Rational Function Approximation

The mathematical problem inherent in the frequency-domain filter design problem is the approximation of a desired complex frequency-response function Hd(z)Hd(z) by a rational transfer function H(z)H(z) with an Mth-degree numerator and an Nth-degree denominator for values of the complex variable zz along the unit circle of z=ejωz=ejω. This approximation is achieved by minimizing an error measure between H(ω)H(ω) and Hd(ω)Hd(ω).

For the digital filter design problem, the mathematics are complicated by the approximation being defined on the unit circle. In terms of zz, frequency is a polar coordinate variable. It is often much easier and clearer to formulate the problem such that frequency is a rectangular coordinate variable, in the way it naturally occurs for analog filters using the Laplace complex variable ss. A particular change of complex variable that converts the polar coordinate variable to a rectangular coordinate variable is the bilinear transformation[4], [5], [3], [2].

The details of the bilinear and alternative transformations are covered elsewhere. For the purposes of this section, it is sufficient to observe[4], [3] that the frequency response of a filter in terms of the new variable is found by evaluating H(s)H(s) along the imaginary axis, i.e., for s=jωs=jω. 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 transfer function, rather than the real and/or imaginary parts of the complex transfer function or the magnitude of the transfer function. The first reason is that the squared-magnitude frequency- response function is an analytic, real-valued function of a real variable, and this considerably simplifies the problem of finding a “best" solution. The second reason is that effects of the signal or interference are often stated in terms of the energy or power that is proportional to the square of the magnitude of the signal or noise.

In order to move back and forth between the transfer function F(s)F(s) and the squared-magnitude frequency response |F(jω)|2|F(jω)|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(jω)F(jω), and the result is an analytic function FF(s)FF(s) with a factor F(s)F(s), which is the desired filter transfer function in terms of the rectangular variable ss. A comparable function can be defined in terms of the digital transfer function using the polar variable zz by defining

which gives the magnitude-squared frequency response when evaluated around the unit circle, i.e., z=ejωz=ejω.

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

Classical Analog Lowpass Filter Approximations

Four basic filter approximations are considered to be standard. They are often developed and presented in terms of a normalized lowpass filter that can be modified to give other versions such as highpass or bandpass filters. These four forms use Taylor's series approximations and Chebyshev approximations in various combinations[5], [6], [1], [7]. 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 approximation problem 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 prototype filters 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 the magnitude squared of the transfer function, which is a function of s=jωs=jω and is illustrated in Figures (Reference) and (Reference).

The Butterworth filter uses a Taylor's series approximation to the ideal at both ω=0ω=0 and ω=∞ω=∞. The Chebyshev filter uses a Chebyshev (min-max) approximation across the passband and a Taylor's series at ω=∞ω=∞. The Inverse or Type-II Chebyshev filter uses a Taylor's series approximation at ω=0ω=0 and a Chebyshev across the stopband. The elliptic-function filter uses a Chebyshev approximation across both the pass and stopbands. The squared- magnitude frequency response for these approximations to the ideal is given in Figure 1, and the design is developed in the following sections.

Figure 1: Frequency Responses of the Four Classical Lowpass IIR Filter Approximations