Clifford-Valued Wave-Packet Transform with Applications to Benchmark Signals

Abstract

In pursuit of efficient time-frequency representation of Clifford-valued signals, we propose a novel integral transform coined as the Clifford-valued wave-packet transform. In the first instance, all the fundamental properties of the Clifford-valued wave-packet transform including the inner product relation, inversion formula and range theorem are studied. This is followed by the formulation of several uncertainty inequalities, such as the Heisenberg–Pauli–Weyl inequality, Pitt’s inequality and logarithmic inequality for the proposed transform. We culminate our study by demonstrating the performance and efficiency of the Clifford-valued wave-packet transform for detection and orientation of pointwise singularities in the context of three-dimensional signal analysis.

1 Introduction

An utter representation of non-transient signals requires frequency analysis that is local in time, resulting in the time-frequency analysis. The major development in the realm of time-frequency analysis came in the form of short-time Fourier transform (STFT) or Gabor transform [1], which is reliant upon analyzing functions determined by the fundamental operations of translation and modulation acting on a given window function. With the aid of this transform, one can analyze the spectral contents of a given signal in the localized neighborhood of time. This astonishing feature of STFT provides the local characteristics of the Fourier transform with time resolution determined by the width of the window function [2]. Although the Gabor representations are quite handy, such representations are not adequate for signals having high frequency components for shorter durations and low frequency components for longer durations, leading to the birth of time-scale integral transform, often known as the wavelet transform [3, 4]. Owing to the lucid nature and close resemblance with the classical Fourier transform, the wavelet transforms have fascinated the mathematical, physical, chemical, biological and engineering communities with their versatile applicability [5]. As the signal analyzing capability of the wavelet transform is limited in the time-frequency plane, it does not serve as an efficient tool for processing those signals whose energy is not well concentrated in the frequency domain. An appropriate redressal of this limitation was given by Torresani [6] in the form of wave-packet transform by incorporating all the three unitary operations: translation, dilation, and modulation to the given window function. As of now, the wave-packet transform has witnessed an ample amount of research in the realm of time-frequency analysis, which include fractional wave-packet transform [7,8,9] linear canonical wave-packet transform [10, 11] and many more.

The Clifford analysis has proved to be a bastion for efficient time-frequency representations of multi-variate signals. It is well known that the multi-variate signals are characterized by certain inherent geometric features; however, the quaternion algebra lacks the geometric properties and is therefore not optimal for an efficient tracking of the edges and corners arising in higher-dimensional signals. On the other hand, the Clifford algebra is elegantly endowed with both the algebraic and geometric properties, which encompasses all dimensions at once, as opposed to the multi-dimensional tensorial approach with tensor products of one-dimensional phenomena [12]. Technically, Clifford algebra is a refinement of exterior algebra using a nonzero quadratic form, which not only inherits the advantages of both the Grossmann’s exterior algebra and Hamilton’s algebra of quaternions but also offers a function theory which is a higher-dimensional analogue of the theory of holomorphic functions of one complex variable. The Clifford algebra is a non-commutative and associative algebra, which is being continuously employed in quantum mechanics, neural computations, computer vision, signal and image processing and robotics [13,14,15,16]. The true multi-dimensional nature underlying the Clifford algebras offers a specific construction of higher-dimensional wavelets and the development of the corresponding continuous wavelet theory [17,18,19,20]. As such, it is quite lucrative to introduce the notion of Clifford-valued wave-packet transform as a pleasant alternative for an efficient representation of higher-dimensional signals. The proposed transform is designed to represent Clifford-valued signals at different scales, locations and orientations. Besides, the fundamental properties of Clifford-valued wave-packet transform, including the inner product and energy preserving relations, inversion formula and range theorem, are also investigated in detail. Moreover, several classes of uncertainty principles are studied by employing the machinery of Clifford-valued Fourier transforms and the operator theory. Nevertheless, the proposed transform is also testified for the detection and orientation of pointwise singularities of benchmark signals.

The layout of the article is as follows: Sect. 2 is completely devoted for the exposition of the preliminaries including the notion of Clifford-valued Fourier and wavelet transforms. In Sect. 3, we formally introduce the Clifford-valued wave-packet transform and obtain all of its basic properties. Some uncertainty inequalities for the Clifford-valued wave packet transform are formulated in Sect. 4. Finally, we extend the scope of the work by employing the new transform for the analysis of benchmark signals.

2 Preliminaries

In this section, we present a brief overview of the Clifford algebras including the formal definition of Clifford-valued Fourier transform, and the various time-frequency analyzing tools in Clifford domains.

2.1 Basics of Clifford Algebra

The Clifford algebra $C\ell _{(0,n)}:=C\ell _{n}$ is defined as a non-commutative algebra generated by the orthonormal basis $\{e_1, e_2, \dots , e_n\}$ of the real n-dimensional Euclidean space ${\mathbb {R}}^n$ governed by the multiplication rule [12]:

$$\begin{aligned} e_ie_j+e_je_i=-2\,\delta _{ij},\qquad i,j=1,2,\dots ,n, \end{aligned}$$

(2.1)

where $\delta _{ij}$ denotes the well-known Kronecker’s delta function. The non-commutative product and the additional axiom of associativity generate the $2^n$-dimensional Clifford geometric algebra $C\ell _{n}$, which can be decomposed as

$$\begin{aligned} C\ell _{n}=\bigoplus _{k=0}^{n} C\ell ^k_{n}, \end{aligned}$$

(2.2)

where $C\ell ^k_{n}$ denotes the space of k-vectors given by

$$\begin{aligned} C\ell ^k_{n}:= \text {span}\Big \{e_{i_{1}},e_{i_{2}},\dots ,e_{i_{k}};\,i_{1}\le i_{2}\le \dots \le i_{k}\Big \}. \end{aligned}$$

Any general element of the Clifford algebra is called a multi-vector and every multi-vector $f \in C\ell _{n}$ can be represented in the following form

$$\begin{aligned} f=\sum _{A} f_A\, e_A=\langle f\rangle _0+\langle f\rangle _1+\dots +\langle f\rangle _n,\quad f_A\in {\mathbb {R}},\,A\subset \big \{1,2,\dots ,n\big \},\, \end{aligned}$$

(2.3)

where $e_A=e_{i_{1}}e_{i_{2}}\dots e_{i_{k}}$ and $i_{1}\le i_{2}\le \dots \le i_{k}$. Moreover, $\langle \cdot \rangle _k$ is called as the grade k-part of f, and $\langle \cdot \rangle _0,\,\langle \cdot \rangle _1,\,\langle \cdot \rangle _2,\dots$, respectively, denote the scalar part, vector part, bi-vector part and so on. The Clifford conjugate of $f\in C\ell _{n}$ is given by

$$\begin{aligned} \overline{{ f}} =\sum _{r=0}^{n}(-1)^{\frac{r(r-1)}{2}} \overline{\langle f\rangle _r}. \end{aligned}$$

(2.4)

We now recall the fundamental notion of integral transformation on the space of functions $L^r({\mathbb {R}}^{n},\,C\ell _{n})$, $1\le r<\infty$ defined as

$$\begin{aligned} L^r({\mathbb {R}}^{n},\,C\ell _{n})=\left\{ f :{\mathbb {R}}^{n}\rightarrow C\ell _{n} :\big \Vert f\big \Vert _{r}=\left( \int _{{\mathbb {R}}^{n}}\big | f(\mathbf{x})\big |^r d^n\mathbf{x}\right) ^{1/r}<\infty \right\} . \end{aligned}$$

It is imperative to mention that any function $f\in L^r({\mathbb {R}}^{n},\,C\ell _{n})$ can be represented as a combination of the real-valued functions $f_A$ and the basis elements $e_A$ as

$$\begin{aligned} f(\mathbf{x})=\sum _A f_A(\mathbf{x})\,e_A. \end{aligned}$$

2.2 Time-Frequency Analysis in Clifford Domains

Due to the non-commutativity of Clifford-valued functions, several analogues of the Clifford Fourier transforms have been introduced in the literature. However, we shall be interested in following definition due to Bahri and Hitzer [17].

Definition 2.1

For any Clifford-valued function $f\in L^2({\mathbb {R}}^n,C\ell _n),\,n=2,3(\hbox {mod}\, 4)$, the Clifford-valued Fourier transform is denoted by ${{\mathcal {F}}_{C\ell }}$ and is given by

$$\begin{aligned} {{\mathcal {F}}_{C\ell }}\big [f(\mathbf{x})\big ](\mathbf{w}) = \int _{{\mathbb {R}}^n} f(\mathbf{x})\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\hbox {d}{} \mathbf{x}, \end{aligned}$$

(2.5)

where $\mathbf{x},\, \mathbf{w}\in {\mathbb {R}}^n,\,$ and $\,\mathbf{i}_n=e_1e_2\dots e_n \in C\ell _n$.

It is pertinent to mention that the Clifford exponential product $e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}$ appearing in (2.5) commutes with every element of $L^2({\mathbb {R}}^n,\,C\ell _n)$, for $n=3(\hbox {mod}\, 4)$ where as it is non-commutative for $n=2(\hbox {mod}\, 4)$. Moreover, the corresponding inversion and Plancherel formulae for CFT are given by

$$\begin{aligned}&f(\mathbf{x})=\dfrac{1}{(2\pi )^n}\int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell }\big [f(\mathbf{x})\big ](\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\hbox {d}{} \mathbf{w},\quad \text {and} \end{aligned}$$

(2.6)

$$\begin{aligned}&\big \langle { f},\,{ g}\big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}=\dfrac{1}{(2\pi )^n} \Big \langle {\mathcal {F}}_{C\ell }\big [f\big ],\,{\mathcal {F}}_{C\ell }\big [{g}\big ]\Big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}, \end{aligned}$$

(2.7)

respectively, where $f,g\in L^2({\mathbb {R}}^n,\,C\ell _n)$.

Note that, in every integral transformation, two types of functions are involved: one the function to be analyzed and the other used as an analyzing function. For instance, in classical Fourier transform the sinusoidal kernel is the analyzing function and in wavelet transform the analyzing function is precisely the mother wavelet. Let ${\mathcal {T}}_\mathbf{b}$, ${\mathcal {D}}_{a}$, ${\mathcal {R}}_\theta$ and ${\mathcal {M}}_\mathbf{w}$ be the translation, dilation, rotation and modulation operators acting on an analyzing function $\psi \in L^2({\mathbb {R}}^n, C\ell _n)$, respectively, given by

$$\begin{aligned} {\mathcal {T}}_{\mathbf{b}} \psi ({\mathbf{x}})&=\psi ({\mathbf{x}}-{\mathbf{b}}),\,\, {\mathcal {D}}_{a}\psi ({\mathbf{x}})=\dfrac{1}{a^{n/2}}\psi \left( \dfrac{\mathbf{x}}{{a}}\right) ,\,\,{\mathcal {R}}_{\theta }^{-1}\psi ({\mathbf{x}}) =\psi \left( R_{\theta }^{-1}{\mathbf{x}}\right) \,\, \text {and}\,\,\\ {\mathcal {M}}_{\mathbf{w}} \psi ({\mathbf{x}})&= e^{{\mathbf{i}}_{n}\,{\mathbf{w}}\cdot {\mathbf{x}}}\,\psi ({\mathbf{x}}), \end{aligned}$$

where $R_\theta$ is a rotation matrix of order n with $\theta \in \text {SO}(n)$, the special orthogonal group of ${\mathbb {R}}^{n}$ generated by the hyperplane reflections. For instance, the action of rotations in ${\mathbb {R}}^2$ is given by $\theta$ and the rotation matrix $R_{\theta }$ is given by

$$\begin{aligned} R_{\theta } = \left[ \begin{array}{cc} \cos \theta &{}\quad -\sin \theta \\ \sin \theta &{}\quad \cos \theta \end{array}\right] , \end{aligned}$$

while as the rotations in ${\mathbb {R}}^3$ are given by

$$\begin{aligned} R_{\theta } = \left[ \begin{array}{ccc} \cos \theta &{}\quad -\sin \theta &{}\quad 0\\ \sin \theta &{}\quad \cos \theta &{}\quad 0\\ 0&{}\quad 0&{}\quad 1\end{array}\right] . \end{aligned}$$

We now recall the definitions of Clifford-valued windowed Fourier and wavelet transforms [18, 19].

