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 Multi-resolution approximation associated with LCT
The theory of multi-resolution 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 multi-resolution 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 multi-resolution 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 2j 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−jπB,+2−jπB] 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 \(\{ \theta (t-k),k \in \mathbb {Z}\}\) forms a Riesz basis of the subspace V0 with θ(t)∈V0 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 \(\{ \theta (t-k),k \in \mathbb {Z}\}\) as a Riesz basis of the subspace of V0, 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 chirp-modulated shift-invariant 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)∈V0.
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)∈W0 to construct mother linear canonical wavelet \(\psi _{M}(t) \in W_{0}^{M}\) such that the dilated, translated, and chirp-modulated 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 \(V_{j-1}^{M}\), 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 \(V_{j-1}^{M}\) 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 two-channel 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 two-channel filter banks associated with LCT
Since both ψM,j−1,k(t) and ϕM,j−1,k(t) form an orthonormal basis for \(W^{M}_{j-1}\) and \(V^{M}_{j-1}\), we can decompose ϕM,j,0(t) and ψM,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 two-scale difference equations belonging to LCWT which reveal the relationship between linear canonical wavelets and linear canonical scale functions in multi-resolution 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), hM,0(k) and hM,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 h0(k) and h1(k) are the conjugate mirror filters in the FT domain. Therefore, according to Zhao [7], hM,0(k) and hM,1(k) actually represent the two-channel 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 H0(u) and H1(u) are the discrete time Fourier transform (DTFT) of h0(k) and h1(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 H0(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 H0(u) and H1(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 \(W^{M}_{j}\), hM,0(k) and hM,1(k) are actually the two-channel 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 \(V_{j-1}^{M}\). 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−jt−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 aM,j+1 and dM,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 aM,j+1 and dM,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 de-modulated with the same chirp rate in different layers, in the process of the multi-resolution 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, f0=0.1, and f1=4.5. Figure 5 shows an example of two-layer DLCWT of this signal x(t) computed using db3 wavelets with M=(2,1,1,1). Note that the initial data aM,−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 dM,1(k) are basically equal to zero, and two frequency components f0 and f1 of x(t) lie in subspaces \(V_{1}^{M}\) and \(W_{0}^{M}\), separately. Signal x(k) is perfectly reconstructed from coefficients dM,0(k), aM,1(k) and dM,1(k), denoted as x′(k).
4.2.3 Computational complexity
Direct computation of (11) would involve O(N2) 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 hM,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].