The DLCWT of x(t) with parameter M=(A,B,C,D) can be defined as
$$\begin{array}{*{20}l} W_{x}^{M}(j,k)&= \left\langle {x(t),{\psi_{M,j,k}}(t)} \right\rangle \\ &= {e^{ i\frac{A}{{2B}}{k^{2}}}}\int_{ \infty }^{+ \infty} {x(t)\psi_{j,k}^{*}(t){e^{i\frac{A}{{2B}}{t^{2}}}}dt} \\ &= {2^{ j/2}}{e^{ i\frac{A}{{2B}}{k^{2}}}} \\ &\quad \times \int_{ \infty}^{+ \infty} {x(t){\psi^{*}}\left({2^{ j}}t  k\right){e^{i\frac{A}{{2B}}{t^{2}}}}dt}, \end{array} $$
(18)
where \(j \in \mathbb {Z}\) and \(k \in \mathbb {Z}\).
4.1 Multiresolution approximation associated with LCT
The theory of multiresolution approximation associated with LCT is first proposed here since it sets the ground for the DLCWT and the construction of orthogonal linear canonical wavelets. According to the definition of multiresolution approximation in [16], we give the following definition.
Definition 1
A sequence of closet subspaces \(\left \{ V_{j}^{M}\right \}, j \in \mathbb {Z}\) of \({ L^{2}}(\mathbb {R})\) is a multiresolution approximation associated with LCT if the following six properties are satisfied:

1)
\(\forall (j,k) \in {\mathbb {Z}^{2}}\),
\(x(t) \in V_{j}^{M} \Leftrightarrow x\left (t  {2^{j}}k\right){e^{ i\frac {A}{B}{2^{j}}k(t  {2^{j}}k)}} \in V_{j}^{M}\);

2)
\(\forall j \in \mathbb {Z}\), \(V_{j}^{M} \supset V_{j + 1}^{M}\);

3)
\(\forall j \in \mathbb {Z}\), \(x(t) \in V_{j}^{M} \Leftrightarrow x\left (\frac {t}{2}\right){e^{ i\frac {{3A}}{{8B}}{t^{2}}}} \in V_{j + 1}^{M}\);

4)
\(\underset {j \to \infty }{\lim } {V_{j}^{M}} = \underset {j=\infty }{\overset {\infty }{\cap }} {V_{j}^{M}} = \{ 0\} \);

5)
\(\underset {j \to  \infty }{\lim } {V_{j}^{M}} = {\text {Closure}}\left (\underset {{j =  \infty }}{\overset {\infty }{\cup }} {V_{j}^{M}}\right) = {L^{2}}(\mathbb {R})\);

6)
There exists a basic function \(\theta (t) \in V_{0} \subset {L^{2}}(\mathbb {R})\) such that \(\left \{ {\theta _{M,0,k}}(t) = \theta (t  k){e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}}, k \in \mathbb {Z}\right \}\) is a Riesz basis of subspace \(V_{0}^{M}\).
Condition (1) means that \(V_{j}^{M}\) is invariant by any translation proportional to the scale 2^{j} together with modulation. Dilating operation and chirping operation in \(V_{j}^{M}\) enlarge the detail and condition (3) guarantees that it defines an approximation at a coarser resolution 2^{−j−1}. The existence of a Riesz basis of \(V_{j}^{M}\) provides a discretization theorem. The theorem below gives the existence condition of Riesz basis in \(V_{j}^{M}\). The following is detailed proof of condition (3). Let XM′(u) denote the LCT of \(x\left ({\frac {t}{2}} \right){e^{ i\frac {{3A}}{{8B}}{t^{2}}}}\). According to the definition of the LCT, one can obtain
