FIR filters with linear phase comprise an important class of LTI sytems. This exercise examines the four types of symmetric impulse responses that will result in linear phase. In addition, constraints on the zeros of linear phase filters will be studied and the group delays will be computed. Knowledge of linear phase systems will then be applied to design FIR, linear phase filters by windowing using both parametric and non-parametric windows.
Filter design by windowing involves first calculating the impulse response for an ideal filter. Since the ideal impulse response will not be time-limited, it must be truncated at some point in order to implement it in a practical system. The truncation is done using both non-parametric and parametric windows. Non-parametric windows have a fixed shape and include rectangular, Bartlett and Hamming. With these windows, there is a fundamentel tradeoff between main lobe width and side lobe height. Parametric windows, such as the Kaiser window, have shapes that are determined by parameters and can be adjusted to satisfy given constraints on the frequency response.
Finally, a filter will be designed using 4 different IIR methods and the results will be compared.
The following 2 blocks from J-DSP will be useful in performing this exercise.
The Window Block. J-DSP contains a Window block under the
basic
blocks menu. This block will be useful in the second and third problems
to truncate the ideal impulse response. This block takes an input signal
and multiplies it with a window of specified length and type.
In the case of the Kaiser
window, the
parameter must also be entered. After changing any of
the settings
in the window block, press the update block for the changes to take effect.
The filter blocks menu in J-DSP also contains a Kaiser block and a FIR block. These blocks will produce filter coefficients based on the window design method used in problems 2 and 3, however, they will only give up to 10th order filters. Since the filters in problems 2 and 3 are of orders larger than 10, use the Window block instead.
The IIR block. The IIR block is also found uder the filter menu
and is used for designing IIR filters. To design an IIR filter, specify
one of four IIR design methods (Butterworth, Chebychev I, Chebychev II or
Elliptic), the filter type(highpass, lowpass, etc.), the cutoff
frequencies
and the tolerances. The cutoff frequencies should be entered as
percentages of pi. For example, a cutoff frequency of
0.5pi should be entered as 0.5. The IIR block can be connected to a
PZ-plot block to see a plot of the filter's poles and zeros and to a
Freq-Resp block to see its frequency response. It can also be
connected to the bottom of a filter block which will set the filter
coefficients of that
block to the coefficients produced by the IIR filter design block.
All reports must be typed and can be upto 20 pages including figures. Do not mix Lab4 and Lab5. Include the following in both the reports.
The report must include the following:
Introduction:
(around ½ page)
State objective of the lab.
Give an introduction to FIR and IIF filters and the concept of Windowing for Lab 4 and introduction to FFT in Lab 5.
Analysis:
For all problems give theoretical justification and give equations as needed.
Experiment:
Give experimental results.
Cut and Paste graphs wherever applicable in your document. You can use Alt+ Print Scrn to port to clipboard. For all figures use figure captions e.g.,
Fig 1: Graph 1:Impulse Response
Quiz Questions:
A set of quiz questions are presented at the end of each problem.The answers for these questions are to be included in the same problem. Take a print out of the quiz questions and mark ur answers in them wherever possible.
Discussion:
Summarize all results and their significance (around ½ page).Discuss the frequency response of different types of filters.
For this lab, use the J-DSP program with Netscape 4.6 (or higher). Push the ``Start'' button below to begin J-DSP.
Problem 1: FIR linear phase systems
Consider the following four impulse responses.
1.1 A sytem with symmetric impulse response has linear phase. Is this statement true or false.
a) True
b) False
1.2 If a sytem has transfer function H(z)=1 - z-1 -
z-2 + z-3 what type of symmetry does its impulse
response have?
a) Type I
b) Type II
c) Type III
d) Type IV
1.3 What is the symmetry condition in the Z-domain for
h1[n]?
a)
H1(z) = z -M H1(z -1) where
M=6.
b) H1(z) = z
-M H1(z -1) where M=7.
c) H1(z) = -z -M H1(z
-1) where M=6.
d)
H1(z) = -z -M H1(z -1) where
M=7.
1.4 What is the symmetry condition in the Z-domain for
h2[n]?
a)
H2(z) = z -M H2(z -1) where
M=6.
b) H2(z) = z
-M H2(z -1) where M=7.
c) H2(z) = -z -M H2(z
-1) where M=6.
d)
H1(z) = -z -M H2(z -1) where
M=7.
1.5 What is the symmetry condition in the Z-domain for
h3[n]?
a)
H3(z) = z -M H3(z -1) where
M=6.
b) H3(z) = z
-M H3(z -1) where M=7.
c) H3(z) = -z -M H3(z
-1) where M=6.
d)
H3(z) = -z -M H3(z -1) where
M=7.
1.6 What is the symmetry condition in the Z-domain for
h4[n]?
a)
H4(z) = z -M H4(z -1) where
M=6.
b) H4(z) = z
-M H4(z -1) where M=7.
c) H4(z) = -z -M H4(z
-1) where M=6.
d)
H4(z) = -z -M H4(z -1) where
M=7.
