Practical filters

An ideal filter simply removes all unwanted frequencies, preserving the remainingfrequencies exactly. This would resemble some sort of a finite rectangle function in frequency. However, a simple, finiterectangle function in frequency is an infinite sinc function in time. This is a problem. A sinc function is an infinite lengthsignal in both the positive and negative directions, making it impossible to create in the real-world. This leads us to aswhat would happen if we just made this sinc function causal by "chopping it off" somewhere. What we find when we do this isthat the frequency domain representation is no longer a perfect rectangle: it now does not fall off immediately and shows somewiggling where it was flat before.

The bands

An ideal filter has two types of bands: the stop band defines the region of frequencies that are eliminated by the filter, while the pass band defines the region of frequencies that the filter allows through. Practical filters add one more, the transition band . This is the area where the filter is moving between the stop band and pass band.

Filter design specifications

In our look at filter design specifications, we will use the example of a lowpass filter. The extension to the other kindsof filters should be fairly straightforward. shows the parameters for a lowpass filter in the frequency domain.

From this image, the passband is the region from $-{}_p$ to ${}_p$ , the transition bands are the regions from $-{}_s$ to $-{}_p$ and from ${}_p$ to ${}_s$ while the stop band is the region less than $-{}_s$ or greater than ${}_s$ .

In the figure above, $e_p$ and $e_s$ represent the acceptable tolerance (or error) around the desired level that the passband and stopband respectively mayvary. The behavior within the transition band is not specified, allowing anything to occur there, as long as thewidth is within specifications.

Beyond lowpass filter design

In the discussion of the different filters (Butterworth, Chebyshev and Elliptical) is common to see explanations basedon lowpass filters. This explanation is very nice when first learning about them, because it is sufficient to understandthe fundamentals of each of them. It is acceptable, because there exist fairly straightforward techniques to convert theselowpass filters into highpass, bandpass or bandstop filters. These techniques are the lowpass to highpass transformation , lowpass to bandpass transformation and lowpass to bandstop transformation .