**1. Introduction**

556 Advances in Wavelet Theory and Their Applications in Engineering, Physics and Technology

Ni, Z.; Shi, Y.Q.; Ansari, N.; Su, W.; Sun, Q. & Lin, X. (2008). Robust lossless image data

Qu, Z.G.; Chen, X.B.; Zhou, X.J.; Niu, X.X. & Yang, Y.X. (2010). Novel quantum

Shih, F.Y. (2008). *Digital watermarking and steganography: fundamentals and techniques*. CRC

Tai, W.L.; Yeh, C.M. & Chang, C.C. (2009). Reversible data hiding based on histogram

Tian, J. (2003). Reversible data embedding using a difference expansion. *IEEE T. Circuits and* 

Wang, S., Yang, B. & Niu, X. (2010). A secure steganography method based on genetic

Wu, H.C.; Lee, C.C.; Tsai, C.S.; Chu, Y.P. & Chen, H.R. (2009). A high capacity reversible

Xiao, D. & Shih, F.Y. (2010). A reversible image authentication scheme based on chaotic

Yamamoto, K. & Iwakiri M. (2010). Real-time audio watermarking based on characteristics

Yang, C.H. & Tsai, M.H. (2010). Improving histogram-based reversible data hiding by interleaving predictors. *IET Image Processing*, Vol. 4, No. 4, pp. 223-234. Yang, C.Y.; Hu, W.C. & Lin, C.H. (2010). Reversible data hiding by coefficient-bias

Yang, C.Y.; Hu, W.C.; Hwang, W.Y. & Cheng, Y.F. (2010). A simple digital watermarking by

Yang, C.Y.; Lin, C.H. & Hu, W.C. (2011). Block-based reversible data hiding," *ICIC Express* 

Zeng, X.T.; Ping, L.D. & Pan, X.Z. (2010). A lossless robust data hiding scheme. *Pattern* 

Zhou, S.; Zhang, Q. & Wei, X. (2010). An image encryption algorithm based on dual DNA sequences for image hiding. *ICIC Express Letters*, Vol. 4, No. 4, pp. 1393-1398. Zou, D.; Shi, Y.Q.; Ni, Z. & Su, W.A. (2006). A semi-fragile lossless digital watermarking

*for Video Technology*, Vol. 18, No. 4, pp. 497-509, 2008.

*Systems for Video Technology*, Vol. 13, No. 8, pp. 890-896.

*Systems and Software*, Vol. 82, pp. 1966-1973

*Control,* Vol. 6 No. 10, pp. 4731-4742.

*Processing*, Vol. 1, No. 2, pp. 59-71.

*and Control, Vol. 6, No. 3,* pp. 1401-1410.

*Technology*, Vol. 16, No. 10, pp. 1294-1300.

*Letters*, Vol. 5, No. 7, pp. 2251-2256.

*Recognition*, Vol. 43, pp. 1656-1667.

2, pp. 91-100.

4782-4786.

Press, FL.

1.

Vol. 19, No. 6, pp. 906-910.

hiding designed for semi-fragile image authentication," *IEEE T. Circuits and Systems* 

steganography with large payload. *Optics Communications*, Vol. 283, No. 23, pp.

modification of pixel differences. *IEEE T. Circuits and Systems for Video Technology*,

algorithm. *Journal of Information Hiding and Multimedia Signal Processing*, Vo1. 1, No.

data hiding scheme with edge prediction and difference expansion. *The Journal of* 

fragile watermark. *International Journal of Innovative Computing, Information and* 

of PCM in digital instrument. *Journal of Information Hiding and Multimedia Signal* 

algorithm. *Journal of Information Hiding and Multimedia Signal Processing*, Vol. 1, No.

the adaptive bit-labeling scheme. *Int. Journal of Innovative Computing, Information* 

scheme based on integer wavelet transform. *IEEE T. Circuits and Systems for Video* 

Discrete-time wavelet transform (DWT) is found to be better than other transforms in the time-varying system analysis, e.g. for time-varying parametric modelling [16], time-varying systems identification [17], time-varying parameter estimation [18] and time domain signal analysis [19]. In the literature the common method to analyze the time-varying system using discrete-time wavelet transform is to model the time-varying system with a time-invariant system firstly, because a general analysis of time-varying discrete-time wavelet transform (TV-DWT) is still missing. To analyze the time-varying system directly using the time-varying discrete-time wavelet transform, we need the theory for the time-varying discrete-time wavelet transform.

The theory of time-invariant discrete-time wavelet transform (DWT) are quite complete [1,2,3]. For time-varying discrete-time wavelet transform, in literature there are some papers related with this topic by studying the changes of two different filter banks [10,11,12]. In [10] the authors analyzed the time-varying wavelet transform through changing the two-band filter banks used in the tree-structured implementation of DWTs with an simple example. In [11] the time-varying wavelet packets built with time-varying cosine-modulated filter banks were investigated. Similar with [10], in [12] the authors studied time-varying wavelet packets more theoretically with changing the two orthogonal two-band filter banks used in tree-structure of DWTs. Generally, in the existed theory of time-varying discrete-time wavelet transform it lacks a basic definition and description of the time-varying discrete-time wavelet transform. A basic analysis of time-varying discrete-time wavelet transform is also missing. The author has studied TV-DWT since some years and has published a series of papers about this topic. In this Chapter we summarize the author's main research results.

In our method the time-varying discrete-time wavelet transform is studied using a time-varying octave-band filter bank with tree structure. With this implementation the analysis of the time-varying discrete-time wavelet transform is equal to the analysis of the time-varying discrete-time octave-band filter bank. Then, the time-varying filter bank theory can be used in TV-DWT analysis. In this chapter we provide some theorems for the time-varying discrete-time wavelet transform with proofs.

Wavelet Transforms 3

Time-Varying Discrete-Time Wavelet Transforms 559

y0(m0)

y1(m1)

↑ 4

<sup>↑</sup> <sup>2</sup><sup>J</sup>(m) <sup>G</sup><sup>M</sup>(m)−<sup>2</sup>(z,m)

G<sup>M</sup>(m)−<sup>1</sup>(z,m)

G1(z,m)

G0(z,m)

x(n)

↑ 2

↓ 4

H<sup>M</sup>(m)−<sup>2</sup>(z,m)

**T**(*j*) *ma* = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ . . . . . . . . . . .

··· . . . . .