Definition 2.2

The Clifford-valued windowed Fourier transform of any Clifford-valued function $f\in L^1({\mathbb {R}}^n,\,C\ell _n)\cap L^2({\mathbb {R}}^n,\,C\ell _n)$ is denoted by ${{\mathcal {G}}_{\phi }}\big [f\big ]$ and is given by

$$\begin{aligned} {{\mathcal {G}}_{\phi }}\big [f(\mathbf{x})\big ](\mathbf{w},\mathbf{y}) = \Big \langle f,\,{\mathcal {M}}_\mathbf{w}{\mathcal {T}}_\mathbf{b} \phi \Big \rangle _{L^2\left( {\mathbb {R}}^{n},\,C\ell _n\right) } =\int _{{\mathbb {R}}^n}f(\mathbf{x})\overline{\phi (\mathbf{x}-\mathbf{y})}\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\hbox {d}{} \mathbf{x}, \end{aligned}$$

(2.8)

where $\mathbf{x}, \mathbf{w}\in {\mathbb {R}}^n$ and $\phi \in L^2({\mathbb {R}}^n,\,C\ell _n)$ is the window function; that is, it is a nonzero function having rapid decay in both the spatial and Clifford Fourier domains.

Definition 2.3

[18]. The Clifford-valued wavelet transform of any Clifford-valued function $f \in L^2\left( {\mathbb {R}}^n,\,C\ell _n\right)$ with respect to an analyzing function $\psi \in L^2\left( {\mathbb {R}}^{n},\,C\ell _n\right)$ is defined by

$$\begin{aligned} {\mathscr {W}}_{\psi }^{{\mathbb {H}}}\big [f\big ](a, \mathbf{b},\theta )&= \Big \langle f,\,{\mathcal {R}}_{\theta } {\mathcal {D}}_{a} {\mathcal {T}}_\mathbf{b} \psi \Big \rangle _{L^2\left( {\mathbb {R}}^{n},\,C\ell _n\right) }= \dfrac{1}{a} \int _{{\mathbb {R}}^{n}} \, f(\mathbf{x})\, \overline{\psi \left( {R_{\theta }} \left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) } \,\hbox {d}{} \mathbf{x},\nonumber \\&\quad a\in {\mathbb {R}}^+, \mathbf{b}\in {\mathbb {R}}^n. \end{aligned}$$

(2.9)

3 The Clifford-Valued Wave-Packet Transform

In this section, we introduce the notion of Clifford-valued wave-packet transform by constructing the Clifford-valued wave-packet systems in $L^2\left( {\mathbb {R}}^n,C\ell _n\right)$ with the aid of translation, dilation, rotation and modulation operators as defined in Sect. 2, acting on a single generator $\psi$. Besides, we shall also investigate some fundamental properties of the proposed transform using the machinery of Clifford-valued Fourier transforms.

For any $\psi \in L^2({\mathbb {R}}^n, C\ell _n),$ we define the collection

$$\begin{aligned} \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})&={\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a}{\mathcal {T}}_\mathbf{b} \psi (\mathbf{x}) =\dfrac{1}{a^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) ,\nonumber \\&\qquad a\in {\mathbb {R}}^{+},\,\mathbf{b},\,\mathbf{w}\in {\mathbb {R}}^{n}. \end{aligned}$$

(3.1)

System (3.1) will be called the Clifford-valued wave-packet system. It is easy to verify that the set ${\mathcal {G}} = {\mathbb {R}}^{+} \times {{\mathbb {R}}^{n}}\times \text {SO}(n)\times {{\mathbb {R}}^{n}}$ constitutes a group under the operations

$$\begin{aligned} \big (a,\mathbf{b},\theta ,\mathbf{w} \big )\odot \big (a^{\prime },\mathbf{b}^{\prime },\theta ^{\prime },\mathbf{w}^{\prime }\big )= \big (aa^{\prime },a^{\prime }{} \mathbf{b}+\mathbf{b}^{\prime },\theta +\theta ^{\prime },\mathbf{w}+R_{\theta }^{-1}\mathbf{w}^{\prime }/a\big ), \end{aligned}$$

(3.2)

where $(1,\mathbf{0},0,\mathbf{0})$ acts as a neutral element of ${\mathcal {G}}$ and $\big (a^{-1},-a^{-1}{} \mathbf{b},-\theta ,a R_{\theta }{} \mathbf{w}\big )$ is the inverse of $\big (a,\mathbf{b},\theta ,\mathbf{w} \big )$ in ${\mathcal {G}}$. Moreover, the left Haar measure on ${\mathcal {G}}$ is given by $\hbox {d}\eta ={\hbox {d}a\,\hbox {d}{} \mathbf{b}\, \hbox {d}\theta }\,d\mathbf{w}/a^{n+1}$, as

$$\begin{aligned}&\int _{{\mathcal {G}}} f\left[ \big (a,\mathbf{b},\theta ,\mathbf{w} \big )\odot \big (a^{\prime },\mathbf{b}^{\prime },\theta ^{\prime },\mathbf{w}^{\prime }\big )\right] \,\hbox {d}\eta \nonumber \\&\quad = \int _{{\mathbb {R}}^{+} \times {{\mathbb {R}}^{n}}\times \text {SO}(n)\times {{\mathbb {R}}^{n}}} f\left[ \big (aa^{\prime },a^{\prime }{} \mathbf{b}+\mathbf{b}^{\prime },\theta +\theta ^{\prime },\mathbf{w}+R_{\theta }^{-1}\mathbf{w}^{\prime }/a\big )\right] \,\dfrac{\text {d}a\,\text {d}{} \mathbf{b}\, \text {d}\theta \,\text {d}\mathbf{w}}{a^{n+1}}. \end{aligned}$$

(3.3)

Taking ${\tilde{a}}:=aa',{\tilde{\varvec{b}}}:=\mathbf{b}'+a'\mathbf{b},\,{\tilde{\theta }}:=\theta +\theta ',\,{\tilde{\varvec{w}}}:=\mathbf{w}+R_{\theta }^{-1}{} \mathbf{w}^{\prime }/a$, that is; $\hbox {d}a={(a')}^{-1}\hbox {d}{\tilde{a}},\,\hbox {d}{} \mathbf{b}={(a')}^{-n}d\tilde{\mathbf{b}},\,\text {d}\theta =\text {d}{\tilde{\theta }},\,\text {d}{} \mathbf{w}=\text {d}{\tilde{\varvec{w}}}$, equation (3.3) becomes

$$\begin{aligned}&\int _{{\mathcal {G}}} f\left[ \big (a,\mathbf{b},\theta ,\mathbf{w} \big )\odot \big (a^{\prime },\mathbf{b}^{\prime },\theta ^{\prime },\mathbf{w}^{\prime }\big )\right] \,\hbox {d}\eta \nonumber \\&\qquad = \int _{{\mathbb {R}}^{+} \times {{\mathbb {R}}^{n}}\times \text {SO}(n)\times {{\mathbb {R}}^{n}}} f\left[ \big ({\tilde{a}},{\tilde{\varvec{b}}},{\tilde{\theta }},{\tilde{\varvec{w}}}\big )\right] \, \dfrac{{(a')}^{-1}\hbox {d}{\tilde{a}}\,{(a')}^{-n}\hbox {d}\tilde{\mathbf{b}}\, \hbox {d}{\tilde{\theta }}\,\hbox {d}{\tilde{\varvec{w}}}}{\left( {\tilde{a}}/a'\right) ^{n+1}}\nonumber \\&\qquad = \int _{{\mathbb {R}}^{+} \times {{\mathbb {R}}^{n}}\times \text {SO}(n)\times {{\mathbb {R}}^{n}}} f\left[ \big ({\tilde{a}},{\tilde{\varvec{b}}},{\tilde{\theta }},{\tilde{\varvec{w}}}\big )\right] \, \dfrac{\hbox {d}{\tilde{a}}\,\hbox {d}\tilde{{\varvec{b}}}\, \hbox {d}{\tilde{\theta }}\,\hbox {d}{\tilde{\varvec{w}}}}{\left( {\tilde{a}}\right) ^{n+1}}. \end{aligned}$$

(3.4)

Hence, $\hbox {d}\eta ={\hbox {d}a\,\hbox {d}{} \mathbf{b}\, \hbox {d}\theta }\,\hbox {d}{} \mathbf{w}/a^{n+1}$ is indeed the left Haar measure on ${\mathcal {G}}$.

Proposition 3.1

For every $\psi \in L^2({\mathbb {R}}^n, C\ell _n)$, the map $\pi : L^2\left( {\mathbb {R}}^{n}, C\ell _n\right) \rightarrow L^2\left( {\mathcal {G}}, C\ell _n\right)$ defined by

$$\begin{aligned} \pi (a,\mathbf{b},\theta ,\mathbf{w})\psi (x)=\psi _{a,\mathbf{b},\mathbf{w}}^\theta (x)={\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a}{\mathcal {T}}_\mathbf{b} \psi (\mathbf{x}). \end{aligned}$$

(3.5)

is a unitary projective group representation of the Clifford-valued wave-packet group ${\mathcal {G}}$ on the n-dimensional Hilbert space of Clifford-valued functions.

Proof

For $\big (a,\mathbf{b},\theta ,\mathbf{w} \big ), \big (a^{\prime },\mathbf{b}^{\prime },\theta ^{\prime },\mathbf{w}^{\prime }\big ) \,\in \,{\mathcal {G}}$, Eq. (3.4) implies that

