1.
Linearity
We have two linear time-invariant
systems,
and
.
(a) What do these systems mean? How
do you define these systems in Matlab?
(b) Show that for any
input x[n], cascade connection of h[n] and g[n] in any order yields the same
output. Does the same property hold for nonlinear systems? If not, give an
example.
2.
Complex plane
(a) Sketch the following numbers in the
complex plane: i) 5, ii) 
, iii) 
. How can you represent these numbers in the form 
?
(b) Assume that these numbers represent
the response of a system to a sine wave. Explain in plain words what the system
does to the input signal.
(c) Express the following numbers
through their magnitude and phase: i) 2–2i, ii) i, iii) -3+2i, iv) -4-i.
3. Sampled signals
(a) In practical
applications, sampled signals are typically represented as arrays of real numbers.
In addition to the sample values themselves, what do you need to know before
the properties of a sampled signal are uniquely defined?
(b) Assume a sampling
frequency of fs = 1000 Hz for this signal, and make it represent a
30 Hz sine wave. Plot both the analog and the digitized (sampled) sine wave
with Matlab (Use the Matlab commands plot and stem). Now make
a frequency vector which contains the frequencies from 0 up to 10000 Hz. Plot
the frequency content of the sampled sine wave for this range of frequencies
(Use the Matlab command freqz(sig,1,f,fs), where f is a frequency vector and fs is the sampling frequency). Based
on the plot, what can you say about the frequency content of a sampled signal?
Note that in Matlab, we always deal with sampled signals.
4.
Convolution
Take the signal
and the system
(a) Filter
with 
and call it
.
(b) Find the
frequency response of and . Then find the product of these frequency responses.
(c) Now compute the
frequency response of. Compare the frequency response of with the product of
the frequency responses of and.
5.
Frequency representation of discrete signals
Take the signals
where 
. Choose a couple of frequencies 
and 
and find 
and 
for k=1, 2. Did you notice any difference?
Based on this, explain what range of the frequency response of a real signal is
enough to give us information about its whole frequency response? (Note that
you also have to deal with the frequencies like -0.7p and 5.3p.)
Designed by Dr. Peyman Arian