Inequalities for the Windowed Linear Canonical Transform of Complex Functions

Abstract

In this paper, we generalize the N-dimensional Heisenberg’s inequalities for the windowed linear canonical transform (WLCT) of a complex function. Firstly, the definition for N-dimensional WLCT of a complex function is given. In addition, the N-dimensional Heisenberg’s inequality for the linear canonical transform (LCT) is derived. It shows that the lower bound is related to the covariance and can be achieved by a complex chirp function with a Gaussian function. Finally, the N-dimensional Heisenberg’s inequality for the WLCT is exploited. In special cases, its corollary can be obtained.

1. Introduction

Inequalities for the Fourier transform (FT) are widely used in mathematics, physics and engineering [1,2,3,4,5,6]. The classical N-dimensional Heisenberg’s inequality of the FT is given by the following formula [7]:

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^{\mathbf{f}})}^2\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}^{\mathbf{f}})}^2\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u}\ge ð{\parallel \mathit{f}\parallel }_{{\mathrm{L}}^2\left({\mathbb{R}}^{\mathbb{N}}\right)}^4,\end{array}\end{array}$$

(1)

where $ð={\left(\frac{N}{4\pi }\right)}^2,$ $\mathbf{t}=(t_1,t_2,\cdots ,t_N)$, $\mathbf{u}=(u_1,u_2,\cdots ,u_N)$. $\widehat{\mathit{f}}\left(\mathbf{u}\right)$ is the FT of any function $f\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \widehat{\mathit{f}}\left(\mathbf{u}\right)=F\left\{f\left(\mathbf{t}\right)\right\}\left(\mathbf{u}\right)=\frac{1}{\sqrt{2\pi }}{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{t}\right)e^{-i\mathbf{t}\mathbf{u}}\mathrm{d}\mathbf{t},\end{array}\end{array}$$

(2)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{t}}^{\mathbf{f}}={\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t},\end{array}\end{array}$$

(3)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{u}}^{\mathbf{f}}={\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u},\end{array}\end{array}$$

(4)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\parallel f\parallel }_{L^2\left({\mathbb{R}}^{\mathbb{N}}\right)}^2={\parallel f\parallel }^2={\int }_{{\mathbb{R}}^{\mathbb{N}}}{\left|f\left(\mathbf{t}\right)\right|}^2\mathrm{d}\mathbf{t},\end{array}\end{array}$$

(5)

Based on Formula (1), Zhang obtained the N-dimensional Heisenberg’s inequality of the fractional Fourier transform (FRFT) [8].

The windowed linear canonical transform (WLCT) [9,10,11] is a generalized integral transform of the FT [12] and the FRFT [13]. In recent years, inequality of the WLCT has become a hot topic. Many scholars [14,15,16,17] have studied different types of inequalities for the WLCT.

The purpose of this paper is to obtain various kinds of N-dimensional inequalities associated with the WLCT.

2. Preliminary

Let any function $f\left(\mathbf{t}\right)=f_1\left(\mathbf{t}\right)e^{i\varphi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$ and window function $0\ne g\left(\mathbf{t}\right)=g_1\left(\mathbf{t}\right)e^{i\phi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$.

Definition 1.

([18]). Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ be a matrix parameter satisfying $a,b,c,d\in \mathbb{R}$ and $ad-bc=1$. For any function $f\left(\mathbf{t}\right)$, the linear canonical transform (LCT) of $f\left(\mathbf{t}\right)$ is defined as

$$\begin{array}{c}\hfill \begin{array}{c}\hfill L_A^f\left(\mathbf{u}\right)=L_A\left[f\left(\mathbf{t}\right)\right]\left(\mathbf{u}\right)=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{t}\right)K_A(\mathbf{t},\mathbf{u})\mathrm{d}\mathbf{t},\hfill & b\ne 0\hfill \\ \sqrt{d}{e}^{i\frac{cd}{2}{\mathbf{u}}^2}f\left(d\mathbf{u}\right)),\hfill & b=0\hfill \end{array}\right.\end{array}\end{array}$$

(6)

where

$$\begin{array}{c}\hfill K_A(\mathbf{t},\mathbf{u})=\frac{1}{\sqrt{i2\pi b}}{e}^{i\frac{a}{2b}{\mathbf{t}}^2-i\frac{1}{b}\mathbf{tu}+i\frac{d}{2b}{\mathbf{u}}^2}.\end{array}$$

(7)

Additionally, the paper [19] presented the following properties:

$$\begin{array}{c}\hfill K_A^{*}(\mathbf{t},\mathbf{u})=K_{A^{-1}}(\mathbf{u},\mathbf{t}),\end{array}$$

(8)

$$\begin{array}{c}\hfill 2\pi \delta \left(\mathbf{x}\right)={\int }_{{\mathbb{R}}^{\mathbb{N}}}{e}^{\pm i\mathbf{ux}}\mathrm{d}\mathbf{u},\end{array}$$

(9)

where $A^{-1}=\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$, $\mathbf{x}=(x_1,x_2,\cdots ,x_N)$.

If $b=0$, then the LCT becomes a kind of scaling and chirp multiplication operations [20]. In this paper, we only consider $b\ne 0$.

