3.7 Conversion of analog to digital transfer functions  (Page 2/5)

The frequency response of the digital filter is a periodically repeated version of the frequency response of the analog filter.This results in an overlapping of the analog response, thus not preserving optimality in the same sense the analog prototypewas optimal. It is a similar phenomenon to the aliasing that occurs when sampling a continuous-time signal to obtain a digitalsignal in A-to-D conversion. If $F\left(j\omega \right)$ is an analog lowpass filter that goes to zero as $\omega$ goes to infinity, the effects of the folding can be made small by high sampling rates (small T).

The impulse-invariant design method can be summarized in the following steps:

1. Design a prototype analog filter with transfer function $F\left(j\omega \right)$ .
2. Make a partial fraction expansion of $F\left(j\omega \right)$ to obtain the $N$ values for $K_i$ and $s_i$ .
3. Form the digital transfer function $H\left(z\right)$ from [link] to give the desired design.

The characteristics of the designed filter are the following:

• It has N poles, the same as the analog filter.
• It is stable if the analog filter was stable. This is seen from the change of variables in the denominator of(6.70) which maps the left-half s-plane inside the unit circle in the z-plane.
• The frequency response is a folded version of the analog filter, and the optimal properties of the analog filterare not preserved.
• The cascade of two impulse-invariant designed filters are not impulse-invariant with the cascade of the twoanalog prototypes. In other words, the filter must be designed in one step.

This method is sometimes used to design digital filters, but because the relation of the analog and digital system isspecified in the time domain, it is more useful in designing a digital simulation of an analog system. Unfortunately, theproperties of this class of filters depend on the input. If a filter is designed so that its impulse response is the sampledimpulse response of the analog filter, its step response will not be the sampled step response of the analog filter.

A step-invariant filter can be designed by first multiplying the analog filter transfer function $F\left(s\right)$ by $1/s$ , which is the Laplace transform of a step function. This product is then expanded in partialfraction just as $F\left(s\right)$ was in [link] and the same substitution made as in [link] giving a z-transform. After the z-transform of a step is removed, the digital filter has the step-invariant property. This ideacan be extended to other input functions, but the impulse-invariant version is the most common. Another modification to the impulse-invariantmethod is known as the matched z transform covered in [link] , but it is less useful.

There can be a problem with the classical impulse-invariant method when the number of finite zeros is too large. This is addressed in [link] , [link] .

An example of a Butterworth lowpass filter used to design a digital filter by the impulse-invariant method can be shown. Note thatthe frequency response does not go to zero at the highest frequency of $w=p$ . It can be made as small as desired by increasing the sampling rate, but this is more expensive to implement. Because the frequency responseof the prototype analog filter for an inverse-Chebyshev or elliptic-function filter does not necessarily go to zero as w goes toinfinity, the effects of folding on the digital frequency response are poor. No amount of sampling rate increase will change this. The sameproblem exists for a highpass filter. This shows the care that must be exercised in using the impulse-invariant design method.