> . . . . . . . . . . .

<sup>H</sup><sup>M</sup>(m)−<sup>1</sup>(z,m) <sup>↓</sup> <sup>2</sup><sup>J</sup>(m)

two-channel time-varying filter bank is perfectly reconstructed.

the *j*-stage two-channel time-varying filter bank shown in Fig. 3 as

H1(z,m)

H0(z,m)

↓ 2

x(n)

y<sup>M</sup>(m)−<sup>1</sup>(m)

↑ 2<sup>J</sup>(m) ↓ 2<sup>J</sup>(m)

The process is ongoing until *J*-stage. In the synthesis side, the signal is merged to generate the reconstructed signal *<sup>x</sup>*�(*n*). From the theorem of time-invariant discrete-time wavelet transform [2], we know that if the individual two-channel filter bank, or each split-merge pair is perfectly reconstructed, the octave-band filter bank is as well. Such statement is also valid for the time-varying octave-band filter bank. Therefore, we have following theorem.

**Theorem 1**: A time-varying discrete-time wavelet transform implemented with a time-varying octave-band filter bank is a biorthogonal time-varying transform if each

We cannot use the method used in the time-invariant case to prove the above theorem because the system is time-varying. To prove theorem 1, we define analysis and synthesis matrices of

. .

. .

. .

··· **<sup>h</sup>**0(*mj*) **<sup>h</sup>**1(*mj*) ··· **<sup>h</sup>***N*(*j*)−1(*mj*) **<sup>h</sup>***N*(*j*)(*mj*) **<sup>h</sup>***N*(*j*)+1(*mj*) ···

··· **0 h**0(*mj*) ··· **<sup>h</sup>***N*(*j*)−2(*mj*) **<sup>h</sup>***N*(*j*)−1(*mj*) **<sup>h</sup>***N*(*j*)(*mj*) ···

··· **0 0** ··· **h**0(*mj*) **h**1(*mj*) **h**2(*mj*) ···

··· **0 0** ··· **0 h**0(*mj* + 1) **h**1(*mj* + 1) ···

.

. ... .

.

.

.

. .

. .

. .

. .

.

. .

. ···

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2)

Fig. 2. Time-varying nonuniform filter bank implementation, where *M*(*m*) = 2*J*(*m*).

y<sup>M</sup>(m)−<sup>2</sup>(m)

#### **2. Formulation of time-varying discrete-time wavelet transforms**

From the point of view of digital signal processing, the time-varying discrete-time wavelet transform can be implemented by a time-varying octave-band filter bank with tree structure. Fig. 1 shows the most general time-varying discrete-time wavelet transform implemented with a time-varying octave-band filter bank, where the lowpass and highpass filter *Hl*(*z*, *m*), *Hu*(*z*, *m*), the stage number of the split-merge *J*(*m*), all are varying with time index *m*. In other words, both the frequency characteristic and the time-frequency tiling of the discrete-time wavelet transform are varying with time. Fig. 2 shows the time-varying nonuniform filter bank implementation. With this implementation the analysis of the time-varying discrete-time wavelet transform is equal to the analysis of the time-varying discrete-time octave-band filter bank.

Note that we define the time-varying discrete-time wavelet transform varying with index *m* which is equivalent to the output index at the last stage of octave-band filter banks. The time indices of the other output are related to *m* by

$$\mathfrak{m}\_{j} = 2^{J(m) - i - j} \cdot m, \; 0 \le j \le J(m) - 2. \tag{1}$$

In the literature there are some papers related with this topic by studying changes between two time-invariant filter banks [10,11,12]. In particular, in [10] the authors have discussed the transition behavior during the change between two time-invariant discrete-time wavelet transforms. Different from the existed publications, in this chapter we analyze the general time-varying discrete-time wavelet transform in detail based on the octave-band filter bank and the nonuniform filter bank implementation.

Fig. 1. Time-varying discrete-time wavelet transform implemented with time-varying octave-band filter banks.

#### **3. Implementation with time-varying octave-band filter Banks**

To make the analysis simple, in the following analysis we suppose that the stage number *J* does not change with time and is a constant. Then we get a *J*-stage time-varying octave-band filter bank. Just as depicted in Fig. 1, a *J*-stage octave-band time-varying filter bank consists of *J* stages of two-channel time-varying filter bank. In the analysis side, the input signal *x*(*n*) is first split by the two-channel time-varying filter bank at the first stage, then the lowpass output is spilt again by the same two-band time-varying filter bank at the second stage. 2 Will-be-set-by-IN-TECH

From the point of view of digital signal processing, the time-varying discrete-time wavelet transform can be implemented by a time-varying octave-band filter bank with tree structure. Fig. 1 shows the most general time-varying discrete-time wavelet transform implemented with a time-varying octave-band filter bank, where the lowpass and highpass filter *Hl*(*z*, *m*), *Hu*(*z*, *m*), the stage number of the split-merge *J*(*m*), all are varying with time index *m*. In other words, both the frequency characteristic and the time-frequency tiling of the discrete-time wavelet transform are varying with time. Fig. 2 shows the time-varying nonuniform filter bank implementation. With this implementation the analysis of the time-varying discrete-time wavelet transform is equal to the analysis of the time-varying discrete-time octave-band filter

Note that we define the time-varying discrete-time wavelet transform varying with index *m* which is equivalent to the output index at the last stage of octave-band filter banks. The time

In the literature there are some papers related with this topic by studying changes between two time-invariant filter banks [10,11,12]. In particular, in [10] the authors have discussed the transition behavior during the change between two time-invariant discrete-time wavelet transforms. Different from the existed publications, in this chapter we analyze the general time-varying discrete-time wavelet transform in detail based on the octave-band filter bank

↓ 2 ↑ 2 Gu(z,m)

↓ 2 ↓ 2

Fig. 1. Time-varying discrete-time wavelet transform implemented with time-varying

x(n) x(n)

y0(m0) y1(m1)

y<sup>J</sup>(m)(m)

To make the analysis simple, in the following analysis we suppose that the stage number *J* does not change with time and is a constant. Then we get a *J*-stage time-varying octave-band filter bank. Just as depicted in Fig. 1, a *J*-stage octave-band time-varying filter bank consists of *J* stages of two-channel time-varying filter bank. In the analysis side, the input signal *x*(*n*) is first split by the two-channel time-varying filter bank at the first stage, then the lowpass output is spilt again by the same two-band time-varying filter bank at the second stage.