The inverse formula of the LCT is given by [19]

$$\begin{array}{c}\hfill f\left(\mathbf{t}\right)={\int }_{{\mathbb{R}}^{\mathbb{N}}}{L}_A^f\left(\mathbf{u}\right)K_{A^{-1}}(\mathbf{u},\mathbf{t})\mathrm{d}\mathbf{u}.\end{array}$$

(10)

Definition 2.

([9]). Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ be a matrix parameter satisfying $a,b,c,d\in \mathbb{R}$ and $ad-bc=1$. The WLCT of function f with respect to g is defined by

$$\begin{array}{cc}\hfill W_g^{A}f(\mathbf{t},\mathbf{u})& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{y}\right)g^{*}(\mathbf{y}-\mathbf{t})K_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{f}_{\mathbf{t}}\left(\mathbf{y}\right)K_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y},\hfill \end{array}$$

(11)

where $\mathbf{y}=(y_1,y_2,\cdots ,y_N)$ and $f_{\mathbf{t}}\left(\mathbf{y}\right)=f\left(\mathbf{y}\right)g^{*}(\mathbf{y}-\mathbf{t})=f_1\left(\mathbf{y}\right)g_1^{*}(\mathbf{y}-\mathbf{t})e^{i\left(\varphi \right(\mathbf{y})-\phi (\mathbf{y}-\mathbf{t}\left)\right)}$.

Next, we will give a lemma.

Lemma 1.

For $f\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$ and $g\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, we have

$$\begin{array}{c}\hfill W_g^{A}f(\mathbf{t},\mathbf{u})={\int }_{{\mathbb{R}}^{\mathbb{N}}}{L}_A^{\mathit{f}}\left(\mathbf{k}\right)Q^{*}\left(\mathbf{k}\right|\mathbf{u},\mathbf{t})\mathrm{d}\mathbf{k},\end{array}$$

(12)

where $A_1=\left[\begin{array}{cc}0& b^{\prime }\\ -\frac{1}{b^{\prime }}& d^{\prime }\end{array}\right]$, $0\ne b^{\prime }=b\in \mathbb{R}$,

$$\begin{array}{c}\hfill Q^{*}\left(\mathbf{k}\right|\mathbf{u},\mathbf{t})=\sqrt{-i2\pi b}{e}^{i\frac{d^{\prime }}{2b}{(\mathbf{k}-\mathbf{u})}^2}{L}_{A_1}^g{(\mathbf{k}-\mathbf{u})}^{*}{K}_A(\mathbf{t},\mathbf{u})K_A^{*}(\mathbf{t},\mathbf{k}).\end{array}$$

Proof. 

According to Definition 2 and Formula (10), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_g^{A}f(\mathbf{t},\mathbf{u})& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{y}\right)\overline{g(\mathbf{y}-\mathbf{t})}{K}_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{L}_A^f\left(\mathbf{k}\right){\int }_{{\mathbb{R}}^{\mathbb{N}}}{K}_{A^{-1}}(\mathbf{k},\mathbf{y})\overline{g(\mathbf{y}-\mathbf{t})}{K}_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y}\mathrm{d}\mathbf{k}.\hfill \end{array}\end{array}$$

(13)

Assume that $Q^{*}\left(\mathbf{k}\right|\mathbf{u},\mathbf{t})={\int }_{{\mathbb{R}}^{\mathbb{N}}}{K}_{A^{-1}}(\mathbf{k},\mathbf{y})\overline{g(\mathbf{y}-\mathbf{t}}{K}_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y}$ and $\mathbf{y}-\mathbf{t}=\mathbf{p}$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill Q^{*}\left(\mathbf{k}\right|\mathbf{u},\mathbf{t})& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{K}_{A^{-1}}(\mathbf{k},\mathbf{y})\overline{g(\mathbf{y}-\mathbf{t}}{K}_A(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{\mathbb{N}}}\overline{g\left(\mathbf{p}\right)}\frac{1}{\sqrt{-i2\pi b}}\frac{1}{\sqrt{i2\pi b}}{e}^{-i\frac{(\mathbf{u}-\mathbf{k})}{b}(\mathbf{p}+\mathbf{t})+i\frac{d}{2b}({\mathbf{u}}^2-{\mathbf{k}}^2)}\mathrm{d}\mathbf{p}\hfill \\ \hfill & =\frac{1}{\sqrt{i2\pi b}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\frac{1}{\sqrt{-i2\pi b}}\overline{g\left(\mathbf{p}\right)}{e}^{i\frac{0}{-2b}{\mathbf{p}}^2-i\frac{(\mathbf{k}-\mathbf{u})}{-b}\mathbf{p}+i\frac{d^{\prime }}{-2b}{(\mathbf{k}-\mathbf{u})}^2}\mathrm{d}\mathbf{p}\hfill \\ \hfill & \times e^{i\frac{d^{\prime }}{2b}{(\mathbf{k}-\mathbf{u})}^2+i\frac{d}{2b}({\mathbf{u}}^2-{\mathbf{k}}^2)-i\frac{(\mathbf{u}-\mathbf{k})}{b}\mathbf{t}}\hfill \\ \hfill & =\frac{1}{\sqrt{i2\pi b}}{e}^{i\frac{d^{\prime }}{2b}{(\mathbf{k}-\mathbf{u})}^2}{L}_{A_1}^g{(\mathbf{k}-\mathbf{u})}^{*}{e}^{-i\frac{\mathbf{ut}}{b}+i\frac{d}{2b}{\mathbf{u}}^2}{e}^{i\frac{\mathbf{k}\mathbf{t}}{b}+i\frac{d}{-2b}{\mathbf{k}}^2}\hfill \\ \hfill & =\sqrt{-i2\pi b}{e}^{i\frac{d^{\prime }}{2b}{(\mathbf{k}-\mathbf{u})}^2}{L}_{A_1}^g{(\mathbf{k}-\mathbf{u})}^{*}{K}_A(\mathbf{t},\mathbf{u})K_A^{*}(\mathbf{t},\mathbf{k}).\hfill \end{array}\end{array}$$

(14)

Hence the Formula (13) becomes (12). □

3. Inequalities Associated with the WLCT

The aim of this section is to obtain the new inequalities for the WLCT by the precise mathematical formulation.

