2. Analysis filter bank section

The analysis filter bank section is made up of the analysis filter banks, and downsamplers or decimators which together act on an input signal to perform a desired function through decomposition of the signal. In this section, we will analyze the mathematical relationship that exists between these two components. To have a thorough understanding of this relationship, it is important to briefly discuss these components separately.

### 2.1 Analysis filter bank

The filters that make up the analysis filter banks could either be low-pass filters, or high-pass filters. Each of these filters, as shown in Figure 2, allows the passage of only a particular frequency component of the input signal y nð Þ. Thus, specific features of the input signal embedded at different frequencies can be individually extracted and investigated using the analysis filter bank [3, 4]. The k-channel filter bank in Figure 2 separates the frequencies of the input signal in the manner presented.

It can be seen from the frequency responses that the output of the filters overlap each other. This is because in practice, the filters are not ideal. However, the overlapping condition can be improved through an optimized design of the filters. Mathematically, the effect of each of the filters in the filter bank on the input signal y nð Þ can be stated as follows:

$$\begin{aligned} U\_0(\mathbf{Z}) &= Y(\mathbf{Z})H\_0(\mathbf{Z}) \\ U\_1(\mathbf{Z}) &= Y(\mathbf{Z})H\_1(\mathbf{Z}) \\ U\_2(\mathbf{Z}) &= Y(\mathbf{Z})H\_2(\mathbf{Z}) \\ U\_{k-1}(\mathbf{Z}) &= Y(\mathbf{Z})H\_{M-1}(\mathbf{Z}) \end{aligned} \tag{1}$$

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

Figure 2.

responses to incoming signals, and how they work together in the derivation of a

The analysis filter bank section is made up of the analysis filter banks, and downsamplers or decimators which together act on an input signal to perform a desired function through decomposition of the signal. In this section, we will analyze the mathematical relationship that exists between these two components. To have a thorough understanding of this relationship, it is important to briefly discuss

The filters that make up the analysis filter banks could either be low-pass filters, or high-pass filters. Each of these filters, as shown in Figure 2, allows the passage of only a particular frequency component of the input signal y nð Þ. Thus, specific features of the input signal embedded at different frequencies can be individually extracted and investigated using the analysis filter bank [3, 4]. The k-channel filter bank in Figure 2 separates the frequencies of the input signal in the manner

It can be seen from the frequency responses that the output of the filters overlap

U0ð Þ¼ Z Y Zð ÞH0ð Þ Z U1ð Þ¼ Z Y Zð ÞH1ð Þ Z U2ð Þ¼ Z Y Zð ÞH2ð Þ Z Uk�<sup>1</sup>ð Þ¼ Z Y Zð ÞHM�<sup>1</sup>ð Þ Z

(1)

each other. This is because in practice, the filters are not ideal. However, the overlapping condition can be improved through an optimized design of the filters. Mathematically, the effect of each of the filters in the filter bank on the input signal

wavelet transform function.

Wavelet Transform and Complexity

k-Channel filter bank [1, 2].

Figure 1.

these components separately.

y nð Þ can be stated as follows:

2.1 Analysis filter bank

presented.

20

2. Analysis filter bank section

Separation of input signals into sub-band frequencies by analysis filter bank.

where Uið Þz is the z-transform of the result from the convolution operation between the z-transform of the input signal Y Zð Þ and the z-transform of the filter Hið Þ Z . The output Uið Þz in Figure 2 is fed into the corresponding downsampler of Figure 1. In the next section, we will analyze the downsampler and state the mathematical operation it performs on a given signal.

### 2.2 Downsampler/decimator

The downsampler shown in Figure 1 downsamples an input signal by a factor of N. This implies that it only retains all the Nth samples in a given sequence. For example, if N ¼ 2, then the downsampler will retain all even samples in a given sequence. Given an input signal x nð Þ, the downsampler with a factor of 2 will downsample the signal as:

$$
\hat{\mathfrak{x}}(n) = \mathfrak{x}(2n), \forall n \in \mathbb{Z} \tag{2}
$$

Figure 3 shows the conceptual depiction of the relationship in Eq. (2).

Mathematically, the output of the decimator in Figure 1 can be expressed as a product of the input sequence uið Þ n and the sequence of unit impulses which are N samples apart, i.e.,

$$w\_i(n) = \sum\_{k \in \mathbb{Z}} u\_i(n)\delta(n - kN), \forall k \in \mathbb{Z} \tag{3}$$

The relationship in Eq. (3) will only select the kNth sample of uið Þ <sup>n</sup> , and the Fourier series expansion of the impulse series can be expressed as [5]:

Figure 3. Decimation by a factor of 2.

$$\sum\_{k \in \mathbb{Z}} \delta(n - kN) = \frac{1}{N} \sum\_{k=0}^{N-1} e^{-j2\pi kn/N} \tag{4}$$

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:

$$\sum\_{k \in \mathbb{Z}} \delta(n - kN) = \frac{1}{N} \sum\_{k=0}^{N-1} W\_N^{-k} \tag{5}$$

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

$$w\_i(n) = \frac{1}{N} \sum\_{k=0}^{N-1} u\_i(n) W\_N^{-k} \tag{6}$$

N ¼ 2, then the upsampler will insert a zero between every two adjacent samples in

Given an input signal við Þ n in Figure 1, an upsampler with a factor of 2 will

Similar to the expression in Eq. (3), the z-transform of the expression in Eq. (9)

Vi ZNW�<sup>k</sup> N

<sup>N</sup> <sup>∑</sup> N�1 k¼0

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.

In wavelet designs, filter banks are required to possess three important properties which are fundamental to the realization of a wavelet function. These properties

This property guarantees that the signal at the output of a given filter bank is a delayed version of the signal at the input of the filter bank. Perfect reconstruction is an important property of a filter bank because it cancels the effect of aliasing of the input signal at the output, caused by the downsamplers and upsamplers. To understand this point, consider a two-channel finite impulse response FIR filter bank shown in Figure 5. The output ^y nð Þ is derived using Eqs. (6) and (10) as follows in terms of the

where the signal\_component and aliasing\_component are defined as:

2

2

Y z ^ ð Þ¼ signal\_component <sup>þ</sup> aliasing\_component (11)

½ � F0ð Þz H0ð Þþ z F1ð Þz H1ð Þz X zð Þ

9 >=

>;

(12)

½ � F0ð Þz H0ð Þþ �z F1ð Þz H1ð Þ �z Xð Þ �z

include: perfect reconstruction, orthogonality, and paraunitary condition.

við Þ n δð Þ n � kN , ∀k∈ ℤ (9)

� � (10)

a given sequence as shown in Figure 4.

Figure 4.

Upsampling by a factor of 2.

upsample the signal using the relationship [7]:

which is an upsampler is stated as follows [8]:

4. Properties of filter banks for wavelet design

signal component and aliasing component as:

signal\_component <sup>¼</sup> <sup>1</sup>

aliasing\_component <sup>¼</sup> <sup>1</sup>

23

4.1 Perfect reconstruction

wið Þ¼ n ∑

Analysis of Wavelet Transform Design via Filter Bank Technique

DOI: http://dx.doi.org/10.5772/intechopen.85051

k∈ ℤ

Wið Þ¼ <sup>Z</sup> <sup>1</sup>

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

$$W\_i(Z) = \frac{1}{N} \sum\_{k=0}^{N-1} U\_i \left( Z^{\frac{1}{N}} W\_N^{-k} \right) \tag{7}$$

Having looked at the analysis filters and downsamplers, we will now turn our attention to synthesis filter bank section of Figure 1.
