1. INTRODUCTION

The Wigner distribution function (WDF) is a time-frequency (phase space) distribution widely used in optics. It allows the interpretation of certain systems and phenomena geometrically, including fractional Fourier transforms (FRTs) [1], paraxial propagation through len systems [2], non-paraxial fields [3] and other propagation problems [4], the Talbot effect [5], the Moiré effect [6], digital holography [7–9], phase retrieval [10], and coupling partially coherent light into optical fibers [11], among others [12,13]. It has been argued that it is an important educational tool for linking ideas in disparate areas of optics [14]. Its cross terms also capture information about coherence [15,16]. More generally, the WDF belongs to Cohen’s class of time-frequency distributions [17] and is widely used in signal processing [18]. Linear canonical transforms (LCTs) are a Li group of transforms that describe paraxial monochromatic propagation in quadratic phase systems [19]. Special cases include the Fresnel integral, the FRT, and the Fourier transform (FT). The WDF may be interpreted as the FT of the autocorrelation function of a wave field. That interpretation has motivated several authors to propose various generalizations of the WDF that make use of the LCT instead of the FT [20–31]. Those papers typically determine properties of the novel distribution; several offer an example application in which a linear frequency modulated signal is analyzed. In this paper, we set out to understand the relationship between these various novel distributions and the WDF, and we to attempt to identify which of them have independent significance.

This paper is organized as follows: In Section 2, the definition of the WDF, LCT, and several novel distributions will be reviewed. Then, we plot their graphs based on a sample signal and identify their relationships with the WDF in Section 3. After that, Section 4 gives the explanation of the cross term of the distributions and focuses on interpreting the cross terms in novel distributions mathematically. In Section 5, some existing properties of the novel distributions will be presented, and some new properties will be proposed and proved. Furthermore, Section 6 discusses the applications of the distributions in partially coherent systems. Finally, Section 7 will summarize the paper.

Table 1. Definitions of the Novel Distributions

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2. DEFINITIONS

The WDF of a function $f(x)$ is defined as follows:

The FT of $f(x)$ is defined to be $F({k_x})$. An alternative definition of the WDF is as a frequency integration by applying the Parseval formula:

The LCT of $f(x)$ for parameters $A = \left({\begin{array}{*{20}{c}}a&b\\c&d\end{array}}\right)$ (which characterises a first-order optical system) is defined as

For $A = \left({\begin{array}{*{20}{c}}0&1\\{- 1}&0\end{array}}\right)$, Eq. (4) reduces to a FT. If $A = \left({\begin{array}{*{20}{c}}1&{\lambda z}\\0&1\end{array}}\right)$, it reduces to a Fresnel integral for wavelength $\lambda$ and propagation distance $z$. $A = \left({\begin{array}{*{20}{c}}{\cos \theta}&{\sin \theta}\\{- \sin \theta}&{\cos \theta}\end{array}}\right)$ is a FRT. When $\theta = 0$, the FRT reduces to imaging. In general, if $A$ is the ABCD matrix of a system, the LCT relates the input and output monochromatic scalar wave fields.

We define the kernel of the transform as follows:

The WDF of ${F_A}(u)$ can be found from ${W_f}(x,{k_x})$ by means of a linear coordinate substitution as follows [2]. This observation may permit the design of spatially varying filters [24]:

A. Novel Distributions

There are many approaches to generalize the WDF, e.g., we might replace the kernel, the function, the autocorrelation, or the shift operation. In Table 1, we collate such generalizations proposed in the literature. For readability, we have named each distribution after the lead author of the paper that to our knowledge first reported it.

Pei and Ding replace $f(x)$ in Eq. (1) with its LCT [2]; we refer to this as the Pei distribution function (PDF). (See also [22,23].)

Bai, Li, and Cheng swap the Fourier kernel for the LCT kernel in Eq. (1) [20]; we refer to this as the Bai distribution function (BDF).