Definition 3.

Let $f\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, then we can define [21]

$$\begin{array}{c}\hfill {\mathbf{t}}^{\mathbf{f}}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t},\end{array}$$

(15)

$$\begin{array}{c}\hfill {\mathbf{u}}^{\mathbf{f}}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(16)

$$\begin{array}{c}\hfill {\mathbf{u}}_{\mathbf{f}}^{\mathbf{A}}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid L_A^{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u}.\end{array}$$

(17)

$$\begin{array}{c}\hfill {\Delta }_{\mathit{f}}^2=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^{\mathbf{f}})}^2\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t},\end{array}$$

(18)

$$\begin{array}{c}\hfill {\Lambda }_{\mathit{f}}^2=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}^{\mathbf{f}})}^2\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(19)

$$\begin{array}{c}\hfill {\Lambda }_{A,f}^2=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_{\mathbf{f}}^{\mathbf{A}})}^2\mid L_A^{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(20)

where

$$\begin{array}{c}\hfill E={\int }_{{\mathbb{R}}^{\mathbb{N}}}\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}={\int }_{{\mathbb{R}}^{\mathbb{N}}}\mid {\mathrm{L}}_{\mathrm{A}}^{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u}={\int }_{{\mathbb{R}}^{\mathbb{N}}}\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(21)

$$\begin{array}{c}\hfill {\mathbf{t}}^{\mathbf{f}}=(t_1^f,t_2^f,\cdots ,t_N^f),\end{array}$$

(22)

$$\begin{array}{c}\hfill t_k^f=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{t}_k\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t},\end{array}$$

(23)

$$\begin{array}{c}\hfill {\mathbf{u}}^{\mathbf{f}}=(u_1^f,u_2^f,\cdots ,u_N^f),\end{array}$$

(24)

$$\begin{array}{c}\hfill u_k^f=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{u}_k\mid \widehat{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u}.\end{array}$$

(25)

Zhang [8] has generalized the N-dimensional Heisenberg’s inequality of the FT for complex function. It can be restated as follows:

Lemma 2.

Let $f\left(\mathbf{t}\right)=f_1\left(\mathbf{t}\right)e^{i\varphi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, for any $1\le \epsilon \le N$, the classical partial derivatives ${\displaystyle \frac{\partial f}{\partial t_{\epsilon }}}$, ${\displaystyle \frac{\partial f_1}{\partial t_{\epsilon }}}$, ${\displaystyle \frac{\partial \varphi }{\partial t_{\epsilon }}}$ exist at any point $\mathbf{t}\in {\mathbb{R}}^{\mathbb{N}}$, then the inequality of the N-dimensional FT can be obtained:

$$\begin{array}{c}\hfill {\Delta }_{\mathit{f}}^2{\Lambda }_{\mathit{f}}^2\ge {\displaystyle \frac{N^2}{16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_f^2,\end{array}$$

(26)

where

$$\begin{array}{c}\hfill CO{V}_f={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{f}}\left|\right|{\varpi }_{\mathbf{t}}\varphi -{\mathbf{u}}^{\mathbf{f}}|f_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(27)

and ${\varpi }_{\mathbf{t}}\varphi =({\displaystyle \frac{\partial \varphi }{\partial t_1}},{\displaystyle \frac{\partial \varphi }{\partial t_2}}\cdots ,{\displaystyle \frac{\partial \varphi }{\partial t_N}})$. If ${\varpi }_{\mathbf{t}}\varphi$ is continuous and $f_1\ne 0$, then the equality holds if and only if $f\left(\mathbf{t}\right)$ is a chirp function, the function is

$$\begin{array}{c}\hfill f\left(\mathbf{t}\right)=e^{-\frac{|\mathbf{t}-{\mathbf{t}}^{\mathbf{f}}{|}^2}{2ϵ}}+\iota e^{2\pi i\left[\frac{1}{2\vartheta }{\sum }_{\kappa =1}^N\varrho \left(t_{\kappa }\right){|t_{\kappa }-t_{\kappa }^f|}^2+\mathbf{t}{\mathbf{u}}^{\mathbf{f}}+{\iota }^{{\prod }_{\sigma =1}^N\varrho \left(t_{\sigma }\right)}\right]},\end{array}$$

