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Unformatted text preview: 54 SIGNALS AND SIGNAL SPACE title(’Amplitude of {\it D}_{\it n}') subplot(212) ;sZ=stem( [—N:N] ,angle(D) ); set(52, ’Linewidth’ ,2); ylabel(’<{\it D}__{\it n} ’ );
tit1e(’Angle of {\it D}_{\it n}'); REFERENCES 1. P. L. Walker, The Theory of Fourier Series and Integrals, WileyInterscience, New York,
1986. 2. R. V. Churchill, and J. W. Brown, Fourier Series and Boundary Value Problems, 3rd ed., McGraw
Hill, New York. 1978. PROBLEMS 2.11 Find the energies of the signals shown in Fig. P2.1—l. Comment on the effect on energy of sign
change, time shift, or doubling of the signal. What is the effect on the energy if the signal is
multiplied by k? Figure P.2.‘l l 1" sint 2.12 (a) Find EX and Ey, the energies of the signals x(t) and y(t) shown in Fig. P2.l—2a. Sketch the
signals x(t) + y(l) and x(t) — y(t) and show that the energy of either of these two signals is
equal to Ex + Ey. Repeat the procedure for signal pair in Fig. P2.1—2b. (b) Now repeat the procedure for signal pair in Fig. P2.1—2c. Are the energies of the signals
x(t) + y(t) and x(t) — y(r) identical in this case? 2.13 Find the power of a sinusoid C cos (wot + 6). 2.14 Show that if col = mg, the power of g(t) = C1 cos(a)1! + 61) + C2 cos(w2t + 02) is [C12 +
C22 + 201 C2 cos(91 — 92)]/2, which is not equal to (C12 + C22)/2. 2.15 Find the power of the periodic signal g(t) shown in Fig. P2. l5. Find also the powers and the rms
values of (a) —g(t) (b) 2g (1‘) (c) eg (1:). Comment. 2.16 Find the power and the rms value for the signals in (a) Fig. P2l6a; (b) Fig. 2.16; (c) Fig. P2 1 —6b;
(d) Fig. P2.74a; (e) Fig. P2.74c. Problems 55 ‘V Figure P.2.'I 2 x(t) ya) l
1 2
0 1 t—>
0 2 1—) —1
(a)
x(t)
1
2n:
0 n t_)
(b)
x(t) 0
yt
1 1
TC
0 n/4 t_) (c) 0 1: t—> Figure P.2.'l 6 2.17 Show that the power of a signal g(t) given by n
g“) = Z Dkeiwkt wi aé wk for all i 75 k k=m is (Parseval’s theorem)
n Pg = Z le12 k=m 56 SIGNALS AND SIGNAL SPACE 2.18 Determine the power and the rms value for each of the following signals: (a) 10 cos (1001 + (d) 10 cos 5: cos 10; (b) 10 cos (100: + + 16 sin (15m + (e) 11) sin 5: cos 10;
(c) (10 + 2 sin 3t) cos 101‘ (f) 6"” cos wot 2.21 Show that an exponential e‘” starting at ——00 is neither an energy nor a power signal for any
real value of a. However, if a is imaginary, it is a power signal with power Pg 2 l regardless of
the value of a. 2.31 In Fig. P2.3—l, the signal g1(t) = g(—t). Express signals g2(z), g3 (t), g4(t), and g5 (I) in terms
of signals g(t), g1(t), and their timeshifted, timescaled, or timeinverted versions. For instance,
g2 (t) = g(t — T) +g1(t — T) for some suitable value of T. Similarly, both g3(t) and g4(t) can be
expressed as g(t — T) + g(t ~— T) for some suitable value of T. In addition, g5 (t) can be expressed
as g(t) timeshifted, time—scaled, and then multiplied by a constant. Figure P.2.3'l 82(1)
1
o z—> 1 2
1.5
33“) 84(1) 85“)
1 1
t.)
—1 [0 t—>1 _L o L 0 1—» 2
2 2 2.32 For the signal g(t) shown in Fig. P2.32, sketch the following signals: (a) g(—t); (b) g(t + 6);
(C) g(3t); (d) g(6  I) Figure P.2.32 2.33 For the signal g(t) shown in Fig. P2.33, sketch (3) g(t — 4); (b) g(t/ 1.5); (c) g(2t — 4); (d)
g(2 — t).
