Describes how to design a general filter from the Laplace Transform and its pole/zero plots.
Introduction
Analog (Continuous-Time) filters are useful for a wide variety of applications, and are especially useful in that they are very simple to build using standard, passive R,L,C components. Having a grounding in basic filter design theory can assist one in solving a wide variety of signal processing problems.
Estimating frequency response from z-plane
One of the motivating factors for analyzing the pole/zero
plots is due to their relationship to the frequency responseof the system. Based on the position of the poles and zeros,
one can quickly determine the frequency response. This is aresult of the correspondence between the frequency response
and the transfer function evaluated on the unit circle in thepole/zero plots. The frequency response, or DTFT, of the
system is defined as:
Next, by factoring the transfer function into poles and zeros
and multiplying the numerator and denominator by
$e^{iw}$ we arrive at the following equations:
From
[link] we have the
frequency response in a form that can be used to interpretphysical characteristics about the filter's frequency
response. The numerator and denominator contain a product ofterms of the form
$\left|e^{iw}-h\right|$ ,
where
$h$ is either a zero, denoted by
${c}_{k}$ or a pole, denoted by
${d}_{k}$ . Vectors are commonly used to represent
the term and its parts on the complex plane. The pole or zero,
$h$ , is a vector from the origin
to its location anywhere on the complex plane and
$e^{iw}$ is a vector from the origin to its
location on the unit circle. The vector connecting these twopoints,
$\left|e^{iw}-h\right|$ , connects the pole or zero location to a
place on the unit circle dependent on the value of
$w$ . From this, we can begin to
understand how the magnitude of the frequency response is aratio of the distances to the poles and zero present in the
z-plane as
$w$ goes from zero to
pi. These characteristics allow us to interpret
$\left|H(w)\right|$ as follows:
In conclusion, using the distances from the unit circle to the
poles and zeros, we can plot the frequency response of thesystem. As
$w$ goes from
$0$ to
$2\pi $ , the following two properties, taken from the above
equations, specify how one should draw
$\left|H(w)\right|$ .
While moving around the unit circle...
if close to a zero, then the magnitude is small. If a
zero is on the unit circle, then the frequency response iszero at that point.
if close to a pole, then the magnitude is large. If a
pole is on the unit circle, then the frequency responsegoes to infinity at that point.
Drawing frequency response from pole/zero plot
Let us now look at several examples of determining the
magnitude of the frequency response from the pole/zero plot ofa z-transform. If you have forgotten or are unfamiliar with
pole/zero plots, please refer back to the
Pole/Zero Plots module.
In this first example we will take a look at the very simple
z-transform shown below:
$$H(z)=z+1=1+z^{-1}$$$$H(w)=1+e^{-(iw)}$$ For this example, some of the vectors represented by
$\left|e^{iw}-h\right|$ , for random values of
$w$ , are explicitly drawn onto
the complex plane shown in the
figure below. These vectors show how the
amplitude of the frequency response changes as
$w$ goes from
$0$ to
$2\pi $ ,
and also show the physical meaning of the terms in
[link] above. One can see that
when
$w=0$ ,
the vector is the longest and thus the frequency responsewill have its largest amplitude here. As
$w$ approaches
$\pi $ , the length of the vectors decrease
as does the amplitude of
$\left|H(w)\right|$ . Since
there are no poles in the transform, there is only this onevector term rather than a ratio as seen in
[link] .
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