Approximating functions
The choice of a transfer function for solving the approximation problem is the transition step from a set of design specifications to the realization of the transfer function by means of a specific filter structure. Accordingly, this is the most fundamental step in filter design because the choice of the transfer function determines the performance of the filter. At the outset, however, it must be emphasized that there is no unique solution to the approximation problem. Rather, we have a set of possible solutions, each with its own distinctive properties.
Basically, the approximation problem is an optimization problem that can be solved only in the context of a specific criterion of optimality. In other words, before we proceed to solve the approximation problem, we have to specify a criterion of optimality in an implicit or explicit sense. Moreover, the choice of that criterion uniquely determines the solution. Two optimality criteria commonly used in filter design are as follows:
- Maximally flat magnitude response
- Equiripple magnitude response
Butterworth filters
A Butterworth function of order K is defined by:
|H(jw)|^2 = 1/ 1+ (w/wc) ^2k, K=1,2,3…….
And a filter so designed is referred to as a Butterworth filter of order K.
Chebyshev filters
The tolerance diagram calls an approximating function that lies between 1 and 1 – E inside the passband range 0 <= Ω <= Ω_{p}. The Butterworth function meets this requirement, but concentrates its approximating ability near Ω = 0 For a given filter order, we can obtain a filter with a reduced transition bandwidth by using an approximating function that exhibits an equiripple characteristic in the passband (i.e., it oscillates uniformly between 1 and 1 – E for 0<= Ω<= Ω n ), and 0.5-dB ripple in the passband. The magnitude responses plotted here satisfy the equiripple criteria described earlier for K odd and K even, respectively. Approximating functions with an equiripple magnitude response are known collectively as Chebyshev functions. A filter designed on this basis is called a Chebyshev filter. The poles of a transfer function H(s) pertaining to a Chebyshev filter lie on an ellipse and s-plane in a manner closely related to those of the corresponding Butterworth filter.
The Chebyshev functions exhibit a monotonic behavior. Alternatively, we may use another class of Chebyshev functions that exhibit monotonic response in the passband, but an equiripple response in the stopband. A filter designed on this basis is called an inverse Chebyshev filter. Unlike a Chebyshev filter, the transfer function of an inverse Chebyshev filter has zeros on the fo-axis of the s-plane in the
The ideas embodied in Chebyshev and inverse Chebyshev filters can be combined further reduce the transition bandwidth by making the approximating function equiripple is op in both the passband and the stopband. Such an approximating function is called an elliptic function, and a filter resulting from its use is called an elliptic filter. An elliptic filter is optimum in the sense that, for a prescribed set of design specifications, the width of the transition band is the smallest that we can achieve. This permits the smallest possible separation between the passband and stopband of the filter. From the standpoint of analysis, however, determining the transfer function H(s) is simplest for a Butterworth filter and most challenging for an elliptic filter. The elliptic filter is able to achieve its optimum behavior by virtue of the fact that its transfer function H(s) has finite zeros in the s-plane, the number of which is uniquely determined by filter order K. In contrast, the transfer function of a Butterworth filter or that of a Chebyshev filter has all of its zeros located at s= H(1).
Up to this point, we have considered the issue of solving the approximation problem for low-pass filters. In that context, it is common practice to speak of a low-pass “prototype” filter, by which we mean a low-pass filter whose cutoff frequency Ω_{c} is normalized to unity. Given that we have found the transfer function of a low-pass prototype filter, we may use it to derive the transfer function of a low-pass filter with an arbitrary cutoff frequency, a high-pass, a band-pass, or a band-stop filter by means of an appropriate transformation of the independent variable. Such a transformation has no effect on the tolerances within which the ideal characteristic of interest is approximated. We considered low-pass to low-pass transformation. In what follows, we consider two other frequency transformations: low pass to high pass and low pass to bandpass.
Frequency transformations
Given we know the transfer function of a low-pass filter with an arbitrary cutoff frequency, a high-pass, a band-pass, or a band-stop filter by means of an appropriate transformation of the independent variable. Such a transformation has no effect on the tolerances within the ideal characteristic of interest is approximated.
Reference