Characterization of Frequency Domains of Bandlimited Frame Multiresolution Analysis

Due to resilience to background noise, stability of sparse reconstruction, and the ability to capture local time-frequency information, framelets are becoming a powerful tool in applied analysis and data science. Let ${\left\{h_n\right\}}_1^{\infty }$ be a sequence in $L^2\left({\mathbb{R}}^d\right)$. If there exist $A,B>0$ such that then, ${\left\{h_n\right\}}_1^{\infty }$ is called a frame for $L^2\left({\mathbb{R}}^d\right)$ with bounds A and B [1,2]. Let ${\left\{{\psi }_{\mathit{\mu }}\right\}}_1^l\subset L^2\left({\mathbb{R}}^d\right)$ and If the affine system $\left\{{\psi }_{\mathit{\mu },j,k}\right\}$ is a frame for $L^2\left({\mathbb{R}}^d\right)$, then, the set ${\left\{{\psi }_{\mathit{\mu }}\right\}}_1^l$ is called a framelet [1,2].

The well-known approach to construct framelets is through frame multiresolution analysis (FMRA) [3,4,5]. In 1998, Benedetto and Li [3] introduced the theory of one-dimensional FMRA, which is a fundamental concept in framelet theory. In 2004, Mu et al. [4] further established the theory of high-dimensional FMRA. The structure of FMRA is as follows:

Let ${\left\{V_m\right\}}_{m\in \mathbb{Z}}$ be a sequence of subspaces of $L^2\left({\mathbb{R}}^d\right)$ such that

(i) $V_m\subset V_{m+1}(m\in \mathbb{Z}),{{\displaystyle \bigcup ^{‾}}}_{m\in \mathbb{Z}}{V}_m=L^2\left({\mathbb{R}}^d\right),{\bigcap }_m{V}_m=\left\{0\right\}$;

(ii) $f\left(\mathbf{t}\right)\in V_m$ if and only if $f\left(2t\right)\in V_{m+1}(m\in \mathbb{Z})$;

(iii) there exists a $\phi \left(\mathbf{t}\right)\in V_0$ such that ${\left\{\phi \left(\mathbf{t}-\mathbf{n}\right)\right\}}_{\mathbf{n}\in {\mathbb{Z}}^d}$ is a frame for $V_0$.

Then $\left\{V_m\right\}$ is called a frame multiresolution analysis (FMRA) and $\phi$ is called a frame scaling function.

If the Fourier transform of a frame scaling function $\phi$ is compactly supported (i.e., $\mathrm{supp}\widehat{\phi }=G$), then $\phi$ is said to be bandlimited. Furthermore, $f\in V_0$, since ${\left\{\phi \left(\mathbf{t}-\mathbf{n}\right)\right\}}_{\mathbf{n}\in {\mathbb{Z}}^d}$ is a frame for $V_0$. When f is expanded into a frame series, $f\left(\mathbf{t}\right)={\sum }_{\mathbf{n}}{d}_{\mathbf{n}}\phi \left(\mathbf{t}-\mathbf{n}\right)$; then, taking Fourier transform of both sides, it follows that It means that $f\in V_0$ is bandlimited. Since $\phi \in V_0$, the support of frequency domain of any function in $V_0$ is contained in G or equal to G. Therefore, such an FMRA is said to be a bandlimited FMRA with the support of frequency domain G.

Various framelets with nice properties are constructed by FMRAs [4,5,6,7,8,9,10]. Mu et al. [4] and Zhang [5] showed that the number of generators in framelets associated with FMRA is determined completely by the frequency domain of FMRAs. Zhang [6] discussed the convergence of a framelet series derived from FMRAs. Chui and He [7], Antolin and Zalin [8], and Atreasa et al. [9] constructed many examples on compactly supported framelets when FMRAs extend into general MRAs. Zhang [10] extended framelets derived by FMRAs into the case of framelet packets with finer time-frequency resolution.

Frequency domain of bandlimited FMRAs plays a key role when derived framelets are applied into narrowband signal processing and data analysis. A suitable frequency domain of bandlimited FMRAs can mitigate the effects of narrowband noises well, so the perfect reconstruction filter bank associated with a bandlimited FMRA can achieve quantization noise reduction simultaneously with reconstruction of a given narrowband signal [3]. This is a unique and key advantage of framelets over traditional wavelets [3]. However, until now, the structure of frequency domain of bandlimited FMRAs has not been investigated.

