1) The di erence equation of a system is given as:
y[n] = x[n] + 3x[n 1] + 7x[n 2] + x[n 3]
a) Show that this system is linear time invariant (LTI).
b) Make a complete signal ow diagram. Signal ow should be from left to right.
c) Obtain an expression for the frequency response function of the system in complex form.
d) Determine the output of the system when x[n] = [n] + [n 1] + [n 2].
e) Determine the output of the system when the input is x[n] = u[n] u[n 3]. Compare
your result with your result in c).
2) Suppose that we have two LTI systems, namely, System 1 and System 2, whose system functions are given as:
H1(z) = 1
2z 1 and H2(z) = 1 z 2
The sequence x[n] = u[n] u[n
2] is given as input to the cascaded arrangement.
a) If x[n] rst passes through System 1, followed by System 2, what will be the output?
b) If x[n] rst passes through System 2, followed by System 1, what will be the output?
c) Compare your results in parts a) and b).
d) Show that your conclusion in c) is true for any pair of LTI systems. (Please provide a rigorous mathematical proof). Is the result generalizable to more than two systems all of which are LTI?
3) In this question, you are given the following cascaded system:
a) If LTI-1 is a 5-point moving averager and LTI-2 is a rst di erence system, deter-mine the frequency response function of the system in complex form.
b) Sketch the magnitude response and phase response functions of the individual sys-
tems and the overall cascade system for !^ .
c) Find v[n] and y[n] if x[n] = 0:5nu[n] 0:1nu[n 1] for < n < 1.
1
4) Forward and inverse z tranformation and the system function:
a) Find the z-transform of the following functions:
i) x[n] = cnu[n], where c is an arbitrary constant.
ii) x[n] = 3 sin(0:5 n)u[n] where u[n] is the unit-step sequence.
b) Find the inverse z-transform of the following functions:
i)
z
ii)
z 2
z a
1 0:9z
1
c) You are given the following system functions:
H1
(z) = 1 3z
1 + 3z 2 z 3
H2
(z) =
1 + 0:75z
1
1 0:25z
2
i) Determine the poles and zeros of H1(z) and H2(z).
ii) Sketch the pole-zero diagrams of H1(z) and H2(z). Label the pole-zero locations and the axes clearly.
iii) Determine the impulse response functions (h1[n] and h2[n]) of the corresponding systems.
5) In this question, you are given the following cascaded system:
In each of the three parts below, determine the impulse response of the overall system and answer the following: Is the overall system causal? Is it stable? Explain.
a) h1(t) = (t + 2);
h2(t) = (t
2), and h3(t) = u(t
2).
b) h1(t) = (t + 2);
h2(t) = (t
2), and h3(t) = u(t).
c) h1(t) = u(t + 2);
h2(t) = u(t
2), and h3(t) = (t
1).
6) You are given the following two signals:
x1(t) =
8
t + 3
for
t 2
[0; 3]
x2(t) =
<
t + 3
for
t 2
[ 3;0)
0
otherwise
:
3 t 3.
a) Sketch x1(t) and x2(t) over the interval
b) Calculate x1(t) x2(t);
x1(t) x1(t), and x2(t) x2(t 3).
1
for t 2 [ 3; 3]
0
otherwise
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