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Unformatted text preview: 333 CHAPTER 20 Chebyshev Filters Chebyshev filters are used to separate one band of frequencies from another. Although they cannot match the performance of the windowed-sinc filter, they are more than adequate for many applications. The primary attribute of Chebyshev filters is their speed, typically more than an order of magnitude faster than the windowed-sinc. This is because they are carried out by recursion rather than convolution. The design of these filters is based on a mathematical technique called the z-transform , discussed in Chapter 33. This chapter presents the information needed to use Chebyshev filters without wading through a mire of advanced mathematics. The Chebyshev and Butterworth Responses The Chebyshev response is a mathematical strategy for achieving a faster roll- off by allowing ripple in the frequency response. Analog and digital filters that use this approach are called Chebyshev filters . For instance, analog Chebyshev filters were used in Chapter 3 for analog-to-digital and digital-to- analog conversion. These filters are named from their use of the Chebyshev polynomials , developed by the Russian mathematician Pafnuti Chebyshev (1821-1894). This name has been translated from Russian and appears in the literature with different spellings, such as: Chebychev, Tschebyscheff, Tchebysheff and Tchebichef. Figure 20-1 shows the frequency response of low-pass Chebyshev filters with passband ripples of: 0%, 0.5% and 20%. As the ripple increases (bad), the roll-off becomes sharper (good). The Chebyshev response is an optimal trade- off between these two parameters. When the ripple is set to 0%, the filter is called a maximally flat or Butterworth filter (after S. Butterworth, a British engineer who described this response in 1930). A ripple of 0.5% is a often good choice for digital filters. This matches the typical precision and accuracy of the analog electronics that the signal has passed through. The Chebyshev filters discussed in this chapter are called type 1 filters, meaning that the ripple is only allowed in the passband . In comparison, The Scientist and Engineer's Guide to Digital Signal Processing 334 Frequency 0.1 0.2 0.3 0.4 0.5 0.0 0.5 1.0 1.5 Ripple 0% 20% 0.5% Amplitude FIGURE 20-1 The Chebyshev response. Chebyshev filters achieve a faster roll-off by allowing ripple in the passband. When the ripple is set to 0%, it is called a maximally flat or Butterworth filter. Consider using a ripple of 0.5% in your designs; this passband unflatness is so small that it cannot be seen in this graph, but the roll-off is much faster than the Butterworth. type 2 Chebyshev filters have ripple only in the stopband . Type 2 filters are seldom used, and we won't discuss them. There is, however, an important design called the elliptic filter , which has ripple in both the passband and the stopband. Elliptic filters provide the fastest roll-off for a given number of poles, but are much harder to design. We won't discuss the elliptic filter here,...
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This document was uploaded on 08/27/2011.
- Summer '09