Carleton University
Department of Systems & Computer Engineering
SYSC5602 Digital Signal Processing
Fall Semester 2007
Final Exam, December 5th, 2007
This examination paper must be handed in with your answer booklet
Answer the following Questions
Question
1
Consider the system given by the following block diagram
x(t)
Ideal
LTI
Ideal
y(
t)
NO
D/A
~
~
Lowpass
?
System
Lowpass
Converter
~
Converter
Prefilter
H(z)
Postfilter
I
r
Is
=
48000
Hz
Is
=
48000
Hz
where the ideal pre and post filters are appropriately chosen such as to avoid aliasing, for a sys
tem that uses a sampling rate
of
Is
=
48000
Hz.
The discrete time LTI system shown is causal
with a transfer function
1
+
Z2
H(z)
=
1 _
0.5z2
a) Draw the polezero diagram for the discretetime filter H(z)
b) List all frequencies that
if present in x(t) would not be passed to yet). Consider the effects of
each block.
c) Determine the output yet)
of the system given the input
x(t)
=
2
+
3cos(
16000nt
+~)
+
4 cos
(24000nt
+~)
Use any information about the system that could simplify determining yet).
5+o.rd
c;{oYl.e.
~sume
that the input signal has the power spectrum shown below.
fXcnl
2
20000
20000
I
Hz
Give a reasonable estimate for the power spectrum
(I
Ycn1
2
)
of the output signal yet). Your answer
may be given in the form
of a reasonable sketch or in an analytical form.
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Consider the following diagram of an unknown LTI filter. With an input x(n)
=
8(n), the filter
zerostate response was found to be
yen)
=
(]rCos(~
+
~)u(n).
x(n)
Unknown
yen)
LTI Filter
"'
a) Is this filter causal? Is it stable? Justify your answers
./' b) Is this filter an FIR
or IIR system? Write a difference equation for the output yen) in terms of
the input x(n).
_ {)
c) Determine the filter's zerostate response if the input is a unit step sequence.
YLo) .
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 Winter '09
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