Digital Signal Processing I
Midterm Exam (B1)
Fall 2015
1) An LTI system with input x(n) and output y (n) has impulse response
h(n) 3 (n) 2 (n 1) (n 2)
a) Find a difference equation that represents the system.
3 y (n) 2 y (n 1) y (n 2) x ( n)
A.
B.
y ( n)

Digital Signal Processing I
Midterm Exam (A2)
Fall 2015
1) An LTI system with input x(n) and output y (n) has impulse response
h(n) 3 (n) 2 (n 1) (n 2)
a) Find a difference equation that represents the system.
3 y (n) 2 y (n 1) y (n 2) x ( n)
A.
B.
y ( n)

Digital Signal Processing I
Midterm Exam (B2)
Fall 2015
1) An LTI system with input x(n) and output y (n) has impulse response
h(n) (n) 2 (n 1) 3 ( n 2)
a) Find a difference equation that represents the system.
y (n) 2 y (n 1) 3 y (n 2) x (n)
A.
B.
y ( n)

EL6113
Solution #6
1.
T
n (t ) k (t ) dt
0
1
T
T
T
1
T
0
e
jn
2
j (n k )
t
T
e
0
2
t
T
1
T
e
jk
2
t
T
1
dt
T
T
e
j (n k )
dt
0
1 ; n k
2 T
j (n k )
t
dt
1
T
e
;
j (n k )2
0
1 ; n k
1
e j ( n k ) 2 1 ;
j ( n k )2
2
t
T
n k
n k
and, since e j ( n k

EL6113
Hwk #3
1) A causal discrete-time system is described by the difference equation,
y (n) x( n) 3 x (n 1) 2 x(n 4)
a) What is the transfer function of the system?
Y ( z ) X ( z ) 3z 1 X ( z ) 2 z 4 X ( z )
Y ( z ) 1 3z 1 2 z 4 X ( z )
H ( z ) 1 3 z
1

Digital Signal Processing I
Midterm Exam (A1)
Fall 2015
1) An LTI system with input x(n) and output y (n) has impulse response
h(n) (n) 2 (n 1) 3 ( n 2)
a) Find a difference equation that represents the system.
y (n) 2 y (n 1) 3 y (n 2) x (n)
A.
B.
y ( n)

Design Techniques; haplei 7
1variance method
function H c (s) of
:11. Is your answer
Td ‘2 Find
gn. Is your answer
r with generalized
id the value of the-‘
aciﬁcations.
window should b-
uecl FIR ﬁlter wit
: ideal real—valued
given by
ciﬁcation? What is
z

EL6113
Hwk #1 Selected Solutions
2.1 a)
y ( n) T cfw_x(n) g (n) x( n)
with g (n) given.
x ( n) M .
To check for stability, let the input be bounded with
y (n) T cfw_x (n) g (n) x(n) g (n) x(n) M g (n) .
Then
We can see that if g (n) is
bounded, say by g (

1) Baseline drift correction using dc notch filter:
In Matlab, load the ECG data file 'ecg_lfn.dat' (on the website) using the command
ecg = load('ecg_lfn.dat')
The sampling frequency is 1000 samples/second. Plot the ECG signal and observe the baseline
dr

Homework #4 Solutions
Additional Problems:
1. This is just running the program.
2. Modify square_pulse.m to calculate 128 samples of the DTFT of the sequences
a) x(n) = e n / 5 [U (n) U (n 21)]
The DTFT of this sequence can be calculated as follows.
X ( e

1.1.6 Make a sketch of each of the following signals
(a)
f (n) =
1
X
( 0.9)
k
(n
3 k)
k=0
(b)
g(n) =
1
X
( 0.9)
|k|
(n
3 k)
k= 1
(c)
x(n) = cos(0.25 n) u(n)
(d)
x(n) = cos(0.5 n) u(n)
1.1.6) Solution
11
1.2.1 A discrete-time system may be classified as fo