$$\begin{array}{*{20}l} {X'_{M}}(u) & = \int_{ \infty}^{+ \infty} {x\left({\frac{t}{2}} \right){e^{ i\frac{{3A}}{{8B}}{t^{2}}}}{K_{M}}(u,t)dt} \\ & = \frac{1}{{\sqrt {i2\pi B}}} \\ &\quad \times \int_{ \infty}^{+ \infty} {x\left({\frac{t}{2}} \right){e^{i\frac{A}{{2B}}{{\left({\frac{t}{2}} \right)}^{2}}}} {e^{i\left( \frac{1}{B}ut + \frac{D}{{2B}}{u^{2}}\right)}}dt} \\ & = \frac{1}{{\sqrt {i2\pi B}}}{e^{ i\frac{D}{{2B}}3{u^{2}}}} \\ &\quad \times \int\limits_{ \infty}^{+ \infty}{x\left({\frac{t}{2}} \right){e^{i\frac{A}{{2B}}{{\left({\frac{t}{2}} \right)}^{2}}}}{e^{ i\frac{1}{B}2u\frac{t}{2}}}{e^{i\frac{D}{{2B}}{{(2u)}^{2}}}}dt} \end{array} $$
(19)
Replacing \({\frac {t}{2}}\) with t^{′} in (19) results in
$$\begin{array}{*{20}l} {X'_{M}}(u) & = \frac{2}{{\sqrt {i2\pi B} }}{e^{ i\frac{D}{{2B}}3{u^{2}}}} \\ &\quad \times \int_{ \infty}^{+ \infty} {x\left(t' \right){e^{i\frac{A}{{2B}}{{t'}^{2}}}}{e^{ i\frac{1}{B}2ut'}}{e^{i\frac{D}{{2B}}{{(2u)}^{2}}}}dt} \\ & = 2{e^{ i\frac{D}{{2B}}3{u^{2}}}}{X_{M}}(2u) \end{array} $$
(20)
where X_{
M
}(u) denotes the LCT of x(t). Since \(V_{j}^{M} \subset {L^{2}}(\mathbb {R})\) and \(V_{j+1}^{M} \subset {L^{2}}(\mathbb {R})\) denote the subspace of all functions bandlimited to the interval [−2^{−j}πB,+2^{−j}πB] and [−2^{−(j+1)}πB,+2^{−(j+1)}πB] in the LCT domain separately, therefore \(x\left ({\frac {t}{2}} \right){e^{ i\frac {{3A}}{{8B}}{t^{2}}}} \in V_{j+1}^{M}\) according to (20).
Theorem 2
\(\{ {\theta _{M,0,k}}(t), k \in \mathbb {Z}\}\) is a Riesz basis of the subspace \(V_{0}^{M}\) if and only if \(\{ \theta (tk),k \in \mathbb {Z}\}\) forms a Riesz basis of the subspace V_{0} with θ(t)∈V_{0} as the basis function.
Proof
if \(\left \{{\theta _{M,0,k}}(t) = \theta (t  k){e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}}, k \in \mathbb {Z}\right \}\) is a Riesz basis of the subspace \(V_{0}^{M}\), then for \(\forall x(t) \in V_{0}^{M}\), we have
$$ x(t) = \sum\limits_{k \in \mathbb{Z}} {{c_{k}}\theta (t  k){e^{ i\frac{A}{{2B}}\left({t^{2}}  {k^{2}}\right)}}}. $$
(21)
After taking the LCT on both sides of (21), we have
$$\begin{array}{*{20}l} {X_{M}}(u) &= \sum\limits_{k \in \mathbb{Z}} {{c_{k}}{{\mathcal{L}}_{M}}\left[{{c_{k}}\theta (t  k){e^{ i\frac{A}{{2B}}\left({t^{2}}  {k^{2}}\right)}}}\right](u)} \\ &= \frac{1}{{\sqrt {i2\pi B}}} \sum\limits_{k \in \mathbb{Z}} {{c_{k}}{e^{i \frac{D}{2B}u^{2}}}{e^{ i\frac{1}{B}uk}}{e^{i\frac{A}{2B}k^{2}}}{\hat {\theta}}\left(\frac{u}{B}\right)} \\ &= \tilde C_{M}(u){\hat {\theta}}\left(\frac{u}{B}\right), \end{array} $$
(22)
where \(\tilde C_{M}(u)\) denotes the DTLCT of c_{
k
} with a period of 2πB, and \(\hat {\theta }\left (\frac {u}{B}\right)\) denotes the FT of θ(t) with its argument scaled by \(\frac {1}{B}\).