In this study, the frequency domain of bandlimited FMRA will be characterized. In Section 2, the necessary condition for G to be the support of frequency domain of bandlimited FMRA is given first. In Section 3, in order to obtain sufficient condition, a fine partition of any bounded region G is presented, satisfying

$G\subset 2G,$${\bigcup }_m{2}^{m}G\cong {\mathbb{R}}^d,$ and $\left(G\backslash \frac{G}{2}\right)\bigcap \left(\frac{G}{2}+2\pi \mathit{\nu }\right)\cong \varnothing (\mathit{\nu }\in {\mathbb{Z}}^d)$.

Based on this partition, in Section 4, a bandlimited FMRA with the support of frequency domain G is directly constructed. With the help of it, for any given narrowband signal, one can choose the most suitable bandlimited FMRA to analyze it and, at the same time, mitigate noise effects.

Let $\left\{V_m\right\}$ be an FMRA and $\phi$ be the corresponding frame scaling function. Since $\phi \in V_0$, $\frac{1}{2^d}\phi \left(\frac{t}{2}\right)\in V_1\subset V_0$, and ${\left\{\phi \left(\mathbf{t}-\mathbf{n}\right)\right\}}_{\mathbf{n}\in {\mathbb{Z}}^d}$ is a frame for $V_0$, $\frac{1}{2^d}\phi \left(\frac{\mathbf{t}}{2}\right)$ can be expanded into a frame series, $\frac{1}{2^d}\phi \left(\frac{\mathbf{t}}{2}\right)={\sum }_{\mathbf{n}}{c}_{\mathbf{n}}\phi \left(\mathbf{t}-\mathbf{n}\right)$. Take the Fourier transform of both sides, it follows that where $H\left(\mathit{\omega }\right)={\sum }_{\mathbf{n}}{c}_{\mathbf{n}}{e}^{-i(\mathbf{n}·\mathit{\omega })}$ is a $2\pi -$periodic function (i.e., $H\in L^2\left(T^d\right)$). From this, we have

In this section, a necessary condition for the support of the frequency domain of bandlimited FMRA is given as follows.

Let G be a bounded closed set in ${\mathbb{R}}^d$, satisfying Theorem 1(i)–(iii). A fine partition of G will be given in this section. This partition will be used to further prove that there exists a bandlimited FMRA with the support of frequency domain G in Section 4, i.e., the converse of Theorem 1 holds.

Some notations are needed as follows: For $m=0,1,...$, Noticing that G is bounded, one can choose a $k\in {\mathbb{Z}}_{+}$ such that

Next, a decomposition of each $G_m^{*}(m\ge 0)$ is given. For arbitrarily, finitely, many distinct points ${\mathit{\nu }}_0,...,{\mathit{\nu }}_r(\mathbf{0}\in {\left\{{\mathit{\nu }}_l\right\}}_{l=0,...,r}\subset {\mathbb{Z}}^d)$, define a point set $F_{{\mathit{\nu }}_0,...,{\mathit{\nu }}_r}^m$:

Let $\mathit{\omega }\in G_m^{*}$. Note that $G_m^{*}\subset G$. Since G is bounded, there only exist finitely many $\mathit{\nu }\in {\mathbb{Z}}^d$ such that $\mathit{\omega }+\pi \mathit{\nu }\in G_m^{*}$. So, $\mathit{\omega }$ must lie in some $F_{{\mathit{\nu }}_0,...,{\mathit{\nu }}_r}^m$, and so, and the right-hand side of (6) is a disjoint union. Let By (1) and $G_m^{*}\subset G$, if some ${\mathit{\nu }}_l$ in ${\mathit{\nu }}_0,...,{\mathit{\nu }}_r(\mathbf{0}\in {\{{\mathit{\nu }}_l\}}_{l=0,...,r}\subset {\mathbb{Z}}^d)$ satisfies $|{\mathit{\nu }}_l|>2^{k+1}\sqrt{d}$, then, for any $\mathit{\omega }\in G_m$, it follows that $\mathit{\omega }+\pi {\mathit{\nu }}_l\notin G_m$. Again, by (5), it follows that $F_{{\mathit{\nu }}_0,...,{\mathit{\nu }}_r}^m=\varnothing$. Therefore, $P^m$ only consists of finitely many point sets.

