MAE107 Homework #4
Prof. M’Closkey
Due Date
The homework is due by Friday, 1PM, February 6, 2009, to the TA at beginning of recitation.
Problem 1
The impulse response of a system is given in the top plot of Fig. 1. The impulse response is zero
for
t<
0and
t
≥
4. It is desired to compute the response of the system to the input shown in
the bottom plot of Fig. 1. Although you can calculate an analytical expression for the output, you
can use a graphical means to determine the system output as well. Thus, it is suggested that you
attempt the “graphical” approach ±rst before the analytical approach.
Carefully
draw the response
of the system to this input –use the bottom graph axes in Fig. 1 and submit it with your solution.
Problem 2
Recall the noncausal “moving average” ±lter that you analyzed in an earlier homework. The impulse
response of the ±lter is shown in Fig. 2. Compute an analytical expression for the frequency response
function of this ±lter from the expression
χ
(
ω
)=
Z
∞
−∞
h
(
t
)
e
−
jωt
dt.
Plot the magnitude and phase of the frequency function. Use a frequency axis extending from
0
.
01 hertz, the magnitude axis extending from 0
.
001 to 10, and the phase axis from
−
20 to 200 de
grees. Study your plots and note that for frequencies below about 1 hertz, the ±lter output is almost
equal to the ±lter input since

χ
(
ω
)
≈
1and
∠
χ
(
ω
)=0
◦
. For frequencies above 1 hertz, though,
the ±lter attenuates the input signal –this is how the “higher frequency” noise is separated from the
lower frequency signal of interest. Also note a series of deep “notches” at
ω
=5
,
10
,
15
,
20
,...,
hertz.
Can you determine why the notches exist (think about driving the ±lter with a 5 hertz sinusoid,
etc.)?
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 Winter '06
 TSAO
 Fourier Series, Signal Processing, LTI system theory, Impulse response

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