According to the Parseval’s relation associated with LCT,
$$\begin{array}{*{20}l} {\left\ {x(t)} \right\^{2}} &= {\left\ {{X_{M}}(u)} \right\^{2}}\\ &= \int_{ \infty}^{+ \infty} {{{\left {\tilde C_{M}(u)} \right}^{2}}{{\left {{\hat{\theta}}\left(\frac{u}{B}\right)} \right}^{2}}du} \\ &= \int_{0}^{2\pi B} {{{\left {\tilde C_{M}(u)} \right}^{2}}\sum\limits_{k =  \infty}^{+ \infty} {{{\left {{\hat{\theta}}\left(\frac{u}{B} + 2k\pi \right)} \right}^{2}}}du}. \end{array} $$
(23)
Since \(x(t) \in {L^{2}}(\mathbb {R})\), it can be easily obtained that
$$\begin{array}{*{20}l} P{\left\ {x(t)} \right\^{2}} &\le \int_{0}^{2\pi B} {{{\left {\tilde C_{M}(u)} \right}^{2}}du} \\ &= \sum\limits_{k =  \infty}^{+ \infty} {{{\left {{c_{k}}} \right}^{2}}} \le Q{\left\ {x(t)} \right\^{2}} \end{array} $$
(24)
and
$$ \frac{1}{Q} \le \sum\limits_{k =  \infty}^{+ \infty} {{{\left{{\hat{\theta}}\left(\frac{u}{B} + 2k\pi\right)} \right}^{2}} \le \frac{1}{P}}. $$
(25)
On the other hand, if (25) holds, then (24) can be obtained. If x(t)=0, then according to (24), for ∀k,c_{
k
}=0. \(\left \{ r (t  k){e^{ i\frac {A}{B}k\left (t  \frac {k}{2}\right)}}, k \in \mathbb {Z}\right \}\) is therefore linear independent with each other. \(\big \{ {\theta _{M,0,k}}(t) = \theta (t  k) {e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}}, k \in \mathbb {Z}\big \}\) is a Riesz basis of the subspace \(V_{0}^{M}\).
This is to say, \(\left \{{\theta _{M,0,k}}(t) = \theta (t  k){e^{ i\frac {A}{{2B}}\left ({t^{2}}  {k^{2}}\right)}},k \in \mathbb {Z}\right \}\) will be a Riesz basis of the subspace \(V_{0}^{M}\), if and only if there exist constants P>0 and Q>0 such that (25) holds. Considering \(\{ \theta (tk),k \in \mathbb {Z}\}\) as a Riesz basis of the subspace of V_{0}, we can deduce that (25) definitely holds due to the inequation
$$ \frac{1}{Q} \le \sum\limits_{k =  \infty}^{\infty} {{{\left {\hat \theta (u' + 2k\pi)} \right}^{2}} \le \frac{1}{P}} $$
(26)
holds (see Theorem 3.4 [23] for details), where u^{′}=u/B∈[−π,π].
The theorem is proved. □
In particular, the family \(\{ {\theta _{M,0,k}}(t), k \in \mathbb {Z}\}\) is an orthonormal basis of the space \(V_{j}^{M}\) if and only if P=Q=1. Theorem 2 implies that \(V_{j}^{M}\) are actually the chirpmodulated shiftinvariant subspaces of \(L^{2}(\mathbb {R})\), because they are spaces in which the generators are modulated by chirps and then translated by integers [24–26].
The following theorem provides the condition to construct an orthogonal basis of each space \(V_{j}^{M}\) by dilating, translating, and chirping the scaling function ϕ(t)∈V_{0}.
Theorem 3
Define \(\left \{ V_{j}^{M}\right \},j \in \mathbb {Z}\) as a sequence of closet subspaces, and \(\{{\phi _{M,j,k}}(t), j,k \in \mathbb {Z}\} \) as a set of scaling functions. If \(\{\phi _{j,k} (t),j,k \in \mathbb {Z}\} \) is an orthonormal basis of the subspace V_{
j
}, then for all \(j \in \mathbb {Z}\), ϕ_{
M,j,k
} forms an orthonormal basis of subspace \(V_{j}^{M}\).