In this section, the converse of Theorem 1 will be proved:

The process is divided into four steps.

Step 1. Define $\widehat{\phi }\left(\mathit{\omega }\right)=0(\mathit{\omega }\notin G)$.

Step 2. Define $\widehat{\phi }\left(\mathit{\omega }\right)$ on $G_k^{*}$ and $H\left(\mathit{\omega }\right)$ on $E_{k+1}^{*}$, in detail;

Define $\widehat{\phi }\left(\mathit{\omega }\right)=1(\mathit{\omega }\in G_k)$. For $\mathit{\omega }\in G_k$, by Lemma 1(b), it follows that $\mathit{\omega }+2\pi \mathit{\nu }\notin G(\mathit{\nu }\ne 0)$. So,

Furthermore, $\widehat{\phi }$ has been defined on $E_k$ and i.e., Formula (**) holds for $\mathit{\omega }\in G_k$.

Define $H\left(\mathit{\omega }\right)=1(\mathit{\omega }\in E_{k+1})$. If $\mathit{\omega }\in G_{k+1}$, then, $2\mathit{\omega }\in G_k$. By $G_{k+1}\subset G_k$ and (8), $\widehat{\phi }\left(2\mathit{\omega }\right)=\widehat{\phi }\left(\mathit{\omega }\right)=1$. Hence, Formula (*) holds for $\mathit{\omega }\in G_{k+1}$. If $\mathit{\omega }\in G_{k+1}+2\pi \mathit{\nu }(\mathit{\nu }\ne \mathbf{0})$, then, $2\mathit{\omega }\in G_k+4\pi \mathit{\nu }(\mathit{\nu }\ne \mathbf{0})$. From this and (8), $\widehat{\phi }\left(2\mathit{\omega }\right)=\widehat{\phi }\left(\mathit{\omega }\right)=0$, Clearly, Formula (*) holds for $\mathit{\omega }\in G_{k+1}+2\pi \mathit{\nu }(\mathit{\nu }\ne \mathbf{0})$. Thus, Formula (*) holds for $\mathit{\omega }\in E_{k+1}$.

Step 3. Based on Step 2, the idea of mathematics induction will be used. For $k\ge m>0$, assume that $\widehat{\phi }\left(\mathit{\omega }\right)$ is defined on $G_m^{*}$ and $H\left(\mathit{\omega }\right)$ is defined on $E_{m+1}^{*}$ such that Formula (*) holds on $E_{m+1}^{*}$, Formula (**) holds on $G_m^{*}$, and where $C_m,D_m,{\tilde{C}}_m$, and ${\tilde{D}}_m$ are constants.

Define $H\left(\mathit{\omega }\right)$ on $G_m^{*}$. By Lemma 3, one only needs to define $H\left(\mathit{\omega }\right)$ on each $A_j^m+\pi {\mathit{\nu }}_l^{\left(j\right)}$.

Since $\widehat{\phi }\left(\mathit{\omega }\right)$ has been defined on $G_m^{*}$ and $C_m\le \widehat{\phi }\left(\mathit{\omega }\right)\le D_m$. Again, by Lemma 3: so, for each $\mathit{\omega }\in A_j^m$ and $l=0,...,r_j$, the values of $\widehat{\phi }(\mathit{\omega }+\pi {\mathit{\nu }}_l^{\left(j\right)})(j=1,...,\lambda )$ have been defined and $C_m\le \widehat{\phi }(\mathit{\omega }+\pi {\mathit{\nu }}_l^{\left(j\right)})\le D_m$. Write Noticing that the point sets $A_j^m+\pi {\mathit{\alpha }}_l^{\left(j\right)}(l=0,1,...,r_j)$ are disjoint, one can define $H(\mathit{\omega }+\pi {\mathit{\alpha }}_l^{\left(j\right)})$ on $A_j^m$ such that for $\mathit{\omega }\in A_j^m$, where $b_l^{\left(j\right)}\left(\mathit{\omega }\right)={|\widehat{\phi }(\mathit{\omega }+\pi {\mathit{\nu }}_l^{\left(j\right)})|}^2$ and where ${\tilde{C}}_{m-1}^{\left(j\right)}$ and ${\tilde{D}}_{m-1}^{\left(j\right)}$ are constants.

