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Unformatted text preview: E R IAIN TIT TE FTE HN L Y
SCHOOL of ELECTRICAL and COMPUTER ENGINEERING E E2 2 mm r2
Problem Set #9 Assigned: Week of 14—Jul—09
Due Date: 20l21—Ju1—09 Please check the T—square “Announcements” daily. All ofﬁcial course announcements will be
posted there. ALL of the STARRED problems should be turned in for grading. PROBLEM 9.1: Try parts (b) and (c) of Problem 11.6 on p. 342 of Signal Processing First. PRQBLEM 9.2: Try Problem 11—8 on p. 343 of Signal Processing First. PRQBLEM 9.3: (a) Do Problem 12.1 on p. 381 of Signal Processing First. (b) Do Problem 12.2 on p. 381 of Signal Processing First. PRQBLEM 9.4: (a) Do Problem 12.4 on p. 381—382 of Signal Processing First. Try to think graphically as much
as possible; you Will ﬁnd that more convenient than wrangling equations. (b) Do Problem 12.5 on p. 382 of Signal Processing First. PRQBLEM 95*: (10 pts) The system in the dashed box below is called a quadrature modulation system. It is
a method of sending two bandlimited signals in the same frequency band in a way that can be
uniquely unscrambled. $105) I w , LTFSystem
H (w) cos wct sin wet u)(t) = $1 (t) cos wot + $2 (t) sin wot
LWLLLJ
Quadrature Modulator Demodulator = Mixer, then Filter Assume that both input signals are bandlimited with highest frequency cum; i.e.7 X1(jw) : 0
for w 2 com and X2(jw) = 0 for w 2 (am, where we >> cum. (a) Determine an expression for the Fourier transform W(jw) in terms of X1(jw) and X2(jw).
Make a sketch of W(jw). Assume simple (“typical”) shapes (each different) for the bandlim—
ited Fourier transforms X1(jw) and X2(jw), and use them in making your sketch of W(jw). (b) From the expression found in part (a) and the sketch that you drew7 you should see that
W(jw) = 0 for w g and and for w 2 tab. Determine cod and cab. You should also observe that
the shifted copies of X1(jw) and X2(jw) are right on top of each other so that we could not
separate them by linear ﬁltering. (c) Given the trigonometric identities 2sin6cos6 = sin 20, 2sin2 9 = (1 7 cos 29), and 2cos2 9 =
(1 + cos 20), show that in the “demodulator” ﬁgure on the right above7 the output of the
signal multiplier (sometimes called a mixer) is: u(t) = éx1(t)(1 + cos 2wct) + %$2(t) sin 2wct Use your assumed “typical bandlimited Fourier transforms” for X1(jw) and X2(jw) with the
above expression for o(t) to construct a “typical” plot of V(jw). (d) The signal u(t) as determined in part (c) is the input to an LTI system. Determine the
frequency response of that system so that its output is y(t) : x1(t). Give your answer as a
carefully labeled plot of H (jw). You should use the plot that you constructed in part (c) as an aid in determining H(jw). (e) Draw a block diagram of a demodulator system whose output will be $2(t) when its input is
w(t). This requires that you change the signal multiplier in the demodulator. PRQBLEM 95*: Consider the following system for discrete—time ﬁltering of a continuous—time signal: Ideal Ideal y(t)
C'tO'D DtoC
Converter Converter (a) Suppose that the discrete—time system is deﬁned by the difference equation = 0.9y[n i 1l+ + min i 2], and the sampling rate of the input is f5 = 400 samples/ second. Determine an expression for
Heg(jw), the overall effective frequency response (versus analog frequency w) of the above
system. Use this result to ﬁnd the output y(t) when the input to the overall system is
$(t) : 2cos(1007rt). (b) Assume that the input signal 30(75) has a bandlimited Fourier transform X (jw) as depicted
below. For this input signal, what is the smallest value of the sampling frequency f5 such that the Fourier transforms of the input and output satisfy the relation Y(jw) = Heg(jw)X(jw)?