Song et al. also swap out the Fourier kernel for the LCT kernel in Eq. (1), and additionally replace the autocorrelation function with generalized correlation autocorrelation function ${\rm GR}(x,\tau) = f({x + \frac{\tau}{2}})f({x - \frac{\tau}{2}}){f^*}({- x + \frac{\tau}{2}})\def\LDeqbreak{}{f^*}({- x - \frac{\tau}{2}})$ [21]; we refer this as to the Song distribution function (SDF).

Zhang’s proposed distribution can be interpreted as multiplying the autocorrelation function by a chirp function, and scaling the Fourier variable by $b$ [28,34]. We refer to this as Zhang’s distribution function (ZDF). The ZDF is defined as

Urynbassarova’s distribution (UBF) is quite similar to ZDF [26].

Zhang also proposed the scaled Wigner distribution (SWDF), which is a scaled version of the WDF as shown below [32]:

Similarly based on SWDF and ZDF, Dar et al. proposed another distribution, which we refer to as the scaled Zhang’s distribution (SZDF) [33],

There are many other related works. Urynbassarova, Li, and Tao also proposed distributions for the offset LCT [35]. M. Younus Bhat proposed a distribution based on SWDF and offset LCT [36]. Mawardi Bahri defined a quaternion WDF [37]. Zhang has also published extensively on the generalization of these distributions, unifying the WDF and ambiguity function, and broadening the scope to include the Cohen class of time-frequency distributions [24,25,27–30]. Wei et al. published a paper on WDF and ambiguity function associated with 2D non-separable LCTs [31]. We omit citation of generalizations of the short-time FT and of the ambiguity function for brevity.

 

Fig. 1. Contour plots of various distributions. Except for (a), all the other graphs present the absolute value of the corresponding distribution. (a) Instantaneous autocorrelation function. (b) Wigner distribution function. (c) Pei’s distribution function. (d) Bai’s distribution function. (e) Song’s distribution function. (f) Zhang’s distribution function. (g) Scaled Wigner distribution. (h) Scaled Zhang’s distribution. (i) Urynbassarova’s distribution. The following LCT parameters are used, $A = \left({\begin{array}{*{20}{c}}1&{0.5}\\0&1\end{array}}\right)$.

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3. PLOTTING THE DISTRIBUTIONS

In Fig. 1, we plot nine distributions evaluated using symbolic integration software. In all of the results, where relevant, the following arbitrary LCT parameters are used: $A = \left({\begin{array}{*{20}{c}}1&{0.5}\\0&1\end{array}}\right)$. We plot these for the test signal $f(x) = \exp ({- {x^2}/\sqrt 2}) \,+\def\LDeqbreak{} \exp ({- {{({x - 4})}^2}/\sqrt 2})$: the sum of two Gaussian beams. This function was chosen for its ease of integration and its distinctive shape. The instantaneous autocorrelation function and WDF of $f(x)$ are plotted in Figs. 1(a) and 1(b). For ${R_f}$, we observe four 2D Gaussian terms. The WDF is the FT of ${R_f}$ in $\tau$ direction (note a pair of 2D Gaussian functions and a rapidly oscillating cross term).

According to Fig. 1(c), the PDF is a skewing in the $x$ direction of the WDF [Fig. 1(b)] consistent with Eq. (6). Note the saturation around (2,0) as the cross term has high amplitude; the saturation has been allowed to better depict the features of the other terms.

Bai’s distribution is depicted in Fig. 1(d). It shows four 2D Gaussian terms with no oscillation observed in the WDF; it is similar to the original ${R_f}$. These four terms will be interpreted in Section 4.

Figure 1(e) shows Song’s distribution function, which only contains one Gaussian term. Compared to the other distributions, it is much simpler. Again, this distribution will be explained in terms of the components of the wave field in Section 4.

Zhang’s distribution is plotted in Fig. 1(f). The absolute value is used because the function is complex. Similarly, the SWDF, SZDF, and UBF—presented in Figs. 1(g)–1(i), respectively—can all be considered as affine transformations of the WDF. As they, therefore, have similar properties and do not provide new perspectives on a signal, they will not be the focus of the remainder of this paper. The results of BDF and SDF show sufficiently different geometrical characteristics from the WDF to merit explanation and discussion below.

