202 CHAPTER EIGHT
Perfect, Brick Wall Anti-alias Filter 0.0
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FIGURE 8-6 A perfect, but impossible to find, antialias filter
antialias filter should pass nothing. Figure 8-6 shows the nature of such a perfect antialias filter. The figure shows the filter’s response versus frequency. We can see that the filter perfectly passes all signals lower in frequency than 0.5 Fs, the sampling frequency, which is 0.5 in this example. Above that point, the filter passes nothing at all. This chart is a typical frequency response chart for a component. The trouble is, it’s impossible to build a filter that can do this. We must make compromises to achieve a suitable antialias filter design. So what problems exist with designing the perfect filter, as shown in the figure?
EXPENSE An ideal antialias filter with an infinitely steep rolloff (defined shortly) like that in the figure cannot be made. Filters are made with real-world components that have definitive, complex impedances. This means the filter will have a transfer function that reduces to differential equations with continuous solutions. This is all a complex way to say that the filter’s frequency transfer chart will not have vertical rolloff lines. The filter must have curves and ramps. The vertical dropoff shown in the ideal filter will actually have to roll off with a less vertical drop. The more vertical the drop, the more expensive and complicated the filter must be. This puts us in a bind. If we want a more perfect filter, our expense goes up. If we want to save money, we will have to settle for a less perfect filter. The typical solution is to put the antialias filter at a frequency a bit lower than the Nyquist Frequency and roll it off at a more gentle (cheaper) angle. A very similar solution is to put an imperfect antialias filter at the Nyquist Frequency and then move the sampling frequency up about 20 percent. We’ll look at filter design shortly.