(28)

where $ϵ,\vartheta >0$ and $\iota ,{\iota }^{{\prod }_{\sigma =1}^N\varrho \left(t_{\sigma }\right)}\in \mathbb{R}$,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \varrho \left(t_{\sigma }\right)=\left\{\begin{array}{cc}1,\hfill & \sigma \in {\mathbf{z}}_{1\tau }\hfill \\ -1,\hfill & \sigma \in {\mathbf{z}}_{2\tau }\hfill \\ sgn(t_{\sigma }-t_{\sigma }^f),\hfill & \sigma \in {\mathbf{z}}_{3\tau }\hfill \\ -sgn(t_{\sigma }-t_{\sigma }^f),\hfill & \sigma \in {\mathbf{z}}_{4\tau }\hfill \end{array}\right.,\end{array}\end{array}$$

(29)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{z}}_{1\tau }=\{z_{11},z_{12},\cdots ,z_{1\tau }\}=\left\{1\le s\le N\mid {\displaystyle \frac{\partial \varphi }{\partial t_s}}=\frac{1}{\vartheta }(t_s-t_s^f)+u_s^f\right\},\end{array}\end{array}$$

(30)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{z}}_{2\tau }=\{z_{21},z_{22},\cdots ,z_{2\tau }\}=\left\{1\le s\le N\mid {\displaystyle \frac{\partial \varphi }{\partial t_s}}=-\frac{1}{\vartheta }(t_s-t_s^f)+u_s^f\right\},\end{array}\end{array}$$

(31)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{z}}_{3\tau }=\{z_{31},z_{32},\cdots ,z_{3\tau }\}=\left\{1\le s\le N\mid {\displaystyle \frac{\partial \varphi }{\partial t_s}}=\left\{\begin{array}{cc}\frac{1}{\vartheta }(t_s-t_s^f)+u_s^f,\hfill & t_s\ge t_s^f\hfill \\ -\frac{1}{\vartheta }(t_s-t_s^f)+u_s^f,\hfill & t_s<t_s^f\hfill \end{array}\right.\right\},\end{array}\end{array}$$

(32)

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\mathbf{z}}_{4\tau }=\{z_{41},z_{42},\cdots ,z_{4\tau }\}=\left\{1\le s\le N\mid {\displaystyle \frac{\partial \varphi }{\partial t_s}}=\left\{\begin{array}{cc}-\frac{1}{\vartheta }(t_s-t_s^f)+u_s^f,\hfill & t_s\ge t_s^f\hfill \\ \frac{1}{\vartheta }(t_s-t_s^f)+u_s^f,\hfill & t_s<t_s^f\hfill \end{array}\right.\right\},\end{array}\end{array}$$

(33)

and ${\bigcup }_{\rho =1}^4{\mathbf{z}}_{\rho \tau }=\{1,2,\cdots ,N\}$, ${\mathbf{z}}_{{\rho }^{\prime }\tau }\cap {\mathbf{z}}_{\rho \tau }=\varnothing$ for $\rho \ne {\rho }^{\prime }$.

Theorem 1.

Let $f\left(\mathbf{t}\right)=f_1\left(\mathbf{t}\right)e^{i\varphi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, $\mathbf{t}f\left(\mathbf{t}\right)\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, for any $1\le \epsilon \le N$ the classical partial derivatives ${\displaystyle \frac{\partial f}{\partial t_{\epsilon }}}$, ${\displaystyle \frac{\partial f_1}{\partial t_{\epsilon }}}$, ${\displaystyle \frac{\partial \varphi }{\partial t_{\epsilon }}}$ exist at any point $\mathbf{t}\in {\mathbb{R}}^{\mathbb{N}}$, $E=1$, then inequality of the N-dimensional LCT can be obtained

$$\begin{array}{c}\hfill {\Delta }_{\mathit{f}}^2{\Lambda }_{A,\mathit{f}}^2\ge {\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,A}^2,\end{array}$$

(34)

where

$$\begin{array}{c}\hfill CO{V}_{f,A}={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{f}}\left|\right|{\varpi }_{\mathbf{t}}{\varphi }^{\prime }-{\mathbf{u}}_{\mathbf{f}}^{\mathbf{A}}|f_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(35)

${\varphi }^{\prime }\left(\mathbf{t}\right)=\varphi \left(\mathbf{t}\right)+\frac{a}{2b}{\mathbf{t}}^{\mathbf{2}}$ and ${\varpi }_{\mathbf{t}}{\varphi }^{\prime }=({\displaystyle \frac{\partial {\varphi }^{\prime }}{\partial t_1}},{\displaystyle \frac{\partial {\varphi }^{\prime }}{\partial t_2}}\cdots ,{\displaystyle \frac{\partial {\varphi }^{\prime }}{\partial t_N}})$, If ${\varpi }_{\mathbf{t}}\varphi$ is continuous and $f_1\ne 0$, then the equality holds if and only if $f\left(\mathbf{t}\right)$ is a chirp function (28).

Proof. 