Hint: Recall that replacing t with t — T delays the signal by T. Thus, g(2t — 4) is g(2t) with t
replaced by t — 2. Similarly, g(2 — t) is g(—t) with I replaced by t —— 2. 234 For an energy signal g(t) with energy Eg, show that the energy of any one of the signals
—g(t), g(——t), and g(t — T) is Eg. Show also that the energy of g(at) as well as g(at — b)
is E g / a. This shows that time inversion and time shifting do not affect signal energy. On the other Problems 57' Figure P.2.33 hand, time compression of a signal by a factor a reduces the energy by the factor a. What is the
effect on signal energy if the signal is (a) time—expanded by a factor a (a > 1) and (b) multiplied
by a constant a?
2.35 Simplify the following expressions: tan t sin 7r(t + 2)
(a) (ztz +1>5(t) (d) 5U — 1)
ja) — 3 cos (m)
(b) (w2+9)8(a)) (e) (1+2 )6(2t+3)
(c) [etcos (3t — n/3)] 30 +11): (0 (S‘nwkw) 8(a)) Hint: Use Eq. (2.10b). For part (1') use L’Hospital’s rule. 2.36 Evaluate the following integrals: (a) ffooo g(t)8(t — m1: (e) [33 5(3 + oet d1 (b) [gauge—0dr (r) , [32(t3+4)a(1—t)dt (c) [3°00 8(t)e‘j“" dr (g) 3°00 g<2 — 06(3 — t) dt (d) flee so — 2) sin 7nd: (h) [3°00 (2061) cos got — 5)6(2x — 3) dx Hint: 8(x) is located atx = 0. For example, 8(1 — t) is located at 1 — t = 0; that is, at t :11; and
so on. 2.37 Prove that 8(at) = Lac)
WI Hence show that 1
8(a)) = $80”) where a) = 21f Hint: Show that 00 1
/ meson) dt = HMO) » 2.41 Derive Eq. (2.19) in an alternate way by observing that e = (g—c x), and M2 = (g— cx)  (g — cx)=1gl2 +£le2 — 2cg  x To minimize lelz, equate its derivative with respect to c to zero. 2.42 For the signals g(t) and x(t) shown in Fig. P2.4—2, ﬁnd the component of the form x(t) contained
in g(t). In other words, ﬁnd the optimum value of c in the approximation g(t) % cx(t) so that the
error signal energy is minimum. What is the resulting error signal energy? 58 SIGNALS AND SIGNAL SPACE Figure P.2.42 Figure P.2.44 x(t) (a) (b) 2.43 For the signals g(t) and x(t) shown in Fig. P2.4—2, ﬁnd the component of the form g(t) contained in x(t). In other words, ﬁnd the optimum value of c in the approximation x(t) % cg (I) so that the
error signal energy is minimum. What is the resulting error signal energy? 2.44 Repeat Prob. 2.42 if x(t) is a sinusoid pulse shown in Fig. P2.4_4. 2.45 The Energies of the two energy signals x(t) and y(t) are E x and Ey, respectively. (a) If x(t) and y(t) are orthogonal, then show that the energy of the signal x(t) + y(t) is identical
to the energy of the signal x(t) — y(t), and is given by Ex + Ey. (b) ifx(t) and y(t) are orthogonal, ﬁnd the energies of signals c1x(t) + c2y(t) and 01x0?) — czy(t).
(c) We deﬁne Exy, the crossenergy of the two energy signals x(t) and y(t), as 00
Exy 2/ x(!)y*(t)dt
—00 If z(t) = x(t) :l: y(t), then show that
it EZ = Ex +Ey :1: (EW +5”) 2.46 Let x10) and x20) be two unit energy signals orthogonal over an interval from t = t1 to t2. Signals x1 (t) and x2 (1‘) are unit energy, orthogonal signals; we can represent them by two unit
length, orthogonal vectors (X1, x2). Consider a signal g(t) where 80) = 611610) + C2x2(t) £1 < t 3 t2 This signal can be represented as a vector g by a point (c1, Q) in the x1 —— x2 plane. (a) Determine the vector representation of the following six signals in this two—dimensional
vector space: (iv) 84(1): X10) + 2x20)
(V) £50) = 2x10) +X2(t)
(vi) 5'60) = 3x10) 0) g1(t) = 2x10) — x2(t)
(ii) g2(t) = —x1(l) + 2x20)
(iii) g3 (t) = —xz(t) (b) Point out pairs of mutually orthogonal vectors among these six vectors. Verify that the pairs
of signals corresponding to these orthogonal vectors are also orthogonal. f
"4
7.