↑ 2 ↑ 2

Stage J(m)

Gl(z,m) Gu(z,m)

y<sup>J</sup>(m)−<sup>1</sup>(m)

Hl(z,m) Hu(z,m)

Stage J(m)

**3. Implementation with time-varying octave-band filter Banks**

*mj* <sup>=</sup> <sup>2</sup>*J*(*m*)−*i*−*<sup>j</sup>* · *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>J</sup>*(*m*) <sup>−</sup> 2. (1)

↑ 2

Gl(z,m)

Stage 2

Gl(z,m) Gu(z,m)

Stage 1

↑ 2

↑ 2

**2. Formulation of time-varying discrete-time wavelet transforms**

indices of the other output are related to *m* by

and the nonuniform filter bank implementation.

↓ 2 ↓ 2

Stage 2

Hu(z,m) Hl(z,m)

bank.

Hu(z,m) Hl(z,m)

↓ 2

octave-band filter banks.

Stage 1

Fig. 2. Time-varying nonuniform filter bank implementation, where *M*(*m*) = 2*J*(*m*).

The process is ongoing until *J*-stage. In the synthesis side, the signal is merged to generate the reconstructed signal *<sup>x</sup>*�(*n*). From the theorem of time-invariant discrete-time wavelet transform [2], we know that if the individual two-channel filter bank, or each split-merge pair is perfectly reconstructed, the octave-band filter bank is as well. Such statement is also valid for the time-varying octave-band filter bank. Therefore, we have following theorem.

**Theorem 1**: A time-varying discrete-time wavelet transform implemented with a time-varying octave-band filter bank is a biorthogonal time-varying transform if each two-channel time-varying filter bank is perfectly reconstructed.

We cannot use the method used in the time-invariant case to prove the above theorem because the system is time-varying. To prove theorem 1, we define analysis and synthesis matrices of the *j*-stage two-channel time-varying filter bank shown in Fig. 3 as

$$\mathbf{T}\_{mn}^{(j)} = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ \cdots & \mathbf{h}\_0(m\_j) \ \mathbf{h}\_1(m\_j) & \cdots & \mathbf{h}\_{N(j)-1}(m\_j) & \mathbf{h}\_{N(j)}(m\_j) & \mathbf{h}\_{N(j)+1}(m\_j) & \cdots\\ \cdots & \mathbf{0} & \mathbf{h}\_0(m\_j) & \cdots & \mathbf{h}\_{N(j)-2}(m\_j) & \mathbf{h}\_{N(j)-1}(m\_j) & \mathbf{h}\_{N(j)}(m\_j) & \cdots\\ \cdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots & \cdots\\ \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{h}\_0(m\_j) & \mathbf{h}\_1(m\_j) & \mathbf{h}\_2(m\_j) & \cdots\\ \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{h}\_0(m\_j+1) & \mathbf{h}\_1(m\_j+1) & \cdots\\ \cdots & \mathbf{0} & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{h}\_0(m\_j+1) & \mathbf{h}\_1(m\_j+1) & \cdots\\ & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \end{bmatrix} \tag{2}$$

$$\mathbf{T}\_{\mathsf{ms}}^{(j)} = \begin{bmatrix} \vdots & \vdots & \vdots & \vdots & \vdots \\ \cdots & \mathbf{g}\_{0}(m\_{j}) & \mathbf{0} & \cdots & \mathbf{0} & \mathbf{0} & \cdots \\ \cdots & \mathbf{g}\_{1}(m\_{j}) & \mathbf{g}\_{0}(m\_{j}) & \cdots & \mathbf{0} & \mathbf{0} & \cdots \\ \cdots & \vdots & \vdots & \ddots & \vdots & \vdots & \cdots \\ \cdots & \mathbf{g}\_{N(j)-1}(m\_{j}) & \mathbf{g}\_{N(j)-2}(m\_{j}) & \cdots & \mathbf{g}\_{0}(m\_{j}) & \mathbf{0} & \cdots \\ \cdots & \mathbf{g}\_{N(j)}(m\_{j}) & \mathbf{g}\_{N(j)-1}(m\_{j}) & \cdots & \mathbf{g}\_{1}(m\_{j}) & \mathbf{g}\_{0}(m\_{j}+1) & \cdots \\ \cdots & \mathbf{g}\_{N(j)+1}(m\_{j}) & \mathbf{g}\_{N(j)}(m\_{j}) & \cdots & \mathbf{g}\_{2}(m\_{j}) & \mathbf{g}\_{1}(m\_{j}+1) & \cdots \\ & \vdots & \vdots & \vdots & & \end{bmatrix},\tag{3}$$

where

$$N(j) = 2^{J-j}, \ j = 1, 2, \cdots, \ \dots \tag{4}$$

Wavelet Transforms 5

Time-Varying Discrete-Time Wavelet Transforms 561

<sup>0</sup> <sup>=</sup> **<sup>Λ</sup>**0**y**(*j*)

<sup>1</sup> <sup>=</sup> **<sup>Λ</sup>**1**y**(*j*)

(*j*) <sup>0</sup> (0) *y*

(*j*) <sup>0</sup> (1)··· *<sup>T</sup>*

(*j*) <sup>1</sup> (1)··· *<sup>T</sup>*

**<sup>x</sup>**(*j*) <sup>=</sup> **<sup>T</sup>**(*j*)

Based on the above matrix definitions we can describe the filter bank at the *j*-th stage showed

*ms* **<sup>T</sup>**(*j*) *ma* **x**(*j*)

After adding the (j+1)-th stage with a biorthogonal time-varying two-channel filter bank