Proof
First, it is easy to find that \({\phi _{M,j,k}} \in V_{j}^{M}\). From it, we have
$$ {\phi_{M,0,0}} = \sum\limits_{k} {{c_{k}}{\theta_{M,0,k}}(t)}. $$
(27)
Taking the LCT with M=(A,B,C,D) on both sides of (27), we have
$$\begin{array}{@{}rcl@{}} \Phi (u/B) &=& \sum\limits_{k \in Z} {{c_{k}}{e^{i\frac{A}{{2B}}{k^{2}}}}{e^{ i\frac{1}{B}ut}}\hat \theta (u/B)} \\ &=& \tilde E(u/B)\hat \theta (u/B), \end{array} $$
(28)
where \({e_{k}} = {c_{k}}{e^{j\frac {A}{{2B}}{k^{2}}}}\), and \(\tilde E({e^{i\omega }})\) is the DTFT of e_{
k
}. Notice that Φ(u) and \(\hat \theta (u)\) are the FT of ϕ(t) and θ(t), respectively.
If \(\left \{ {\phi _{M,0,k}}(t) = \phi (t  k){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}},k \in \mathbb {Z}\right \} \) forms an orthonormal basis of \(V_{0}^{M}\), according to Theorem 2, we have
$$ \sum\limits_{k =  \infty}^{\infty} {{{\left{\Phi (u/B + 2k\pi)} \right}^{2}}} = 1. $$
(29)
Applying (28) and (29), we can obtain
$$ {\left {\tilde E(u/B)} \right^{2}}\sum\limits_{k =  \infty }^{\infty} {{{\left {\hat \theta (u/B + 2k\pi)} \right}^{2}}} = 1. $$
(30)
As \(\sum \limits _{k =  \infty }^{\infty } {{{\left  {\hat \theta (u/B + 2k\pi)} \right }^{2}}}\) is limited, combining (28) and (30) yields
$$ \Phi (u/B) = \frac{{\hat \theta (u/B)}}{{{{\left[\sum\limits_{k =  \infty }^{\infty} {{{\left {\hat \theta (u/B + 2k\pi)} \right}^{2}}} \right]}^{1/2}}}}. $$
(31)
If \(\{ {2^{ j/2}}\phi ({2^{ j}}t  k),k \in \mathbb {Z} \} \) is an orthonormal basis of the subspace V_{
j
}, then the FT of ϕ(t) definitely makes (31) hold. Therefore, \(\left \{{\phi _{M,0,k}}(t) = \phi (t  k){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}},k \in \mathbb {Z}\right \}\) forms an orthonormal basis of the subspace \(V_{0}^{M}\).
Moreover, it is easy to prove that for \(\forall j,{k_{1}},{k_{2}} \in \mathbb {Z}\),
$$ \left\langle {{\phi_{M,j,{k_{1}}}}(t),{\phi_{M,j,{k_{2}}}}(t)} \right\rangle = \delta ({k_{1}}  {k_{2}}). $$
(32)
The theorem is proved. □
Thus, according to Theorems 2 and 3, one can use the mother wavelet ψ(t)∈W_{0} to construct mother linear canonical wavelet \(\psi _{M}(t) \in W_{0}^{M}\) such that the dilated, translated, and chirpmodulated family
$$ \left\{{\psi_{M,j,k}}(t) = {2^{ j/2}}\psi ({2^{ j}}t  k){e^{ i\frac{A}{{2B}}({t^{2}}  {k^{2}})}},k \in \mathbb{Z}\right\} $$
(33)
is an orthonormal basis of \(W_{j}^{M}\). As \(W_{j}^{M}\) is the orthogonal complement of \(V_{j}^{M}\) in \(V_{j1}^{M}\), i.e.,
$$ W_{j}^{M} \perp V_{j}^{M} $$
(34)
and
$$ V_{j  1}^{M} = V_{j}^{M} \oplus W_{j}^{M}, $$
(35)
the orthogonal projection of input signal x on \(V_{j1}^{M}\) can be decomposed as the sum of orthogonal projections on \(V_{j}^{M}\) and \(W_{j}^{M}\).
4.2 Discrete orthogonal LCWT and its fast algorithm
In this section, we will give the relationship between the DOLCWT and the conjugate mirror filter banks associated with LCT, and the condition to construct the orthonormal linear canonical wavelets. These twochannel filter banks implement a fast computation of DOLCWT which only has O(N) computational complexity for signals of length N.
