Practical Filters 2.5 2001/06/21 2002/06/14 Richard G. Baraniuk richb@rice.edu Justin Romberg jrom@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu all pass bandpass bandstop highpass lowpass pass band signals stop band systems transition band Describes how real-world filters behave. An ideal filter simply removes all unwanted frequencies, preserving the remaining frequencies exactly. This would resemble some sort of a finite rectangle function in frequency. However, a simple, finite rectangle function in frequency is an infinite sinc function in time. This is a problem. A sinc function is an infinite length signal in both the positive and negative directions, making it impossible to create in the real-world. This leads us to as what would happen if we just made this sinc function causal by "chopping it off" somewhere. What we find when we do this is that the frequency domain representation is no longer a perfect rectangle: it now does not fall off immediately and shows some wiggling where it was flat before.

A truncated sinc function in time... ... leads to a smooth, wiggly, "rectangle" in frequency.

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 kinds of 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 ω p to ω p , the transition bands are the regions from ω s to ω p and from ω p to ω s while the stop band is the region less than ω s or greater than ω 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 may vary. The behavior within the transition band is not specified, allowing anything to occur there, as long as the width is within specifications. This example will look at a moving average system.

Time domain representation of the moving average. The frequency domain representation of the moving average system is a sinc function. H ω ω T 2 ω T 2 ω T 2

Some notes about this system: It is lowpass It has linear phase with jumps of radians when the sinc function changes sign The duration of the filter is inversely proportional to its bandwidth This filter is finite impulse response (FIR) It cannot be built with passive R, L, C circuits We do not have independent control over all four design specifications We are going to design a moving average filter with the following design specs: ω p 100 , e s 0.1 , e p 0.1

With this specification, we are allowing ω s to be a dependant variable (since we need one). We can now find the equation for this moving average system. We begin with H ω 1.1 ω T 2 ω T 2 We will now solve for T with H 100 1.1 50 T 50 T 0.9 For these specs, T 0.007 . This means that H ω does not stay below e s 0.1 until ω s 771 .

A graphical look at the transfer function of this lowpass filter with the passband and stopband noted.

It is very clear from this representation that the transition band is huge ( 671 ). This is a very bad filter, especially when you consider that it cannot be implemented with passive circuitry. Fortunately better filters (e.g. Butterworth, Chebyshev and Elliptical) do exist.

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