**5. Conclusion**

8 Will-be-set-by-IN-TECH

··· *yJ*−1(−1) *yJ*−1(0) *yJ*−1(1)··· *<sup>T</sup>*, (29)

··· *yJ*(−1) *yJ*(0) *yJ*(1)··· *<sup>T</sup>*. (30)

**<sup>y</sup>***j*−1, (31)

*ma* to **<sup>T</sup>**(*J*) *ma*

<sup>0</sup> **<sup>y</sup>***J*−<sup>1</sup> = **<sup>G</sup>***J*−<sup>1</sup> **<sup>y</sup>***J*−1, (32)

<sup>1</sup> **y***<sup>J</sup>* = **G***<sup>J</sup>* **y***J*, (33)

**G***<sup>j</sup>* **H***j*. (35)

**T***ms* **T***ma* = **I**, (36)

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

At synthesis side, we have similar definitions as

**<sup>x</sup>***J*−<sup>1</sup> <sup>=</sup> **<sup>T</sup>**(1)

**<sup>x</sup>***<sup>J</sup>* <sup>=</sup> **<sup>T</sup>**(1)

time-varying filter bank in Fig. 5 as

the production **T**(1)

and (17), we get

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

**y** =

*ma***T**(*J*)

**<sup>y</sup>***J*−<sup>1</sup> <sup>=</sup>

**y***<sup>J</sup>* =

**<sup>x</sup>***j*−<sup>1</sup> <sup>=</sup> **<sup>T</sup>**(1)

*ms* **Λ***<sup>T</sup>*

*ms* **Λ***<sup>T</sup>*

*ma* can be expressed by

coefficients in Fig. 2, then we have following equation

*ms* **Λ***<sup>T</sup>*

<sup>1</sup> **<sup>T</sup>**(2)

<sup>1</sup> **<sup>T</sup>**(2)

<sup>1</sup> **<sup>T</sup>**(2)

*ms* ··· **<sup>Λ</sup>***<sup>T</sup>*

*ms* ··· **<sup>Λ</sup>***<sup>T</sup>*

*ms* ··· **<sup>Λ</sup>***<sup>T</sup>*

<sup>1</sup> **<sup>T</sup>**(*J*) *ms* **Λ***<sup>T</sup>*

<sup>1</sup> **<sup>T</sup>**(*J*) *ms* **Λ***<sup>T</sup>*

Now, based on the definition in (23), we can build the analysis output vector for th

Suppose that **T***ma* and **T***ms* are the analysis and synthesis matrices for the time-varying filter

with same time index *m*, **T***ms* is built with similar way, but interleaving the columns. Then,

*J* ∑ *j*=0

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

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

*< gi*(*n* − *kM*, *m* + *r*), *hj*(*n* − *lM*, *m* + *s*) *>*= *δ*(*k* − *l*) *δ*(*i* − *j*) *δ*(*r* − *s*), (37)

. The proof of equation (37) can be simply got by using the PR condition in (36).

bank in Fig. 5. Referencing (34), **<sup>T</sup>***ma* is constructed by interleaving the rows from **<sup>T</sup>**(1)

**T***ms* **T***ma* =

 **G***j*−<sup>1</sup>

<sup>1</sup> **<sup>T</sup>**(*j*) *ms* **Λ***<sup>T</sup>* 0

··· **<sup>y</sup>**0(−1) ··· **<sup>y</sup>***J*(−1) **<sup>y</sup>**0(0) ··· **<sup>y</sup>***J*(0) **<sup>y</sup>**0(1) ··· *<sup>T</sup>*. (34)

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 in practice.
