Signals and Systems/Filter Implementations

Filter design mostly bases on a limited set of widely used transfer functions. Optimization methods allow to design other types of filters, but the functions listed here have been studied extensively, and designs for these filters (including circuit designs to implement them) are readily available. The filter functions presented here are of lowpass type, and transformation methods allow to obtain other common filter types such as highpass, bandpass or bandstop.

Butterworth Filters

Plot of the amplitude response of the normalized Butterworth lowpass transfer function, for orders 1 to 5

The Butterworth filter function has been designed to provide a maximally flat amplitude response. This is obtained by the fact that all the derivatives up to the filter order minus one are zero at DC. The amplitude response has no ripple in the passband. It is given by:

$A(j\omega) = |H(j\omega)| = \sqrt{ \frac{1}{1+\omega^{2n}} }$

It should be noted that whilst the amplitude response is very smooth, the step response shows noticeable overshoots. They are due to the phase response wich is not linear or, in other words, to the group delay which is not constant.

The amplitude response plot shows that the slope is 20n dB/decade, where n is the filter order. This is the general case for all-pole lowpass filters. Zeroes in the transfer function can accentuate the slope close to their frequency, thus masking this general rule for zero-pole lowpass filters.

The plot also shows that whatever the order of the filter, all the amplitudes cross the same point at , which coresponds to approximatively -3 db. This -3 db reference is often used to specify the cutoff frequency of other kinds of filters.

Butterworth filters don't have a particularly steep drop-off but, together with Chebyshev type I filters, they are of all-pole kind. This particularity results in reduced hardware (or software, depending on the implementation method), which means that for a similar complexity, higher order Butterworth filters can be implemented, compared to functions with a steeper drop-off such as elliptic filters.

Zeroes of the Butterworth function

Poles of a 4^th order Butterworth filter

The normalized Butterworth function is indirectly defined by:

$H(s) \cdot H(-s) = \frac{1}{1+\omega^{2n}}$

This functions has zeroes regularily placed on the unit circle. Knowing that a stable filter has all of its poles on the left half s-plane, it is clear that the left half poles on the unit circle belong to H(s), whilst the right half poles on the right belong to H( − s).

The normalized Butterworth function has a cutoff frequency at f_c = ω_c / (2π) = 1 / (2π)[Hz]. A different cutoff frequency is achieved by scaling the circle radius to ω_c = 2πf_c.

Butterworth Transfer Function

The transfer function of a Butterworth filter is of the form:

$H(s) = \frac{1}{den(s)}$

It can also be written as a function of the poles:

$H(s) = \frac{k}{\prod_{i=1}^N (s-p_i)}$

With this, the denominator ploynom is found from the values of the poles.

Chebyshev Filters

In comparison to Butterworth filters, Chebyshev filters have much steeper roll-off, but at the same time suffer from a rippleing effect in the passband that can cause unspecified results. Also, Chebyshev equations utilize a complex string of trigonomic and arc-trigonometric functions, which can be difficult to deal with mathematically.

Chebyshev filters can be divided into two types: Type I and Type II Chebyshev filters.

Ripple Effect

Chebyshev filters exhibit a ripple-shape in the frequency response of the filter. The locations of the ripples varies with the type of filter (discussed below).

Chebyshev Type I

Chebyshev Type I filters have ripples in the passband.

Chebyshev Type II

Chebyshev Type II filters have ripples in the stopband.

Elliptic Filters

Elliptic filters, also called Cauer filters, suffer from a ripple effect like Chebyshev filters. However, unlike the type 1 and Type 2 Chebyshev filters, Elliptic filters have ripples in both the passband and the stopband. To counteract this limitation, Elliptic filters have a very aggressive rolloff, which often more than makes up for the ripples.

Comparison

The following image shows a comparison between 5^th order Butterworth, Chebyshev and elliptic filter amplitude responses.

Filter Design

Using what we've learned so far about filters, this chapter will discuss filter design, and will show how to make decisions as to the type of filter (Butterworth, Chebyshev, Elliptic), and will help to show how to set parameters to acheive a set of specifications.