By (9), for $j=1,...,\lambda$, define Due to the Lemma 3, $H\left(\mathit{\omega }\right)$ is defined on $G_m^{*}$ and where ${\tilde{C}}_{m-1}={min}_{1\le j\le \lambda }\{{\tilde{C}}_{m-1}^{\left(j\right)}\}$ and ${\tilde{D}}_{m-1}={max}_{1\le j\le \lambda }\{{\tilde{D}}_{m-1}^{\left(j\right)}\}$.

Below, we will prove that if $\mathit{\omega }\in G_m^{*}$ and $\mathit{\omega }+2\pi \tilde{\mathit{\nu }}\in G_m^{*}$ for some $\tilde{\mathit{\nu }}\in {\mathbb{Z}}^d$, then $H(\mathit{\omega }+2\pi \tilde{\mathit{\nu }})=H\left(\mathit{\omega }\right)$.

By $\mathit{\omega }\in G_m^{*}$ and Lemma 3, there exist j and n such that where ${\mathit{\omega }}_0\in A_j^m$. Since $\mathit{\omega }+2\pi \tilde{\mathit{\nu }}={\mathit{\omega }}_0+\pi (2\tilde{\mathit{\nu }}+{\mathit{\nu }}_n^{\left(j\right)})\in G_m^{*}$ and ${\mathit{\omega }}_0\in A_j^m$, there is some $s(0\le s\le r_j)$ such that By (12)–(14), it follows that By uniqueness of decomposition in (9), it follows that ${\mathit{\alpha }}_n^{\left(j\right)}={\mathit{\alpha }}_s^{\left(j\right)}$. It means that $H\left(\mathit{\omega }\right)=H(\mathit{\omega }+2\pi \tilde{\mathit{\nu }})$ when $\mathit{\omega }\in G_m^{*}$ and $\mathit{\omega }+2\pi \tilde{\mathit{\nu }}\in G_m^{*}$ for some $\tilde{\mathit{\nu }}\in {\mathbb{Z}}^d$.

Based on this fact, one can further define $H(\mathit{\omega }+2\pi \mathit{\nu })=H\left(\mathit{\omega }\right)(\mathit{\omega }\in G_m^{*},\mathit{\nu }\in {\mathbb{Z}}^d)$. So, $H\left(\mathit{\omega }\right)$ is well-defined on $E_m^{*}$ and ${\tilde{C}}_{m-1}\le H\left(\mathit{\omega }\right)\le {\tilde{D}}_{m-1}(\mathit{\omega }\in E_m^{*})$.

Now, define $\widehat{\phi }\left(\mathit{\omega }\right)$ on $G_{m-1}^{*}$. Let Then, $\widehat{\phi }\left(\mathit{\omega }\right)$ is defined on $G_{m-1}^{*}(G_{m-1}^{*}=2G_m^{*})$ and By Step 1 and Lemma 1(a), it follows that $\widehat{\phi }\left(\mathit{\omega }\right)=0(\mathit{\omega }\in {\mathit{E}}_{\mathit{m}-\mathit{1}}^{*}\backslash {\mathit{G}}_{\mathit{m}-\mathit{1}}^{*})$. So, $\widehat{\phi }\left(\mathit{\omega }\right)$ is defined on $E_{m-1}^{*}$.

This will prove that Formula (*) holds on $E_m^{*}$: If $\mathit{\omega }\in G_m^{*}$, by (16), Formula (*) holds. If $\mathit{\omega }\notin G_m^{*}$, then, $\mathit{\omega }\in E_m^{*}\backslash G_m^{*}$. By Lemma 1(a), it follows that $\mathit{\omega }\notin G$. By $G\subset 2G$, we get $2\mathit{\omega }\notin G$. So, $\widehat{\phi }\left(2\mathit{\omega }\right)=\widehat{\phi }\left(\mathit{\omega }\right)=0$. Formula (*) also holds.