1.3.7 Discrete-time signals f and g are defined as:
f (n) = an u(n)
g(n) = f ( n) = a
n
u( n)
Find the convolution:
x(n) = (f g)(n)
Plot f , g, and x when a = 0.9. You may use a computer for plotting.
1.3.7) Solution
1
X
x(n) =
=
k= 1
1
X
f (k)g(n
k)
ak u

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EL6113
Hwk #2
Textbook Problems: from Scanned Homework 2 document. Put high priority on the
problems in red.
3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.8, 3.9, 3.12, 3.19, 3.20, 3.21, 3.22, 3.27, 3.28, 3.31
Additional Problems:
1. For the z transform
F ( z) =
3z 3 +

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Homework #4 Solutions
Additional Problems:
1. This is just running the program.
2. Modify square_pulse.m to calculate 128 samples of the DTFT of the sequences
a) x(n) e n / 5 U (n) U (n 21)
The DTFT of this sequence can be calculated as follows.
X (e j )

EL6113
Homework 11
1. Suppose a function, f (t ) , has a fourier transform F ( ) . Then f (t ) can be
represented by the inverse transform formula as
1
f (t ) =
2
F ( ) e
j t
d .
Now suppose we want to approximate f (t ) by another function, g (t ) , whi

1.6.9 Echo Canceler. A recorded discrete-time signal r(n) is distorted due to an echo. The echo has a lag of 10
samples and an amplitude of 2/3. That means
r(n) = x(n) +
2
x(n
3
10)
where x(n) is the original signal. Design an LTI system with impulse resp

1.6.1 A causal discrete-time system is described by the dierence equation,
y(n) = x(n) + 3 x(n
1) + 2 x(n
4)
(a) What is the transfer function of the system?
(b) Sketch the impulse response of the system.
1.6.1) Solution
H(z) = 1 + 3 z
1
+ 2z
4
h(n)
4
3
2

1) Baseline drift correction using dc notch filter:
In Matlab, load the ECG data file 'ecg_lfn.dat' (on the website) using the command
ecg = load('ecg_lfn.dat')
The sampling frequency is 1000 samples/second. Plot the ECG signal and observe the baseline
dr

Homework 9
Scanned Textbook Problems:
7.8, 7.35, 7.36
Additional Problems:
1. For each of the low pass filter specifications given below use the posted matlab
program op_filt.m to find a filter that meets the specifications. Assume the maximum
error in ea

% find 128 samples (between 0 and 2*pi) of DTFT of the sequence x(n) consisting
of ones from n=-5 to n=5.
% generate vector representing the aliased version, xbar(n). xbar(n) is
periodic with period 128, so
% the base period from n=0 to n=128 has six ones

EL6113
Hwk 10
Textbook Problems: from Scanned Homework 5 document.
5.12, 5.13, 5.17, 5.18, 5.38
Additional Problems:
1. By using the bilinear transformation we wish to design an IIR digital filter low pass
filter with the following specifications:
1 H ( e

EL 6113 Midterm Exam
Fall 2016
DTFT
X f (!) =
Name:
x(n) e
jn!
n
SOLUTION
Inverse DFT
ID Number:
1. Closed book, closed notes.
2. Non-graphing calculators permitted.
3. Show your work. Simplify your answers.
4. No cell phones.
Cell phones must be turned o

1
Exercises in Digital Signal Processing
Ivan W. Selesnick
The Discrete Fourier Transform
1.1 Compute the DFT of the 2-point signal by hand (without a calculator or
computer).
January 29, 2015
x = [20,
5]
SOLUTIONS
Solution
Contents
1 The Discrete Fourier

6.3 What is the eect of the width of the zero-weighted transition band? As
the zero-weighted transition band becomes wider, what happens to the
frequency response? Show examples using Matlab to design several filters
to illustrate what happens.
Solution
A

10.2 You should do this problem by hand (no computer).
in the z-plane. Similarly, poles in the RHS of the imaginary axis in the
s-plane must go to the outside of the unit circle in the z-plane.
(a) Suppose you have a stable causal analog filter. Prove why