4.2.1 Relationship between DOLCWT and twochannel filter banks associated with LCT
Since both ψ_{M,j−1,k}(t) and ϕ_{M,j−1,k}(t) form an orthonormal basis for \(W^{M}_{j1}\) and \(V^{M}_{j1}\), we can decompose ϕ_{M,j,0}(t) and ψ_{M,j,0}(t) as
$$\begin{array}{*{20}l} {\phi_{M,j,0}}(t) = \sum\limits_{k =  \infty }^{\infty} {{h_{M,0}}(k){\phi_{M,j  1,k}}(t)} \end{array} $$
(36a)
and
$$\begin{array}{*{20}l} {\psi_{M,j,0}}(t) = \sum\limits_{k =  \infty }^{\infty} {{h_{M,1}}(k){\phi_{M,j  1,k}}(t)} \end{array} $$
(36b)
with
$$\begin{array}{*{20}l} {h_{M,0}}(k) = {h_{0}}(k){e^{ i\frac{A}{{2B}}{k^{2}}}} \end{array} $$
(37a)
and
$$\begin{array}{*{20}l} {h_{M,1}}(k) = {h_{1}}(k){e^{ i\frac{A}{{2B}}{k^{2}}}}. \end{array} $$
(37b)
Equation (36) are the twoscale difference equations belonging to LCWT which reveal the relationship between linear canonical wavelets and linear canonical scale functions in multiresolution approximation analysis associated with LCT. Moreover, due to the orthogonality between {ϕ_{
M,j,k
}(t)} and {ψ_{
M,j,k
}(t)}, we have
$$\begin{array}{*{20}l} {h_{M,0}}(k) = \left\langle {{\phi_{M,j,0}}(\cdot),{\phi_{M,j  1,k}}(\cdot)} \right\rangle \end{array} $$
(38a)
and
$$\begin{array}{*{20}l} {h_{M,1}}(k) = \left\langle {{\psi_{M,j,0}}(\cdot),{\phi_{M,j  1,k}}(\cdot)} \right\rangle. \end{array} $$
(38b)
As can be seen from (38), h_{M,0}(k) and h_{M,1}(k) are irrelevant to j, because of the complex amplitude we multiply to the mother linear canonical wavelet (see (8)). Moreover, it should be noticed that the sequence h_{0}(k) and h_{1}(k) are the conjugate mirror filters in the FT domain. Therefore, according to Zhao [7], h_{M,0}(k) and h_{M,1}(k) actually represent the twochannel filter banks in the LCT domain.
Assume that j=1. By taking the LCT of both sides of (36), we obtain
$$\begin{array}{*{20}l} \Phi (u/B) = \frac{1}{{\sqrt 2}}{H_{0}}(u/2B)\Phi (u/2B) \end{array} $$
(39a)
and
$$\begin{array}{*{20}l} \Psi (u/B) = \frac{1}{{\sqrt 2}}{H_{1}}(u/2B)\Phi (u/2B), \end{array} $$
(39b)
where H_{0}(u) and H_{1}(u) are the discrete time Fourier transform (DTFT) of h_{0}(k) and h_{1}(k), respectively.
According to the orthogonality of \(\{{\phi _{M,0,k}}(t),k \in \mathbb {Z}\}\), we have
$$ \sum\limits_{k =  \infty}^{\infty} {{{\left {{H_{0}}(u/2B + k\pi)}\right}^{2}}{{\left {\Phi (u/2B + k\pi)} \right}^{2}}} = 2. $$
(40)
Since H_{0}(u) is 2π periodic, splitting k into odd and even parts, i.e., substituting k=2p and k=2p+1, \(p \in \mathbb {Z}\) into (40) yields
$$\begin{array}{*{20}l} & {\left {{H_{0}}(u/B)} \right^{2}} \sum\limits_{p =  \infty }^{\infty} {{{\left {\Phi (u/B + 2p\pi)} \right}^{2}}} + \\ & {\left {{H_{0}}(u/B + \pi)} \right^{2}}\sum\limits_{p =  \infty }^{\infty} {{{\left {\Phi (u/B + 2p\pi + \pi)} \right}^{2}}} = 2. \end{array} $$
(41)
Notice that \(\sum \limits _{p =  \infty }^{\infty } {{{\left  {\Phi (u/B + 2p\pi)} \right }^{2}}} = 1\) and \(\sum \limits _{p =  \infty }^{\infty } {{{\left  {\Phi (u/B + 2p\pi + \pi)} \right }^{2}}} = 1\), it is easy to find that
$$\begin{array}{*{20}l} {\left {{H_{0}}(u/B)} \right^{2}} + {\left {{H_{0}}(u/B + \pi)} \right^{2}} = 2. \end{array} $$
(42a)
Similar with \(\{ {\phi _{M,0,k}}(t),k \in \mathbb {Z}\}\), the relationship
$$\begin{array}{*{20}l} {\left {{H_{1}}(u/B)} \right^{2}} + {\left {{H_{1}}(u/B + \pi)} \right^{2}} = 2 \end{array} $$
(42b)
holds.