X (W)
A —507r 5071' 22 (0) Assume that the discrete—time system has frequency response H (6”) deﬁned by the following plot:
H(ejw) A : i >
_‘ _‘ 71' w
2 2 271' Now, if f5 : 400 samples/sec , make a carefully labeled plot of Heg(jw), the effective fre— quency response of the overall system. Also plot Y(jw), the Fourier transform of the output
y(t), when the input has Fourier transform X ( jw) as depicted in the graph of part il27l' —I7T 7 (d) For the input in part (b) and the system in part (c)7 what is the smallest sampling rate such
that the input signal passes through the lowpass ﬁlter unaltered; i.e., what is the minimum
f5 such that Y(jw) : X(jw)? PRQBLEM 9.7*: Suppose a continuous—time LTI system has an impulse response given by hot) : sin(16007rt) (1) mt (a) Find the frequency response (or, equivalently, the Fourier transform) H ( jw) of this system. (b) In an earlier lecture, it was determined that the periodic function $(t) (with fundamental
period To) deﬁned on a single period by 30(75) 2 sin for 7T0/2 g t g T0/2
0
bag! :‘eI‘jei1 , 4k k,
: — —1
ak: .77T(4k2 _ ( ) Suppose T0 2 1/300 seconds, i.e., the fundamental frequency is (do = 60077 radians/ second.
If this signal 30(75) is input to the LTI system with impulse response h(t) given in 1, what
is the resulting output y(t)? Write your answer in terms of a sum of cosines with correct
amplitudes, frequencies, and phases. PRQBLEM 9.8: The derivation of the Sampling Theorem involves the operations of impulse train sampling and
reconstruction as shown in the following system: ‘T H) LTI System w
HTUW) pa): f (Kt—7m) The LTI system is the ideal bandlimited reconstruction ﬁlter with frequency response given by . i Ts lwl S 7T/T'S
HTW) *{ 0 M > 7r/Ts. The “typical” bandlimited Fourier transform of the input is depicted below: ‘ was)
A F
69710597; (n (a) For the input with Fourier transform depicted above, use the Sampling Theorem to choose
the sampling rate ws : 27r/T5 so that $T(t) : Plot Xs(jw) for the value of ws : 27r/TS
that is equal to the Nyquist rate.1 (b) If (us 2 27T/TS = 10071' in the above system and X(jw) is as depicted above, plot the Fourier
transform X S ( jw) and show that aliasing occurs. There will be an inﬁnite number of shifted
copies of X ( jw), so indicate what the pattern is versus w. (c) For the conditions of part (b), determine and sketch the Fourier transform of the output Xrljwl. 1Remember that the Nyquist rate is the lowest possible sampling rate that does not cause aliasing. PRQBLEM 93*: (10 pts) The input signal for the above sampling/ reconstruction system is
$(t) : 4cos(407rt) + 8cos(1007rt — 7r/4) — 00 < t < 00
and the frequency response of the lovvpass reconstruction ﬁlter is . _ Ts M < TF/Ts
BTW) —{ 0 w > 7r/TS Where TS is the sampling period. (a) Determine the Fourier transform X ( jw) and plot the Fourier transform X s (jw) for 727T/TS < w < 27r/TS for the case where 27r/TS : 5007. Carefully label your sketch to receive full credit.
What is the output m (t) in this case? (b) Now assume that (us 2 27T/TS = 1807?. Plot the Fourier transform Xs(jw) for *27r/Ts < w < 27T/Ts for the case Where 27T/TS = 18071 Carefully label your sketch to receive full credit.
What is the output air (t) in this case? (0) Is it possible to choose the sampling rate so that
mot) : A + B cos(407rt) Where A and B are constant? If so What is the value of T5 and What are the numerical values
of A and B? Is it unique? ...
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 Spring '08
 JUANG
 Digital Signal Processing, Signal Processing, sampling rate

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