Due 24 February 2016 at 5:00 PM
ECE 438
1.
Assignment No. 4
Spring 2016
For the signals below, do the following: (i) sketch the signal, (ii) find the ZT if it
exists, (iii) sketch the ROC for the ZT, showing all poles and zeros.
n
a.
æ 1ö
x[n] = ç ÷ sin(p n / 3)u[n]
è 2ø
b.
ì 1 n
ï æç ö÷ , n ³ 0
x[n] = í è 2 ø
ï
n
n £ -1
î 2 ,
c.
ì 2n ,
n³0
ï
x[n] = í æ 1 ö n
ï çè 2 ÷ø , n £ -1
î
d.
x[n] = n ( -1) u[n]
n
Hint: You should, as much as possible, use known signal-ZT pairs and ZT
properties, rather than evaluating the ZT sum directly.
2.
Consider the ZT
H (z) =
1
æ 1 -1 ö æ 4 -1 ö
çè 1- z ÷ø çè 1- z ÷ø
3
3
Sketch the 3 different ROC’s that are possible for this ZT; and for each ROC, find
the corresponding signal x[n] . Simplify your answers as much as possible. Also, for
each of the three signals x[n] , state whether or not the DTFT X(w ) exists for that
signal.
3.
Consider a causal LTI system with transfer function
1- z -1
H (z) =
æ 1 -2 ö
çè 1+ z ÷ø
4
a.
Sketch the locations of the poles and zeros.
b.
Use the graphical approach to determine the magnitude and phase of the
frequency response H( ) , for w = 0, p / 6, p / 2, 5p / 6, p . Based on these
2
values, sketch the magnitude and phase of the frequency response for
-p £ w £ p . (Be sure to show your work.)
4.
c.
Is the system stable, Explain why or why not?
d.
Find the difference equation for y[n] in terms of x[n] , corresponding to this
transfer function H(z) .
Consider a DT LTI system described by the following non-recursive difference
equation (moving average filter)
y[n] =
1
{ x[n] + x[n -1] + ...+ x[n - 6] + x[n - 7]} .
8
a.
Find the impulse response h[n] for this filter. Is it of finite or infinite
duration?
b.
Find the transfer function H(z) for this filter.
c.
Sketch the locations of poles and zeros in the complex z-plane.
Hint: To factor H(z) , use the geometric series and the fact that the roots of the
polynomial z N - p0 = 0 are given by
zk = p0
5.
1/ N
e[
j (arg p 0 ) / N + 2pk / N ]
, k = 0,… , N -1
Consider a DT LTI system described by the following recursive difference equation
y[n] =
1
{ x[n] - x[n - 8]} + y[n -1].
8
a.
Find the transfer function H(z) for this filter.
b.
Sketch the locations of poles and zeros in the complex z-plane.
Hint: See Part c of Problem 4.
c.
Find the impulse response h[n] for this filter by computing the inverse ZT of
H(z) . Is it of finite or infinite duration?