$$\begin{aligned}&{\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a}{\mathcal {T}}_\mathbf{b} \left( {\mathcal {M}}_{\mathbf{w}'} {\mathcal {R}}_{\theta '}^{-1} {\mathcal {D}}_{a'}{\mathcal {T}}_{\mathbf{b}'}\right) \psi (\mathbf{x})\\&\quad ={\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a}{\mathcal {T}}_\mathbf{b} \left[ \dfrac{1}{a'^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}'\cdot \mathbf{x}}\,\psi \left( R_{\theta '}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'}{a'}\right) \right) \right] \\&\quad ={\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a} \left[ \dfrac{1}{a'^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}'\cdot \mathbf{x}-\mathbf{b}}\,\psi \left( R_{\theta '}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'}{a'}-\mathbf{b}\right) \right) \right] \\&\quad =\dfrac{1}{a'^{n/2}}{\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} \left[ \dfrac{1}{a^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}'\cdot (\mathbf{x}-\mathbf{b})/a}\,\psi \left( R_{\theta '}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'-a'{} \mathbf{b}}{aa'}\right) \right) \right] \\&\quad =\dfrac{1}{(aa')^{n/2}}{\mathcal {M}}_\mathbf{w} \left[ e^{\mathbf{i}_n\,R_{\theta }^{-1}{} \mathbf{w}'\cdot (\mathbf{x}-\mathbf{b})/a}\,\psi \left( R_{\theta }^{-1}R_{\theta '}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'-a'{} \mathbf{b}}{aa'}\right) \right) \right] \\&\quad =\dfrac{1}{(aa')^{n/2}}{\mathcal {M}}_\mathbf{w} \left[ e^{\mathbf{i}_n\,R_{\theta }^{-1}{} \mathbf{w}'\cdot (\mathbf{x}-\mathbf{b})/a}\,\psi \left( (R_{\theta '}R_{\theta })^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'-a'{} \mathbf{b}}{aa'}\right) \right) \right] \\&\quad =\dfrac{1}{(aa')^{n/2}} \left[ e^{\mathbf{i}_n\big (\mathbf{wx}+R_{\theta }^{-1}{} \mathbf{w}'\cdot (\mathbf{x}-\mathbf{b})/a \,\big )}\,\psi \left( (R_{\theta '}R_{\theta })^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'-a'{} \mathbf{b}}{aa'}\right) \right) \right] \\&\quad =e^{-\mathbf{i}_nR_{\theta }^{-1}{} \mathbf{w}'\mathbf{b}/a}\,\dfrac{1}{(aa')^{n/2}} \left[ e^{\mathbf{i}_n\big (\mathbf{wx}+R_{\theta }^{-1}{} \mathbf{w}'\cdot \mathbf{x}/a \,\big )}\,\psi \left( (R_{\theta '}R_{\theta })^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}'-a'{} \mathbf{b}}{aa'}\right) \right) \right] \\&\quad =e^{-\mathbf{i}_nR_{\theta }^{-1}{} \mathbf{w}'\mathbf{b}/a}\,{\mathcal {M}}_{\mathbf{w}+a^{-1}R_{\theta }^{-1}{} \mathbf{w}^{\prime }} {\mathcal {R}}_{\theta '+\theta }^{-1} {\mathcal {D}}_{aa'}{\mathcal {T}}_{a'{} \mathbf{b}+\mathbf{b}'} \psi (\mathbf{x})\\&\quad =\gamma _{a,\mathbf{b},\mathbf{w}'}^{\theta }\,{\mathcal {M}}_{\mathbf{w}+a^{-1}R_{\theta }^{-1}\mathbf{w}^{\prime }} {\mathcal {R}}_{\theta '+\theta }^{-1} {\mathcal {D}}_{aa'}{\mathcal {T}}_{a'{} \mathbf{b}+\mathbf{b}'} \psi (\mathbf{x}), \end{aligned}$$

where $\gamma _{a,\mathbf{b},\mathbf{w}'}^{\theta }=e^{-\mathbf{i}_nR_{\theta }^{-1}{} \mathbf{w}'{} \mathbf{b}/a},$ which in turn implies that $\pi$ is a projective unitary group representation of the Clifford-valued wave-packet group ${\mathcal {G}}$.

We now formally introduce the notion of Clifford-valued wave-packet transform of Clifford-valued signals. $\square$

Definition 3.2

The Clifford-valued wave-packet transform of a multi-vector signal $f \in L^2\left( {\mathbb {R}}^{n}, C\ell _n\right)$ with respect to a nonzero $\psi \in L^2\left( {\mathbb {R}}^{n}, C\ell _n\right)$ is denoted by ${\mathcal {WP}}_{\psi }\big [f\big ]$ and defined by

$$\begin{aligned} {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})&=\Big \langle f,\,{\mathcal {M}}_\mathbf{w} {\mathcal {R}}_{\theta }^{-1} {\mathcal {D}}_{a}{\mathcal {T}}_\mathbf{b} \psi (\mathbf{x})\Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}\nonumber \\&= \Big \langle f,\,\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}, \end{aligned}$$

(3.6)

where $\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})$ is given by (3.1).

More explicitly, (3.6) can also be expressed as

$$\begin{aligned} {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w}) =\dfrac{1}{a^{n/2}}\, \int _{{\mathbb {R}}^n} f(\mathbf{x})\,\overline{\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) } \,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\, \hbox {d}\mathbf{x}. \end{aligned}$$

(3.7)

Definition 3.2 allows us to make the following comments:

• The terms appearing in the integrand of (3.7) cannot be interchanged due to the non-commutativity of Clifford-valued functions.

• For $a=1$ and $R_{\theta }=I_{n\times n}$, Definition 3.2 boils down to the Clifford-valued windowed Fourier transform (2.8).

• For $\mathbf{w}=(w_1,w_2,\dots ,w_n)=(0,0,\dots ,0)$, Definition 3.2 reduces to the conventional Clifford-valued wavelet transform (2.9).

• For $f,\psi \in L^2\left( {\mathbb {R}}^{2}, C\ell _2\right)$, Definition 3.2 reduces to the quaternion-valued wave-packet transform [21].

• The spectral representation of the Clifford-valued wave-packet transform is given by

  $$\begin{aligned}&{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\nonumber \\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\, \int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u})\,e^{i_n\mathbf{u}\cdot \mathbf{b}}\, \overline{{\mathcal {F}}_{C\ell }\left[ \,e^{i_nR_{\theta }\mathbf{w}\cdot a(\mathbf{x})}\,\psi (\mathbf{x})\right] \left( R_{\theta }^{-1}a\mathbf{u}\right) }\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\, \hbox {d}{} \mathbf{u}. \end{aligned}$$

  (3.8)

The Clifford-valued wave-packet transform (3.6) satisfies the following properties:

• (i) Linearity: For any $f_1,f_2\in L^2\left( {\mathbb {R}}^{n}, C\ell _n\right)$ and $\alpha _1,\alpha _2 \in C\ell _n$, we have

  $$\begin{aligned} {\mathcal {WP}}_{\psi }\big [\alpha _1 f_1+\alpha _2f_2\big ](a,\theta ,\mathbf{b},\mathbf{w})&= \alpha _1 {\mathcal {WP}}_{\psi }\big [f_1\big ](a,\theta ,\mathbf{b},\mathbf{w})\\&\quad +\alpha _2 {\mathcal {WP}}_{\psi }\big [f_2\big ](a,\theta ,\mathbf{b},\mathbf{w}). \end{aligned}$$

• (ii) Translation covariance:

  $$\begin{aligned} {\mathcal {WP}}_{\psi }\Big ({\mathcal {T}}_\mathbf{t}\big [ f(\mathbf{x})\big ]\Big )(a,\theta ,\mathbf{b},\mathbf{w})={\mathcal {WP}}_{{\psi }}\big [f\big ](a,\theta ,\mathbf{b}-\mathbf{t},\mathbf{w})\, e^{-i_n\mathbf{w}\cdot \mathbf{t}}. \end{aligned}$$

• (iii) Dilation covariance: For $\lambda \in {\mathbb {R}}\setminus \{0\}$, we have

  $$\begin{aligned} {\mathcal {WP}}_{\psi }\big [{\mathcal {D}}_{\lambda } f( \mathbf{x})\big ](a,\theta ,\mathbf{b},\mathbf{w})={\lambda }^n\, {\mathcal {WP}}_{\psi }\left[ f(\mathbf{x})\right] \left( \lambda a,\theta ,\lambda \mathbf{b},\frac{\mathbf{w}}{\lambda }\right) . \end{aligned}$$

• (iv) Modulation covariance:

  $$\begin{aligned} {\mathcal {WP}}_{\psi }\Big ({M}_{\mathbf{w^{\prime }}}\big [ f(\mathbf{x})\big ]\Big )(a,\theta ,\mathbf{b},\mathbf{w})={\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w}-\mathbf{w}^{\prime }). \end{aligned}$$

• (v) Rotation covariance:

  $$\begin{aligned} {\mathcal {WP}}_{\psi }\Big (\mathcal {R}_{{\theta }^{\prime }}[ f(\mathbf{x})]\Big )(a,\theta ,\mathbf{b},\mathbf{w})={\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ^{\prime \prime },R_{\theta }\mathbf{b},R_{{\theta }^{\prime }}^{-1}{} \mathbf{w}), \end{aligned}$$

  where $\mathcal {R}_{{\theta }^{\prime }}[ f(\mathbf{x})]=f\left( R_{{\theta }^{\prime }}{} \mathbf{x}\right)$, and $R_{\theta ^{\prime \prime }}=R_{\theta ^{\prime }} \cdot R_{\theta }.$

• (v) Parity:

  $$\begin{aligned} {\mathcal {WP}}_{P\psi }\Big [Pf(\mathbf{x})\Big ](a,\mathbf{y},\theta ,\mathbf{w})&=(-1)^n\,{\mathcal {WP}}_{\psi }\Big [f(\mathbf{x})\Big ](a,\theta ,-\mathbf{b},-\mathbf{w}),\\ P f(\mathbf{x})&=f(-\mathbf{x}). \end{aligned}$$

• (vi) Translation in $\psi$:

  $$\begin{aligned} {\mathcal {WP}}_{{\mathcal {T}}_\mathbf{k} \big [\psi \big ]}^{{\mathbb {H}}}\Big [f(\mathbf{x})\Big ](a,\theta ,\mathbf{b},\mathbf{w})={\mathcal {WP}}_{\psi }^{{\mathbb {H}}}\Big [f(\mathbf{x})\Big ](a,\theta ,\mathbf{b}+aR_{-\theta }^{-1}{} \mathbf{k},\mathbf{w}). \end{aligned}$$

• (vii) Dilation in $\psi$:

  $$\begin{aligned} {\mathcal {WP}}_{{\mathcal {D}}_{\lambda }\big [\psi \big ]}\left[ f( \mathbf{x})\right] (a,\theta ,\mathbf{b},\mathbf{w})=\lambda ^n\cdot {\mathcal {WP}}_{\psi }\left[ f(\mathbf{x})\right] \left( a\lambda ,\theta ,\mathbf{b},\mathbf{w}\right) ,\,\lambda \in {\mathbb {R}}\setminus \{0\}. \end{aligned}$$

• (viii) Modulation in $\psi$:

  $$\begin{aligned} {\mathcal {WP}}_{\mathcal {M}_{\mathbf{w^{\prime \prime }}}\big [\psi \big ]}\big [ f(\mathbf{x})\big ](a,\theta ,\mathbf{b},\mathbf{w})={\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w}+\mathbf{w}^{\prime \prime }). \end{aligned}$$

Definition 3.3

(Admissibility condition) A Clifford-valued function $\psi \in L^2({\mathbb {R}}^n,\, C\ell _n)$ is said to be admissible if

$$\begin{aligned} {\mathcal {C}}_\psi =\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {\mathbb {R}}^{n}}\Big |{\mathcal {F}}_{C\ell }\big [\psi \big ]\big (R_{\theta }^{-1}a\mathbf{w}\big )\Big |^2\,\dfrac{\text {d}{} \mathbf{w}\,\text {d}\theta \,\text {d}a }{a}, \end{aligned}$$

(3.9)

is an invertible multivector constant and finite.

We now establish an important relationship between two Clifford-valued functions and their respective Clifford-valued wave-packet transforms. As a consequence, we can deduce the energy conservation for the Clifford-valued wave-packet transform (3.6).

Theorem 3.4

(Plancherel theorem) Let ${\mathcal {WP}}_{\psi }\big [f\big ]$ and ${\mathcal {WP}}_{\psi }\big [g\big ]$ be the Clifford-valued wave-packet transforms of the Clifford-valued functions f and g, respectively. Then, we have

$$\begin{aligned} \Big \langle {\mathcal {WP}}_{\psi }\big [ f\big ],\,{\mathcal {WP}}_{\psi }\big [ g\big ]\Big \rangle _{L^2({\mathcal {G}},\,C\ell _n)}= \Big \langle f\, {\mathcal {C}}_{\psi },\,g\Big \rangle _{L^2({\mathbb {R}}^2,\,C\ell _n)}, \end{aligned}$$

(3.10)

where ${\mathcal {C}}_{\psi }$ is given by (3.9) with $|\psi |^2$ real-valued.