4. PROPERTIES OF CROSS TERMS

In Fig. 1(b), the cross terms are the oscillation. They complicate the WDF of multi-component signals, so they are suppressed in audio signal processing. In optics, their physical meaning is in interference effects. The WDF can be readily extended to partial coherence [38] when the cross terms would disappear if the two Gaussian beams were mutually incoherent, while intermediate cases appear as a smoothing of the distribution [15] and are relevant in numerical wave optics [16]. This section focuses on interpreting the cross terms of the WDF, BDF, and SDF mathematically. We start with the WDF in order to provide a point of comparison for the other two distributions.

A. WDF Cross Terms

We assume an example composite signal $f = {f_1} + {f_2}$ given by

Based on Eq. (2), we can find the relationships between the instantaneous autocorrelation functions of ${f_1}$, ${f_2}$, and ${f}$, respectively, ${R_{{f_1}}}$, ${R_{{f_2}}}$, and ${R_f}$. That relationship is as follows:

We observe that two terms appear in the equation above, ${R_{{f_1}{f_2}}} = {f_1}({t + \frac{\tau}{2}})f_2^*({t + \frac{\tau}{2}})$ and ${R_{{f_2}{f_1}}} = {f_2}({t + \frac{\tau}{2}})f_1^*({t + \frac{\tau}{2}})$. These are the cross terms of ${R_f}$. Similarly, ${R_{{f_1}}}$ and ${R_{{f_2}}}$ are called auto terms. The WDF of the signal $f$ can be obtained by calculating the FT of ${R_f}$ on $\tau$ direction, giving

The cross terms are given by

In order to examine the cross terms in the WDF, the first cross term is presented as follows:

Similarly, the second cross term, ${R_{{f_2}{f_1}}}$, is given by

It can be observed that the two cross terms are located at the same $x$ coordinate and are symmetrical around $\tau = 0$. Then, we assume the expression of a signal $g(x,\tau)$ is

The cosine leads to the oscillation in Fig. 1(a).

 

Fig. 2. Contour plots of BDF with FRT matrix. (a) $\theta = 0.0001\,\pi$. (b) $\theta = 0.2\,\pi$. (c) $\theta = 0.3\,\pi$. (d) $\theta = 0.4\,\pi$. (e) $\theta = 0.45\,\pi$. (f) $\theta = 0.5\,\pi$.

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B. BDF Cross Terms

To explain the BDF as depicted in Fig. 1(d), we again consider the composite signal $f = {f_1} + {f_2}$ defined in Eq. (11). The cross terms in the BDF are the LCT of ${R_{{f_1}{f_2}}} + {R_{{f_2}{f_1}}}$ in the $\tau$ direction, given by

By substituting Eqs. (15) and (16) into Eq. (19), the cross terms can be written as

Again, we assume that the LCT of the signal $g(x,\tau)$ on $\tau$ direction is ${F_A}(x,{u_x})$. By applying the time shift property of LCT, the cross term can be calculated using the LCT’s shift theorem,

Based on Eqs. (17), (20), and (21), the cross terms of the BDF could be simplified to

It indicates that the location and interference of the resultant cross terms are only related to three parameters: $a$, $c$, and ${x_0}$. We substitute the matrix $A = \left({\begin{array}{*{20}{c}}1&{0.5}\\0&1\end{array}}\right)$ when ${x_0} = 4$, which are the parameters used in Fig. 1(d).

We see that as $c = 0$ (large $a$ can produce a similar result) the cross terms do not interact with each other to produce oscillation as in the WDF case.

