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Unformatted text preview: 34 CHAPTER 2 SINUSOIDS P2.17 Deﬁne x(t) as
x(t) = 5cos(a)t + £71) + 4cos(wt + %H)
+ 4cos(a)t + ﬁr) (3) Express x(t) in the form x(t) = A cos(wt + 45) by
ﬁnding the numerical values of A and d). (b) Plot all the phasors used to solve the problem in (a)
in the complex plane. P2.18 Solve the following simultaneous equations by using the phasor method. Is the answer for A1, A2, $1,
$2 unique? Provide a geometrical diagram to explain the
answer. cos(a)0t) = A1 cos(a)ot + 451) + A2 cos(woz + 432)
sin(w0t) = 2A1cos(wot + (151) + A2 cos(a)ot + 452) P2.19 Solve the following equation for M and 1p.
Obtain all possible answers. Use the phasor method, and
provide a geometrical diagram to explain the answer. 5 cos(w0t) = M cos(wot — 7r/6) + 5 cos(a)ot + W) Hint: Describe the ﬁgure in the zplane given by the set
{z : z = 561‘” — 5} whereO f w 5 2n. P2.20 Let x[n] be the complex exponential sequence X[n] : 7ej(0.227rn70.257r) deﬁned for n 2 —oo,... ,—”l,0, 1,2,... ,00. If we deﬁne a new sequence y[n] to be the second difference
y[n] =x[n+ 1]—2x[n]+x[n— l] for alln, it is possible to express y[n] in the form ym] : Aej (éon+¢) Determine the numerical values of A, ¢, and cbo. P2.21 In a mobile radio system (e.g., cell phones),
there is one type of degradation that can be modeled
easily with sinusoids. This is the case of multipath
fading caused by reﬂections of the radio waves interfering
destructiver at some locations. Suppose that a
transmitting tower sends a sinusoidal signal, and a mobile
user receives not one but two copies of the transmitted
signal: a directpath transmission and a reﬂectedpath
signal (e.g., from a large building) as depicted in Fig.
P2.21. (Oadr) E Figure P2.21 The received signal is the sum of the two copies, and
since they travel different distances they have different time delays. If the transmitted signal is s(t), then the
received signal10 is 70) = 30 — t1) +S(t —12) In a mobile phone scenario, the distance between
the mobile user and the transmitting tower is always 10For simplicity we are ignoring propagation losses: When a radio
signal propagates over a distance R, its amplitude will be reduced by
an amount that is proportional to 1/ R 2. ! 2 10 PROBLEMS 35
changing. Suppose that the direct—path distance is Determine the received signal when x = 0. Prove
that the received signal is a sinusoid and ﬁnd its d1 = v x2 + 106 (meterS) amplitude, phase, and frequency when x = 0. where x is the position of a mobile user who is moving (0) The amplitude of the received signal is a measure of along the xaxis. Assume that the reﬂectedpath distance its strength. Show that as the mobile user moves, it is is possible to ﬁnd positions where the signal strength is zero. Find one such location. d2 = x/ (x — 55)2 + 106 + 55 (meters) (d) If you have access to MATLAB write a script that (a) The amount of the delay (in seconds) can be will plot signal strength versus position x, thus computed for both propagation paths, using the fact demonstrating that there are numerous locations that the time delay is the distance divided by the Where DC signal iS rCCCiVCd. Use )6 in the range speed of light (3 X 108 m/s). Determine 11 and t; as —100 é x é a function of the mobile’s position (x). (b) Assume that the transmitted signal is s(t) = c0s(300x1067rt) ...
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 Fall '08
 JUANG
 Signal Processing

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