Proof

Using the definition of Clifford-valued wave-packet transform (3.6) and the well-known Fubini’s theorem, we have

$$\begin{aligned}&\Big \langle {\mathcal {WP}}_{\psi }\big [ f\big ],\,{\mathcal {WP}}_{\psi }\big [ g\big ]\Big \rangle _{L^2({\mathcal {G}},\,C\ell _n)}\\&\quad = \int _{{\mathcal {G}} } {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,\overline{{\mathcal {WP}}_{\psi }\big [g\big ](a,\theta ,\mathbf{b},\mathbf{w})}\,\hbox {d}\eta \\&\quad = \int _{{\mathcal {G}} } \Big \langle f,\,\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}\,\overline{\Big \langle g,\,\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\Big \rangle }_{L^2({\mathbb {R}}^n,C\ell _n)}\,\hbox {d}\eta \\&\quad =\int _{{\mathcal {G}} }\dfrac{1}{(2\pi )^{2n}} \Big \langle {\mathcal {F}}_{C\ell } \big [f\big ],\,{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\right] \Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)} \end{aligned}$$

$$\begin{aligned}&\qquad \overline{\Big \langle {\mathcal {F}}_{C\ell } \big [g\big ],\,{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\right] \Big \rangle }_{L^2({\mathbb {R}}^n,C\ell _n)}\,\hbox {d}\eta \\&\quad = \dfrac{1}{(2\pi )^{2n}}\int _{{\mathcal {G}} } \int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u})\overline{{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\right] (\mathbf{u})}\hbox {d}\mathbf{u}\\&\qquad \quad \,\overline{\int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })\overline{{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{y})\right] (\mathbf{u}^{\prime })}}\hbox {d}{\mathbf{u}}^{\prime }\,\hbox {d}\eta \\&\quad =\dfrac{1}{(2\pi )^{2n}}\int _{{\mathcal {G}} } \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u})\,\overline{{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})\right] (\mathbf{u})}\,\,{\mathcal {F}}_{C\ell } \left[ \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{y})\right] (\mathbf{u}^{\prime })\\&\qquad \,\overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\hbox {d}\mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\hbox {d}\eta \\&\quad = \dfrac{1}{(2\pi )^{2n}a^n}\int _{{\mathcal {G}} } \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,\overline{\int _{{\mathbb {R}}^n} \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,e^{-i_{n}{} \mathbf{u}\cdot \mathbf{x}\,}\hbox {d}{} \mathbf{x}}\\&\qquad \times \int _{{\mathbb {R}}^n} \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{y}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{y}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{y}}\,\hbox {d}{} \mathbf{y}\,\overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\hbox {d}{} \mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\hbox {d}\eta \\&\quad =\dfrac{1}{(2\pi )^{2n}a^n}\int _{{\mathcal {G}} } \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\overline{ \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) }\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\\&\qquad \times \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{y}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{y}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{y}}\,\overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\hbox {d}{} \mathbf{y}\,\hbox {d}{} \mathbf{x}\,\hbox {d}{} \mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\hbox {d}\eta \end{aligned}$$

$$\begin{aligned}&\quad =\dfrac{1}{(2\pi )^{2n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\\&\qquad \overline{\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) }\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot (\mathbf{x}-\mathbf{y})}\,\times \,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{y}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{y}}\\&\qquad \overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\text {d}{} \mathbf{y}\,\text {d}{} \mathbf{x}\,\text {d}{} \mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{2n+1}} \\&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\\&\qquad \overline{ \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) }\dfrac{1}{(2\pi )^{n}}\,\int _{{\mathbb {R}}^{n}}e^{-\mathbf{i}_n\,\mathbf{w}\cdot (\mathbf{x}-\mathbf{y})}\,\hbox {d}{} \mathbf{w}\\&\qquad \times \,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{y}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{y}}\,\overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\text {d}{} \mathbf{y}\,\text {d}{} \mathbf{x}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\hbox {d}{} \mathbf{b}}{a^{2n+1}} \\&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\\&\qquad \overline{ \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) }\delta (\mathbf{x}-\mathbf{y})\times \,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{y}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{y}}\\&\qquad \overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{y})](\mathbf{u}^{\prime })}\,\text {d}{} \mathbf{y}\,\text {d}{} \mathbf{x}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}}{a^{2n+1}} \end{aligned}$$

$$\begin{aligned}&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\overline{ \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) }\\&\qquad \times \,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{x}}\,\overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{x})](\mathbf{u}^{\prime })}\,\text {d}{} \mathbf{x}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}}{a^{2n+1}} \\&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \,e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\left| \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \right| ^2\\&\qquad e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{x}} \times \overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{x})](\mathbf{u}^{\prime })}\,\text {d}\mathbf{x}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}\mathbf{b}}{a^{2n+1}} \\&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \left| \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \right| ^2\\&\qquad e^{\mathbf{i}_n\mathbf{u}\cdot \mathbf{x}\,}\,e^{-\mathbf{i}_n\mathbf{u}^{\prime }\cdot \mathbf{x}}\times \overline{ {\mathcal {F}}_{C\ell } [g(\mathbf{x})](\mathbf{u}^{\prime })}\,\text {d}{} \mathbf{x}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}}{a^{2n+1}} \end{aligned}$$

$$\begin{aligned}&\quad = \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \Big | \psi \left( R_{\theta }^{-1}{} \mathbf{z}\right) \Big |^2\,\,e^{\mathbf{i}_n(\mathbf{u}-\mathbf{u}^{\prime })\cdot (a\mathbf{z}+\mathbf{b})\,}\\&\qquad \times \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u}^{\prime })}\,a^n\,\text {d}{} \mathbf{z}\,\text {d}{} \mathbf{u}\,\text {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}}{a^{2n+1}} \\&\quad = \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \Big | \psi \left( R_{\theta }^{-1}{} \mathbf{z}\right) \Big |^2\,\,e^{\mathbf{i}_n(\mathbf{u}-\mathbf{u}^{\prime })\cdot a\mathbf{z}}\\&\qquad \dfrac{1}{(2\pi )^{n}}\int _{{\mathbb {R}}^n}\,e^{\mathbf{i}_n(\mathbf{u}-\mathbf{u}^{\prime })\cdot \mathbf{b}\,}\,\hbox {d}{} \mathbf{b}\times \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u}^{\prime })}\,\hbox {d}{} \mathbf{z}\,\hbox {d}{} \mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \\&\quad = \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n\times {\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \Big | \psi \left( R_{\theta }^{-1}{} \mathbf{z}\right) \Big |^2\,\,e^{\mathbf{i}_n(\mathbf{u}-\mathbf{u}^{\prime })\cdot a\mathbf{z}}\,\delta (\mathbf{u}-\mathbf{u}^{\prime })\\&\qquad \times \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u}^{\prime })}\,\hbox {d}\mathbf{z}\,\hbox {d}{} \mathbf{u}\,\hbox {d}{\mathbf{u}}^{\prime }\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \\&\quad = \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \Big | \psi \left( R_{\theta }^{-1}\mathbf{z}\right) \Big |^2\,\overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\text {d}{} \mathbf{z}\,\text {d}{} \mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \\&\quad = \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \int _{{\mathbb {R}}^n}\Big | \psi \left( R_{\theta }^{-1}\mathbf{z}\right) \Big |^2\,\text {d}{} \mathbf{z}\,\overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\text {d}{} \mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \end{aligned}$$

$$\begin{aligned}&\quad =\dfrac{1}{(2\pi )^n} \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \int _{{\mathbb {R}}^n}\Big | {\mathcal {F}}_{C\ell } \big [\psi \big ]\left( R_{\theta }^{-1}\mathbf{u}^{\prime \prime }\right) \Big |^2\,\hbox {d}{} \mathbf{u}^{\prime \prime }\,\\&\qquad \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\hbox {d}\mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \\&\quad =\dfrac{1}{(2\pi )^n}\int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \int _{{\mathbb {R}}^n}\Big | {\mathcal {F}}_{C\ell } \big [\psi \big ]\left( R_{\theta }^{-1}a\mathbf{w}\right) \Big |^2\,a^n\,\text {d}{} \mathbf{w}\\&\qquad \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\text {d}{} \mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta }{a^{n+1}} \\&\quad = \dfrac{1}{(2\pi )^n}\int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u}) \int _{{\mathbb {R}}^{+}\times \text {SO(n)}} \int _{{\mathbb {R}}^n}\Big | {\mathcal {F}}_{C\ell } \big [\psi \big ]\left( R_{\theta }^{-1}a\mathbf{w}\right) \Big |^2\,\hbox {d}{} \mathbf{w}\,\dfrac{\text {d}a\,\text {d}\theta }{a}\\&\qquad \overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\,\text {d}{} \mathbf{u}\, \\&\quad = \dfrac{1}{(2\pi )^n}\int _{{\mathbb {R}}^n} {\mathcal {F}}_{C\ell } \big [f\big ](\mathbf{u})\,{\mathcal {C}}_{\psi } \,\overline{ {\mathcal {F}}_{C\ell } \big [g\big ](\mathbf{u})}\,\,\text {d}{} \mathbf{u}\, \\&\quad = \int _{{\mathbb {R}}^{n}} f(\mathbf{x})\,{\mathcal {C}}_{\psi }\, \overline{g(\mathbf{x})}\, \text {d}{} \mathbf{x}\\&\quad = \Big \langle f\,{\mathcal {C}}_{\psi }, g\Big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}, \end{aligned}$$

where ${\mathcal {C}}_\psi$ is given by (3.9). This completes the proof of Theorem 3.4. $\square$

Corollary 3.5

(Energy conservation) For $f=g$, we have the following identity:

$$\begin{aligned} \int _{{\mathbb {R}}^{+}\times \text {SO(2)} \times {\mathbb {R}}^n \times {\mathbb {R}}^n }\Big |{\mathcal {WP}}_{\psi }\big [ f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\dfrac{\mathrm{d}a\,\mathrm{d}\theta \,\mathrm{d}\mathbf{b}\, \mathrm{d}{} \mathbf{w}}{a^{n+1}}=\Big \langle f\,{\mathcal {C}}_{\psi }, \, f\Big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

(3.11)

Remark 3.6

Except the factor ${\mathcal {C}}_{\psi }$, the Clifford-valued wave-packet transform is an isometry from the space $L^2({\mathbb {R}}^n,\,C\ell _n)$ to the space of transformations $L^2({\mathbb {R}}^{+}\times {\mathbb {R}}^n\times \text {SO(n)}\times {\mathbb {R}}^n\times {\mathbb {R}}^n,\,C\ell _n)$.

The next theorem guarantees the reconstruction of the input Clifford-valued signal from the corresponding Clifford-valued wave-packet transform.