<sup>1</sup> **<sup>y</sup>**(*j*) 1 

*ma***x**(*j*) + **Λ***<sup>T</sup>*

Because we suppose that the added two-channel filter bank is biorthogonal, we have

<sup>0</sup> **<sup>Λ</sup>**<sup>0</sup> + **<sup>Λ</sup>***<sup>T</sup>*

<sup>0</sup> **<sup>Λ</sup>**<sup>0</sup> + **<sup>Λ</sup>***<sup>T</sup>*

which means that the time-varying octave-band filter bank is still perfectly reconstructed after adding next stage of time-varying biorthogonal two-channel filter bank. In other words,

<sup>1</sup> **Λ**<sup>1</sup> **T**(*j*) *ma* **x**(*j*)

**Λ***T*

*ms* **Λ***T*

**<sup>x</sup>**(*j*) <sup>=</sup> **<sup>T</sup>**(*j*)

= **<sup>T</sup>**(*j*) *ms* **<sup>T</sup>**(*j*) *ma* **x**(*j*) (*j*) <sup>1</sup> (0) *y*

(*j*) <sup>0</sup> (1) *y*

<sup>1</sup> **<sup>T</sup>**(*j*+1) *ms* **<sup>T</sup>**(*j*+1) *ma* **<sup>Λ</sup>**1**T**(*j*)

(*j*) <sup>1</sup> (1)··· *<sup>T</sup>*

, (9)

, (10)

, (12)

. (13)

. (14)

*ma***x**(*j*) 

<sup>1</sup> **Λ**<sup>1</sup> = **I**. (17)

**<sup>T</sup>**(*j*+1) *ms* **<sup>T</sup>**(*j*+1) *ma* <sup>=</sup> **<sup>I</sup>**, (16)

= **x**(*j*) (18)

, (11)

. (15)

**y**(*j*)

**y**(*j*)

(*j*) <sup>1</sup> (−1) *y*

(*j*) <sup>0</sup> (0) *y*

(*j*) <sup>1</sup> (0) *y*

to extract the lowpass and highpass output like

**y**(*j*) = ··· *y* (*j*) <sup>0</sup> (−1) *y*

**y**(*j*) <sup>0</sup> = ··· *y* (*j*) <sup>0</sup> (−1) *y*

**y**(*j*) <sup>1</sup> = ··· *y* (*j*) <sup>1</sup> (−1) *y*

shown in Fig. 4, we have

Then, we can rewrite (15) as

theorem 1 is correct.

**<sup>x</sup>**(*j*) <sup>=</sup> **<sup>T</sup>**(*j*)

= **<sup>T</sup>**(*j*) *ms* **Λ***T* <sup>0</sup> **<sup>y</sup>**(*j*) <sup>0</sup> <sup>+</sup> **<sup>Λ</sup>***<sup>T</sup>*

= **<sup>T</sup>**(*j*) *ms* **Λ***T* <sup>0</sup> **<sup>Λ</sup>**0**T**(*j*)

*ms* **y**(*j*)

where

in Fig. 3 as

and

$$\mathbf{h}\_{l}(m\_{j}) = \begin{bmatrix} h\_{l}(L-2\mathrm{i}-1, m\_{j}) \ h\_{l}(L-2\mathrm{i}-2, m\_{j})\\ h\_{l}(L-2\mathrm{i}-1, m\_{j}) \ h\_{l}(L-2\mathrm{i}-2, m\_{j}) \end{bmatrix} \tag{5}$$

$$\mathbf{g}\_{l}(m\_{j}) = \begin{bmatrix} g\_{u}(2i, m\_{j}) & g\_{l}(2i, m\_{j}) \\ g\_{u}(2i+1, m\_{j}) \ g\_{l}(2i+1, m\_{j}) \end{bmatrix} \tag{6}$$

where *L* is the filter length.

Fig. 3. The *j*-th stage two-channel time-varying filter bank.

Furthermore, we define two special matrices **Λ**<sup>0</sup> and **Λ**<sup>1</sup>

$$\mathbf{A}\_{0} = \begin{bmatrix} \vdots \vdots \vdots \vdots \vdots \\ \cdots 1 \ 0 \ 0 \ 0 \ 0 \ \cdots \ \vdots \\ \cdots 0 \ 0 \ 1 \ 0 \ 0 \ \cdots \ \vdots \\ \vdots \vdots \vdots \vdots \vdots \vdots \\ \vdots \vdots \vdots \vdots \vdots \vdots \\ \cdots 0 \ 1 \ 0 \ 0 \ 0 \ \cdots \ \vdots \\ \cdots 0 \ 0 \ 0 \ 1 \ 0 \ \cdots \ \vdots \\ \vdots \vdots \vdots \vdots \vdots \end{bmatrix} \tag{8}$$

to extract the lowpass and highpass output like

$$\mathbf{y}\_0^{(j)} = \mathbf{A}\_0 \mathbf{y}^{(j)},\tag{9}$$

$$\mathbf{y}\_1^{(j)} = \mathbf{A}\_1 \mathbf{y}^{(j)},\tag{10}$$

where

4 Will-be-set-by-IN-TECH

··· **g**0(*mj*) **0** ··· **0 0** ··· ··· **g**1(*mj*) **g**0(*mj*) ··· **0 0** ···

··· **<sup>g</sup>***N*(*j*)−1(*mj*) **<sup>g</sup>***N*(*j*)−2(*mj*) ··· **<sup>g</sup>**0(*mj*) **<sup>0</sup>** ··· ··· **<sup>g</sup>***N*(*j*)(*mj*) **<sup>g</sup>***N*(*j*)−1(*mj*) ··· **<sup>g</sup>**1(*mj*) **<sup>g</sup>**0(*mj* <sup>+</sup> <sup>1</sup>) ··· ··· **g***N*(*j*)+1(*mj*) **g***N*(*j*)(*mj*) ··· **g**2(*mj*) **g**1(*mj* + 1) ···

> *hu*(*L* − 2*i* − 1, *mj*) *hu*(*L* − 2*i* − 2, *mj*) *hl*(*L* − 2*i* − 1, *mj*) *hl*(*L* − 2*i* − 2, *mj*)

> > *gu*(2*i*, *mj*) *gl*(2*i*, *mj*) *gu*(2*i* + 1, *mj*) *gl*(2*i* + 1, *mj*)

(n) x(j)

y(j) <sup>0</sup> (m<sup>j</sup> )

y(j) <sup>1</sup> (m<sup>j</sup> )

↓ 2 ↑ 2

↓ 2 ↑ 2

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

··· 10000 ··· ··· 00100 ··· . . . . . . . . . . . . . . .

