barcode lib ssrs Thus, the system function of the seventh-order Butterworth filter is in Software

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Thus, the system function of the seventh-order Butterworth filter is
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Let apand 52, be the desired passband and stopband cutoff frequencies of an analog low-pass filter, and let 6, and 6, be the passband and stopband ripples. Show that the order of the Butterworth filter required to meet these s~ecifications is log d N Z log k with the 3-dB cutoff frequency a, being any value within the range
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The squared magnitude of the frequency response of the Butterworth filter is
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FILTER DESIGN
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[CHAP. 9
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Because I H,(jQ)l is monotonically decreasing, the maximum error in the passband and stopband occurs at the band edges, Q, and a,, respectively. Therefore, we want
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and From the first equation, we have
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and from the second.
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Dividing these two equations, we have
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and taking the logarithm gives Nlog( ) Dividing by log 2 = log k log d log k
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5 log d
("n)
we have
(note that the inequality is reversed because log k < 0). Because the right side of this equation will not generally be an integer, the order N is taken to be the smallest integer larger than (logd)/(log k ) . Finally, once the order N is fixed, it follows from Eqs. (9.20) and (9.21) that a,, may be any value in the range
Suppose that w e would like to design an analog Chebyshev low-pass filter s o that
1 - Jp 5
IHa(jQ)l I 1
I H a ( j Q ) l 5 J,T
IQI 1 Qp Qs I IQI
Find an expression for the required filter order, N , as a function of
a a,, tip,and ,
For a Chebyshev filter, the magnitude of the frequency response squared is
where is an Nth-order Chebyshev polynomial. Over the passband, In1 < Q,, the magnitude of the frequency response oscillates between 1 and ( l c2)-'I2. Therefore, the ripple amplitude, 6,. is
6, = 1
+c ~ ) - ' / ~
CHAP. 91
or At the stopband frequency R , we have
FILTER DESIGN
= (1 - a,)-'
I I H ~ ( ~ Q= ~ ) I ~ . I c~T,:(R~I
which we want to be less than or equal to 8::
Therefore, )), Because (Q,/R,) > I, then TN(R,/i2,) = cosh(N ~ o s h - ~ ( ~ , / R , and we have
or which is the desired expression.
cosh-'(I I d ) C O S ~ - I (QJR,)
- cosh-'( l / d ) COS~-~(I,X)
If Ha(s) is a third-order type I Chebyshev low-pass filter with a cutoff frequency
find Ha(s)H,(-s).
= I and
= 0. I ,
The magnitude of the frequency response squared for an Nth-order type 1 Chebyshev filter is
where TN(x)is an Nth-order Chebyshev polynomial that is defined recursively as follows,
with To(x) = 1 and Tl(x) = x. Therefore, to find the third-order Chebyshev polynomial, we first find T2(.u) as follows, T2(x)= 2xTl(x) - To(x) = 2x2 - I and then we have T3(x) = 2xT2(x) - Tl(x) = 4x3 - 2x - x = x(4x2 - 3) Thus, the denominator polynomial in I Hu(jR)12is
and we have
we make the substitution R = s / j in 1H,(jC2)l2as follows:
398 9.28
FILTER DESIGN
[CHAP. 9
Show that the bilinear transformation maps the jQ-axis in the s-plane onto the unit circle, lzl = 1, and maps the left-half s-plane. Re(s) < 0 insidc the unit circle, l z ( < 1.
To investigate the characteristics of the bilinear transformation, let z = reJ"' and s = a transformation may then be written as
+ JR.
The bilinear
='(T ,
Therefore,
3r sin w r2- l I +r2+2rcosw +'1+r2+2rcosw 2
=T, I
+ rr' + 2r cosw
r2 - I
and Note that if r < I, then o < 0, and if r 1, then 0 z 0. Consequently, the left-half s-plane is mapped inside the unit circle. and the right-half s-plane is mapped outside the unit circle. If 1, = I, then a = 0, and
Thus, the JR-axis is mapped onto the unit circle. Using trigonometric identities, thismay be written in the equivalent
Let H&) be an all-pole filter with no zeros in the finite s-plane,
If H,,(s) is mapped into a digital filter using the bilinear transformation, will H ( z ) be an all-pole filter
With T, = 2, the bilinear transformation is
-:-I
+z-'
and the system function of the digital filter is
This may be written in the more conventional form as follows,
where
CHAP. 9 1
FILTER DESIGN
and Therefore, H ( z ) has p poles (inside the unit circle if Re(sd) < 0) and p zeros at z = - I . Note that rhese zeros come from the p zeros in HJs) at s = m. which are mapped to z = - I by the bilinear transformation. Thus. H ( : ) will nor be an all-pole filter.
Shown in the figure below is the magnitude of the frequency response of a low-pass tilter that was designed by mapping a type I analog Chebyshev filter into a discrete-lime filter using the bilinear transformation.
Find the filter order (i.e., the number of poles and zeros in H(::)).
The magnitude-squared response of a type I analog Chebyshev filter is
where is an Nth-order Chebyshev polynomial. Over the passband, i R,, the magnitude of the frequency response oscillates between I and ( 1 c ' ) - ' / ~ . AS the frequency varies from I2 = O to R = R,,, H = N cos-'(R/R,,) varies from 8 = N 7r/2 to 0 = 0. Therefore.
reaches its maximum or oscillates between zero and 1 N $ 1 times over the interval [O, Q,] [i.e., T,~(R/Q,,) I times]. The bilinear transformation is a one-to-one mapping of the jR-axis onto the unit minimum value N I times belween I and I/([ 6') over the interval 10, w , ) , where circle. Therefore. I H ( ~ J ' " ) ~ alternate N will ~
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