Digital Signal Processing. The Scientist and Engineers Guide. Steven W. Smith

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Two parameters must be selected before using these equations: f, the center frequency, and BW, the bandwidth (measured at an amplitude of 0.707). Both of these are expressed as a fraction of the sampling frequency, and therefore must be between 0 and 0.5. From these two specified values, calculate the intermediate variables: R and K, and then the recursion coefficients.

As shown in (a), the band-pass filter has relatively large tails extending from the main peak. This can be improved by cascading several stages. Since the design equations are quite long, it is simpler to implement this cascade by filtering the signal several times, rather than trying to find the coefficients needed for a single filter.

Figure (b) shows examples of the band-reject filter. The narrowest bandwidth that can be obtain with single precision is about 0.0003 of the sampling frequency. When pushed beyond this limit, the attenuation of the notch will degrade. Figure (c) shows the step response of the band-reject filter. There is noticeable overshoot and ringing, but its amplitude is quite small. This allows the filter to remove narrowband interference (60 Hz and the like) with only a minor distortion to the time domain waveform.

Analog electronic circuits have this same problem with the phase response. Imagine a circuit composed of resistors and capacitors sitting on your desk. If the input has always been zero, the output will also have always been zero. When an impulse is applied to the input, the capacitors quickly charge to some value and then begin to exponentially decay through the resistors. The impulse response (i.e., the output signal) is a combination of these various decaying exponentials. The impulse response cannot be symmetrical, because the output was zero before the impulse, and the exponential decay never quite reaches a value of zero again. Analog filter designers attack this problem with the Bessel filter, presented in Chapter 3. The Bessel filter is designed to have as linear phase as possible; however, it is far below the performance of digital filters. The ability to provide an exact linear phase is a clear advantage of digital filters.

Fortunately, there is a simple way to modify recursive filters to obtain a zero phase. Figure 19-8 shows an example of how this works. The input signal to be filtered is shown in (a). Figure (b) shows the signal after it has been filtered by a single pole low-pass filter. Since this is a nonlinear phase filter, the left and right edges do not look the same; they are inverted versions of each other. As previously described, this recursive filter is implemented by starting at sample 0 and working toward sample 150, calculating each sample along the way.

Now, suppose that instead of moving from sample 0 toward sample 150, we start at sample 150 and move toward sample 0. In other words, each sample in the output signal is calculated from input and output samples to the right of the sample being worked on. This means that the recursion equation, Eq. 19-1, is changed to:

Figure (c) shows the result of this reverse filtering. This is analogous to passing an analog signal through an electronic RC circuit while running time backwards. !esrevinu eht pu-wercs nac lasrever emit -noituaC

Filtering in the reverse direction does not produce any benefit in itself; the filtered signal still has left and right edges that do not look alike. The magic happens when forward and reverse filtering are combined. Figure (d) results from filtering the signal in the forward direction and then filtering again in the reverse direction. Voila! This produces a zero phase recursive filter. In fact, any recursive filter can be converted to zero phase with this bidirectional filtering technique. The only penalty for this improved performance is a factor of two in execution time and program complexity.

How do you find the impulse and frequency responses of the overall filter? The magnitude of the frequency response is the same for each direction, while the phases are opposite in sign. When the two directions are combined, the magnitude becomes squared, while the phase cancels to zero. In the time domain, this corresponds to convolving the original impulse response with a left-for-right flipped version of itself. For instance, the impulse response of a

CHAPTER

Chebyshev Filters

20

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 rolloff 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 tradeoff 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,

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Frequency

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, but be aware that it is frequently the first choice of professional filter designers, both in analog electronics and DSP. If you need this level of performance, buy a software package for designing digital filters.

Designing the Filter

You must select four parameters to design a Chebyshev filter: (1) a high-pass or low-pass response, (2) the cutoff frequency, (3) the percent ripple in the passband, and (4) the number of poles. Just what is a pole? Here are two answers. If you don't like one, maybe the other will help:

Answer 1- The Laplace transform and z-transform are mathematical ways of breaking an impulse response into sinusoids and decaying exponentials. This is done by expressing the system's characteristics as one complex polynomial divided by another complex polynomial. The roots of the numerator are called zeros, while the roots of the denominator are called poles. Since poles and zeros can be complex numbers, it is common to say they have a "location" in the complex plane. Elaborate systems have more poles and zeros than simple ones. Recursive filters are designed by first selecting the location of the poles and zeros, and then finding the appropriate recursion coefficients (or analog components). For example, Butterworth filters have poles that lie on a circle in the complex plane, while in a Chebyshev filter they lie on an ellipse. This is the topic of Chapters 32 and 33.

Answer 2- Poles are containers filled with magic powder. The more poles in a filter, the better the filter works.

FIGURE 20-2

Chebyshev frequency responses. Figures (a) and (b) show the frequency responses of low-pass Chebyshev filters with 0.5% ripple, while (c) and (d) show the corresponding high-pass filter responses.

Kidding aside, the point is that you can use these filters very effectively without knowing the nasty mathematics behind them. Filter design is a specialty. In actual practice, more engineers, scientists and programmers think in terms of answer 2, than answer 1.

Figure 20-2 shows the frequency response of several Chebyshev filters with 0.5% ripple. For the method used here, the number of poles must be even. The cutoff frequency of each filter is measured where the amplitude crosses 0.707 (-3dB). Filters with a cutoff frequency near 0 or 0.5 have a sharper roll-off than filters in the center of the frequency range. For example, a two pole filter at fC ' 0.05 has about the same roll-off as a four pole filter at fC ' 0.25 . This is fortunate; fewer poles can be used near 0 and 0.5 because of round-off noise. More about this later.

There are two ways of finding the recursion coefficients without using the z- transform. First, the cowards way: use a table. Tables 20-1 and 20-2 provide the recursion coefficients for low-pass and high-pass filters with 0.5% passband ripple. If you only need a quick and dirty design, copy the appropriate coefficients into your program, and you're done.