··· 01000 ··· ··· 00010 ··· . . . . . . . . . . . . . . .

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

. ... .

. . . . . ···

, *j* = 1, 2, ··· , , (4)

�

�

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

, (3)

, (5)

, (6)

(n)

(7)

Gu(z,m)

Gl(z,m)

, (8)

. . . . . . . . . . .

.

. . . . .

**T**(*j*) *ms* =

where *L* is the filter length.

Hu(z,m)

Hl(z,m)

where

and

x(j)

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ . .

. .

�

**g***i*(*mj*) =

Fig. 3. The *j*-th stage two-channel time-varying filter bank. Furthermore, we define two special matrices **Λ**<sup>0</sup> and **Λ**<sup>1</sup>

**Λ**<sup>0</sup> =

**Λ**<sup>1</sup> =

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

�

··· . .

**h***i*(*mj*) =

. .

. .

. .

*N*(*j*) = 2*J*−*<sup>j</sup>*

$$\mathbf{y}^{(j)} = \left[ \cdots y\_0^{(j)}(-1) \ y\_1^{(j)}(-1) \ y\_0^{(j)}(0) \ y\_1^{(j)}(0) \ y\_0^{(j)}(1) \ y\_1^{(j)}(1) \ \cdots \right]^T,\tag{11}$$

$$\mathbf{y}\_0^{(j)} = \left[ \cdots y\_0^{(j)}(-1) \ y\_0^{(j)}(0) \ y\_0^{(j)}(1) \cdots \right]^T,\tag{12}$$

$$\mathbf{y}\_1^{(j)} = \left[ \cdots y\_1^{(j)} (-1) \ y\_1^{(j)} (0) \ y\_1^{(j)} (1) \cdots \right]^T. \tag{13}$$

Based on the above matrix definitions we can describe the filter bank at the *j*-th stage showed in Fig. 3 as

$$
\hat{\mathbf{x}}^{(j)} = \mathbf{T}\_{ms}^{(j)} \mathbf{T}\_{ma}^{(j)} \mathbf{x}^{(j)}.\tag{14}
$$

After adding the (j+1)-th stage with a biorthogonal time-varying two-channel filter bank shown in Fig. 4, we have

$$\begin{aligned} \hat{\mathbf{x}}^{(j)} &= \mathbf{T}\_{ms}^{(j)} \mathbf{y}^{(j)} \\ &= \mathbf{T}\_{ms}^{(j)} \left\{ \mathbf{A}\_0^T \mathbf{y}\_0^{(j)} + \mathbf{A}\_1^T \mathbf{y}\_1^{(j)} \right\} \\ &= \mathbf{T}\_{ms}^{(j)} \left\{ \mathbf{A}\_0^T \mathbf{A}\_0 \mathbf{T}\_{ma}^{(j)} \mathbf{x}^{(j)} + \mathbf{A}\_1^T \mathbf{T}\_{ms}^{(j+1)} \mathbf{T}\_{ma}^{(j+1)} \mathbf{A}\_1 \mathbf{T}\_{ma}^{(j)} \mathbf{x}^{(j)} \right\}. \end{aligned} \tag{15}$$

Because we suppose that the added two-channel filter bank is biorthogonal, we have

$$\mathbf{T}\_{ms}^{(j+1)} \cdot \mathbf{T}\_{ma}^{(j+1)} = \mathbf{I}\_{\prime} \tag{16}$$

$$
\boldsymbol{\Lambda}\_0^T \boldsymbol{\Lambda}\_0 + \boldsymbol{\Lambda}\_1^T \boldsymbol{\Lambda}\_1 = \mathbf{I}.\tag{17}
$$

Then, we can rewrite (15) as

$$
\hat{\mathbf{x}}^{(j)} = \mathbf{T}\_{ms}^{(j)} \left\{ \mathbf{A}\_0^T \mathbf{A}\_0 + \mathbf{A}\_1^T \mathbf{A}\_1 \right\} \mathbf{T}\_{ma}^{(j)} \mathbf{x}^{(j)}
$$

$$
= \mathbf{T}\_{ms}^{(j)} \mathbf{T}\_{ma}^{(j)} \mathbf{x}^{(j)}
$$

$$
= \mathbf{x}^{(j)} \tag{18}
$$

which means that the time-varying octave-band filter bank is still perfectly reconstructed after adding next stage of time-varying biorthogonal two-channel filter bank. In other words, theorem 1 is correct.

Wavelet Transforms 7

Time-Varying Discrete-Time Wavelet Transforms 563

y<sup>0</sup>,<sup>2</sup>(m)

y<sup>0</sup>,K(m)

y<sup>1</sup>,<sup>1</sup>(m)

y<sup>J</sup>−<sup>1</sup>(m)

y<sup>J</sup> (m)

**<sup>y</sup>**<sup>0</sup> <sup>=</sup> **<sup>Λ</sup>**<sup>0</sup> **<sup>T</sup>**(1)

**y**<sup>0</sup> = [ ··· **y**0(−1) **y**0(0) **y**0(1)··· ]

*ma* **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(*j*−1) *ma* ··· **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(1)

 **H***j*−<sup>1</sup>

*ma* **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(*J*−1) *ma* ··· **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(1)

*ma* **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(*J*−1) *ma* ··· **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(1)

··· **<sup>y</sup>***j*−1(−1) **<sup>y</sup>***j*−1(0) **<sup>y</sup>***j*−1(1)··· *<sup>T</sup>*

*yj*−1,1(*m*) *yj*−1,2(*m*) ··· *yj*−1,*K*(*m*)

*ma*

**y**0(*m*) = [ *y*0,1(*m*) *y*0,2(*m*) ··· *y*0,*K*(*m*) ]

y<sup>0</sup>,<sup>1</sup>(m)

↓ M

↓ M

↓ M

↓ M

↓ M

↓ M

Fig. 5. The equivalent *M*-channel time-varying uniform filter bank.