According to the Formulas (2) and (6), we have

$$\begin{array}{c}\hfill L_A\left[f\left(\mathbf{t}\right)\right]\left(\mathbf{u}\right)=\frac{1}{\sqrt{ib}}F\left\{f\left(\mathbf{t}\right)e^{i\frac{a}{2b}{\mathbf{t}}^2}\right\}\left(\frac{\mathbf{u}}{b}\right)e^{i\frac{d}{2b}{\mathbf{u}}^2},\end{array}$$

(36)

let ${\mathbf{u}}^{\prime }=\frac{\mathbf{u}}{b}$ and ${\mathit{f}}^{\prime }\left(\mathbf{t}\right)=f\left(\mathbf{t}\right)e^{i\frac{a}{2b}{\mathbf{t}}^2}$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Delta }_{\mathit{f}}^2{\Lambda }_{A,\mathit{f}}^2& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^{\mathbf{f}})}^2\mid f\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_{\mathbf{f}}^{\mathbf{A}})}^2\mid {\mathrm{L}}_{\mathrm{A}}^{\mathit{f}}\left(\mathbf{u}\right){\mid }^2\mathrm{d}\mathbf{u}\hfill \\ & =\frac{1}{ib}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^{\mathbf{f}})}^2\mid f^{\prime }\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_{\mathbf{f}}^{\mathbf{A}})}^2\mid \mathit{F}\left\{{\mathit{f}}^{\prime }\left(\mathbf{t}\right)\right\}\left(\frac{\mathbf{u}}{\mathrm{b}}\right){\mid }^2\mathrm{d}\mathbf{u}\hfill \\ & =\frac{b^2}{i}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^{\mathbf{f}})}^2\mid f^{\prime }\left(\mathbf{t}\right){\mid }^2\mathrm{d}\mathbf{t}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{({\mathbf{u}}^{\prime }-{\mathbf{u}}_{\mathbf{f}}^{\prime \mathbf{A}})}^2\mid \mathit{F}\left\{{\mathit{f}}^{\prime }\left(\mathbf{t}\right)\right\}\left({\mathbf{u}}^{\prime }\right){\mid }^2\mathrm{d}{\mathbf{u}}^{\prime }.\hfill \end{array}\end{array}$$

(37)

By the Formula (26), we have

$$\begin{array}{c}\hfill \begin{array}{c}\hfill {\Delta }_{\mathit{f}}^2{\Lambda }_{A,\mathit{f}}^2\ge {\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,A}^2.\end{array}\end{array}$$

(38)

□

Corollary 1.

When $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, the above theorem can become the Lemma 2.

Corollary 2.

When $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, the above theorem can reduce the N-dimensional Heisenberg’s inequality of the FRFT for complex function [8].

Definition 4.

Let $f,g\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, then we can give the definition [11]

$$\begin{array}{c}\hfill {\mathbf{t}}_A^W=\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(39)

$$\begin{array}{c}\hfill {\mathbf{u}}_A^W=\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(40)

$$\begin{array}{c}\hfill {\Phi }_{A,\mathit{W}}^2=\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}_A^W)}^2\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(41)

$$\begin{array}{c}\hfill {\Psi }_{A,\mathit{W}}^2=\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_A^W)}^2\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(42)

Next, the N-dimensional Heisenberg’s inequality of the WLCT will be obtained.

Theorem 2.

Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ be a matrix parameter satisfying $a,b,c,d\in \mathbb{R}$ and $ad-bc=1$. For $f\left(\mathbf{t}\right)=f_1\left(\mathbf{t}\right)e^{i\varphi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, $g\left(\mathbf{t}\right)=g_1\left(\mathbf{t}\right)e^{i\phi \left(\mathbf{t}\right)}\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$,$\mathbf{t}f\left(\mathbf{t}\right)\in L^2\left({\mathbb{R}}^{\mathbb{N}}\right)$, we have

$$\begin{array}{cc}\hfill {\Phi }_{A,W}^2{\Psi }_{A,W}^2& \ge {\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,A}^2+{\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel g\parallel }^2+CO{V}_{g,A_1}^2\hfill \end{array}$$

(43)

$$\begin{array}{cc}& +2{\left(\left({\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,A}^2\right)\left({\displaystyle \frac{{\left(bN\right)}^2}{i16{\pi }^2}}{\parallel g\parallel }^2+CO{V}_{g,A_1}^2\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(44)

where $A_1=\left[\begin{array}{cc}0& b^{\prime }\\ -\frac{1}{b^{\prime }}& d^{\prime }\end{array}\right]$, $0\ne b^{\prime }=b\in \mathbb{R}$, the equality holds if and only if $f\left(\mathbf{t}\right)$ is a chirp function (28).

Proof. 

On the one hand, according to Lemma 1 and the Formula (9), we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\left|W_g^{A}f(\mathbf{t},\mathbf{u})\right|}^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}[{\int }_{{\mathbb{R}}^{\mathbb{N}}}{L}_A^f\left(\mathbf{m}\right)\sqrt{-i2\pi b}{e}^{i\frac{d^{\prime }}{2b}{(\mathbf{m}-\mathbf{u})}^2}\hfill \\ \hfill & \times L_{A_1}^g{(\mathbf{m}-\mathbf{u})}^{*}{K}_A(\mathbf{t},\mathbf{u})K_A^{*}(\mathbf{t},\mathbf{m})\mathrm{d}\mathbf{m}]\hfill \\ \hfill & \times [{\int }_{{\mathbb{R}}^{\mathbb{N}}}{L}_A^f\left(\mathbf{n}\right)\sqrt{-i2\pi b}{e}^{i\frac{d^{\prime }}{2b}{(\mathbf{n}-\mathbf{u})}^2}\hfill \\ \hfill & \times L_{A_1}^g{(\mathbf{n}-\mathbf{u})}^{*}{K}_A(\mathbf{t},\mathbf{u})K_A^{*}(\mathbf{t},\mathbf{n})\mathrm{d}\mathbf{n}{]}^{*}\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u}\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}|L_A^f{\left(\mathbf{m}\right)|}^2{\left|L_{A_1}^g(\mathbf{m}-\mathbf{u})\right|}^2\mathrm{d}\mathbf{m}\mathrm{d}\mathbf{u}.\hfill \end{array}\end{array}$$

