3. Synthesis filter bank section

The synthesis filter bank section is made of the upsamplers and synthesis filter banks. These components work together to perform the opposite operation performed by the analysis filter bank section shown in Figure 1. In this section, we will make an analysis of the mathematical relationship that governs the operation of the synthesis filters and upsamplers.

### 3.1 Synthesis filter bank

Similar to the analysis filter bank, the synthesis filter bank is made of low-pass and high-pass filters. The output of these filters as shown in Figure 1, are summed to a common output. In typical filter bank applications, the frequency responses of these filters are typically matched to those of the analysis filters shown in Figure 2. The mathematical expression for the effect each of these filters has on the corresponding input signal wið Þ n is as stated below [6]:

$$\begin{aligned} P\_0(\mathbf{Z}) &= W\_0(\mathbf{Z}) \mathbf{G}\_0(\mathbf{Z}) \\ P\_1(\mathbf{Z}) &= W\_1(\mathbf{Z}) \mathbf{G}\_1(\mathbf{Z}) \\ P\_2(\mathbf{Z}) &= W\_2(\mathbf{Z}) \mathbf{G}\_2(\mathbf{Z}) \\ P\_{k-1}(\mathbf{Z}) &= W\_{k-1}(\mathbf{Z}) \mathbf{G}\_{M-1}(\mathbf{Z}) \end{aligned} \tag{8}$$

In Figure 2, the input to the synthesis filter bank is upsamplers or expanders. The next section gives a brief review of the upsamplers.

### 3.2 Upsampler/expander

The upsampler expands an input signal by a factor N. It does this by inserting zeros at every nth position in the sequence of the input signal. For example, if

Analysis of Wavelet Transform Design via Filter Bank Technique DOI: http://dx.doi.org/10.5772/intechopen.85051

Figure 4. Upsampling by a factor of 2.

∑ k∈ ℤ

> ∑ k∈ ℤ

> > við Þ¼ n

Við Þ¼ <sup>Z</sup> <sup>1</sup>

attention to synthesis filter bank section of Figure 1.

3. Synthesis filter bank section

the synthesis filters and upsamplers.

3.1 Synthesis filter bank

3.2 Upsampler/expander

22

Substituting Eq. (5) into (3) yields:

Wavelet Transform and Complexity

<sup>δ</sup>ð Þ¼ <sup>n</sup> � kN <sup>1</sup>

Setting WN <sup>¼</sup> <sup>e</sup>�j2π=<sup>N</sup> and <sup>n</sup> <sup>¼</sup> 1, the relationship in Eq. (4) becomes:

<sup>δ</sup>ð Þ¼ <sup>n</sup> � kN <sup>1</sup>

1 <sup>N</sup> <sup>∑</sup> N�1 k¼0

<sup>N</sup> <sup>∑</sup> N�1 k¼0

In terms of z-transformation, the relationship in Eq. (6) can be expressed as:

Having looked at the analysis filters and downsamplers, we will now turn our

The synthesis filter bank section is made of the upsamplers and synthesis filter

performed by the analysis filter bank section shown in Figure 1. In this section, we will make an analysis of the mathematical relationship that governs the operation of

Similar to the analysis filter bank, the synthesis filter bank is made of low-pass and high-pass filters. The output of these filters as shown in Figure 1, are summed to a common output. In typical filter bank applications, the frequency responses of these filters are typically matched to those of the analysis filters shown in Figure 2.

banks. These components work together to perform the opposite operation

The mathematical expression for the effect each of these filters has on the

P0ð Þ¼ Z W0ð Þ Z G0ð Þ Z P1ð Þ¼ Z W1ð Þ Z G1ð Þ Z P2ð Þ¼ Z W2ð Þ Z G2ð Þ Z

Pk�<sup>1</sup>ð Þ¼ Z Wk�<sup>1</sup>ð Þ Z GM�<sup>1</sup>ð Þ Z

In Figure 2, the input to the synthesis filter bank is upsamplers or expanders.

The upsampler expands an input signal by a factor N. It does this by inserting zeros at every nth position in the sequence of the input signal. For example, if

corresponding input signal wið Þ n is as stated below [6]:

The next section gives a brief review of the upsamplers.

<sup>N</sup> <sup>∑</sup> N�1 k¼0 e

> <sup>N</sup> <sup>∑</sup> N�1 k¼0

uið Þ <sup>n</sup> <sup>W</sup>�<sup>k</sup>

Ui Z<sup>1</sup>

NW�<sup>k</sup> N 

W�<sup>k</sup>

�j2πkn=<sup>N</sup> (4)

<sup>N</sup> (5)

<sup>N</sup> (6)

(7)

(8)

N ¼ 2, then the upsampler will insert a zero between every two adjacent samples in a given sequence as shown in Figure 4.

Given an input signal við Þ n in Figure 1, an upsampler with a factor of 2 will upsample the signal using the relationship [7]:

$$\omega\_i(n) = \sum\_{k \in \mathbb{Z}} v\_i(n)\delta(n - kN), \forall k \in \mathbb{Z} \tag{9}$$

Similar to the expression in Eq. (3), the z-transform of the expression in Eq. (9) which is an upsampler is stated as follows [8]:

$$W\_i(Z) = \frac{1}{N} \sum\_{k=0}^{N-1} V\_i \left( Z^N W\_N^{-k} \right) \tag{10}$$

To be useful in wavelet designs, filter banks must be designed to have certain characteristics which guarantee that a signal at the input of a filter bank will be received accurately at the output of the filter bank. In the next section, we will examine the properties of filter banks and how these properties influence the design of wavelets.