Theorem 3.7

(Reconstruction formula) Let${\mathcal {WP}}_{\psi }\big [f\big ]$ be the Clifford-valued wave-packet transform of any $f\in L^2({\mathbb {R}}^n,\,C\ell _n)$. Then, we have

$$\begin{aligned} f(\mathbf{x}) = \int _{{\mathcal {G}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\,\dfrac{\mathrm{d}a\,\mathrm{d}\theta \,\mathrm{d}{} \mathbf{b}\,\mathrm{d}\mathbf{w}}{a^{\frac{3n}{2}+1}},\quad \hbox {a.e.} \end{aligned}$$

(3.12)

Proof

Implication of Plancherel theorem for any $g\in L^2({\mathbb {R}}^n,C\ell _n)$ implies that

$$\begin{aligned}&\Big \langle f \,{\mathcal {C}}_{\psi },\,g \Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}\\&\quad =\Big \langle {\mathcal {WP}}_{\psi }\big [f\big ],\,{\mathcal {WP}}_{\psi }\big [g\big ] \Big \rangle _{L^2({\mathcal {G}},C\ell _n)}\\&\quad =\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\overline{{\mathcal {WP}}_{\psi }\big [g\big ](a,\theta ,\mathbf{b},\mathbf{w})}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{n+1}}\\&\quad =\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\\&\qquad \overline{\dfrac{1}{a^{n/2}}\, \int _{{\mathbb {R}}^n} g(\mathbf{x})\overline{\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) } \,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\, \hbox {d}\mathbf{x}}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{n+1}}\\&\quad =\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}\int _{{\mathbb {R}}^n}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \\&\quad \quad \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,\overline{g(\mathbf{x})} \, \hbox {d}\mathbf{x}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{\frac{3n}{2}+1}}\\&=\int _{{\mathbb {R}}^n}\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \\&\qquad \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}\mathbf{w}}{a^{\frac{3n}{2}+1}}\,\overline{g(\mathbf{x})}\, \hbox {d}{} \mathbf{x}\\&=\Bigg \langle \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \\&\quad \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}\mathbf{w}}{a^{\frac{3n}{2}+1}},\,g \Bigg \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}. \end{aligned}$$

Since $g\in L^2({\mathbb {R}}^n,C\ell _n)$ is arbitrary, so we have

$$\begin{aligned} f(x)\, {\mathcal {C}}_{\psi }&= \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\\&\quad \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}\mathbf{w}}{a^{\frac{3n}{2}+1}}. \end{aligned}$$

Or equivalently, we can write

$$\begin{aligned} f(x) = \int _{{\mathcal {G}}} {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{\frac{3n}{2}+1}}. \end{aligned}$$

This completes the proof of Theorem 3.7. $\square$

Next, we present the characterization of the range of the Clifford-valued wave-packet transform (3.6).

Theorem 3.8

If $\psi \in L^2({\mathbb {R}}^n,\,C\ell _n)$ satisfies admissibility (3.9), then the range of the Clifford-valued wave-packet transform ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ is a reproducing kernel in $L^2\left( {\mathbb {R}}^{+}\times \text {SO(n)}\right.$ $\left. \times {\mathbb {R}}^n\times {\mathbb {R}}^n,\,C\ell _n\right)$, where kernel is given by

$$\begin{aligned} K_{\psi }(a,\theta ,\mathbf{b},\mathbf{w}; a^{\prime },\theta ^{\prime },\mathbf{b}^{\prime }, \mathbf{w}^{\prime } )&= \dfrac{1}{\big (\sqrt{aa^{\prime }}\big )^n} \left\langle e^{\mathbf{i}_n\mathbf{w}\cdot \mathbf{x}} \,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,{\mathcal {C}}_{\psi }^{-1},\right. \nonumber \\&\quad \left. e^{\mathbf{i}_n\mathbf{w}^{\prime }\cdot \mathbf{x}} \,\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \right\rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

(3.13)

Moreover, we have

$$\begin{aligned} \Big |K_{\psi }(a,\theta ,\mathbf{b},\mathbf{w}; a^{\prime },\theta ^{\prime },\mathbf{b}^{\prime }, \mathbf{w}^{\prime } )\Big |\,\le \, \dfrac{1}{\sqrt{\left| {\mathcal {C}}_{\psi }\right| }}\, \big \Vert \psi \big \Vert _{L^1({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

(3.14)

Proof

Invoking the reconstruction formula (3.12) together with (3.6), we have

$$\begin{aligned}&{\mathcal {WP}}_{\psi }\big [f\big ](a^{\prime },\theta ^{\prime },\mathbf{b}^{\prime },\mathbf{w}^{\prime })\\&\quad =\dfrac{1}{(\sqrt{a^{\prime }}\big )^{n}}\, \int _{{\mathbb {R}}^n} f(\mathbf{x})\overline{\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) } \,e^{-\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\, \hbox {d}{} \mathbf{x}\\&\quad =\dfrac{1}{\big (\sqrt{a^{\prime }}\big )^{n}}\, \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\\&\qquad \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\times \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}\mathbf{w}}{a^{\frac{3n}{2}+1}}\,\overline{\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) } \,e^{-\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\, \hbox {d}{} \mathbf{x}\\&=\dfrac{1}{\big (\sqrt{a^{\prime }}\big )^{n}}\, \int _{{\mathbb {R}}^n} \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\\&\qquad \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\times \overline{\,e^{\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\,\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) } \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}\mathbf{b}\,\text {d}{} \mathbf{w}}{a^{\frac{3n}{2}+1}}\, \hbox {d}{} \mathbf{x}\\&=\dfrac{1}{\big (\sqrt{a^{\prime }}\big )^{n}} \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\, \int _{{\mathbb {R}}^n} \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\\&\qquad \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\times \,\overline{\,e^{\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\,\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) }\, \hbox {d}{} \mathbf{x} \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{\frac{3n}{2}+1}}\\&=\dfrac{1}{\big (\sqrt{aa^{\prime }}\big )^{n}} \int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\times \left\langle \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\right. \\&\quad \left. \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1},\,e^{\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\,\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \,\right\rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)} \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{n+1}}\\&{=}\int _{{\mathbb {R}}^{+}\times \text {SO}(n) {\times }{{\mathbb {R}}^{n}} \times {{\mathbb {R}}^{n}}}{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\,K_{\psi }(a,\theta ,\mathbf{b},\mathbf{w}; a^{\prime },\theta ^{\prime },\mathbf{b}^{\prime }, \mathbf{w}^{\prime } ) \,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{b}\,\text {d}{} \mathbf{w}}{a^{n{+}1}}, \end{aligned}$$

where $K_{\psi }$ is given by (3.13), which completes the proof of first assertion.

Also, we have

$$\begin{aligned}&\Big |K_{\psi }(a,\theta ,\mathbf{b},\mathbf{w}; a^{\prime },\theta ^{\prime },\mathbf{b}^{\prime }, \mathbf{w}^{\prime } )\Big | = \left| \dfrac{1}{\big (\sqrt{aa^{\prime }}\big )^{n}} \left\langle \,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1},\right. \right. \\&\qquad \left. \left. e^{\mathbf{i}_n\,\mathbf{w}^{\prime }\cdot \mathbf{x}}\,\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \,\right\rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)} \right| \\&\quad \le \,\dfrac{1}{\left| \big (\sqrt{aa^{\prime }}\big )^{n}\right| } \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\right| \,\left| \overline{\psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) } \right| \hbox {d}{} \mathbf{x}\\&\qquad \le \,\dfrac{1}{\left| \big (\sqrt{aa^{\prime }}\big )^{n}\right| } \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\right| \, \hbox {d}\mathbf{x}\right\} ^{1/2}\\&\qquad \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \right| \,\hbox {d}\mathbf{x}\right\} ^{1/2}\\&\quad =\,\dfrac{1}{\left| \big (\sqrt{aa^{\prime }}\big )^{n}\right| } \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}}{a}-\dfrac{\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\right| \, \hbox {d}\mathbf{x}\right\} ^{1/2}\\&\qquad \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta ^{\prime }}^{-1}\left( \dfrac{\mathbf{x}}{a^{\prime }}-\dfrac{\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \right| \,\hbox {d}{} \mathbf{x}\right\} ^{1/2}\\&\qquad =\,\dfrac{1}{\left| \big (\sqrt{aa^{\prime }}\big )^{n}\right| } \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta }^{-1}\left( \mathbf{y}-\dfrac{\mathbf{b}}{a}\right) \right) \, {\mathcal {C}}_{\psi }^{-1}\right| \,a^n \hbox {d}\mathbf{y}\right\} ^{1/2}\\&\qquad \left\{ \int _{{\mathbb {R}}^{n}} \left| \psi \left( R_{\theta ^{\prime }}^{-1}\left( \mathbf{y}^{\prime }-\dfrac{\mathbf{b}^{\prime }}{a^{\prime }}\right) \right) \right| \,{a^{\prime }}^n\,\hbox {d}{} \mathbf{y}^{\prime }\right\} ^{1/2}\\&\quad =\, \dfrac{1}{\sqrt{\left| {\mathcal {C}}_{\psi }\right| }}\,\left\{ \int _{{\mathbb {R}}^{n}} \Big |\psi \left( R_{\theta }^{-1}\left( \mathbf{y}\right) \right) \,\Big | \hbox {d}{} \mathbf{y}\right\} ^{1/2}\,\left\{ \int _{{\mathbb {R}}^{n}} \Big |\psi \left( R_{\theta ^{\prime }}^{-1}\left( \mathbf{y}^{\prime }\right) \right) \Big | \,\hbox {d}\mathbf{y}^{\prime }\right\} ^{1/2}\\&\quad = \dfrac{1}{\sqrt{\left| {\mathcal {C}}_{\psi }\right| }}\,\left\{ \big \Vert \psi \big \Vert _{L^1({\mathbb {R}}^n,\,C\ell _n)} \right\} ^{1/2}\,\left\{ \big \Vert \psi \big \Vert _{L^1({\mathbb {R}}^n,\,C\ell _n)}\right\} ^{1/2}\\&\quad =\, \dfrac{1}{\sqrt{\left| {\mathcal {C}}_{\psi }\right| }}\, \big \Vert \psi \big \Vert _{L^1({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

This completes the proof of Theorem 3.7. $\square$

4 Uncertainty Principles for Clifford Wave Packet Transform

Heisenberg’s uncertainty principle in harmonic analysis is of central importance in time-frequency analysis as it provides a lower bound for optimal simultaneous resolution in the time and frequency domains [23]. With the advent of time-frequency analysis, this principle has been extended to several directions, including the integral transformations beyond the Fourier transform. In this section, we shall establish several uncertainty inequalities including Heisenberg–Pauli–Weyl uncertainty, Pitt’s and logarithmic-type inequalities for the Clifford-valued wave-packet transform as defined by (3.6). Prior to that, we have the following lemma:

Lemma 4.1

Let ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ be the Clifford-valued wave-packet transform of any $f\in L^2({\mathbb {R}}^n, C\ell _n)$ with ${\mathcal {F}}_{C\ell }\left[ \psi \right] \in L^2({\mathbb {R}}^n)$. Then, we have

$$\begin{aligned} {\mathcal {F}}_{C\ell }\Big [{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ](\mathbf{u})=a^{n/2}\, \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\,\, {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,\, e^{-\mathbf{i}_n\mathbf{w}\cdot \mathbf{b}}. \, \end{aligned}$$

(4.1)