Moreover, because \(W_{0}^{M}\) and \(V_{0}^{M}\) are orthogonal with each other, \(\left \{ {{\psi _{M,0,k}}(t),k \in \mathbb {Z}} \right \}\) and \(\left \{ {{\phi _{M,0,k}}(t),k \in \mathbb {Z}} \right \}\) are orthogonal, i.e.,
$$ \left\langle {{\phi_{M,0,{k_{1}}}}(t),{\psi_{M,0,{k_{2}}}}(t)} \right\rangle = 0 $$
(43)
for \(\forall {k_{1}},{k_{2}} \in \mathbb {Z}\), and it is easy to verify that
$$ \sum\limits_{k =  \infty}^{\infty} {\Phi (u/B + 2k\pi){\Psi^{*}}(u/B + 2k\pi)} = 0. $$
(44)
Therefore, substituting (39a) and (39b) into (44), we have
$$\begin{array}{*{20}l} \sum \limits_{k =  \infty}^{\infty} & {{H_{0}}(u/B + k\pi) \Phi (u/B + k\pi)} \\ & \times H_{1}^{*}(u/B + k\pi){\Phi^{*}}(u/B + k\pi) = 0. \end{array} $$
(45)
Similarly, since H_{0}(u) and H_{1}(u) are both 2π periodic, splitting k into odd and even parts, i.e., substituting k=2p and k=2p+1, \(p \in \mathbb {Z}\) into (45) gives
$$ {H_{0}}(u/B)H_{1}^{*}(u/B) + {H_{0}}(u/B + \pi)H_{1}^{*}(u/B + \pi) = 0. $$
(46)
Equations (42a), (42b), and (46) together indicate that if \({\psi _{M,j,k}}(t) = {2^{ j/2}}\psi ({2^{ j}}t  k){e^{ i\frac {A}{{2B}}({t^{2}}  {k^{2}})}}\) is an orthonormal basis for \(W^{M}_{j}\), then
$$ \mathbf{M} \cdot {\mathbf{M}^{\dag}} = 2I, $$
(47)
where † denotes conjugate transpose, I is identity matrix, and
$$ \mathbf{M} = \left[{\begin{array}{*{20}{c}} {{H_{0}}(u/B)}&{{H_{0}}(u/B + \pi)}\\ {{H_{1}}(u/B)}&{{H_{1}}(u/B + \pi)} \end{array}} \right]. $$
(48)
Equation (47) indicates that when \(\{ {\psi _{M,j,k}}(t),k \in \mathbb {Z}\}\) forms an orthonormal basis for \(W^{M}_{j}\), h_{M,0}(k) and h_{M,1}(k) are actually the twochannel conjugate orthogonal mirror filter banks associated with the LCT.
Overall, the construction of the orthonormal linear canonical wavelets can be summarized in the following theorem.
Theorem 4
Define \(\left \{ V_{j}^{M}\right \},j \in \mathbb {Z}\) as a sequence of closet subspaces. \(W_{j}^{M}\) is the orthogonal complement of \(V_{j}^{M}\) in \(V_{j1}^{M}\). If \(\{ {\phi _{M,j,k}}(t), j,k \in \mathbb {Z}\}\) is a set of orthonormal basis of \(V_{j}^{M}\), then \(\{{\psi _{M,j,k}}(t), j,k \in \mathbb {Z} \}\) is a set of orthonormal basis of \(W_{j}^{M}\) if and only if M satisfy (47), i.e., \(\{ \psi _{j,k}(t), j,k \in \mathbb {Z} \}\) is a set of orthonormal basis of W_{
j
}.