(45)

Let $\mathbf{m}-\mathbf{u}=\mathbf{v}$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2& ={\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}|L_A^f{\left(\mathbf{m}\right)|}^2{\left|L_{A_1}^g\left(\mathbf{v}\right)\right|}^2\mathrm{d}\mathbf{m}\mathrm{d}\mathbf{v}\hfill \\ \hfill & =\parallel L_A^f{\left(\mathbf{m}\right)\parallel }^2{\parallel L_{A_1}^g\left(\mathbf{v}\right)\parallel }^2.\hfill \end{array}\end{array}$$

(46)

Moreover, we obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{t}}_A^W& =\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u}\hfill \\ \hfill & =\frac{1}{\parallel L_A^f{\left(\mathbf{m}\right)\parallel }^2{\parallel L_{A_1}^g\left(\mathbf{v}\right)\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\left[{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left({\mathbf{m}}^{\prime }\right)\overline{g({\mathbf{m}}^{\prime }-\mathbf{t})}{K}_A({\mathbf{m}}^{\prime },\mathbf{u})\mathrm{d}{\mathbf{m}}^{\prime }\right]\hfill \\ \hfill & \times {\left[{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left({\mathbf{n}}^{\prime }\right)\overline{g({\mathbf{n}}^{\prime }-\mathbf{t})}{K}_A({\mathbf{n}}^{\prime },\mathbf{u})\mathrm{d}{\mathbf{n}}^{\prime }\right]}^{*}\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u}\hfill \\ \hfill & =\frac{1}{\parallel L_A^f{\left(\mathbf{m}\right)\parallel }^2{\parallel L_{A_1}^g\left(\mathbf{v}\right)\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}|f\left({\mathbf{m}}^{\prime }\right){|}^2{\left|g({\mathbf{m}}^{\prime }-\mathbf{t})\right|}^2\mathrm{d}{\mathbf{m}}^{\prime }\mathrm{d}\mathbf{t}.\hfill \end{array}\end{array}$$

(47)

Let ${\mathbf{m}}^{\prime }-\mathbf{t}=\mathbf{r}$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathbf{t}}_A^W& =\frac{1}{\parallel L_A^f{\left(\mathbf{m}\right)\parallel }^2{\parallel L_{A_1}^g\left(\mathbf{v}\right)\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}({\mathbf{m}}^{\prime }-\mathbf{r})|f\left({\mathbf{m}}^{\prime }\right){|}^2{|g\left(\mathbf{r}\right)|}^2\mathrm{d}{\mathbf{m}}^{\prime }\mathrm{d}\mathbf{r}\hfill \\ \hfill & =\frac{1}{\parallel L_A^f\left({\mathbf{m}}^{\prime }\right){\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\mathbf{m}}^{\prime }|f\left({\mathbf{m}}^{\prime }\right){|}^2|\mathrm{d}{\mathbf{m}}^{\prime }-\frac{1}{\parallel {\mathit{L}}_{{\mathrm{A}}_1}^{\mathrm{g}}{\left(\mathbf{v}\right)\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{r}{\left|\mathrm{g}\left(\mathbf{r}\right)\right|}^2\mathrm{d}\mathbf{r}\hfill \\ \hfill & ={\mathbf{t}}^{\mathbf{f}}-{\mathbf{t}}^{\mathbf{g}}.\hfill \end{array}\end{array}$$

(48)

Using the same method, we can obtain

$$\begin{array}{c}\hfill {\mathbf{u}}_A^W={\mathbf{u}}_f^A-{\mathbf{u}}_g^{A_1}.\end{array}$$

(49)

From the Formula (46), then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Psi }_{A,W}^2& =\frac{1}{\parallel W_g^A{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_A^W)}^2\mid W_g^{A}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u}\hfill \\ \hfill & =\frac{1}{\parallel L_A^f{\left(\mathbf{m}\right)\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{({\mathbf{m}}^{\prime }-{\mathbf{u}}_f^A)}^2{\left|L_A^{\mathit{f}}\left({\mathbf{m}}^{\prime }\right)\right|}^2\mathrm{d}{\mathbf{m}}^{\prime }+\frac{1}{\parallel {\mathit{L}}_{{\mathrm{A}}_1}^{\mathrm{g}}{\left(\mathbf{v}\right)\parallel }^2}\hfill \\ & \times {\int }_{{\mathbb{R}}^{\mathbb{N}}}{({\mathbf{v}}^{\prime }-{\mathbf{u}}_g^{A_1})}^2{\left|L_{A_1}^g\left({\mathbf{v}}^{\prime }\right)\right|}^2\mathrm{d}{\mathbf{v}}^{\prime }-2\frac{1}{\parallel {\mathit{L}}_{\mathrm{A}}^{\mathit{f}}{\left(\mathbf{m}\right)\parallel }^2}\hfill \\ \hfill & \times {\int }_{{\mathbb{R}}^{\mathbb{N}}}({\mathbf{m}}^{\prime }-{\mathbf{u}}_f^A){\left|L_A^{\mathit{f}}\left({\mathbf{m}}^{\prime }\right)\right|}^2\mathrm{d}{\mathbf{m}}^{\prime }\frac{1}{\parallel {\mathit{L}}_{{\mathrm{A}}_1}^{\mathrm{g}}{\left(\mathbf{v}\right)\parallel }^2}\hfill \\ & \times {\int }_{{\mathbb{R}}^{\mathbb{N}}}({\mathbf{v}}^{\prime }-{\mathbf{u}}_g^{A_1}){\left|L_{A_1}^g\left({\mathbf{v}}^{\prime }\right)\right|}^2\mathrm{d}{\mathbf{v}}^{\prime }\hfill \\ \hfill & ={\Lambda }_{A,f}^2+{\Lambda }_{A_1,g}^2.\hfill \end{array}\end{array}$$