Proof

From the definition of Clifford-valued wave-packet transform (3.6), we have

$$\begin{aligned}&{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\\&\quad = \Big \langle f,\,\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }\Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}\\&\quad = \dfrac{1}{(2\pi )^n}\,\Big \langle {\mathcal {F}}_{C\ell }\big [f\big ],\,{\mathcal {F}}_{C\ell }[\psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})]\Big \rangle _{L^2({\mathbb {R}}^n,C\ell _n)}\\&\quad =\dfrac{1}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\, \overline{{\mathcal {F}}_{C\ell }\left[ \dfrac{1}{a^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \right] (\mathbf{u})}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{1}{(2\pi )^n\,a^{n/2}}\,\int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\, \overline{\int _{{\mathbb {R}}^n}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\psi \left( R_{\theta }^{-1}\left( \dfrac{\mathbf{x}-\mathbf{b}}{a}\right) \right) \,e^{-\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{x}}\,\,\hbox {d}\mathbf{x}}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{1}{(2\pi )^n\,a^{n/2}}\,\int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\, \overline{\int _{{\mathbb {R}}^n}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot (a\mathbf{y}+\mathbf{b})}\,\psi \left( R_{\theta }^{-1}(\mathbf{y})\right) \,e^{-\mathbf{i}_n\,\mathbf{u}\cdot (a\mathbf{y}+\mathbf{b})}}\,a^n\,\hbox {d}{} \mathbf{y}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\, \overline{\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\,\int _{{\mathbb {R}}^n}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot (a\mathbf{y})}\,\psi \left( R_{\theta }^{-1}(\mathbf{y})\right) \,e^{-\mathbf{i}_n\,\mathbf{u}\cdot (a\mathbf{y})}\,e^{-\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{b}}}\,\hbox {d}\mathbf{y}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,e^{\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{b}}\, \overline{\int _{{\mathbb {R}}^n}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot (a\mathbf{y})}\,\psi \left( R_{\theta }^{-1}(\mathbf{y})\right) \,e^{-\mathbf{i}_n\,\mathbf{u}\cdot (a\mathbf{y})}}\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\,\hbox {d}\mathbf{y}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}\,{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,e^{\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{b}}\,\, \overline{{\mathcal {F}}_{C\ell }\Big [e^{\mathbf{i}_n\,\mathbf{w}\cdot (a\mathbf{y})}\,\psi \left( R_{\theta }^{-1}(\mathbf{y})\right) \Big ](a\mathbf{u})}\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}\,{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,e^{\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{b}}\,\,\overline{{\mathcal {F}}_{C\ell }\big [\psi (\mathbf{y})\big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\,\hbox {d}{} \mathbf{u}\\&\quad =\dfrac{a^{n/2}}{(2\pi )^n}\,\int _{{\mathbb {R}}^n}\overline{{\mathcal {F}}_{C\ell }\big [\psi (\mathbf{y})\big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\,{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\,e^{\mathbf{i}_n\,\mathbf{u}\cdot \mathbf{b}}\,\hbox {d}{} \mathbf{u}\\&\quad ={\mathcal {F}}_{C\ell }^{-1}\left( a^{n/2}\overline{{\mathcal {F}}_{C\ell }\big [\psi (\mathbf{y})\big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\,{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\,e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{b}}\right) (\mathbf{b}). \end{aligned}$$

This completes the proof of Lemma 4.1. $\square$

We are now ready to establish the Heisenberg-type inequalities for the proposed Clifford-valued wave-packet transform (3.68).

Theorem 4.2

Let ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ be the Clifford-valued wave-packet transform of any Clifford-valued function $f\in L^2({\mathbb {R}}^n, C\ell _n))$. Then, we have

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\mathrm{d}{} \mathbf{b}\dfrac{\mathrm{d}a\,\mathrm{d}\theta \,\mathrm{d}{} \mathbf{w} }{a^{n+1}}\right\} }^{1/2} \nonumber \\&\quad \times \left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\, {\mathcal {C}}_\psi \, \Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |\,\mathrm{d}\mathbf{u}\right\} ^{1/2}\,\ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\Big \langle f\, {\mathcal {C}}_{\psi },\, f\Big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

(4.2)

Proof

For any Clifford-valued function $f\in L^2({\mathbb {R}}^n, C\ell _n)$, the Heisenberg–Paul–Weyl inequality for the Clifford-valued Fourier transform is given by [17]

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\big |f(\mathbf{b})\big |^{2}\,\hbox {d}\mathbf{b}\right\} }^{1/2}\,\left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,{\big |{{\mathcal {F}}_{C\ell }[ f]}(\mathbf{u})\big |^{2}}\,\hbox {d}\mathbf{u}\right\} ^{1/2}\nonumber \\&\quad \ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\int _{{\mathbb {R}}^{n}}\big |f(b)\big |^{2}\,\hbox {d}\mathbf{b}.\, \end{aligned}$$

(4.3)

Considering ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ as a function of $\mathbf{b}$ and replacing f by ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ in (4.3), we obtain

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\mathbf{b}\right\} }^{1/2} \,\left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,{\Big |{{\mathcal {F}}_{C\ell }\Big [ {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ]}(\mathbf{u})\Big |^{2}}\,\hbox {d}{} \mathbf{u}\right\} ^{1/2}\\&\qquad \ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\mathbf{b}. \end{aligned}$$

We now integrate the above inequality with respect to measure ${\hbox {d}a\,\hbox {d}\theta \,\hbox {d}{} \mathbf{w} }/{a^{n+1}}$ and using the classical Cauchy–Schwartz’s inequality, we obtain

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\mathrm{d}{} \mathbf{b}\dfrac{\mathrm{d}a\,\mathrm{d}\theta \,\mathrm{d}{} \mathbf{w} }{a^{n+1}}\right\} }^{1/2} \\&\qquad \times \left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,{\Big |{{\mathcal {F}}_{C\ell }\Big [ {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ]}(\mathbf{u})\Big |^{2}}\,\hbox {d}{} \mathbf{u}\,\dfrac{\hbox {d}a\,\hbox {d}\theta \,\hbox {d}{} \mathbf{w} }{a^{n+1}}\right\} ^{1/2}\\&\qquad \ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}{} \mathbf{b} \dfrac{\hbox {d}a\,\hbox {d}\theta \,\hbox {d}{} \mathbf{w} }{a^{n+1}}. \end{aligned}$$

Applying the Lemma 4.1, we obtain

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}\right\} }^{1/2} \\&\quad \times \left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,{\Big |a^{n/2} \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\Big |^{2}} \Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^{2}\,\hbox {d}\mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}\right\} ^{1/2}\\&\qquad \ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\int _{{\mathbb {R}}^{+}\times \text {SO}(n) \times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}{} \mathbf{b} \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}. \end{aligned}$$

Moreover, by virtue of Fubini theorem, we have

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}\right\} }^{1/2} \\&\quad \times \left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}{\Big |{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{w})\big )\Big |^{2}}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a}\, \Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^{2}\,\text {d}\mathbf{u}\right\} ^{1/2}\\&\quad \ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\text {d}{} \mathbf{b} \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}. \end{aligned}$$

Using the admissibility condition (3.9) along with the energy preserving relation (3.11), we have

$$\begin{aligned}&{\left\{ \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w} }{a^{n+1}}\right\} }^{1/2} \\&\qquad \times \left\{ \int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2}\,{\mathcal {C}}_\psi \, \Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^{2}\,\hbox {d}\mathbf{u}\right\} ^{1/2}\,\ge \,\dfrac{\sqrt{n}\,(2\pi )^{n/2}}{2}\,\Big \langle f\, {\mathcal {C}}_{\psi },\, f\Big \rangle _{L^2({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

This completes the proof of Theorem 4.2. $\square$

Remark 4.3

For a real-valued ${\mathcal {C}}_{\psi }$, Theorem 4.2 boils down to

$$\begin{aligned}&\int _{{\mathcal {G}}}|\mathbf{b}|^{2}\,\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\eta \,\int _{{\mathbb {R}}^{n}}|\mathbf{u}|^{2} \Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^{2}\,\hbox {d}{} \mathbf{u}\nonumber \\&\quad \ge \,\dfrac{n\,{\mathcal {C}}_{\psi }(2\pi )^{n}}{4}\,\big \Vert f\big \Vert ^4_{L^2({\mathbb {R}}^n,\,C\ell _n)}. \end{aligned}$$

(4.4)

The classical Pitt’s inequality expresses a fundamental relationship between a sufficiently smooth function and the corresponding Fourier transform [22]. We establish the Pitt’s inequality for Clifford-valued wave-packet transform (3.6). The Schwartz class in $L^2({\mathbb {R}}^n,\,C\ell _n)$ is denoted by ${\mathbb {S}}({\mathbb {R}}^n,C\ell _n)$ and is defined by

$$\begin{aligned} {\mathbb {S}}\left( {\mathbb {R}}^n,C\ell _n\right) =\left\{ f\in C^{\infty }({\mathbb {R}}^n,C\ell _n): \sup _{t\in {\mathbb {R}}^n}\left| t^{\alpha }\partial _{t}^{\beta }f(t)\right| <\infty \right\} ,\, \end{aligned}$$

(4.5)

where $C^{\infty }({\mathbb {R}}^n,C\ell _n)$ is the class of smooth functions, $\alpha ,\beta$ denote multi-indices, and ${\partial }_{t}$ denotes the usual partial differential operator.

Theorem 4.4

(Pitt’s Inequality) For every $f\in {\mathbb {S}}({\mathbb {R}}^n,C\ell _n)$ and the admissible Clifford-valued function $\psi$, the Pitt’s inequality for the Clifford-valued wave-packet transform (3.6) is given by

$$\begin{aligned}&\int _{{\mathbb {R}}^{n}}\left| \mathbf{u}\right| ^{-\lambda }\,{\mathcal {C}}_{\psi }\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^2\,\mathrm{d}{} \mathbf{u}\nonumber \\&\quad \le P_{\lambda }\int _{{\mathbb {R}}^{+}\times {\mathbb {R}}^{n-1}\times \text {SO}(n)}\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \,\mathrm{d}\mathbf{b}\, \dfrac{\mathrm{d}a\,\mathrm{d}\theta \,\mathrm{d}{} \mathbf{w}}{a^{n+1}}, \end{aligned}$$

(4.6)

where ${\mathcal {C}}_{\psi }$ is given by (3.9) and

$$\begin{aligned} P_{\lambda }=\pi ^{\lambda }\left[ \Gamma \left( \frac{n-\lambda }{4}\right) \Big / \Gamma \left( \frac{n+\lambda }{4}\right) \right] ^2,\quad 0\le \lambda <n,\, \end{aligned}$$

(4.7)

with $\Gamma (\cdot )$ denotes the well-known Euler’s gamma function.

Proof

Considering ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\in {\mathbb {S}}({\mathbb {R}}^n,\,C\ell _n)$ as a function of the translation variable $\mathbf{b}$, the Pitt’s inequality in the Clifford-valued Fourier domain ([24], Theorem 4) implies that

$$\begin{aligned} \int _{{\mathbb {R}}^n}\left| \mathbf{u}\right| ^{-\lambda }\left| {\mathcal {F}}_{C\ell }\Big [{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ](\mathbf{u})\right| ^2\hbox {d}{} \mathbf{u}\le P_{\lambda }\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \hbox {d}{} \mathbf{b}, \end{aligned}$$

which upon integration with respect to the measure ${\hbox {d}a\,\hbox {d}\theta \,\hbox {d}{} \mathbf{w}}/{a^{n+1}}$ yields

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\left| \mathbf{u}\right| ^{-\lambda }\Big |{\mathcal {F}}_{C\ell }\Big [{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ](\mathbf{u})\Big |^2\,\hbox {d}{} \mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}\nonumber \\&\quad \le P_{\lambda }\int _{{\mathbb {R}}^{+}\times {\mathbb {R}}^{n-1}\times \text {SO}(n)}\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \,\hbox {d}\mathbf{b}\, \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

(4.8)

Using Lemma 4.1, we can express inequality (4.8) in the following manner:

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\left| \mathbf{u}\right| ^{-\lambda }\Big |a^{n/2} \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\\&\qquad \cdot {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\cdot e^{-\mathbf{i}_n\mathbf{w}\cdot \mathbf{b}}\Big |^2\,\hbox {d}{} \mathbf{u}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}\mathbf{w}}{a^{n+1}}\\&\quad \le P_{\lambda }\int _{{\mathbb {R}}^{+}\times {\mathbb {R}}^{n-1}\times \text {SO}(n)}\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \,\hbox {d}\mathbf{b}\, \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

Equivalently, we can have

$$\begin{aligned}&\int _{{\mathbb {R}}^{n}}\left| \mathbf{u}\right| ^{-\lambda }\,\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\Big | \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\Big |^2\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a}\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^2\,\hbox {d}{} \mathbf{u}\nonumber \\&\quad \le P_{\lambda }\int _{{\mathbb {R}}^{+}\times {\mathbb {R}}^{n-1}\times \text {SO}(n)}\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \,\hbox {d}\mathbf{b}\, \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

(4.9)

Consequently, inequality (4.9) reduces to

$$\begin{aligned}&\int _{{\mathbb {R}}^{n}}\left| \mathbf{u}\right| ^{-\lambda }\,{\mathcal {C}}_{\psi }\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u}) \Big |^2\,\hbox {d}{} \mathbf{u}\\&\quad \le P_{\lambda }\int _{{\mathbb {R}}^{+}\times {\mathbb {R}}^{n-1}\times \text {SO}(n)}\int _{{\mathbb {R}}^n}\left| \mathbf{b}\right| ^{\lambda }\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^2 \,\hbox {d}\mathbf{b}\, \dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

which establishes the desired inequality for the Clifford-valued wave-packet transform. $\square$

Remark 4.5

For $\lambda =0$, inequality (4.6) reduces to the energy conservation relation (3.11).