4.2.2 Fast algorithm
Since \(\{ {\phi _{M,j,k}}(t),j,k \in \mathbb {Z}\}\) and \(\{ {\psi _{M,j,k}}(t),j,k \in \mathbb {Z}\}\) are orthonormal bases of V_{
M,j
} and W_{
M,j
}, the projection in these spaces can be characterized by
$$\begin{array}{*{20}l} {a_{M,j}}(k) = \left\langle {x(t),{\phi_{M,j,k}}(t)} \right\rangle \end{array} $$
(49a)
and
$$\begin{array}{*{20}l} {d_{M,j}}(k) = \left\langle {x(t),{\psi_{M,j,k}}(t)} \right\rangle. \end{array} $$
(49b)
An actual implementation of the MAR of LCWT requires computation of the inner products shown above, which is computationally rather involved. Therefore, in this section, we develop a fast filter bank algorithm associated with the LCT that computes the orthogonal linear canonical wavelet coefficients of a signal measured at a finite resolution.
From the orthogonormal functions \({\phi _{M,j + 1,k}} \in V_{j + 1}^{M}\), \({\phi _{M,j,k}} \in V_{j}^{M}\), and \(V_{j + 1}^{M} \subset V_{j}^{M}\), we get
$$ {\phi_{M,j + 1,k}}(t) = \sum\limits_{n =  \infty}^{\infty} {{c_{n}}{\phi_{M,j,n}}(t)}. $$
(50)
With the change of variable t^{′}=2^{−j}t−2k, we obtain
$$\begin{array}{*{20}l} {c_{n}} &= \left\langle {{\phi_{M,j + 1,k}}(t),{\phi_{M,j,n}}(t)} \right\rangle \\ &= {2^{ j  \frac{1}{2}}}\int_{ \infty}^{+ \infty} {\phi\left(\frac{t}{{{2^{j + 1}}}}  k\right){e^{i\frac{A}{{2B}}{k^{2}}}}} \\ & \quad \times {\phi^{*}}\left(\frac{t}{{{2^{j}}}}  n\right){e^{ i\frac{A}{{2B}}{n^{2}}}}dt \\ &= {2^{ \frac{1}{2}}}\int_{ \infty}^{+ \infty} {\phi\left(\frac{{t}}{2}\right){e^{i\frac{A}{{2B}}{k^{2}}}}} \\ & \quad \times {\phi^{*}}(t  n + 2k){e^{ i\frac{A}{{2B}}{n^{2}}}}dt \\ &= {e^{i\frac{A}{{2B}}(5{k^{2}}  4nk)}}\left\langle {{\phi_{M,1,0}}(t),{\phi_{M,0,n  2k}}(t)} \right\rangle \\ &= {e^{i\frac{A}{{2B}}(5{k^{2}}  4nk)}}{h_{M,0}}(n  2k). \end{array} $$
(51)
Equation (50) implies that
$$\begin{array}{*{20}l} {\phi_{M,j + 1,k}}(t) &= \sum\limits_{n =  \infty}^{\infty} {{e^{i\frac{A}{{2B}}(5{k^{2}}  4nk)}}} \\ & \qquad \qquad \times {h_{M,0}}(n  2k){\phi_{M,j,n}}(t). \end{array} $$
(52)
Taking the inner product by x(t) on both sides of (52) yields
$$\begin{array}{*{20}l} {a_{M,j + 1}}(k) = {a_{M,j}}(k)\Theta {\bar h_{M,0}}(2k). \end{array} $$
(53a)
From the orthogonal functions \({\psi _{M,j + 1,k}} \in W_{j + 1}^{M}\), \({\phi _{M,j,k}} \in V_{j}^{M}\), and \(W_{j + 1}^{M} \subset V_{j}^{M}\), we have
$$\begin{array}{*{20}l} {d_{M,j + 1}}(k) = {a_{M,j}}(k)\Theta {\bar h_{M,1}}(2k), \end{array} $$
(53b)
where \(\bar h(k) = h( k)\).
