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Butterworth Filter
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A low-pass Butterworth filter is an all-pole filter with a squared magnitude response given by
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FILTER DESIGN
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[CHAP. 9
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The parameter N is the order of the filter (number of poles in the system function), and Q,. is the 3-dB cutoff frequency. The magnitude of the frequency response may also be written as
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where The frequency response of the Butterworth filter decreases monotonically with increasing 0, and as the filter order increases, the transition band becomes narrower. These properties are illustrated in Fig. 9-6, which shows IH,(jQ)l for Butterworth filters of orders N = 2 , 4 , 8 , and 12. Because
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from the magnitude-squared function, we may write
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Fig. 9-6. The magnitude of the frequency response for Butterworth filters of orders N = 2.4, 8.
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Therefore, the poles of G,(s) are located at 2N equally spaced points around a circle of radius Q,.,
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= ( - I ) ~ / ' ~ ( ~ R , Q, expIj =)
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and are symmetrically located about the jR-axis. Figure 9-7 shows these pole positions for N = 6 and N = 7. The system function, H,(s), is then formed from the N roots of H,(s)H,(-s) that lie in the left-half s-plane. For a iiormuli-.ed Butterworth filter with Q,. = 1 , the system function has the form
I H,(s) = -A N ( ~ ) sN alsN-'
Table 9-4 lists the coefficients of AN(s) for I a Butterworth filter are as follows:
N 5 8. Given Q p , Q,7,S,,, and S,, the steps involved in designing
Find the values for the selectivity factor, k , and the discrimination factor, d. from the filter specifications. Determine the order of the filter required to meet the specifications using the design formula NZlog d log k
2N +2k)n
k = 0 . 1,
.... 2N-1
(9.7)
+ . . . + aN-Is + a N
(9.8)
CHAP. 91
FILTER DESIGN
(u) Order N = 6. ( h ) Order N = 7. Fig. 9-7. The poles of H,(a)H,(-s) a Butterworth filter of order N = 6 and N = 7. for
Table 9-4 The Coefficients in the System Function of a Normalized Butterworth Filter (a, I) for = Orders I 5 N 5 8
1 .om 2.6131 5.2361 9.1416 14.59 18 2 1.8462
1 .OOOO 3.2361 7.4641 14.59 I8 25.6884
1.OW0 3.8637 10.0978 2 1.8462
3. Set the 3-dB cutoff frequency, Q,, to any value in the range
n,[(i - J,)-~ - i ~ - 5~Q,/ 5~c2,[q2~
1 1 - l ' ~ ~
Synthesize the system function of the Butterworth filter from the poles of Ga(s) = Ha(s)Ha(-s) = that lie in the left-half s-plane. Thus,
1 (S/~Q,)~~
where
EXAMPLE 9.4.1
Let us design a low-pass Butterworth filter to meet the following specifications:
f, =6kHz f, = 10kHz
6, = 6 , = 0 . l
FTLTER DESIGN
[CHAP. 9
First, we compute the discrimination and selectivity factors:
Because
N Z = 5.92 log k
log d
it follows that the minimum filter order is N = 6. With fp[(l
- s,,)-' - 11-'IZN = 6770
f, [6;' - I ] - . " ~ ~68 19 =
the center frequency, f,,may be any value in the range 6770
f,. 5 6819
The system function of the Butterworth filter may then be found using Eq. (9.8) by first constructing a sixth-order normalized Butterworth filter from Table 9-4, Hu(s) = and then replacing s with s/ R,
sb 3.8637s5
+ 7.4641s4 + 9.1416s' + 7.4641s2 + 3.8637s + I
SO that
the cutoff frequency is 52, instead of unity (see Sec. 9.4.3).
Chebyshev Filters
Chebyshev filters are defined in terms of the Chebyshev polynomials:
These polynomials may be generated recursively as follows,
= = with To(x) 1 and Tl(x) x. The following properties of the Chebyshev polynomials follow from Eq. (9.9):
For 1x1 5 1 the polynomials are bounded by 1 in magnitude, ITN(x)l 1, and oscillate between f1. 5 For 1x1 > I , the polynomials increase monotonically with x.
2. TN(l) I for all N. = 3. TN(0) fl for N even, and TN(0) 0 for N odd. = =
All of the roots of TN(x) in the interval are
1 _< x I1.
There are two types of Chebyshev filters. A type I Chebyshev filter i s all-pole with an equiripple passband and a monotonically decreasing stopband. The magnitude of the frequency response is
where N is the order of the filter, Q, is the passband cutoff frequency, and 6 is a parameter that controls the passband ripple amplitude. Because T ~ ( Q / Q ~ ) between 0 and 1 for varies < Q,,, I H , ( ~ R ) I ~ oscillates between I and 1/(1 e2). As the order of the filter increases, the number of oscillations (ripples) in the passband increases, and the transition width between the passband and stopband becomes narrower. Examples are given in Fig. 9-8 for N = 5,6.
CHAP. 91
FILTER DESIGN
( a ) Odd order (N = 5).
( 6 ) Even order (N = 6).
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