\_{\mathbb{R}} e^{i(\eta\_{1} - \eta\_{2})t^{2}} e^{i\left[Bu + k\left(\eta\_{1} +$$

similarly

$$\begin{split} & \|SAF\_{\nu\_{2}(t),\mu\_{1}(t)}^{\Omega}(t,u) \\ &= \frac{1}{kb} \frac{1}{\sqrt{\pi(\eta\_{2}-\eta\_{1})}} e^{i\left[\frac{\eta\_{2}-\eta\_{1}}{4}k^{2}\tau^{2} + \frac{\xi\_{1}+\xi\_{2}+2\delta}{2}k\tau - Eu\right]} e^{-i\frac{\left[Ru+k\left(\eta\_{2}+\eta\_{1}+2\delta\right)-\left(\xi\_{2}-\xi\_{1}\right)\right]^{2}}{4\left(\eta\_{2}-\eta\_{1}\right)}}. \end{split}$$

Hence the SAFQ of a bi-component signal *ω*ðÞ¼ *t ω*1ðÞþ*t ω*2ð Þ*t* is given by

$$\begin{split} \mathcal{S}AF^{\Omega}\_{w(t)}(\tau,u) &= \mathcal{S}AF^{\Omega}\_{w\_{1}(t)+w\_{2}(t)}(\tau,u) \\ &= e^{i[k(\xi\_{1}+D)\tau+Eu]}\delta[2k(\eta\_{1}+A)\tau+Bu] \\ &+ e^{i[k(\xi\_{1}+D)\tau+Eu]}\delta[2k(\eta\_{2}+A)\tau+Bu] \\ &+ \frac{B}{k}\frac{1}{\sqrt{\pi(\eta\_{1}-\eta\_{2})}}e^{i\left[\frac{\eta\_{1}-\eta\_{2}}{4}k^{2}\tau^{2}+\frac{\zeta\_{1}+\zeta\_{2}+2\delta}{2}k\tau+Eu\right]}e^{-i\frac{[\eta\_{1}+k(\eta\_{1}+\eta\_{2}+2\lambda)-(\zeta\_{1}-\zeta\_{2})]^{2}}{4(\eta\_{1}-\eta\_{2})}} \\ &+ \frac{1}{kb}\frac{1}{\sqrt{\pi(\eta\_{2}-\eta\_{1})}}e^{i\left[\frac{\eta\_{2}-\eta\_{1}}{4}k^{2}\tau^{2}+\frac{\zeta\_{1}+\zeta\_{2}+2\delta}{2}k\tau-Eu\right]}e^{-i\frac{[\ln k + \delta\{\eta\_{1}+\eta\_{2}+2\lambda\}-(\zeta\_{1}-\zeta\_{2})]^{2}}{4(\eta\_{2}-\eta\_{1})}}. \end{split} \tag{24}$$

It is clear from (24) a that the first two auto-terms are able to generate impulses which the cross terms cannot generate, and therefore, although the existence of cross terms has a certain influence on the detection performance, but the bi-component LFM signal still can be detected. This indicates that the scaled AFQ is also useful and powerful for detecting bi-component LFM signals. Moreover for an adequate value of *k* and matrix parameter Ω, the scaled AFQ benefits in cross-term reduction while maintaining a perfect time-frequency resolution with clear auto terms angle resolution.

#### **5. Conclusion**

Motivated by degree of freedom corresponding to the choice of a factor *k* in the fractional instantaneous auto-correlation and the extra degree of freedom present in QPFT, we proposed novel scaled AFQ. First, we studied the fundamental properties of the proposed distributions, including the time marginal, conjugate symmetry, non-linearity, time shift, frequency shift, frequency marginal, scaling, inverse and Moyal formula. Finally to show the of advantage of the theory, we provided the applications of the scaled AFQ in the detection of single-component and bi-component linear- frequency-modulated (LFM) signal.

#### **Acknowledgements**

This work is supported by Research project(JKSTIC/SRE/J/357-60) provided by DST Government of Jammu and Kashmir, India.

#### **Author contributions**

Both the authors contributed equally in the paper.

#### **Conflict of interest**

The authors declare no potential conflict of interests.

#### **Classification**

#### **2000 Mathematics subject classification:**

42C40; 81S30; 11R52; 44A35

*Scaled Ambiguity Function Associated with Quadratic-Phase Fourier Transform DOI: http://dx.doi.org/10.5772/intechopen.108668*

#### **Author details**

Mohammad Younus Bhat<sup>1</sup> \*, Aamir Hamid Dar<sup>1</sup> , Altaf Ahmad Bhat2 and Deepak Kumar Jain<sup>3</sup>

1 Department of Mathematical Sciences, Islamic University of Science and Technology, Kashmir, India

2 University of Technology and Applied Sciences, Salalah, Oman

3 Madhav Institute of Technology and Science, Gwalior, India

\*Address all correspondence to: gyounusg@gmail.com

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Section 2 Applications

#### **Chapter 4**