Finally, it will prove that Formula (**) holds on $G_{m-1}^{*}$.

Let $\mathit{\omega }\in A_j^m=F_{{\mathit{\nu }}_0^{\left(j\right)},...,{\mathit{\nu }}_{r_j}^{\left(j\right)}}^m$. Then, By $2G_m^{*}=G_{m-1}^{*}$, it follows that $2\mathit{\omega }\in G_{m-1}^{*}$ and then, $2\mathit{\omega }+2\pi \mathit{\nu }\in E_{m-1}^{*}(\mathit{\nu }\in {\mathbb{Z}}^d)$. Again, by (17), it follows that $2\mathit{\omega }+2\pi \mathit{\nu }\notin G_{m-1}^{*}(\mathit{\nu }\notin {\{{\mathit{\nu }}_l^{\left(j\right)}\}}_{l=0,...,r_j})$. So, $2\mathit{\omega }+2\pi \mathit{\nu }\in E_{m-1}^{*}\backslash G_{m-1}^{*}(\mathit{\nu }\notin {\left\{{\mathit{\nu }}_l\right\}}_{l=0,...,r_j})$. By Lemma 1(a), it means that $2\mathit{\omega }+2\pi \mathit{\nu }\notin G(\mathit{\nu }\notin {\left\{{\mathit{\nu }}_l\right\}}_{l=0,...,r_j})$. So, $\widehat{\phi }(2\mathit{\omega }+2\pi \mathit{\nu })=0(\mathit{\nu }\notin {\{{\mathit{\nu }}_l^{\left(j\right)}\}}_{l=0,...,r_j})$, and so, By (9) and (16), it follows that Again by (18) and (10), it follows that for $\mathit{\omega }\in A_j^m$, Similar to the above process, it follows that (19) also holds for $\mathit{\omega }\in A_j^m+\pi {\mathit{\nu }}_l^{\left(j\right)}$. Again, by Lemma 3, (19) holds for $\mathit{\omega }\in G_m^{*}$. Noticing that $2G_m^{*}=G_{m-1}^{*}$, it means that ${\sum }_{\mathit{\nu }}{\left|\widehat{\phi }(\mathit{\omega }+2\pi \mathit{\nu })\right|}^2=1$ holds on $G_{m-1}^{*}$.

Step 4. From Steps 1–3, $\widehat{\phi }(\mathit{\omega })$ is defined on G and $H(\mathit{\omega })$ is defined on $\frac{1}{2}G+2\pi {\mathbb{Z}}^d$. Now, define $H(\mathit{\omega })=0(\mathit{\omega }\in G_0^{*}+2\pi {\mathbb{Z}}^d)$ and $H(\mathit{\omega })=0(\mathit{\omega }\in {\mathbb{R}}^d\backslash \mathsf{\Omega })$. By assumption (iii) in Theorem 2, it follows that It means that H has been well-defined on ${\mathbb{R}}^d$ and $H\in L^{\infty }(T^d)$.

From Steps 1–3, Formula (*) holds on $\frac{G}{2}+2\pi {\mathbb{Z}}^d$. For $\mathit{\omega }\in G_0^{*}+2\pi {\mathbb{Z}}^d$, it follows that $2\mathit{\omega }\in 2G_0^{*}+4\pi {\mathbb{Z}}^d$. By (20), it follows that $(2G_0^{*}+4\pi {\mathbb{Z}}^d)\bigcap G\cong \varnothing$. Hence, $\widehat{\phi }(2\mathit{\omega })=0$. Again, by $H(\mathit{\omega })=0(\mathit{\omega }\in G_0^{*}+2\pi {\mathbb{Z}}^d)$, Formula (*) holds also on $G_0^{*}+2\pi {\mathbb{Z}}^d$. Thus, by (20), Formula (*) holds for $\mathit{\omega }\in \Omega$. When $\mathit{\omega }\notin \Omega$, by assumption (i) in Theorem 2, it follows that $2\mathit{\omega }\notin \Omega$, which means that Formula (*) also holds for $\mathit{\omega }\notin \Omega$. Therefore, Formula (*) holds on ${\mathbb{R}}^d$.