↑ M

↑ M

↑ M

z−2<sup>J</sup>

G0(z,m)

G1(z,m)

G0(z,m)

z−<sup>2</sup>G0(z,m)

G<sup>J</sup> (z,m)

*ma* **x**, (21)

*<sup>T</sup>*, (22)

*<sup>T</sup>*, (23)

**x**, (24)

*ma* **<sup>x</sup>** = **<sup>H</sup>***J*−<sup>1</sup> **<sup>x</sup>**, (25)

*ma* **x** = **H***<sup>J</sup>* **x**, (26)

*T*

, (27)

, (28)

G<sup>J</sup>−<sup>1</sup>(z,m)

x(n)

↑ M

↑ M

↑ M

H<sup>J</sup> (z,m)

<sup>−</sup><sup>1</sup> <sup>−</sup> 1. In general, we have

**<sup>y</sup>***j*−<sup>1</sup> <sup>=</sup> **<sup>Λ</sup>**<sup>0</sup> **<sup>T</sup>**(*j*)

**<sup>y</sup>***J*−<sup>1</sup> <sup>=</sup> **<sup>Λ</sup>**<sup>0</sup> **<sup>T</sup>**(*J*)

**<sup>y</sup>***<sup>J</sup>* <sup>=</sup> **<sup>Λ</sup>**<sup>1</sup> **<sup>T</sup>**(*J*)

**<sup>y</sup>***j*−<sup>1</sup> =

**<sup>y</sup>***j*−1(*m*) =

**y**0(*m*0) can be expressed as

where

and *K* = 22*<sup>J</sup>*

where

where 1 ≤ *j* ≤ *J* − 1, and

H<sup>J</sup>−<sup>1</sup>(z,m)

H1(z,m)

H0(z,m)

z−2<sup>J</sup>

H0(z,m)

z−<sup>2</sup>H0(z,m)

x(n)

Fig. 4. Adding the (*j* + 1)-th stage.

#### **4. Implementation with time-varying nonuniform filter banks**

Fig. 2 shows another implementation of a time-varying wavelet transform with (*J*(*m*) + 1)-channel time-varying nonuniform filter bank. To make the analysis easy we suppose that *J*(*m*) does not change with time and is equal to constant *J*. For analysis of the (*J* + 1)-channel time-varying nonuniform filter bank we first reconstruct the nonuniform filter bank to a time-varying uniform filter bank through adding following filters between *Hi*(*z*, *m*) and *Hi*<sup>+</sup>1(*z*, *m*) (0 ≤ *j J* − 1)

$$H\_{i,k}(z,m) = z^{-k \cdot 2^{l+1}} H\_l(z,m), \quad 1 \le k \le 2^{l-i-1} - 1. \tag{19}$$

After adding additional filters in the nonuniform filter bank in Fig. 2 the filter bank becomes *M*-channel time-varying uniform filter filter bank. The number of channel *M* is calculated by

$$\begin{aligned} \mathcal{M} &= \sum\_{i=0}^{I-2} (2^{I-i-1} - 1) + (f+1) \\ &= 2^{I-1} \sum\_{i=0}^{I-2} 2^{-i} + 2 \\ &= 2^I (1 - 2 \cdot 2^{-I}) + 2 \\ &= 2^I. \end{aligned}$$

For the time-varying system in Fig. 5 we have following theorem.

**Theorem 2**: A time-varying discrete-time wavelet transform implemented with a time-varying nonuniform filter bank is biorthogonal if each two-channel time-varying filter bank in its tree-structured implementation is perfectly reconstructed.

To prove theorem 2, we need to describe the filter *Hi*(*z*, *m*) in Fig. 5 based on the tree structure in Fig. 1. In the time-invariant discrete-time wavelet transform the description of such filters can be simply got using the convolution role in the transform-domain. However, in the time-varying case, we cannot describe *Hi*(*z*, *m*) as product of functions in the previous stages, like *H*0(*z*, *m*)*H*1(*z*2, *m*), because the system is time-varying and the convolution role does not exist. Referencing to definitions of **T**(*j*) *ma* and **<sup>T</sup>**(*j*) *ms* in (2) and (3), we find that the analysis output

Fig. 5. The equivalent *M*-channel time-varying uniform filter bank.

**y**0(*m*0) can be expressed as

$$\mathbf{y}\_0 = \mathbf{A}\_0 \ \mathbf{T}\_{ma}^{(1)} \ \mathbf{x}\_{\prime} \tag{21}$$

where

6 Will-be-set-by-IN-TECH

<sup>G</sup>u(z,m) <sup>x</sup>(n) <sup>x</sup>(j)

y(j) <sup>0</sup> (mj)

<sup>x</sup>(j+1)(n) <sup>y</sup>(j)

(j+1) stage

Hl(z,m) Gl(z,m)

Fig. 2 shows another implementation of a time-varying wavelet transform with (*J*(*m*) + 1)-channel time-varying nonuniform filter bank. To make the analysis easy we suppose that *J*(*m*) does not change with time and is equal to constant *J*. For analysis of the (*J* + 1)-channel time-varying nonuniform filter bank we first reconstruct the nonuniform filter bank to a time-varying uniform filter bank through adding following filters between *Hi*(*z*, *m*) and

After adding additional filters in the nonuniform filter bank in Fig. 2 the filter bank becomes *M*-channel time-varying uniform filter filter bank. The number of channel *M* is calculated by

(2*J*−*i*−<sup>1</sup> <sup>−</sup> <sup>1</sup>)+(*<sup>J</sup>* <sup>+</sup> <sup>1</sup>)

) + 2

2−*<sup>i</sup>* + 2

↓ 2 Hu(z,m) ↑ 2

↑ 2

↑ 2 Gu(z,m)

Hu(z,m) ↑ 2

↓ 2

Gl(z,m)

j stage

<sup>1</sup> (m<sup>j</sup> )

