3.1 Design of infinite impulse response (iir) filters by frequency  (Page 2/2)

The details of the bilinear and alternative transformations are covered elsewhere. For the purposes of this section, it issufficient to observe [link] , [link] that the frequency response of a filter in terms of the new variable is found by evaluating $H\left(s\right)$ along the imaginary axis, i.e., for $s=j\omega$ . This is exactly how the frequency response of analog filters is obtained.

There are two reasons that the approximation process is often formulated in terms of the square of the magnitude of the transferfunction, rather than the real and/or imaginary parts of the complex transfer function or the magnitude of the transferfunction. The first reason is that the squared-magnitude frequency- response function is an analytic, real-valued function of a realvariable, and this considerably simplifies the problem of finding a “best" solution. The second reason is that effects of the signal orinterference are often stated in terms of the energy or power that is proportional to the square of the magnitude of the signalor noise.

In order to move back and forth between the transfer function $F\left(s\right)$ and the squared-magnitude frequency response ${\left|F\left(j\omega \right)\right|}^2$ , an intermediate function is defined. The analytic complex-valued function of the complex variable s is defined by

which is related to the squared magnitude by

If

then

In this context, the approximation is arrived at in terms of $F\left(j\omega \right)$ , and the result is an analytic function $FF\left(s\right)$ with a factor $F\left(s\right)$ , which is the desired filter transfer function in terms of the rectangular variable $s$ . A comparable function can be defined in terms of the digital transfer function using the polarvariable $z$ by defining

which gives the magnitude-squared frequency response when evaluated around the unit circle, i.e., $z=e^{j\omega }$ .

The next section develops four useful approximations using the continuous-time Laplace transform formulation in s. These will betransformed into digital transfer functions by techniques covered in another module. They can also be used directly for analog filterdesign.

Classical analog lowpass filter approximations

Four basic filter approximations are considered to be standard. They are often developed and presented in terms of a normalizedlowpass filter that can be modified to give other versions such as highpass or bandpass filters. These four forms use Taylor's seriesapproximations and Chebyshev approximations in various combinations [link] , [link] , [link] , [link] . It is interesting to note that none of these are defined in terms of a mean-squared error measure.Although it would be an interesting error criterion, the reason is that there is no closed-form solution to the LS-error approximationproblem which is nonlinear for the IIR filter.

This section develops the four classical approximations in terms of the Laplace transform variable s. They can be used as prototypefilters to be converted into digital filters or used directly for analog filter design.

The desired lowpass filter frequency response is similar to the case for the FIR filter. Here it is expressed in terms of themagnitude squared of the transfer function, which is a function of $s=j\omega$ and is illustrated in Figure 8 from FIR Digital Filters and Figure 1 from Least Squared Error Design of FIR Filters .

The Butterworth filter uses a Taylor's series approximation to the ideal at both $\omega =0$ and $\omega =\infty$ . The Chebyshev filter uses a Chebyshev (min-max) approximation across the passband and aTaylor's series at $\omega =\infty$ . The Inverse or Type-II Chebyshev filter uses a Taylor's series approximation at $\omega =0$ and a Chebyshev across the stopband. The elliptic-function filter uses aChebyshev approximation across both the pass and stopbands. The squared- magnitude frequency response for these approximations to the ideal isgiven in [link] , and the design is developed in the following sections.