From Steps 1–3, Formula (**) holds on G. Since the sum ${\sum }_{\mathit{\nu }}{\left|\widehat{\phi }(\mathit{\omega }+2\pi \mathit{\nu })\right|}^2$ is a $2\pi {\mathbb{Z}}^d-$periodic function, ${\sum }_{\mathit{\nu }}{\left|\widehat{\phi }(\mathit{\omega }+2\pi \mathit{\nu })\right|}^2=1$ holds on $G+2\pi {\mathbb{Z}}^d=\Omega$. If $\mathit{\omega }\notin \Omega$, then, for any $\mathit{\nu }\in {\mathbb{Z}}^d$, it follows that $\mathit{\omega }+2\pi \mathit{\nu }\notin G$. So, ${\sum }_{\mathit{\nu }}{\left|\widehat{\phi }(\mathit{\omega }+2\pi \mathit{\nu })\right|}^2=0$ on ${\mathbb{R}}^d\backslash \Omega$, and so, ${\sum }_{\mathit{\nu }}{\left|\widehat{\phi }(\mathit{\omega }+2\pi \mathit{\nu })\right|}^2={\chi }_{\Omega }\left(\mathit{\omega }\right)(\mathit{\omega }\in {\mathbb{R}}^d)$, i.e., Formula (**) holds on ${\mathbb{R}}^d$.

Up to now, the constructed $\phi \in L^2\left({\mathbb{R}}^d\right)$ and $H\in L^{\infty }\left(T^d\right)$ satisfy Formulas (*) and (**). Let $V_m=\overline{\mathrm{span}}\{\phi (2^m\mathbf{t}-\mathbf{n}),\mathbf{n}\in {\mathbb{Z}}^d\}(m\in \mathbb{Z})$. Since H is a $2\pi {\mathbb{Z}}^d-$periodic bounded function, $H\left(\mathit{\omega }\right)$ can be expanded into a Fourier series: $H\left(\mathit{\omega }\right)={\sum }_{\mathbf{k}}{c}_{\mathbf{k}}{e}^{i(\mathbf{k}·\mathit{\omega })}$. By Formula (*), it follows that Taking the inverse Fourier transform on both sides, we have and so, $\phi (2^m\mathbf{t}-\mathbf{n})=2^d{\sum }_{\mathbf{k}}{c}_{\mathbf{k}}\phi (2^{m+1}\mathbf{t}-(2\mathbf{n}+\mathbf{k}))\subset V_{m+1}$. This implies that $V_m\subset V_{m+1}$.

By a known results in [1-2], Formula (**) implies the system ${\left\{\phi (·-\mathbf{n})\right\}}_{\mathbf{n}\in {\mathbb{Z}}^d}$ is a frame for $V_0=\overline{\mathrm{span}}\{\phi (·-\mathbf{n}),\mathbf{n}\in {\mathbb{Z}}^d\}$. For any $f\in V_m$, it follows that $f(2^{-m}·)\in V_0$. It can be expanded into a frame series with respect to ${\left\{\phi (·-\mathbf{n})\right\}}_{\mathbf{n}}$: Taking the Fourier transform on both sides, we have i.e., $\widehat{f}\left(\mathit{\omega }\right)=\tau \left(\mathit{\omega }\right)\widehat{\phi }\left(2^{-m}\mathit{\omega }\right)$, where $\tau \left(\mathit{\omega }\right)=2^{-m}{\sum }_{\mathbf{n}}{d}_{\mathbf{n}}{e}^{-i(\mathbf{n}·\mathit{\omega })/2^m}$. So, $\mathrm{supp}\widehat{f}\subset \mathrm{supp}\widehat{\phi }(2^{-m}·)$ for $f\in V_m$. Since $\phi (2^m·)\in V_m$, it implies that By Assumption (ii), it follows that By a known result in [1,2], we have ${{\displaystyle \bigcup ^{‾}}}_m{V}_m=L^2({\mathbb{R}}^d)$. Therefore, $\{V_m\}$ is a bandlimited FMRA with frequency domain G.