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FILTER DESIGN To determine whether or not this system is stable, it is necessary to find the denominator polynomial,

Fig. 7-25

[CHAP, 9

If a tangent and a secant intersect outside a circle, the tangent is the mean proportional between the secant and its external segment.

and check to see whether or not the roots of A(z) lie inside the unit circle. Given that H(z) has two poles and two zeros, we may use the Pad6 approximation method to find the denominator coefficients:

Fig. 2-4

42) = 1 3

PRINCIPLE 2:

iz-'

PRINCIPLE 3:

+ !z-2

If two secants intersect outside a circle, the product of the lengths of one of the secants and its external segment equals the product of the lengths of the other secant and its external segment.

What type of window(s) may be used to design a low-pass filter with a passband cutoff frequency w , = 0.35n, a , transition width A o = 0 . 0 2 5 ~and a maximum stopband deviation of 8, = 0.003 Use the window design method to design a minimum-order low-pass filter with a passband cutoff frequency o, = 0.457, a stopband cutoff frequency w, = OSn, and a maximum stopband deviation 6, = 0.005. We would like to design a bandstop filter to satisfy the following specifications:

RULE 3 (Product Rule).

Fig. 7-26

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( b ) What are the minimum and maximum numbers of extremal frequencies that a type I filter of order N = 128 must have

Fig. 7-27

Suppose that we would like todesign a low-pass filter oforder N = 128 with apassband cutoff frequency w, = 0 . 4 8 ~ and a stopband cutoff frequency of w, = 0 . 5 2 ~ .

( b ) If an equiripple filter were designed so that it had a passband ripple equal to that of the Kaiser window design found in part ( a ) , how small would the stopband ripple be

Fig. 7-28

Dx ( f (x) g(x)) = f (x) Dx g(x) + g(x) Dx f (x)

We would like to design an equiripple low-pass filter of order N = 30. For a type I filter of order N , what is the minimum number of alternations that this filter may have. and what is the maximum number For a low-pass filter with ti,, = ti,, what is the difference in the stopband attenuation in decibels between a Kaiser window design and an equiripple filter if both filters have the same transition width

(a) ED (b) AE (c) CE 4. Then 16x EB 4(12), so that 16x 8(2), so x2 x. Then x2 48 or x 16 and x 27(3) or x2 3. 4. 81, and x 9.

Find the minimum order and the 3-dB cutoff frequency of a continuous-time Butterworth filter that will satisfy the following Frequency response constraints:

Fig. 7-29

Use the bilinear transformation to design a first-order low-pass Butterworth filter that has a 3-dB cutoff frequency w,. 0 . 5 ~ . = Use the bilinear transformation to design a second-order bandpass Butterworth filter that has 3-dB cutoff frequencies w,= 0 . 4 and w,, = 0 . 6 ~ . ~ If the specifications for an analog low-pass filter are to have a I-dB cutoff frequency of 1 kHz and a maximum stopband ripple 6, = 0.01 for If I > 5 kHz, determine the required filter order for the following:

1 72.

HAjQ)ln=o = 1 ( a ) If a discrete-time filter is designed using the impulse invariance method, is it necessarily true that

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