.l. 1', Figure 9.2.51 Problems 59 2.51 Find the correlation coefﬁcient c,, of signal x(t) and each of the four pulses g1 (t), g2 (t), g3 (t), and
g4(t) shown in Fig. P2.51. To provide maximum margin against the noise along the transmission
path, which pair of pulses would you select for a binary communication? (a) g1(t) (b) (c)
1 sin 4m —sin 2m
1
t —> 0 l —> z —> 2.71 (3) Sketch the signal g(l) = t2 and ﬁnd the exponential Fourier series to represent g(t) over the
interval (—1, 1). Sketch the Fourier series goo”) for all values of t. (b) Verify Parseval’s theorem [Eq. (2.68a)] for this case, given that 001 7T4
272E n=l n 2.72 (3) Sketch the signal g(t) = l and ﬁnd the exponential Fourier series to represent g(t) over the
interval (—7r, 71). Sketch the Fourier series go(t) for all values of t. (b) Verify Parseval’s theorem [Eq. (26821)] for this case, given that 2.73 If a periodic signal satisﬁes certain symmetry conditions, the evaluation of the Fourier series
coefﬁcients is somewhat simpliﬁed. ’ (a) Show that if g(t) = g(—t) (even symmetry), then the coefﬁcients of the exponential Fourier
series are real. (b) Show that if g(t) = —g(—t) (odd symmetry), the coefﬁcients of the exponential Fourier
series are imaginary. (c) Show that in each case, the Fourier coefﬁcients can be evaluated by integrating the periodic
signal over the halfcycle only. This is because the entire information of one cycle is implicit
in a halfcycle owing to symmetry. Hint: If ge (t) and go (2) are even and odd functions, respectively, of t, then (assuming no impulse
or its derivative at the origin), a 2a a
/ 39(1) (11‘ = / ge(t) dz and / g0(l) dt = O
—a 0 —a 60 SIGNALS AND SGNAL SPACE Also, the product of an even and an odd function is an odd function, the product of two odd
functions is an even function, and the product of two even functions is an even function. 2.74 For each of the periodic signals shown in Fig. P2.7—4, ﬁnd‘ the exponential Fourier series and
sketch the amplitude and phase spectra. Note any symmetric property. ‘ Figure P.2.74 1 mil at... *ZO‘E —101t #‘N 11 lOrl: 207E t a 2.7 5 (a) Show that an arbitrary function g(t) can be expressed as a sum of an even function ge (t) and
an odd function g0 (t): g0) = ge(t) + go(t) . 1 1
Hmt: gm = E[g<r) + g<—r>] + 5[g(t> — g(—r)]
—v—__—I —,—/ geU) 800) Problems 61 (b) Determine the odd and even components of the following functions: (i) u(t); (ii) e_a’u(t);
(iii) e” .
2.76 (a) If the two halves of one period of a periodic signal are of identical shape except that one is the negative of the other, the periodic signal is said to have'a halfwave symmetry. If a periodic
signal g(t) with a period T 0 satisﬁes the halfwave symmetry condition, then g (t — = g(t) In this case, show that all the evennumbered harmonics (coefﬁcients) vanish. (b) Use this result to ﬁnd the Fourier series for the periodic signals in Fig. P2.7—6. Figure P.2.76 (a) (b) 1) 2.81 A periodic signal g(t) is expressed by the following Fourier series: 2
g(t) = 35in t+cos <3; — + 2cos(8t+ %) (a) By applying Euler’s identities on the signal g(t) directly, write the exponential Fourier series
for g (t). (b) By applying Euler’s identities 0n the signal g (t) directly, sketch the exponential Fourier series
spectra. " ...
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