We now derive the logarithmic uncertainty principle for the Clifford-valued wave-packet transform ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ as defined by (3.6).

Theorem 4.6

(Logarithmic inequality) Given any Clifford-valued signal $f\in {\mathbb {S}}({\mathbb {R}}^n,C\ell _n)$ and the admissible $\psi$, the Clifford-valued wave-packet transform ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ satisfies the following logarithmic inequality:

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}\mathbf{w}}{a^{n+1}}\nonumber \\&\quad +\int _{{\mathbb {R}}^{n}}\,{\mathcal {C}}_{\psi }\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}{} \mathbf{u}\nonumber \\&\qquad \ge \left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \Big \langle f\,{\mathcal {C}}_{\psi }, \,f\Big \rangle _{L^2(R^n,C\ell _n)},\, \end{aligned}$$

(4.10)

provided the L.H.S of (4.10) is well-defined.

Proof

For any Clifford-valued function $f\in {\mathbb {S}}({\mathbb {R}}^n,C\ell _n)$, the logarithmic inequality in the Clifford Fourier domain is given by [24]:

$$\begin{aligned} \int _{{\mathbb {R}}^{n}}\big |f(\mathbf{b})\big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}\mathbf{b}\,+\,\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}\mathbf{u}\ge \,\left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \int _{{\mathbb {R}}^{n}}\big |f(\mathbf{b})\big |^{2}\,\hbox {d}{} \mathbf{b}. \end{aligned}$$

Replacing $f(\mathbf{b})$ by ${\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})$ in the above inequality, we obtain

$$\begin{aligned}&\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}\mathbf{b}+\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {F}}_{C\ell }\Big [{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big ](\mathbf{u})\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}{} \mathbf{u}\nonumber \\&\qquad \ge \,\left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\mathbf{b}. \end{aligned}$$

(4.11)

Integrating (4.11) with respect to measure ${ \hbox {d}a\,\hbox {d}\theta \,\hbox {d}\mathbf{w}}/{a^{n+1}}$ and employing Lemma 4.1 in the second integral of the left hand side, we obtain

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}\nonumber \\&\qquad +\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |a^{n/2} \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\nonumber \\&\quad \quad \,\cdot {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\cdot e^{-\mathbf{i}_n\mathbf{w}\cdot \mathbf{b}}\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}{} \mathbf{u}\,\dfrac{ \text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}\nonumber \\&\quad \ge \,\left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\mathbf{b}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

(4.12)

Moreover, the Fubini’s theorem implies that

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}\\&\qquad +\int _{{\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\Big | \overline{{\mathcal {F}}_{C\ell } \big [\psi \big ]\big (R_{\theta }^{-1}a(\mathbf{u}-\mathbf{w})\big )}\Big |^{2}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a}\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}{} \mathbf{u}\\&\quad \ge \,\left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\hbox {d}\mathbf{b}\,\dfrac{\text {d}a\,\text {d}\theta \,\text {d}{} \mathbf{w}}{a^{n+1}}. \end{aligned}$$

Finally, the desired inequality is obtained by using the admissibility condition (3.9) together with the energy preserving relation (3.11) as

$$\begin{aligned}&\int _{{\mathbb {R}}^{+}\times \text {SO}(n)\times {\mathbb {R}}^{n}}\int _{{\mathbb {R}}^{n}}\Big |{\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})\Big |^{2}\,\ln |\mathbf{b}|\,\hbox {d}{} \mathbf{b}\dfrac{\text {d}a\,\text {d}\theta \,\text {d}\mathbf{w}}{a^{n+1}}\\&\qquad +\int _{{\mathbb {R}}^{n}}\,{\mathcal {C}}_{\psi }\,\Big | {\mathcal {F}}_{C\ell }\big [f\big ](\mathbf{u})\Big |^{2}\,\ln |\mathbf{u}|\,\hbox {d}{} \mathbf{u}\\&\quad \ge \left( \dfrac{\Gamma ^{\prime }(n/4)}{\Gamma (n/4)}-\ln \pi \right) \Big \langle f\,{\mathcal {C}}_{\psi }, \,f\Big \rangle _{L^2(R^n,C\ell _n)}. \end{aligned}$$

This completes the proof of Theorem 4.4$\square$ .

5 Applications to the Benchmark Signals

To broaden the scope of the present study, we shall validate the proposed wave-packet transform via illustrative examples for detecting the direction and point-wise analysis of benchmark signals.

For the difference of Gaussian functions

$$\begin{aligned} \psi (\mathbf{x})=\lambda ^{-2} \exp \left\{ \frac{-|\mathbf{x}|^2}{2\lambda ^2}\right\} -\exp \left\{ \frac{-|\mathbf{x}|^2}{2}\right\} ,\quad 0<\lambda <1, \end{aligned}$$

(5.1)

the Clifford-valued wave-packet system is given by

$$\begin{aligned} \psi _{a,\mathbf{b},\mathbf{w}}^{\theta }(\mathbf{x})=\dfrac{1}{a^{n/2}}\,e^{\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\left( \dfrac{1}{\lambda ^{2}}\,e^{\frac{-|\mathbf{x}-\mathbf{b}|^2}{2a^2\lambda ^2}}\,-\,e^{\frac{-|\mathbf{x}-\mathbf{b}|^2}{2a^2}}\right) . \end{aligned}$$

(5.2)

Consequently, the Clifford-valued wave-packet transform (3.6) of any Clifford-valued signal $f(\mathbf{x})$ is given by

$$\begin{aligned} {\mathcal {WP}}_{\psi }\big [f\big ](a,\theta ,\mathbf{b},\mathbf{w})&= \dfrac{1}{\lambda ^{2}\,a^{n/2}}\int _{{\mathbb {R}}^{n}}f(\mathbf{x})\, e^{\frac{-|\mathbf{x}-\mathbf{b}|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\hbox {d}\mathbf{x}\nonumber \\&\quad -\dfrac{1}{a^{n/2}}\int _{{\mathbb {R}}^{n}}f(\mathbf{x})\,e^{\frac{-|\mathbf{x}-\mathbf{b}|^2}{2a^2}}\, e^{-\mathbf{i}_n\,\mathbf{w}\cdot \mathbf{x}}\,\hbox {d}{} \mathbf{x}, \end{aligned}$$

(5.3)

which is independent of the rotation parameter $\theta$.

In order to check the efficiency and applicability of the Clifford-valued wave-packet transform (5.2) for the point-wise analysis and direction detections of the scalar-valued signals of infinite, semi-infinite and finite lengths, we shall confine ourselves to the three dimension, that is; $n=3$.

Case-I: Consider the rod $f_{1}$ of infinite length laying along $x_1$-axis

$$\begin{aligned} f_{1}(x_1,x_2,x_3)=\delta (x_2)\,\delta (x_3),\,\,b_2=b_3=0,\,\,w_2=w_3=0. \end{aligned}$$

(5.4)

Then, the Clifford-valued wave-packet transform (5.3) yields

$$\begin{aligned}&{\mathcal {WP}}_{\psi }\big [f_{ 1}\big ](a,\theta ,\mathbf{b},\mathbf{w})\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}^{3}}\delta (x_2)\,\delta (x_3)\, e^{\frac{-|(x_1-b_1)+(x_2-b_2)+(x_3-b_3)|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,(w_1x_1+w_2x_2+w_3x_3)}\,\hbox {d}x_1\hbox {d}x_2\hbox {d}x_3\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}^{3}}\delta (x_2)\,\delta (x_3)\, e^{\frac{-|(x_1-b_1)+(x_2-b_2)+(x_3-b_3)|^2}{2a^2}}\, e^{-\mathbf{i}_n\,(w_1x_1+w_2x_2+w_3x_3)}\,\hbox {d}x_1\hbox {d}x_2\hbox {d}x_3\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}} e^{-\frac{|x_1-b_1|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\,-\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}}\, e^{-\frac{|x_1-b_1|^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}} e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\,-\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}}\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\,\sqrt{2\pi a^2{\lambda }^2} e^{-\frac{w_1^2a^2{\lambda }^2}{2}} \cdot e^{-\mathbf{i}_n\,w_1b_1}\,-\dfrac{1}{a^{3/2}}\,\sqrt{2\pi a^2} e^{-\frac{w_1^2a^2}{2}}\cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad = \dfrac{\sqrt{2\pi }}{\sqrt{a}}\left( \dfrac{1}{\lambda }\,e^{-\frac{w_1^2a^2{\lambda }^2}{2}} \,-\, e^{-\frac{w_1^2a^2}{2}}\right) e^{-\mathbf{i}_n\,w_1b_1}, \end{aligned}$$

(5.5)

which vanishes as $w_1\rightarrow \infty$. It is pertinent to mention that any singularity in the transformed domain can easily be detected by choosing different values of a and $\lambda$ as shown in Fig. 1 and tabulated in Table 1. Moreover, the peaks of the signal at the origin can also be efficiently analyzed by taking the smaller values of a, whereas the directionality of the transformed signal can be detected by adjusting the translation parameter $b_1$ as depicted in Fig. 2.