(50)

From the same method, we can obtain

$$\begin{array}{c}\hfill {\Phi }_{A,W}^2={\Delta }_f^2+{\Delta }_g^2,\end{array}$$

(51)

On the other hand, using the Formulas (48)–(51), we can obtain

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Phi }_{A,W}^2{\Psi }_{A,W}^2& =({\Delta }_f^2+{\Delta }_g^2)({\Lambda }_{A,f}^2+{\Lambda }_{A_1,g}^2)\hfill \\ & ={\Delta }_f^2{\Lambda }_{A,f}^2+{\Delta }_g^2{\Lambda }_{A_1,g}^2+{\Delta }_f^2{\Lambda }_{A_1,g}^2+{\Delta }_g^2{\Lambda }_{A,f}^2.\hfill \end{array}\end{array}$$

(52)

According to the fact: $n^2+m^2\ge 2nm$, for $\forall n,m\in \mathbb{R}$, then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\Phi }_{A,W}^2{\Psi }_{A,W}^2& ={\Delta }_f^2{\Lambda }_{A,f}^2+{\Delta }_g^2{\Lambda }_{A_1,g}^2+{\Delta }_f^2{\Lambda }_{A_1,g}^2+{\Delta }_g^2{\Lambda }_{A,f}^2\hfill \\ & \ge {\Delta }_f^2{\Lambda }_{A,f}^2+{\Delta }_g^2{\Lambda }_{A_1,g}^2+2\sqrt{{\Delta }_f^2{\Lambda }_{A,f}^2{\Delta }_g^2{\Lambda }_{A_1,g}^2}.\hfill \end{array}\end{array}$$

(53)

From the Formula (34), we can obtain the result. □

Corollary 3.

When $A=\left[\begin{array}{cc}cos\alpha & sin\alpha \\ -sin\alpha & cos\alpha \end{array}\right]$, the N-dimensional Heisenberg’s inequality of the windowed fractional Fourier transform (WFRFT) [22] for the complex function can be obtained:

$$\begin{array}{cc}\hfill {\Phi }_{\alpha ,W}^2{\Psi }_{\alpha ,W}^2& \ge {\displaystyle \frac{{(sin\alpha N)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,\alpha }^2+{\displaystyle \frac{{(sin\alpha N)}^2}{i16{\pi }^2}}{\parallel g\parallel }^2+CO{V}_{g,{\alpha }_1}^2\hfill \end{array}$$

(54)

$$\begin{array}{cc}& +2{\left(\left({\displaystyle \frac{{(sin\alpha N)}^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_{f,\alpha }^2\right)\left({\displaystyle \frac{{(sin\alpha N)}^2}{i16{\pi }^2}}{\parallel g\parallel }^2+CO{V}_{g,{\alpha }_1}^2\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(55)

where

$$\begin{array}{c}\hfill {\Phi }_{\alpha ,\mathit{W}}^2=\frac{1}{\parallel W_g^{\alpha }{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}_{\alpha }^W)}^2\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(56)

$$\begin{array}{c}\hfill {\Psi }_{\alpha ,\mathit{W}}^2=\frac{1}{\parallel W_g^{\alpha }{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}_{\alpha }^W)}^2\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(57)

$$\begin{array}{c}\hfill {\mathbf{t}}_{\alpha }^W=\frac{1}{\parallel W_g^{\alpha }{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(58)

$$\begin{array}{c}\hfill {\mathbf{u}}_{\alpha }^W=\frac{1}{\parallel W_g^{\alpha }{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(59)

$$\begin{array}{c}\hfill CO{V}_{f,\alpha }={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{f}}\left|\right|{\varpi }_{\mathbf{t}}{\varphi }^{\prime }-{\mathbf{u}}_{f,W}^{\alpha }|f_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(60)

$$\begin{array}{c}\hfill CO{V}_{g,{\alpha }^{\prime }}={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{g}}\left|\right|{\varpi }_{\mathbf{t}}{\varphi }^{\prime }-{\mathbf{u}}_{g,W}^{{\alpha }^{\prime }}|g_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(61)

$$\begin{array}{c}\hfill {\mathbf{u}}_{f,W}^{\alpha }=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(62)

$$\begin{array}{c}\hfill {\mathbf{u}}_{g,W}^{{\alpha }^{\prime }}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_g^{\alpha }f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(63)

and $W_g^{\alpha }f(\mathbf{t},\mathbf{u})$ is the WFRFT of complex function

$$\begin{array}{c}\hfill \begin{array}{c}\hfill W_g^{\alpha }f(\mathbf{t},\mathbf{u})=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{y}\right)g^{*}(\mathbf{y}-\mathbf{t})K_{\alpha }(\mathbf{y},\mathbf{u})\mathrm{d}\mathbf{y},\hfill & \alpha \ne n\pi \hfill \\ f\left(\mathbf{u}\right),\hfill & \alpha =2n\pi \hfill \\ -f\left(\mathbf{u}\right),\hfill & \alpha =(2n+1)\pi \hfill \end{array}\right.,\end{array}\end{array}$$

