Home Assignment Solution 2
Exercise 1
Determine whether or not each of the following signals is periodic. If a signal is
periodic, determine its fundamental period:
(a) x1 (t) = 2 cos(3 t) + 3 sin(4t),
(b) x2 [n] = 4 cos(0.1 n),
(c) x3 (t) = 3 sin(3000 t)
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Solution to Home Assignment 1
Exercise 1
h[k ]x[n k ]
y [n] =
k=
where sequences are as follows
h[k ] =
k, 0 k 7
0, otherwise
(1)
x[ k ] =
1, 1 k 11
0, otherwise
(2)
Flip x[k ]. Then one will have the following cases
1. n 0, y [n] = 0
n1
k=0
2. 1 n 7, y
ECE 413 Tutorial 9
July 9th, 2013
Exercise 1
You are given a combined (i.e. discrete & continuous) signal processing system:
Figure 1: Pertaining to Problem 1
The system is fed with a continuous time DC signal x(t) = 1. The signal is sampled with
sampling
ECE 413 Tutorial 8
July 2nd, 2013
Exercise
Consider the system in Figure 1 with the following relations:
Xc (j ) = 0, | 2 104
x[n] = xc (nT )
n
y [n] = T
x[k ]
k=
Figure 1:
1. For this system, what is the maximum allowable value of T to avoid
aliasing?
2.
University of Waterloo
Department of Electrical and Computer Engineering
ECE 413 Digital Signal Processing
Midterm Exam, Spring 2010
June 10th, 2010, 5:30-6:50 PM
Instructor: Dr. Oleg Michailovich
Students name:
Students ID #:
Instructions:
This exam has
University of Waterloo
Department of Electrical and Computer Engineering
ECE 413 Digital Signal Processing
Midterm Exam, Spring 2011
June 22nd, 2010, 5:30-7:00 PM
Instructor: Dr. Oleg Michailovich
Instructions:
This exam has 3 pages.
No books and lectur
ECE 413 Tutorial 1
May 14, 2013
Exercise
Consider the system whose input-output system is given by
max(n,n0 )
T (x[n]) =
x[ k ]
k=min(n,n0 )
where n0 is some xed integer. Is the system (1) stable, (2) causal, (3) linear,
(4) time invariant?
Solution
(1)
A
ECE 413 Tutorial 2
May 21, 2013
Exercise
Find the Z transform of the sequence
x[ n ] =
1
3
n
1
2
u[n]
n
u[n 1]
(1)
Solution
1
2
1
3
n
1
Z
u[n]
n
1+
1 1 ,
z
3
1
Z
u[n 1]
1
1 1 ,
z
2
ROC:|z | >
1
3
ROC:|z | <
By linearity of the Z transform,
X (z ) =
1
1
ECE 413 Tutorial 3
May 28, 2012
Exercise
Let x[n] be a causal stable sequence with z-transform X (z ). The complex
cepstrum x[n] is dened as the inverse transform of the logarithm of X (z ),
i.e.
Z
X (z ) = log(X (z ) x[n]
(1)
where the ROC of X (z ) incl
ECE 413 Tutorial 5
June 11th, 2013
Exercise
Consider a discrete-time LTI system with frequency response H (ej ) and
corresponding impulse response h[n]. We are given the following two clues
about the system:
(i) The system is causal
(ii) The DTFT of the s
ECE 413 Tutorial 6
June 18th, 2013
Exercise
Consider the transfer function of a stable system
H (z ) =
z D
,
1 az D
1<a<1
(a) Determine the impulse response h[n] as a function of D and a
(b) Determine the magnitude response of the IIT filter and show that
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