We wanted to investigate the similarity between the BDF of the example function and the instantaneous autocorrelation function of the same function. In Fig. 2, we plotted the BDF for a range of a single parameter, using the FRT as a one-parameter special case of the LCT. Its parameter is fractional order, or equivalently the rotation angle in phase space $\theta$; these range from 0 to 1 and from 0 to $\pi /2$, respectively, and are proportional. Six distributions for $0 \lt \theta \le \pi /2$ are shown in Figs. 2(a)–2(f). Substituting the FRT parameters into Eq. (22), the new cross terms are

As $\theta$ increases, the value of $\cos \theta$ reduces so that the cross terms move toward one another. Where they overlap, the complex exponential acts to create oscillations. The two terms completely coincide when $\cos \theta = 0$. The expression can be simplified to $2{{\rm G}_\theta}(x,{u_x})\cos (c{x_0}t) = {W_f}(x,{u_x})$. We conclude that Fig. 2 shows how the BDF can interpolate between the instantaneous autocorrelation function and the WDF. As both are significant in partial coherence, we suggest that the BDF may have applications in that field. In general, the BDF embeds the instantaneous autocorrelation function and the WDF in a 3D parameter space.

C. SDF Cross Terms

Depicted in Fig. 1(e) is the SDF. In contrast with the other distributions, only one term appears. Recall, in the SDF, the instantaneous autocorrelation function in the WDF is replaced by a generalized instantaneous autocorrelation function, specifically,

To explain the figure, we separate ${\rm GR}(x,\tau)$ into ${{\rm GR}_1}(x,\tau) = f({x + \frac{\tau}{2}})f({x - \frac{\tau}{2}})$ and ${{\rm GR}_2}(x,\tau) = {f^*}({- (x + \frac{\tau}{2}})){f^*}({- (x - \frac{\tau}{2}}))$. For $f(t) \in \mathbb{R}$, ${\rm GR_1}$ and ${\rm GR_2}$ are instantaneous autocorrelation functions of $f(t)$ and $f(- t)$, respectively; these are symmetrical about $x = 0$. When multiplying ${\rm GR_1}$ and ${\rm GR_2}$ together, the non-overlapping parts cancel out. In this example, only one term in these two instantaneous autocorrelation functions overlaps, which leads to the result shown in Fig. 1(e).

This multiplication generally leads to information loss, complicating the interpretation of the resultant distribution. For an example, the SDF of signal

 

Fig. 3. SDF of a polynomial Gaussian signal.

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5. OTHER PROPERTIES OF THE DISTRIBUTIONS

In this section, we collect the properties of the distributions (WDF, BDF, SDF; some others in Section 5.A); this knowledge facilitates analysis of the distributions and their potential applications. Some properties of the novel distributions have been investigated in the papers that proposed them; others are derived here for the first time. There is also scope for further work. All properties apply for all ABCD parameters except where noted.

A. Relationship between Distributions of a Signal and Its Transformation

As noted, the relationship between the WDF of a function and the WDF of the LCT of that function is given by Eq. (6).

We have established the relationships between ZDF, UBF, and WDF which can be expressed as follows:

The proof is as follows:

The relationship between the WDF, SWDF, and SZDF was given by Dar and Bhat [33],

According to Eqs. (27) and (28), the ZDF, UBF, SWDF, and SZDF are affine transforms of the WDF. Consequently, they will not be considered further in this paper.

When $A = \left({\begin{array}{*{20}{c}}0&1\\{- 1}&0\end{array}}\right)$, the BDF reduces to the WDF [20],

We can obtain a similar result for the SDF providing $f \in \mathbb{R}$. In that case, and if $A = \left({\begin{array}{*{20}{c}}0&1\\{- 1}&0\end{array}}\right)$, we have established that the relationship between WDF and SDF is as follows:

The proof is as follows:

B. Support

If a signal $f(x) = 0$ for $x \gt {x_0}$, then the WDF is zero for the same range. If the signal’s FT, $F({k_x})$, is zero for $|{k_x}| \gt {k_0}$, then the WDF is zero for the same range. Both of these things cannot be true at the same time. Various other marginals have similar results [39]. We can observe similar properties for the BDF and SDF.

