LINEAR-PHASE FIR FILTER DESIGN BY LINEAR
PROGRAMMING
Often it is desirable that an FIR lter be designed to minimize
the Chebyshev error subject to linear constraints that the ParksMcClellan algorithm does not allow. An example described by Rabiner include
EL 713: Digital Signal Processing
HW 5 Solution
5.14 The Spectrogram.
Shown below are eight spectrograms of the same signal. (The signal is one of the signals from the
previous problem.) Each spectrogram is computed using a dierent set of parameters.
R cf
EL 713: Digital Signal Processing
HW 8 Solution
10.1 Consider the design of an analog lter to meet the following requirements.
0.96 |Ha (j )| 1
|Ha (j )| 0.03
for
for
| | 4
(141)
| | 4.6
(142)
(a) Design an analog Chebyshev-I lter of minimal degree to mee
EL 713: Digital Signal Processing
HW 6 Solution
7.1 Consider the design of a type I FIR lter of minimal length satisfying the following specications:
1 p A( ) 1 + p ,
s A( ) s ,
0 p
(96)
s
(97)
where
p = 0.018,
s = 0.004,
(98)
p = 0.31,
s = 0.38.
(99)
Wh
EL 713: Digital Signal Processing
HW 1 Solution
1.1 Compute the DFT of the 2-point signal by hand (without a calculator or computer).
x = [20,
5]
Solution:
For this problem, note that W2 = 1 so
X (0) = x(0) + x(1) = 25
X (1) = x(0) x(1) = 15
3
EL 713: Dig
11.7 Can you change the order of an up-sampler and a down-sampler with out change the total system? In
other words, are the following two systems equivalent?
System A:
M L
System B:
L M
Consider the following cases:
(a) M = 2, L = 2.
(b) M = 2, L = 3.
EL 713: Digital Signal Processing
HW 9 Solution
11.1 The signal x(n)
x(n) = cfw_. . . , 0, 0, 1, 2, 1, 0, 1, 0, 0, . . .
where 1 represents x(0) is applied as the input to the following system.
x(n) 2 H (z 2 ) y (n)
If the impulse response h(n) is given
EL 713: Digital Signal Processing
HW 7 Solution
8.1 For the lter, H (z ), whose zeros are shown in the following diagram, make a sketch of the zero diagram
of each spectral factorization of H (z ).
ZPLANE(h)
1
Imaginary Part
2
0.5
2
0
0.5
2
1
2
1
0
1
Real
Halfband Filter Design Exercise
Ivan W. Selesnick
April 28, 2009
Consider the problem of interpolating a signal x(n) so as to increase the rate of the signal by a
factor of two. This can be done by rst up-sampling the signal and then ltering it with a lte
LINEAR-PHASE FIR FILTERS
1. The amplitude response
2. Why linear-phase?
3. The four types of linear-phase FIR lter
4. Amplitude response characteristics
5. Evaluating the amplitude response
6. Zero locations of linear-phase lters
7. Automatic zeros
8. Des
MINIMUM-PHASE FIR FILTER DESIGN
The Lifting Procedure
1. INTRODUCTION
2. PROBLEM FORMULATION
3. THE SQUARE MAGNITUDE
4. A TRANSFORMATION
5. MEETING SPECIFICATIONS
6. SUMMARY OF THE LIFTING PROCEDURE
7. MEETING SPECIFICATIONS EXAMPLE
8. PLOTTING THE PHASE
FINITE PRECISION EFFECTS
1. FLOATING POINT VERSUS FIXED POINT
2. WHEN IS FIXED POINT NEEDED?
3. TYPES OF FINITE PRECISION EFFECTS
4. FACTORS INFLUENCING FINITE PRECISION EFFECTS
5. FINITE PRECISION EFFECTS: FIR VERSUS IIR FILTERS
6. DIRECT FIR FILTER STRU
Short-Time Fourier Transform and Its Inverse
Ivan W. Selesnick
April 14, 2009
1
Introduction
The short-time Fourier transform (STFT) of a signal consists of the Fourier transform of overlapping
windowed blocks of the signal. In this note, we assume the ov
EL 7133: Digital Signal Processing
Instructor: Ivan Selesnick
Spring 2010
2. DFT. Find the DFT of the N -point discrete-time signal x(n),
x(n) = [1, 1, . . . , 1, 0, 0, . . . , 0],
n = 0, 1, . . . , N 1.
M terms
Final Exam
6 single-sided pages of notes a
EL 7133: Digital Signal Processing
Instructor: Ivan Selesnick
Spring 2011
D() DESIRED FREQUENCY RESPONSE
1
0.8
Final Exam
0.6
SOLUTION
0.4
0.2
0
6 single-sided pages of notes are allowed. Otherwise the test is
closed notes.
0.66
0
0.66
0.66 0.83
W() W
EL 7133: Digital Signal Processing
Spring 2008
Polytechnic University, Brooklyn, NY
Instructor: Ivan Selesnick
Midterm Exam
SOLUTION
3 Single-sided pages of notes are allowed. Otherwise the test
is closed notes.
Closed book.
Show your work!
Calculator
Halfband Filter Design Exercise
Ivan W. Selesnick
May 5, 2009
Consider the problem of interpolating a signal x(n) so as to increase the rate of the signal by a
factor of two. This can be done by rst up-sampling the signal and then ltering it with a lter
H
EL 713: Digital Signal Processing
HW 3 Solution
1.7 What is y after running the following MATLAB commands?
clear
x = [1 2 3 4];
g = [5 6 7 8];
X = fft([x 0 0 0]);
G = fft([g 0 0]);
Y = X.*G;
y = ifft(Y);
Solution:
This causes a point by point multiplicati
EL 713: Digital Signal Processing
HW 4 Solution
4.13 Show a derivation for each of your answers. Do not use MATLAB to answer any part of this question.
(a) The transfer function of an FIR digital lter is
H1 (z ) = (1 + z 1 )3
Which of the four types of li