1
|
- Butterworth, Chebyshev, and Elliptic Filters
|
2
|
- 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
|
3
|
- 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)
|
4
|
- |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.
|
5
|
- |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
|
6
|
- 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
|
7
|
- 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
|
8
|
|
9
|
- 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
|
10
|
|
11
|
- 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
|
12
|
|
13
|
- 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.
|
14
|
- 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.
|