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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:
(a) What weighting function W(eJU)should be used to design this filter
CHAP. 91
FILTER DESIGN
( b ) What are the minimum and maximum numbers of extremal frequencies that a type I filter of order N = 128 must have
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 ~ .
(a) Find the approximate passband and stopband ripple if we were to use a Kaiser window design.
( 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
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
IIR Filter Design 9.60
Find the minimum order and the 3-dB cutoff frequency of a continuous-time Butterworth filter that will satisfy the following Frequency response constraints:
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:
( a ) Butterworth filter
(h) Type 1 Chebyshev filter
(c) Type I1 Chebyshev filter
(d) Elliptic filter
Let H,(jQ) be an analog filter with
HAjQ)ln=o = 1 ( a ) If a discrete-time filter is designed using the impulse invariance method, is it necessarily true that
(b) Repeat part ( a ) for the bilinear transformation
Consider a causal and stable continuous-time filter that has a system function
If a discrete-time filter is designed using impulse invariance with T, = 1, find H ( z ) .
The system function of a digital filter is
FILTER DESIGN
[CHAP 9
( a ) Assuming that this filter was designed using impulse invariance with T, = 2, find the system function of two different analog filters that could have been the analog filter prototype. ( b ) If this filter was designed using the bilinear transformation with T, = 2, find the analog filter that was used as the prototype. 9.67
Determine the characteristics of the s-plane-to-z-plane mapping
The system function of an analog filter Ha(s)may be expressed as a parallel connection of two lower-order systems
If Ha(s),Hal(s), Ha2(s)are mapped into digital filters using the impulse invariance technique, will it be true that and
What about with the bilinear transformation
If an analog filter has an equiripple passband, will the digital filter designed using the impulse invariance method have an equiripple passband Will it have an equiripple passband if the bilinear transformation is used Can an analog allpass filter be mapped to a digital allpass filter using the bilinear transformation An IIR low-pass digital filter is to be designed to meet the following specifications: Passband cutoff frequency of 0.221~ a passband ripple less than 0.01 with Stopband cutoff frequency of 0.241~ a stopband attenuation greater than 40 dB with
(a) Determine the filter order required to meet these specifications if a digital Butterworth filter is designed using the bilinear transformation.
9.70 9.71
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