(64)

and $K_{\alpha }(\mathbf{y},\mathbf{u})={(1-icot\alpha )}^{\frac{N}{2}}{e}^{{\pi i\left(\right|\mathbf{y}|}^2+|\mathbf{u}{|}^2)cot\alpha -2\pi i\mathbf{y}\mathbf{u}csc\alpha }$.

Corollary 4.

When $A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$, the N-dimensional Heisenberg’s inequality of the windowed Fourier transform (WFT) [23] for the complex function can be obtained:

$$\begin{array}{cc}\hfill {\Phi }_W^2{\Psi }_W^2& \ge {\displaystyle \frac{N^2}{i16{\pi }^2}}{(\parallel f\parallel }^2+{\parallel g\parallel }^2)+CO{V}_f^2+CO{V}_g^2\hfill \end{array}$$

(65)

$$\begin{array}{cc}& +2{\left(\left({\displaystyle \frac{N^2}{i16{\pi }^2}}{\parallel f\parallel }^2+CO{V}_f^2\right)\left({\displaystyle \frac{N^2}{i16{\pi }^2}}{\parallel g\parallel }^2+CO{V}_g^2\right)\right)}^{\frac{1}{2}},\hfill \end{array}$$

(66)

where

$$\begin{array}{c}\hfill {\Phi }_{\mathit{W}}^2=\frac{1}{\parallel W_g{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{t}-{\mathbf{t}}^W)}^2\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(67)

$$\begin{array}{c}\hfill {\Psi }_{\mathit{W}}^2=\frac{1}{\parallel W_g{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{(\mathbf{u}-{\mathbf{u}}^W)}^2\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(68)

$$\begin{array}{c}\hfill {\mathbf{t}}^W=\frac{1}{\parallel W_g{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{t}\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(69)

$$\begin{array}{c}\hfill {\mathbf{u}}^W=\frac{1}{\parallel W_g{f(\mathbf{t},\mathbf{u})\parallel }^2}{\int }_{{\mathbb{R}}^{\mathbb{N}}}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{t}\mathrm{d}\mathbf{u},\end{array}$$

(70)

$$\begin{array}{c}\hfill CO{V}_f={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{f}}\left|\right|{\varpi }_{\mathbf{t}}{\varphi }^{\prime }-{\mathbf{u}}_{\mathbf{f},\mathbf{W}}|f_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(71)

$$\begin{array}{c}\hfill CO{V}_g={\int }_{{\mathbb{R}}^{\mathbb{N}}}|\mathbf{t}-{\mathbf{t}}^{\mathbf{g}}\left|\right|{\varpi }_{\mathbf{t}}{\varphi }^{\prime }-{\mathbf{u}}_{\mathbf{g},\mathbf{W}}|g_1^2\left(\mathbf{t}\right)\mathrm{d}\mathbf{t},\end{array}$$

(72)

$$\begin{array}{c}\hfill {\mathbf{u}}_{\mathbf{f},\mathbf{W}}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(73)

$$\begin{array}{c}\hfill {\mathbf{u}}_{\mathbf{g},\mathbf{W}}=\frac{1}{E}{\int }_{{\mathbb{R}}^{\mathbb{N}}}\mathbf{u}\mid W_{g}f(\mathbf{t},\mathbf{u}){\mid }^2\mathrm{d}\mathbf{u},\end{array}$$

(74)

and $W_{g}f(\mathbf{t},\mathbf{u})$ is the WFT of the complex function

$$\begin{array}{c}\hfill \begin{array}{c}\hfill W_{g}f(\mathbf{t},\mathbf{u})=\left\{\begin{array}{cc}{\int }_{{\mathbb{R}}^{\mathbb{N}}}f\left(\mathbf{y}\right)g^{*}(\mathbf{y}-\mathbf{t})e^{-i\mathbf{y}\mathbf{u}}\mathrm{d}\mathbf{y},\hfill & \alpha \ne n\pi \hfill \\ f\left(\mathbf{u}\right),\hfill & \alpha =2n\pi \hfill \\ -f\left(\mathbf{u}\right),\hfill & \alpha =(2n+1)\pi \hfill \end{array}\right..\end{array}\end{array}$$

(75)

4. Conclusions

In this paper, by the N-dimensional Heisenberg’s inequality of the FT, the N-dimensional Heisenberg’s inequalities for the WLCT of a complex function are generalized. Firstly, the definition for N-dimensional WLCT of a complex function is given. In addition, according to the second-order moment of the LCT, the N-dimensional Heisenberg’s inequality for the linear canonical transform (LCT) is derived. It shows that the lower bound is related to the covariance and can be achieved by a complex chirp function with a Gaussian function. Finally, the second-order moment of the WLCT is given, the relationship between the LCT and WLCT is obtained, and the N-dimensional Heisenberg’s inequality for the WLCT is exploited. In special cases, its corollaries can be obtained.