If $f(x) = 0$ for $|x| \gt {x_0}$, then ${B_{f,A}}(x,{u_x}) = 0$ for the same range. If the signal’s LCT, ${F_{f,A}}({u_x})$, is zero for $|{u_x}| \gt {u_0}$, then the BDF is zero for the same range. Both of these things cannot be true at the same time.

A more limited property holds for the SDF. If $f(x)f(- x) = 0$ for $|x| \gt {x_0}$, then ${S_{f,A}}(x,{u_x}) = 0$ for the same range. If ${F_{f,A}}({u_x}){L_{f,A}}(- {u_x})$ is zero for $|{u_x}| \gt {u_0}$, then the SDF is zero for the same range. Both of these things cannot be true at the same time.

C. Real and Conjugate Properties

The WDF is real for all signals,

Similarly, Bai et al. claim a conjugate symmetry property [20],

We can show that the SDF has a similar symmetry and conjugation property as well,

The proof is as follows:

D. Shift Theorems

If $g(x) = f(x - {x_0})$, the time shift property of the WDF is

If $g(x) = f(x){e^{\textit{jwx}}}$, the phase shift property of the WDF is given by

Bai et al. give this: for $g(x) = f(x - {x_0})$, the time shift property of BDF is

Similarly, for $g(x) = f(x - {x_0})$, the time shift property of the SDF is given by

The proof of this property is very simple, so it will not be presented here. Due to the complexity of the SDF, it is difficult to find a frequency shift property.

E. Multiplication and Convolution

The product and convolution properties of the WDF are shown, respectively, as follows:

For BDF, Bahri et al. defined a special convolution operator to simplify the expression of the BDF convolution theorem [23],

The proof is as follows:

Substituting Eq. (46) into (45),

By modifying the convolution theorem proposed by Bahri et al. using the novel operator, it can be written in the same form as the product theorem [23],

For SDF, we also established the product theorem, similar to that of the BDF,

However, due to the complexity of ${{\rm GR}_f}$, the convolution theorem of SDF cannot be expressed using the same form. The proof for the product theorem of the SDF is basically the same as the BDF, so it will not be presented again.

F. Projection

The marginals of the WDF are readily interpreted,

For BDF, we propose the $x$ marginal, which is given by

To obtain the ${u_x}$ marginal of the BDF, we first establish the alternative expression for the BDF in frequency domain when $a = 0$, which is given as follows:

Using the alternative expression, the ${u_x}$ marginal of the BDF when $a = 0$ is written as follows:

For SDF, we established the $x$ marginal properties as well. The proofs of Eqs. (52)–(55) are similar enough to the WDF, so they are omitted,

G. Moyal Formula

For the WDF, the Moyal formula can be expressed as

Bai et al. [20] give

We established the Moyal formula for the SDF, which is given by

The proof is as follows:

If the instantaneous autocorrelation function is symmetrical about $x = 0$, then

H. Recovery

The recovery property for WDF could be written as follows [2]:

The recovery property of the BDF proposed by Bai et al. is as follows [20]:

We establish an alternative recovery property when $a = 0$ for the BDF in frequency domain, which is shown below. By using the alternative expression in Section 5.F, the proof is similar to Bai’s method, so it will not be presented again,

The SDF is not generally invertible.

I. Moments

For the WDF, the moments property can be written as

The moments property in the time domain of the BDF is established in this paper as shown below:

Furthermore, the moments property in the frequency domain of the BDF is also proposed when $a = 0$, which is given by

Finally, we established the moments property for the SDF as well. Due to the similarity between the moments and projection property, we omit the proof,

6. BDF APPLIED TO PARTIAL COHERENCE

In this section, we analyze the propagation of the cross-spectral density of a partially coherent wave field in free space. We see that, under the Fresnel approximations, the BDF can link the position variables in the source and observation planes, analogous to how the WDF links input position and angle in the Fraunhofer regime.

