1. Introduction

Filter banks can be defined as the cascaded arrangement of filters, i.e., low-pass, high-pass, and band-pass filters connected by sampling operators in such a manner as to achieve the decomposition and recomposition of a signal from a spectrum perspective. The sampling operators could either be downsamplers or upsamplers. The downsamplers are called decimators while the upsamplers are called expanders. The technique of filter banks plays an important role in most digital systems that rely on signal processing for their operations. Using this technique, any signal feature can be reliably extracted and analyzed; hence filter banks have wide applications in digital signal processing systems. A filter bank as shown in Figure 1 [1, 2] consists of different parts, which collectively execute a desired function.

As can be seen in Figure 1, the filter bank is made of two sections: the analysis filter bank section (composed of analysis filters and downsamplers), and the synthesis filter bank section (composed of upsamplers and synthesis filters). In this chapter, we will discuss the analysis and synthesis filter bank sections, their

### Figure 1. k-Channel filter bank [1, 2].

responses to incoming signals, and how they work together in the derivation of a wavelet transform function.

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

The downsampler shown in Figure 1 downsamples an input signal by a factor of

x n ^ð Þ¼ xð Þ 2n , ∀n ∈ ℤ (2)

uið Þ n δð Þ n � kN , ∀k∈ ℤ (3)

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

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

við Þ¼ n ∑

k∈ ℤ

Fourier series expansion of the impulse series can be expressed as [5]:

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

The relationship in Eq. (3) will only select the kNth sample of uið Þ <sup>n</sup> , and the

mathematical operation it performs on a given signal.

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

Analysis of Wavelet Transform Design via Filter Bank Technique

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

2.2 Downsampler/decimator

Figure 2.

downsample the signal as:

samples apart, i.e.,

Figure 3.

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Decimation by a factor of 2.
