	Although DSP has long been considered an EE topic, recent developments have also generated signifi cant interest from the computer science community. DSP applications in the consumer market, such as bioinformatics, the MP3 audio format, and MPEG-based cable/satellite television have fueled a desire to understand this technology outside of hardware circles. Designed for upper division engineering and computer science students as well as practicing engineers, Digital Signal Processing Using MATLAB and Wavelets emphasizes the practical applications of signal processing. Over 100 MATLAB examples and wavelet techniques provide the latest applications of DSP, including image processing, games, fi lters, transforms, networking, parallel processing, and sound. The book also provides the mathematical processes and techniques needed to ensure an understanding of DSP theory. Designed to be incremental in diffi culty, the book will benefi t readers who are unfamiliar with complex mathematical topics or those limited in programming experience. Beginning with an introduction to MATLAB programming, it moves through filters, sinusoids, sampling, the Fourier transform, the z-transform and other key topics. An entire chapter is dedicated to the discussion of wavelets and their applications. A CD-ROM (platform independent) accompanies the book and contains source code, projects for each chapter, and the fi gures contained in the book. FEATURES: Contains over 100 short examples in MATLAB used throughout the book Includes an entire chapter on the wavelet transform Designed for the reader who does not have extensive math and programming experience Accompanied by a CD-ROM containing MATLAB examples, source code, projects, and fi gures from the book Contains modern applications of DSP and MATLAB project ideas

Digital Signal Processing Using MATLAB and Wavelets

By Michael Weeks

Chapter 9.7 - Wavelet Filter Design--Filters with Four Coefficients

Here we will examine designing a general filter bank with four taps per filter, very similar to that in the last section. However, one difference is that we use the z-transform in this section, which actually makes the analysis a bit easier. Figure 9.21 shows what this would look like. We specify values $a, b, c, d$

and

for coefficients in one channel, with the understanding that the coefficients for the other channel are a reversed and sign-changed version of these. That is, in the other channel we would have the coefficients $d, -c, b, -a$

and

, respectively.

We will use $H_0(z)$ and $H_1(z)$ for the lowpass and highpass filters' transfer functions, respectively, while $F_0(z)$ and $F_1(z)$ are for the corresponding inverse filters. and go with the bottom channel in Figure 9.21.

Examining the individual filter's responses, we see

$\begin{eqnarray*} F_0(z) &=& d + c z^{-1} + b z^{-2} + a z^{-3} \H_0(z) &=&... ...} \H_1(z) &=& F_0(-z) = d - c z^{-1} + b z^{-2} - a z^{-3} . \end{eqnarray*}$

Using $P_0(z)$ and $P_0(-z)$ notation to represent the product of two sequential filters, for the top channel and the bottom channel, respectively, we get the following equations for a CQF with four coefficients per filter.

$\begin{eqnarray*} P_0(z) &=& H_0(z) F_0(z) \P_0(z) &=& (a + b z^{-1} + c z^... ... bd ) z^{-1} + 2 (aa + bb + cc + dd) z^{-3} + 2 (ac + bd) z^{-5} \end{eqnarray*}$

Note that $ - P_0(-z) = H_1(z) F_1(z) $ , but we do not have to calculate this explicitly. Instead, we just repeat the expression for $P_0$ with $-z$ as the argument. As we saw in section 8.7, $-z$ raised to a negative power gives a negative value for an odd power, but returns a positive value for an even one.

**Figure 9.21:** Designing a filter bank with four taps each.

That is, we can determine the filters' responses and see how the terms cancel each other out. Then we can divide by two, which is the same effect that down/up-samplers have.

Without down/up-samplers:

$\begin{displaymath}P_0(z) - P_0(-z) = 2 ( ac + bd ) z^{-1} + 2 (aa + bb + cc + dd) z^{-3} + (ac + bd) z^{-5} = 2 z^{-L} . \end{displaymath}$

With down/up-samplers:

$\begin{displaymath}P_0(z) - P_0(-z) = ( ac + bd ) z^{-1} + (aa + bb + cc + dd) z^{-3} + (ac + bd) z^{-5} = z^{-L} . \end{displaymath}$

We are left with the output in terms of three delayed versions of the input: $z^{-1}$ , $z^{-3}$ , and $z^{-5}$ . Two of these must be eliminated, which can be accomplished by setting $ac+bd=0$ . Also, we do not want the result to be a scaled version of the input, so $aa + bb + cc + dd = 1$ . It is also desirable that the sum of the highpass filter coefficients be zero, i.e., $d - c + b - a = 0$ , for this to be a wavelet transform.