A temporally stationary random process or space-time realization $V({\boldsymbol r},t)$ can be used to describe a partially coherent light field. ${\boldsymbol r}$ is a 2D position vector. The ensemble average of ${V^*}({{\boldsymbol r}_1},t)V({{\boldsymbol r}_2},t - \tau)$ is the mutual coherence function at positions ${{\boldsymbol r}_1}$ and ${{\boldsymbol r}_2}$ given by

The angular brackets stand for the ensemble average. $\tau$ is a time delay. The FT of the mutual coherence function is the cross-spectral density,

As shown in Fig. 4, the light field is diffracted by an aperture on the source plane and propagates from the source to the observation plane. The relationship between the cross-spectral density on these two planes can be expressed using diffraction theory. If the space-frequency realization at the point ${P_0}$ on the source plane is ${U_0}({\boldsymbol \rho},\omega) = {U_0}({x_0},{y_0},\omega)$ and at the point P on the observation plane is $U({\boldsymbol r},\omega) = U(x,y,\omega)$, then $U(x,y,\omega)$ can be expressed in terms of ${U_0}({x_0},{y_0},\omega)$ using the Fresnel–Kirchoff diffraction integral,

 

Fig. 4. Optical diffraction setup.

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Fig. 5. Two approximation methods.

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Here, we apply another approximation and extend the concept proposed by Bastiaans. Under the paraxial approximation, we approximate $R$ on the denominator of the term $\frac{{{e^{\textit{jkR}}}}}{R}$ as $z$ (the length of the red line in Fig. 5) and the phase term as $z\Big[{1 + \frac{{{{({x - {x_0}})}^2}}}{{2{z^2}}} + \frac{{{{({y - {y_0}})}^2}}}{{2{z^2}}}}\Big]$. Equation (71) becomes a Fresnel diffraction integral,

By expanding the quadratic terms and using vectors ${\boldsymbol r}$ and ${\boldsymbol \rho}$ to represent the positions, we can obtain a Fresnel transform multiplied by constant terms,

7. CONCLUSION

A strength of the WDF is that heuristic explanations of the features of the distribution allow us to relate them to the components of the signal, e.g., the WDF is bilinear: in the example used in Fig. 1(b), we can clearly see contributions from each of the two Gaussian beams and an interference term between them. The interference term has also lead to Cohen class filtered approaches to describe partial coherence. Shift, product, and convolution theorems and marginals offer additional routes to interpreting and using WDFs. In this paper, we evaluated a number of novel distributions associated with the WDF in order to ascertain which of them might have independent significance and applications.

In the first three sections, we introduced the definitions of the WDF and seven distributions associated with it and plotted them in order to present their geometrical properties and relationships. Some of the new distributions, including PDF, ZDF, UBF, SWDF, and SZDF [2,26,28,32,33], are just linear coordinate transformations of the WDF, and their relationships are shown in Section 5.A. Therefore, they should have similar effects when analyzing signals compared with the WDF; hence, this paper does not focus on them much.

The remaining distributions, including BDF and SDF [20,21], have different geometrical and mathematical properties. In Section 4, the cross terms of the WDF, BDF, and SDF are interpreted. For BDF, we calculate the general expression of its cross terms and determine the effects of each parameter. By adjusting the parameter in the LCT matrix $A$, we can not only obtain more flexibility in signal analysis but also impact the interference or oscillations of the cross terms. We have also shown how the BDF can interpolate between the instantaneous autocorrelation function and the WDF. Song et al. have demonstrated that the SDF is effective in the QFM signal parameter estimation.

In Section 5, we proposed several properties of the BDF and SDF for the first time. By replacing the FT kernel with LCT kernel, the properties of the BDF are more complicated than the WDF, especially for the multiplication and convolution theorem. Due to the mathematical complexity of the SDF, it is also significantly difficult to obtain general expressions for some of its properties. These can cause limitations when applying the BDF and SDF.

Finally, we generalize the WDF of cross-spectral density defined by Basstiaans using the BDF to link the ray concept through the entire propagation process from the source plane to the far field. It can present the energy distribution from each point on the source plane to each point on each observation plane as well as the coherence property. Hence, we have shown one role the BDF may have in the analysis of partially coherent light.