*Hi*(*z*, *<sup>m</sup>*), 1 <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> <sup>2</sup>*J*−*i*−<sup>1</sup> <sup>−</sup> 1. (19)

. (20)

*ms* in (2) and (3), we find that the analysis output

(n)

↓ 2

Fig. 4. Adding the (*j* + 1)-th stage.

j stage <sup>↓</sup> <sup>2</sup>

**4. Implementation with time-varying nonuniform filter banks**

*Hi*,*k*(*z*, *<sup>m</sup>*) = *<sup>z</sup>*−*k*·2*i*+<sup>1</sup>

*M* =

*J*−2 ∑ *i*=0

= 2*J*−<sup>1</sup>

= 2*<sup>J</sup>*

= 2*<sup>J</sup>*

For the time-varying system in Fig. 5 we have following theorem.

bank in its tree-structured implementation is perfectly reconstructed.

exist. Referencing to definitions of **T**(*j*)

*J*−2 ∑ *i*=0

(<sup>1</sup> <sup>−</sup> <sup>2</sup> · <sup>2</sup>−*<sup>J</sup>*

**Theorem 2**: A time-varying discrete-time wavelet transform implemented with a time-varying nonuniform filter bank is biorthogonal if each two-channel time-varying filter

To prove theorem 2, we need to describe the filter *Hi*(*z*, *m*) in Fig. 5 based on the tree structure in Fig. 1. In the time-invariant discrete-time wavelet transform the description of such filters can be simply got using the convolution role in the transform-domain. However, in the time-varying case, we cannot describe *Hi*(*z*, *m*) as product of functions in the previous stages, like *H*0(*z*, *m*)*H*1(*z*2, *m*), because the system is time-varying and the convolution role does not

*ma* and **<sup>T</sup>**(*j*)

Hl(z,m)

*Hi*<sup>+</sup>1(*z*, *m*) (0 ≤ *j J* − 1)

$$\mathbf{y}\_0 = \begin{bmatrix} \cdots \mathbf{y}\_0(-1) \ \mathbf{y}\_0(0) \ \mathbf{y}\_0(1) \cdots \end{bmatrix}^T,\tag{22}$$

$$\mathbf{y}\_0(m) = \begin{bmatrix} y\_{0,1}(m) \ y\_{0,2}(m) \ \cdots \ y\_{0,K}(m) \ \end{bmatrix}^T,\tag{23}$$

and *K* = 22*<sup>J</sup>* <sup>−</sup><sup>1</sup> <sup>−</sup> 1. In general, we have

$$\mathbf{y}\_{j-1} = \underbrace{\mathbf{A}\_0 \mathbf{T}\_{ma}^{(j)} \mathbf{A}\_1 \mathbf{T}\_{ma}^{(j-1)} \cdots \mathbf{A}\_1 \mathbf{T}\_{ma}^{(1)}}\_{\mathbf{H}\_{j-1}} \mathbf{x}\_{\mathbf{y}} \tag{24}$$

where 1 ≤ *j* ≤ *J* − 1, and

$$\mathbf{y}\_{f-1} = \mathbf{A}\_0 \mathbf{T}\_{ma}^{(f)} \mathbf{A}\_1 \mathbf{T}\_{ma}^{(f-1)} \cdots \mathbf{A}\_1 \mathbf{T}\_{ma}^{(1)} \mathbf{x} = \mathbf{H}\_{f-1} \mathbf{x},\tag{25}$$

$$\mathbf{y}\_{l} = \mathbf{A}\_{1} \ \mathbf{T}\_{ma}^{(l)} \ \mathbf{A}\_{1} \ \mathbf{T}\_{ma}^{(l-1)} \ \cdots \ \mathbf{A}\_{1} \ \mathbf{T}\_{ma}^{(1)} \ \mathbf{x} = \mathbf{H}\_{l} \ \mathbf{x}\_{l} \tag{26}$$

where

$$\mathbf{y}\_{j-1} = \left[ \cdots \mathbf{y}\_{j-1}(-1) \; \mathbf{y}\_{j-1}(0) \; \mathbf{y}\_{j-1}(1) \cdots \right]^T \; \tag{27}$$

$$\mathbf{y}\_{j-1}(m) = \begin{bmatrix} y\_{j-1,1}(m) \ y\_{j-1,2}(m) \ \cdots \ y\_{j-1,K}(m) \end{bmatrix}^T \tag{28}$$

for *K* = 22*<sup>J</sup>* <sup>−</sup>*<sup>j</sup>* <sup>−</sup> 1 and 1 <sup>≤</sup> *<sup>j</sup>* <sup>≤</sup> *<sup>J</sup>* <sup>−</sup> 2, and

$$\mathbf{y}\_{I-1} = \begin{bmatrix} \cdots \ y\_{I-1}(-1) \ y\_{I-1}(0) \ y\_{I-1}(1) \cdots \end{bmatrix}^T \tag{29}$$

$$\mathbf{y}\_I = \begin{bmatrix} \cdots \ y\_I(-1) \ y\_I(0) \ y\_I(1) \cdots \end{bmatrix}^T. \tag{30}$$

Wavelet Transforms 9

Time-Varying Discrete-Time Wavelet Transforms 565

In the theory of discrete-time signal expansion, the wavelet transform is very important. In this chapter, we defined the general discrete time-varying dyadic wavelet transform and analyzed its properties in detail. Some theorems describing properties of time-varying discrete-time wavelet transforms were presented. The conditions for a biorthogonal time-varying discrete-time wavelet transform were given. The theory and algorithms presented in this chapter can be used in design of time-varying discrete-time signal expansion

This work is supported by the National Natural Science Foundation of China (No. 61071195)

[1] H.S. Malvar, *Signal Processing with Lapped Transforms.* Boston, MA: Artech House, 1992. [2] M. Vetterli and J. Kovacevic, *Wavelets and Subband Coding,* Englewood Cliffs, NJ: Prentice

[3] Alfred Mertins, *Signal Analysis, Wavelets, Filter Banks, Time-Frequency Transforms and*