Since \(V_{j}^{M} = V_{j + 1}^{M} \oplus W_{j + 1}^{M}\), \({\phi _{M,j + 1,k}}(t) \in V_{j + 1}^{M}\), \({\psi _{M,j + 1,k}} (t)\in W_{j + 1}^{M}\), and ϕ_{
M,j,k
}(t) can be decomposed as
$$\begin{array}{*{20}l} & {\phi_{M,j,k}}(t) \\ & = \sum\limits_{n =  \infty }^{\infty} {\left\langle {{\phi_{M,j,k}}(t),{\phi_{M,j + 1,n}}(t)} \right\rangle {\phi_{M,j + 1,n}}(t)} \\ &\quad + \sum\limits_{n =  \infty }^{\infty} {\left\langle {{\phi_{M,j,k}}(t),{\psi_{M,j + 1,n}}(t)} \right\rangle {\psi_{M,j + 1,n}}(t)}. \end{array} $$
(54)
Combining (52), we obtain
$$\begin{array}{*{20}l} \left\langle {{\phi_{M,j,k}}(t),{\phi_{M,j + 1,n}}(t)} \right\rangle & = {h_{M,0}}(k  2n) \\ & \quad \times {e^{i\frac{A}{{2B}}(5{n^{2}}  4nk)}} \end{array} $$
(55a)
and
$$\begin{array}{*{20}l} \left\langle {{\phi_{M,j,k}}(t),{\psi_{M,j + 1,n}}(t)} \right\rangle & = {h_{M,1}}(k  2n) \\ & \quad \times {e^{i\frac{A}{{2B}}(5{n^{2}}  4nk)}}. \end{array} $$
(55b)
Substituting (55) into (54) yields
$$\begin{array}{*{20}l} & {\phi_{M,j,k}}(t) \\ & = \sum\limits_{n =  \infty}^{\infty} {{\phi_{M,j + 1,n}}(t){h_{M,0}}(k  2n){e^{i\frac{A}{{2B}}(5{n^{2}}  4nk)}}} \\ &\quad + \sum\limits_{n =  \infty}^{\infty} {{\psi_{M,j + 1,n}}(t){h_{M,1}}(k  2n){e^{i\frac{A}{{2B}}(5{n^{2}}  4nk)}}}. \end{array} $$
(56)
Taking the inner product by x(t) on both sides of (56) yields
$$ {a_{M,j}}(k) = {a_{M,j + 1}}(k)\Theta {h_{M,0}}(2k) + {d_{M,j + 1}}(k)\Theta {h_{M,1}}(2k). $$
(57)
Equations (53a) and (53b) prove that a_{M,j+1} and d_{M,j+1} can be obtained by taking every other sample of the linear canonical convolution of a_{
M,j
} with \({\bar h_{M,0}}(k)\) and \({\bar h_{M,1}}(k)\), respectively, as illustrated by Fig. 4a. The reconstruction (57) is an interpolation that inserts zeroes to expand a_{M,j+1} and d_{M,j+1} and filters these signals in the LCT domain, as shown in Fig. 4b. Compared to the structure shown in Fig. 1, the coefficients of each layer can be chirp modulated and demodulated with the same chirp rate in different layers, in the process of the multiresolution analysis of the DLCWT.
The following is an example showing decompositions and reconstructions of 1D signal utilizing the DLCWT. We observe a chirp signal given by
$$ x(t) = \left({\sin (2\pi {f_{0}}t) + \sin (2\pi {f_{1}}t)} \right){e^{ i\frac{k}{2}{t^{2}}}} $$
(58)
where k=2, f_{0}=0.1, and f_{1}=4.5. Figure 5 shows an example of twolayer DLCWT of this signal x(t) computed using db3 wavelets with M=(2,1,1,1). Note that the initial data a_{M,−1}(k)=x(k) where x(k) denote samples of continuous signal x(t) with sampling rate Δt=0.1. As shown in Fig. 5, the coefficients d_{M,1}(k) are basically equal to zero, and two frequency components f_{0} and f_{1} of x(t) lie in subspaces \(V_{1}^{M}\) and \(W_{0}^{M}\), separately. Signal x(k) is perfectly reconstructed from coefficients d_{M,0}(k), a_{M,1}(k) and d_{M,1}(k), denoted as x^{′}(k).
4.2.3 Computational complexity
Direct computation of (11) would involve O(N^{2}) operations per scale with N as the length of the input sequence. However, when using the fast algorithm shown in Fig. 4, the DOLCWT’s computational complexity depends on that of the linear canonical convolution. According to (4), (53a), and (53b), each takes O(N) time at the first level. Then, the downsampling operation splits the signal into two branches of size N/2. But the filter bank only recursively splits one branch convolved with h_{M,0}(n). This leads to a recurrence relation which conduces to an O(N) time for the entire operation. Furthermore, because the proposed fast filter bank algorithm can inherit the conventional lifting scheme, the computational complexity could be halved for long filters [27].