1.7 What is the group delay in problem 1 for system h1[n] and
h3[n]?
a) 3
b) 3.5
c)
171.8
d) 200.5
1.8 What is the group delay in problem 1 for system h2[n] and
h4[n]?
a) 3
b) 3.5
c)
171.8
d) 200.5
1.9 The zeros of H2(z) and H4(z) are symmetric
about
a) the imaginary axis
only.
b) the real axis only.
c) the real axis and the imaginary axis.
d) neither the real axis nor the imaginary
axis.
1.10 The zeros of H1(z) and H3(z) are symmetric
about
a) the imaginary axis
only.
b) the real axis only.
c) the real axis and the imaginary axis.
d) neither the real axis nor the imaginary
axis.
1.11 If a causal sytem has a symmetric impulse response which is nonzero
up to n=M, the group delay of the system will be
a) M.
b)
M/2.
c) (M-1)/2.
d) M-1.
1.12 A filter with type II symmetry as in part b will always have a zero
at
a) 0 radians.
b) pi radians.
c)
pi/2 radians.
d) 3*pi/2 radians.
1.13 A filter with type III symmetry, as in part c, will always have a
zero or zeros at what frequencies?
a) 0 and pi radians
b) 0 radians
only
c) pi radians only
Problem 2: FIR Filter Design by Windowing
Let
be the ideal impulse response of a lowpass filter. Design a FIR filter with generalized linear phase by truncating this ideal impulse to 60 samples.
For all parts of this problem, use a signal generator block with the following settings:
2.1 After truncating the sinc function in problem 2, which window type
gave the narrowest transition region(sharpest cutoff) in the frequency
domain?
a) rectangular
b) bartlett
c) hamming
2.2 After truncating the sinc function in problem 2, which window type
gave the largest side lobes in the frequency domain?
a) rectangular
b)
bartlett
c) hamming
2.3 After truncating the sinc function in problem 2, which window type
gave the wider transition region in the frequency domain?
a) rectangular
b) hamming
Problem 3: Filter design using the Kaiser window method
Design a highpass filter with generalized linear phase using the Kaiser window method.
Use the following specifications:
for a filter that meets the specifications above.
3.1 What is the correct value of M in problem 3?
a) 50
b) 52
c) 25
d) 26
3.2 Why is it necessary to round M up to the next highest even integer
instead of down to the next lowest even integer?
a) It ensures the filter will have linear phase.
b) By doing so, we guarantee that the filter
specifications will be met or exceeded in both the passband and the
stopband.
c) None of the above
3.3 Why must M be an even integer in this problem instead of
odd?
a) Linear phase can only be
accomplished if M is even.
b) If M were
odd, we would force a zero at pi radians thus destroying our highpass
filter.
c) The impulse response would not
have been symmetric had M been odd.
3.4 What is the group delay for the filter in this problem?
a) 26
b)
25
c) 52
d) 50
3.5 What is the expression for the analytical impulse response of the
filter in problem 3?
a) h[n] =
0.4*sinc(0.4*pi(n-26))
b) h[n] =
sinc(pi(n-26))
c) h[n] = sinc(pi(n-26))
- 0.4*sinc(0.4*pi(n-26))
d) h[n] =
0.4*sinc(pi(n-26)) - 0.4*cos(0.4*pi)
3.6 What is the value of the Kaiser filter parameter beta that meets the
specifications in problem 3?
a)
2.31
b) 1.92
c) 1.51
d) 4.0
Problem 4: IIR Filter Design
In this exercise, you will design an IIR filter with JDSP. The filter will be designed using four different IIR methods (Butterworth, Chebychev I, Chebychev II and Elliptic) so that results of the 4 different methods can be compared. The spefications for the filter are shown below.
Use J-DSP's IIR block to design the filter using each one of the four IIR methods mentioned above. You may want to create all 4 of the filters simultaneously with J-DSP so you can compare the frequency response and pole-zero plot of each one. To determine whether the filter is monotonic or equiripple in the stopband, view the frequency response on a dB scale. To determine whether the filter is monotonic or equiripple in the passband, view the frequency response on a linear scale.
4.1 Which filter design method requires the highest order to meet the
specifications?
a)
Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.2 Which filter design method requires the lowest order to meet the
specifications?
a)
Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.3 Which of the 4 design methods produces a stable filter?
a) Butterworth
b) Chebychev I
c) Chebychev
II
d) Elliptic
e) All of the above
4.4 Filters designed with these 4 IIR methods have linear
phase?
a) True
b) False
4.5 Which filter is equiripple in the stopband and monotonic in the
passband?
a) Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.6 Which filter is monotonic in the stopband and equiripple in the
passband?
a) Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.7 Which filter is equiripple in the both the passband and the
stopband?
a) Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.8 Which filter is monotonic in the both the passband and the
stopband?
a) Butterworth
b) Chebychev I
c) Chebychev II
d) Elliptic
4.9 For the filters which are monotonic in the stopband, where are all the
zeros located?
a) z=1
b) z=exp(j*pi/2)
c) z= -1
d)
z=exp(-j*pi/2)
Copyright 2000 Andreas Spanias, MIDL, Arizona State University JDSP and Report Submission Software Developed by ASU-MIDL For questions contact Prof. Spanias spanias@asu.edu.