[4] V. DeBrunner, W. Lou, and J. Thripuraneni, "Multiple Transform Algorithms for Time-Varying Signal Representation," *IEEE Trans. Circuits and Systems-II: Analog and*

[5] M. Morhac and V. Matousek, "New Adaptive Cosine-Walsh Transform and its Application to Nuclear Data Compression,"*IEEE Trans. Signal Processing,* vol. 48, No. 9,

[6] N. Prelcic, A. Pena, "An Adaptive Tiling of the Time-Frequency Plane with Application to Multiresolution-Based Perceptive Audio Coding," *Signal Processing,* vol. 81, pp. 301-319,

[7] P. Srinivasan and L. Jamieson, "High-Quality Audio Compression Using an Adaptive Wavelet Packet Decomposition and Psychoacoustic Modeling," *IEEE Trans. Signal*

[8] ISO/IEC JTCI/SC29, "Information technology-coding of moving pictures and associated audio for digital storage media at up to about 1.5 Mbit/s-IS 11172-3 (audio)," 1992. [9] C. Guillemot, P. Rault, and P. Onno, "Time-invariant and time-varying multirate filter banks: application to image coding ," *Annales des Telecommunications-Annals of*

[10] I. Sodagar, K. Nayebi, and T.P Barnwell,III, "Time-varying filter banks and wavelets,"

[11] R.L. de Queiroz, K.R. Rao, "Time-varying lapped transform and wavelet packets," *IEEE*

[12] C. Herley, M. Vetterli, "Orthogonal Time-Varying Filter Banks and Wavelet Packets,"

**5. Conclusion**

in practice.

**7. References**

Hall, 1995.

Feb. 2001.

**6. Acknowledgments**

and Sino-Finland Cooperation Project (No. 1018).

*Application* England: John Wiley & Sons, 1999.

*Processing,* vol. 46, No. 4, pp. 1085-1093, April 1998.

*Telecommunications*, vol. 53, No.5-6, pp.192-218, May-Jun 1998.

*Trans. Signal Processing,* vol. 41, No. 12, pp. 3293-3305, Dec. 1993.

*IEEE Trans. Signal Processing,* vol. 42, No. 11, pp. 2983-2996, Nov. 1994.

*IEEE Trans. Signal Processing,* vol. 42, No.10, pp. 2650-2663, Oct. 1994.

pp. 2663-2696, September 2000.

*Digital Signal Processing,* vol. 44, No. 8, pp. 663-667, Aug. 1997.

At synthesis side, we have similar definitions as

$$\widehat{\mathbf{x}}\_{j-1} = \underbrace{\mathbf{T}\_{ms}^{(1)} \, \mathbf{A}\_1^T \, \mathbf{T}\_{ms}^{(2)} \, \cdots \, \mathbf{A}\_1^T \, \mathbf{T}\_{ms}^{(j)} \, \mathbf{A}\_0^T \, \mathbf{y}\_{j-1}}\_{\mathbf{G}\_{j-1}} \, \mathbf{y}\_{j-1} \tag{31}$$

$$\widehat{\mathbf{x}}\_{I-1} = \mathbf{T}\_{\mathrm{ms}}^{(1)} \boldsymbol{\Lambda}\_1^T \, \mathbf{T}\_{\mathrm{ms}}^{(2)} \, \cdots \, \boldsymbol{\Lambda}\_1^T \, \mathbf{T}\_{\mathrm{ms}}^{(I)} \, \boldsymbol{\Lambda}\_0^T \, \mathbf{y}\_{I-1} = \mathbf{G}\_{I-1} \, \mathbf{y}\_{I-1} \tag{32}$$

$$\hat{\mathbf{x}}\_{I} = \mathbf{T}\_{ms}^{(1)} \,\mathbf{A}\_{1}^{T} \,\mathbf{T}\_{ms}^{(2)} \,\cdots \,\mathbf{A}\_{1}^{T} \,\mathbf{T}\_{ms}^{(I)} \,\mathbf{A}\_{1}^{T} \,\mathbf{y}\_{J} = \mathbf{G}\_{I} \,\mathbf{y}\_{I'} \tag{33}$$

Now, based on the definition in (23), we can build the analysis output vector for th time-varying filter bank in Fig. 5 as

$$\mathbf{y} = \begin{bmatrix} \cdots \mathbf{y}\_0(-1) \ \cdots \mathbf{y}\_I(-1) \ \mathbf{y}\_0(0) \ \cdots \ \mathbf{y}\_I(0) \ \mathbf{y}\_0(1) \ \cdots \ \end{bmatrix}^T. \tag{34}$$

Suppose that **T***ma* and **T***ms* are the analysis and synthesis matrices for the time-varying filter bank in Fig. 5. Referencing (34), **<sup>T</sup>***ma* is constructed by interleaving the rows from **<sup>T</sup>**(1) *ma* to **<sup>T</sup>**(*J*) *ma* with same time index *m*, **T***ms* is built with similar way, but interleaving the columns. Then, the production **T**(1) *ma***T**(*J*) *ma* can be expressed by

$$\mathbf{T}\_{ms}\ \mathbf{T}\_{ma} = \sum\_{j=0}^{J} \mathbf{G}\_{j}\ \mathbf{H}\_{j}.\tag{35}$$

Substituting **H***<sup>i</sup>* and **G***<sup>i</sup>* defined in (24)-(26) and (31)-(33) into (35), and using properties in (16) and (17), we get

$$\mathbf{T}\_{ms}\,\mathbf{T}\_{ma} = \mathbf{I}\_{\prime} \tag{36}$$

which means that the time-varying nonuniform filter bank in Fig. 2 is perfectly reconstructed.

Finally, we give another property related with filter coefficients of the time-varying filter bank in Fig. 2. Suppose that *hi*(*n*, *m*) and *gi*(*n*, *m*) represent the analysis and synthesis filter coefficients in Fig. 2, then we have following equation

$$<\langle g\_i(n - kM, m + r), \ h\_j(n - lM, m + s)> = \delta(k - l)\,\delta(i - j)\,\delta(r - s),\tag{37}$$

where *M* = 2*<sup>J</sup>* . The proof of equation (37) can be simply got by using the PR condition in (36).