Fig. 1

Clifford-valued wave-packet transforms of $f_1$ for different values of a and $\lambda$

Table 1 Clifford-valued wave-packet transform of $f_{1}$ for different values of scaling parameter a, translation parameter $\mathbf{b}$ and $\lambda$

Case-II: Consider the thin-plate $f_2$ of the form

$$\begin{aligned} f_{ 2}(x_1,x_2,x_3)=\delta (x_3),\,\,b_3=0,\,\,w_3=0. \end{aligned}$$

(5.6)

Then, the corresponding Clifford-valued wave-packet transform given by (5.2) yields

$$\begin{aligned}&{\mathcal {WP}}_{\psi }\big [f_{2}\big ](a,\theta ,\mathbf{b},\mathbf{w})\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}^{3}}\delta (x_3)\, e^{\frac{-|(x_1-b_1)+(x_2-b_2)+(x_3-b_3)|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,(w_1x_1+w_2x_2+w_3x_3)}\,\hbox {d}x_1\hbox {d}x_2\hbox {d}x_3\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}^{3}}\delta (x_2)\delta (x_3)\, e^{\frac{-|(x_1-b_1)+(x_2-b_2)+(x_3-b_3)|^2}{2a^2}}\, e^{-\mathbf{i}_n\,(w_1x_1+w_2x_2+w_3x_3)}\,\hbox {d}x_1\hbox {d}x_2\hbox {d}x_3\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}} e^{\frac{-|(x_1-b_1)|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,(w_1x_1)}\,\hbox {d}x_1\,\int _{{\mathbb {R}}} e^{\frac{-|(x_2-b_2)|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,(w_2x_2)}\,\hbox {d}x_2\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}} e^{\frac{-|(x_1-b_1)|^2}{2a^2}}\, e^{-\mathbf{i}_n\,(w_1x_1)}\,\hbox {d}x_1\,\int _{{\mathbb {R}}} e^{\frac{-|(x_2-b_2)|^2}{2a^2}}\, e^{-\mathbf{i}_n\,(w_2x_2)}\,\hbox {d}x_2\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{{\mathbb {R}}} e^{-\frac{y_1^2}{2a^2\lambda ^2}}\cdot e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\,\int _{{\mathbb {R}}} e^{-\frac{y_2^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_2}\,\hbox {d}y_2\cdot e^{-\mathbf{i}_n\,(w_1b_1+w_2b_2)}\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{{\mathbb {R}}}\, e^{-\frac{y_1^2}{2a^2}}\cdot e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\,\int _{{\mathbb {R}}} e^{-\frac{y_2^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_2}\,\hbox {d}y_2\cdot e^{-\mathbf{i}_n\,(w_1b_1+w_2b_2)}\nonumber \\&\quad = \left( \dfrac{1}{\lambda ^{2}\,a^{3/2}}\,2\pi a^2{\lambda }^2 e^{-\frac{(w_1^2+w_2^2)a^2{\lambda }^2}{2}} \,-\dfrac{1}{a^{3/2}}\,2\pi a^2 e^{-\frac{(w_1^2+w_2^2)a^2}{2}}\right) e^{-\mathbf{i}_n(w_1b_1+w_2b_2)}\,\nonumber \\&\quad = 2\pi \sqrt{a}\left( \,e^{-\frac{(w_1^2+w_2^2)a^2{\lambda }^2}{2}} \,-\,e^{-\frac{(w_1^2+w_2^2)a^2}{2}}\right) e^{-\mathbf{i}_n(w_1b_1+w_2b_2)}, \end{aligned}$$

(5.7)

which vanishes as $(w_1,w_2)\rightarrow \infty$. The Clifford-valued wave-packet transform of $f_2$ for different values of a, $\mathbf{b}$ and $\lambda$ is listed in Table 2.

Case-III: We now consider the semi-infinite rod $f_3$ laying along the positive $x_1$-axis

$$\begin{aligned} f_{ 3}(x_1,x_2,x_3)=\left\{ \begin{array}{ccc} \delta (x_2)\,\delta (x_3),&{}\quad b_2=b_3=0, \,\,w_2=w_3=0 \\ \\ 0, &{}\quad x_1\in (-\infty ,0). \end{array}\right. \end{aligned}$$

(5.8)

Fig. 2

Clifford-valued wave-packet transform of $f_1$ for different values of $b_1$

Table 2 Clifford-valued wave-packet transform of $f_{2}$ for different values of scaling parameter a, translation parameter $\mathbf{b}$ and $\lambda$

Then, the Clifford-valued wave-packet transform (5.2) becomes

$$\begin{aligned}&{\mathcal {WP}}_{\psi }[f_{3}](a,\theta ,\mathbf{b},\mathbf{w})\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{0}^{\infty } e^{-\frac{|x_1-b_1|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\,-\dfrac{1}{a^{3/2}}\int _{0}^{\infty }\, e^{-\frac{|x_1-b_1|^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{-b_1}^{\infty }\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{-b_1}^{\infty }\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad = \dfrac{1}{a^{3/2}}\left( \dfrac{1}{\lambda ^{2}}\int _{-b_1}^{\infty }\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\,-\,\int _{-b_1}^{\infty }\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\right) \cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad = \dfrac{1}{a^{3/2}}\left( \dfrac{1}{\lambda ^{2}}\int _{-b_1}^{0}\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\,+\,\int _{0}^{\infty }\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\right. \nonumber \\&\qquad \left. -\,\int _{-b_1}^{0}\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\,-\,\int _{0}^{\infty }\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\right) \cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad =\dfrac{1}{a^{3/2}}\left\{ \dfrac{1}{\lambda ^2}\left( -\sqrt{\dfrac{\pi }{2}} \,a\lambda \,\,e^{\frac{-(w_1a\lambda )^2}{2}}\left( \text {erf}\left( \dfrac{-b_1+\mathbf{i}_nw_1a^2\lambda ^2}{a\lambda \sqrt{2}}\right) +\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a\lambda }{\sqrt{2}}\right) \right) \right. \right. \nonumber \\&\qquad \left. +\sqrt{\dfrac{\pi }{2}}\,a\lambda \,e^{\frac{-(w_1a\lambda )^2}{2}}\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a\lambda }{\sqrt{2}}+1\right) \right) \nonumber \\&\qquad +\sqrt{\dfrac{\pi }{2}}\,a\,e^{\frac{-(w_1a)^2}{2}}\left( \text {erf}\left( \dfrac{-b_1+\mathbf{i}_nw_1a^2}{a\sqrt{2}}\right) +\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a}{\sqrt{2}}\right) \right) \nonumber \\&\quad \qquad \qquad \left. -\sqrt{\dfrac{\pi }{2}}\,a\,e^{\frac{-(w_1a)^2}{2}}\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a}{\sqrt{2}}+1\right) \right\} \cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad =\sqrt{\dfrac{\pi }{2a}}\left\{ \dfrac{-1}{\lambda }\,e^{\frac{-(w_1a\lambda )^2}{2}} \left( \text {erf}\left( \dfrac{-b_1+\mathbf{i}_nw_1a^2\lambda ^2}{a\lambda \sqrt{2}}\right) +\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a\lambda }{\sqrt{2}}\right) \right. \right. \nonumber \\&\qquad \left. -\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a\lambda }{\sqrt{2}}+1\right) \right) +\,e^{\frac{-w_1^2a^2}{2}}\left( \text {erf}\left( \dfrac{-b_1+\mathbf{i}_nw_1a^2}{a\sqrt{2}}\right) +\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a}{\sqrt{2}}\right) \right. \nonumber \\&\qquad \left. \left. -\text {erf}\left( \dfrac{-\mathbf{i}_nw_1a}{\sqrt{2}}+1\right) \right) \right\} \cdot e^{-\mathbf{i}_n\,w_1b_1}, \end{aligned}$$

(5.9)

where $\text {erf}(\cdot )$ is error function For different choices of $a, \mathbf{b}$ and $\lambda$, the corresponding forms of expression (5.9) are given in Table 3.

Fig. 3

Clifford-valued wave-packet transform of $f_4$ for different values of a and $\lambda$

Table 3 Clifford-valued wave packet transformations a signal $f_{3}$, for particular choices of $a,\,\lambda ,\,$ and $b_1$

Table 4 Clifford-valued wave-packet transform of $f_{4}$ for different values of $a,\,\lambda ,$ and $w_1$

Case-IV: Finally, we consider a rod $f_4$ of finite length laying along the interval $[-1,1]$ on the $x_1$-axis

$$\begin{aligned} f_{4}(x_1,x_2,x_3)=\left\{ \begin{array}{lll} \delta (x_2)\,\delta (x_3),&{}\quad b_2=b_3=0, \,\,w_2=w_3=0 \\ 0, &{}\quad x_1\in (-\infty ,1)\cap (1,\infty ). \end{array}\right. \end{aligned}$$

(5.10)

Consequently, the Clifford-valued wave-packet transform (5.2) yields

$$\begin{aligned}&{\mathcal {WP}}_{\psi }[f_{4}](a,\theta ,\mathbf{b},\mathbf{w})\nonumber \\&\quad =\dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{-1}^{1}\, e^{-\frac{|x_1-b_1|^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\,-\dfrac{1}{a^{3/2}}\int _{-1}^{1}\, e^{-\frac{|x_1-b_1|^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1x_1}\,\hbox {d}x_1\nonumber \\&\quad = \dfrac{1}{\lambda ^{2}\,a^{3/2}}\int _{-(1+b_1)}^{1-b_1}\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\,\nonumber \\&\qquad -\dfrac{1}{a^{3/2}}\int _{-(1+b_1)}^{1-b_1}\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\quad = \dfrac{1}{a^{3/2}}\left( \dfrac{1}{\lambda ^{2}}\int _{-(1+b_1)}^{1-b_1}\, e^{-\frac{y_1^2}{2a^2\lambda ^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1-\int _{-(1+b_1)}^{1-b_1}\, e^{-\frac{y_1^2}{2a^2}}\, e^{-\mathbf{i}_n\,w_1y_1}\,\hbox {d}y_1\right) \cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\qquad = \dfrac{1}{a^{3/2}}\left\{ \dfrac{a}{\lambda }\sqrt{\dfrac{\pi }{2}}\,e^{\frac{-(w_1a\lambda )^2}{2}} \left( \text {erf}\left( \dfrac{1-b_1-\mathbf{i}_n w_1 a^2\lambda ^2}{a\lambda \sqrt{2}}\right) +\text {erf}\left( \dfrac{-(1+b_1)-\mathbf{i}_n w_1 a^2\lambda ^2}{a\lambda \sqrt{2}}\right) \right) -a\right. \nonumber \\&\left. \qquad \,\sqrt{\dfrac{\pi }{2}}\,e^{\frac{-(w_1a)^2}{2}}\left( \text {erf}\left( \dfrac{1-b_1-\mathbf{i}_n w_1 a^2}{a\sqrt{2}}\right) +\text {erf}\left( \dfrac{-(1+b_1)-\mathbf{i}_n w_1 a^2}{a\sqrt{2}}\right) \right) \right\} \cdot e^{-\mathbf{i}_n\,w_1b_1}\nonumber \\&\qquad = \sqrt{\dfrac{\pi }{2a}}\left\{ \dfrac{1}{\lambda }\,e^{\frac{-(w_1a\lambda )^2}{2}} \left( \text {erf}\left( \dfrac{1-b_1-\mathbf{i}_n w_1 a^2\lambda ^2}{a\lambda \sqrt{2}}\right) +\text {erf}\left( \dfrac{-(1+b_1)-\mathbf{i}_n w_1 a^2\lambda ^2}{a\lambda \sqrt{2}}\right) \right) \right. \nonumber \\&\qquad \left. -\,e^{\frac{-(w_1a)^2}{2}}\left( \text {erf}\left( \dfrac{1-b_1-\mathbf{i}_n w_1 a^2}{a\sqrt{2}}\right) +\text {erf}\left( \dfrac{-(1+b_1)-\mathbf{i}_n w_1 a^2}{a\sqrt{2}}\right) \right) \right\} \cdot e^{-\mathbf{i}_n\,w_1b_1}. \end{aligned}$$

(5.11)

From (5.11), it is quite evident that the Clifford-valued wave-packet transform vanishes as $w_1\rightarrow \infty$. The numerical values of the transform for different values of $a,\,\lambda$ and $\omega _1$ are tabulated in Table 4 and depicted in Fig. 3.

Form the above cases, we conclude that the Clifford-valued wave packet transform indeed shows all the characteristics needed to qualifying it as an efficient tool for a pointwise analysis and direction analysis of signals in three dimensions.

6 Conclusion

To represent Clifford-valued signals more efficiently, we proposed a novel transform called Clifford-valued wave-packet transform in the context of higher-dimensional time-frequency analysis. Firstly, we studied some fundamental properties of the Clifford-valued wave-packet transform by means of Clifford Fourier transform. Secondly, we established some analogues of the classical Heisenberg–Pauli–Weyl, logarithmic and Pitt’s inequalities for the Clifford-valued wave-packet transform. Finally, we broaden the scope of the study by showing that the proposed transform is quite efficient for detecting point-wise singularity and direction analysis of benchmark signals. It is hoped that the proposed transform can be very handy in multi-dimensional color video processing, crystallography, aerospace engineering, oil exploration and for the solution of several types of differential equations.