Filter Approximation Theory

	Butterworth, Chebyshev, and Elliptic Filters

Approximation Polynomials

	Every physically realizable circuit has a transfer function that is a rational polynomial in s
	We want to determine classes of rational polynomials that approximate the “Ideal” low-pass filter response (high-pass band-pass and band-stop filters can be derived from a low pass design)
	Four well known approximations are discussed here:
		Butterworth: Steven Butterworth,"On the Theory of Filter Amplifiers", Wireless Engineer (also called Experimental Wireless and the Radio Engineer), vol. 7, 1930, pp. 536-541
		Chebyshev: Pafnuty Lvovich Chebyshev (1821-1894) - Russia Cyrillic alphabet - Spelled many ways
		Elliptic Function: Wilhelm Cauer (1900-1945) - Germany U.S. patents 1,958,742 (1934), 1,989,545 (1935), 2,048,426 (1936)
		Bessel: Friedrich Wilhelm Bessel, 1784 - 1846

Definitions

	Let |H(w)|2 be the approximation to the ideal low-pass filter response |I(w)|2
	Where wc is the ideal filter cutoff frequency (it is normalized to one for convenience)

Definitions - 2

	|H(w)|2 can be written as
	Where F(w) is the “Characteristic Function” which attempts to approximate:
		This cannot be done with a finite order polynomial
		e provides flexibility for the degree of error in the passband or stopband.

Filter Specification

	|H(w)|2 must stay within the shaded region
	Note that this is an incomplete specification.  The phase response and transient response are also important and need to be appropriate for the filter application

Butterworth

	F(w) = wn and e = 1 and
	Characteristics
		Smooth transfer function (no ripple)
		Maximally flat and Linear phase (in the pass-band)
		Slow cutoff L

Butterworth Continued

	Pole locations in the s-plane at: w2n = -1 or w = (-1)(1/2n)
		Poles are equally spaced on the unit circle at q=kp/2n.
		H(s) only uses the n poles in the left half plane for stability.
		There are no zeros

Butterworth Filter
|H(s)| for n=4

Chebyshev – Type 1

	F(w) = Tn(w)  so
		T1(w) = w and Tn(w) = 2 wTn(w) – Tn-1(w)
	Characteristics
		Controlled equiripple in the pass-band
		Sharper cutoff than Butterworth
		Non-linear phase (Group delay distortion) L

Chebyshev
|H(s)| for n=4, r=1 (Type 1)

Elliptic Function

	F(w) = Un(w) – the Jacobian elliptic function
	S-Plane
		Poles approximately on an ellipse
		Zeros on the jw-axis
	Characteristics
		Separately controlled equiripple in the pass-band and stop-band
		Sharper cutoff than Chebyshev (optimal transition band)
		Non-linear phase (Group delay distortion) L

Elliptic Function
H(s) for n=4, r_p=3, r_s=50

Bessel Filter

	Butterworth and Chebyshev filters with sharp cutoffs (high order) carry a penalty that is evident from the positions of their poles in the s plane. Bringing the poles closer to the jw axis increases their Q, which degrades the filter's transient response. Overshoot or ringing at the response edges can result.
	The Bessel filter represents a trade-off in the opposite direction from the Butterworth. The Bessel's poles lie on a locus further from the jw axis. Transient response is improved, but at the expense of a less steep cutoff in the stop-band.

Practical Filter Design

	Use a tool to establish a prototype design
		MatLab is a great choice
		See http://doctord.webhop.net/courses/Topics/Matlab/index.htm for a Matlab tutorial by Dr. Bouzid Aliane; Chapter 5 is on filter design.
	Check your design for ringing/overshoot.
		If detrimental, increase the filter order and redesign to exceed the frequency response specifications
		Move poles near the jw-axis to the left to reduce their Q
		Check the resulting filter against your specifications
			Moving poles to the left will reduce ringing/overshoot, but degrade the transition region.