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For mathematical convenience, the four classical IIR filter transfer functions were developed in terms of the Laplace transform ratherthan the z-transform. The prototype Laplace-transform transfer functions are descriptions of analog filters. In this section they are converted toz-transform transfer functions for implementation as IIR digital filters.
There have been several different methods of converting analog systems to digital described over the history of digitalfilters. Two have proven to be useful for most applications. The first is called the impulse-invariant method and results in adigital filter with an impulse response exactly equal to samples of the prototype analog filter. The second method uses afrequency mapping to convert the analog filter to a digital filter. It has the desirable property of preserving theoptimality of the four classical approximations developed in the last section. This section will develop the theory and designformulas to implement both of these conversion approaches.
Although the transfer functions in Continuous Frequency Definition of Error were designed with criteria in the frequency domain, the impulse-invariant method willconvert them into digital transfer functions using a time-domain constraint [link] , [link] , [link] . The digital filter designed by the impulse-invariant method is required to have an impulse response that isexactly equal to equally spaced samples of the impulse response of the prototype analog filter. If the analog filter has a transfer function $F\left(s\right)$ with an impulse response $f\left(t\right)$ , the impulse response of the digital filter $h\left(n\right)$ is required to match the samples of $f\left(t\right)$ . For samples at $T$ second intervals, the impulse response is
The transfer function of the digital filter is the z-transform of the impulse response of the filter, which is given by
The transfer function of the prototype analog filter is always a rational function written as
where $B\left(s\right)$ is the numerator polynomial with roots that are the zeros of $F\left(s\right)$ , and $A\left(s\right)$ is the denominator with roots that are the poles of $F\left(s\right)$ . If $F\left(s\right)$ is expanded in terms of partial fractions, it can be written as
The impulse response of this filter is the inverse-Laplace transform of [link] , which is
Sampling this impulse response every $T$ seconds gives
The basic requirement of [link] gives
which is clearly a rational function of $z$ and is the transfer function of the digital filter, which has samples of the prototype analog filter asits impulse response.
This method has its requirements set in the time domain, but the frequency response is important. In most cases, the prototype analog filter is oneof the classical types, which is optimal in the frequency domain. If the frequency response of the analog filter is denoted by $F\left(j\omega \right)$ and the frequency response of the digital filter designed by the impulse-invariant method is $H\left(\omega \right)$ , it can be shown in a development similar to that used for the sampling theorem
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