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Unformatted text preview: Name: Solutions Student ID: ECE 366 EXAM 2
November 17, 2006 N0 textbooks, notes or HW solutions.
One page of hand—written notes.
Calculators are allowed. Exam is 50 minutes. To maximize your score on this exam, read the questions carefully and write
legibly. For those problems that allow partial credit, show your work clearly. Good luck. 1. [20] State whether the following statements are true or false. No partial credit will
be given so you do not have to provide any explanation. a) b) d) e) D An audio signal with 101112 bandwidth is transmitted over a channel using AM
modulation. The minimum bandwidth of the channel should be 20Hz. .1. The Nyquist frequency of sampling for x(t) == sin C(SI) is 20 rad/sec. Xiuﬂ 1: JIFQCT i (NS: \OY'ad/sFc ‘11:. 5'
The signal x[n] = cos(3n) + cos(§7m) is periodic.
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For a real and even periodic signal, the Fourier series coefﬁcients, Ck , are always real. T The power of u[n] is 1.
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The spectrum of a periodic signal is always periodic. ii’r’s discrete, ~—
hot’ponociic 1“ g) The ideal lowpass ﬁlter is not used in practice because it is not causal. T h) If a periodic signal x(t) has Fouﬁer series coefﬁcients C k , the Fourier series coefﬁcients for the signal x(—r) will also be equal to Ck . "F:
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SxHScH: Xl0\é i§> w 2. [40] Consider the ideal sampling system given below for the continuous time signal
with the spectrum given in Figure l (a). Let Q = 8x,a)2 = 1071'. a) [5] What is the Nyquist frequency of sampling for this signal? b) [6] Assume that this signal is sampled using the ideal sampler shown below with T = 0.5 . Derive an expression for the spectrum of the sampled signal, X p ((0) , in terms of X ((0) .
c) [10] Sketch the spectrum of the sampled signal, X p ((0) , you found in part (b). Draw at least the ﬁrst 5 terms in the summation, i.e. k = 0,:t1,i2,i3,i4. Make sure to label the frequency and the amplitude axes.
cl) [6] Show that you can reconstruct the original signal from the sampled one. What are the values of A, a)“ , cab for the bandpass ﬁlter shown in the Figure? e) [7] If the sampling rate T is now increased to 1 sec, can you still reconstruct the
original signal? Sketch the spectrum for the sampled signal, and discuss Whether you
can recover the original signal. l) [6] Show that the sampling frequency used in part (b) is less than the Nyquist
sampling frequency. Explain Why we can recover the original signal even though we
sampled at a rate that is less than the Nyquist rate. XI 11.!) p(t) = goat—M) n=w (t) plt)
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bom d p083, gxgw" . 01$ {GIT . mmmmmmmmmmmmm v—uWW.WWWW—_ 3. [40] Consider the system shown in the Figure below. Assume that
sin(1 0721:) x(t) = sin C(ZOmf), g(t) = cos(87rt) + cos(16m‘) and Mt) = 72': a) [6] Compute and. sketch X ((0) .
b) [6] Compute and sketch G(a)).
c) [6] Compute and sketch H (w) .
d) [6] Compute and sketch Z (w) .
e) [6] Compute and sketch W ((0) .
f) [10] Compute and sketch Y(a)).
sin(x) Hint: sin 00:) =
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 Digital Signal Processing, Fourier Series, Original signal

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