CH04 Digital Filtering

# CH04 Digital Filtering - CHAPTER 4 Digital Filtering...

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C H A P T E R 4 Digital Filtering Filtering of digital signals is a fundamental concept in digital signal processing. Here, it is assumed that the reader has already taken a theory course in digital signal processing or is already familiar with Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filter design methods. In this chapter, the structure of digital filters is brieﬂy mentioned, followed by a discussion on the LabVIEW Digital Filter Design (DFD) toolkit. This toolkit provides various tools for the design, analysis, and simulation of digital filters. 4.1 Digital Filtering 4.1.1 Difference Equations As a difference equation, an FIR filter is expressed as y ½ n ¼ X N k ¼ 0 b k x ½ n ± k (4.1) where b ’s denote the filter coefficients and N the number of zeros or filter order. As described by this equation, an FIR filter operates on a current input x [ n ] and a number of previous inputs x [ n ± k ] to generate a current output y [ n ]. The equi-ripple method, also known as the Remez algorithm, is normally used to produce an optimal FIR filter [ 1 ]. Figure 4-1 shows the filter responses using the avail- able design methods consisting of equi-ripple, Kaiser window, and Dolph-Chebyshev window. Among these methods, the equi-ripple method generates a response whose deviation from the desired response is evenly distributed across the passband and stopband [ 2 ]. 93

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The difference equation of an IIR filter is given by y ½ n ¼ X N k ¼ 0 b k x ½ n ± k ± X M k ¼ 1 a k y ½ n ± k (4.2) where b ’s and a ’s denote the filter coefficients and N and M the number of zeros and poles, respectively. As indicated by Equation (4.2) , an IIR filter uses a number of previous outputs y [ n ± k ] as well as a current and a number of previous inputs to generate a current output y [ n ]. Several methods are widely used to design IIR filters. They include Butterworth, Chebyshev, Inverse Chebyshev, and Elliptic methods. Figure 4-2 shows the magni- tude response of an IIR filter designed by these methods having the same order for comparison purposes. For example, the elliptic method generates a relatively narrower transition band and more ripples in passband and stopband, whereas the Butterworth method generates a monotonic type response [ 2 ]. Table 4-1 summarizes the characteristics of these design methods. 5 −80 −70 −60 −50 −40 −30 −20 −10 0 Normalized Frequency 0.5 0 0.1 0.2 0.3 0.4 Kaiser Dolph-Chebyshev Equi-Ripple Magnitude (dB) Figure 4-1: Responses of different FIR filter design methods. 94 Chapter 4
4.1.2 Stability and Structure In general, as compared to IIR filters, FIR filters require less precision and are computationally more stable. The stability of an IIR filter depends on whether its poles are located inside the unit circle in the complex plane. Consequently, when an IIR filter is implemented on a fixed-point processor, its stability can be affected. Table 4-2 provides a summary of the differences between the attributes of FIR and IIR filters.

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