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Unformatted text preview: Chapter 1 Answers 1.1. Converting from polar to Cartesian coordinates: is?” = ems" =_~_%e . #73" = ewe1r); —% .
ail: cosx(%) +Jsm = J, (’51: cos  Jsm = 1
e155 _=eJe =1, \/—2—e37_=\/§(cos.(§)+jsin(%))=1+j
ﬁe¥=¢§e¥=1+L ﬁeﬂF=ﬁe=F=1—j
x/ieI‘F = 1 —j
1.2. Converting from Cartesian to polar coordinates: _
5 = 5e10, —2 = 2e“, 33' = 3617
in“? =e'ji, 1+j = x/ie’i, (1 52')? =2e'ji
j(1_j)=er, alien, {2T~;;7~5’2=e—m m
1.3. (a) Eco =/ e'4‘dt = i, P0° = 0, because Eco < oo
o co m
12(t)2dt = / dt = 00, P00 = m w (b) z2(t) = ejm‘li'), z2(t) = 1. Therefore, Eco = / Tam Tam Tax '1‘ T
lim ill—,Lleﬁtszt = lim TIT/Jot = lim 1 =1
00 00
(c) z3(t) = cos(t). Therefore, E0° = / z3(t)2dt = / c052(t)dt = 00,
—m w
1 T 2 1 T l+cos(2t) 1
= I — = I — _.————— = —
Pee T1320 2T 4°08 W“ T1220” 2 2 (d) min] = (%)"uln],lz1[n]2 = G)" ulnl. Therefore, Em = 55 min“? e 2 it)” = %« nz—oo n '0
P00 = 0, because Em < oo. 00
(e) 2:2[n] = ej(%+%), :1:2[n]2 = 1. Therefore, E0° = Z x2[n]2 = 00, ‘ nz—oo
1 N 1 N
= ' 2= ' 1:],
R” 13520 2N + 1 2 Mn“ 131?; 2N +1 2
n=—N n=—N
W 00 7r
(f) 13M] 2 cos(§n). Therefore, E},o = Z: l2:;;[1n]2 = Z 0032(Zn) = 00,
n=—oo n=oo
N N n
. 1 2 1r _ . 1 l+cos(§n)) _}
Pw‘AlT‘w2N+1";NC°S(4")’nl§“m2N+1n;N( 2 ‘2 1.4. (a) The signal :r[n] is shifted by 3 to the right. The shifted signal will be zero for n < 1
and n > 7. (b) The signal :c[n] is shifted by 4 to the left. The shifted signal will be zero for n < 6
and n > 0. 1.5., 1.6. 1.7. (c) The signal :r[n] is ﬂipped. The ﬂipped signal will be zero for n < —4 and n > 2. (d) The signal $[n] is ﬂipped and the ﬂipped signal is shifted by 2 to the right. This new
signal will be zero for n < 2 and n > 4. (e) The signal :r[n] is ﬂipped and the ﬂipped signal is shifted by 2 to the left. This new
signal will be zero for n <_ —6 and n > 0. (a) 1(1 — t) is obtained by ﬂipping :z:(t) and shifting the ﬂipped signal by 1 to the right.
Therefore, :r(l — t) will be zero for t > —2. (b) From (a), we know that :c(1—t) is zero fort > —2. Similarly, 2(2—1‘.) is zero for t > —1.
Therefore, a:(1 — t) + a:(2 — t) will be zero for t > —2. (c) $(3t) is obtained by linearly compressing 1(t) by a factor of 3. Therefore, :r(3t) will be
zero for t < 1. (d) 1(t / 3) is obtained by linearly stretching a:(t) by a factor of 3. Therefore, 9305/ 3) will be
zero for t < 9. (a) z1(t) is not periodic because it is zero for t < 0. (b) 2:2[n] = 1 for all n. Therefore, it is periodic with a fundamental period of 1. (c) z3[n] is as shown in the Figure 31.6. Figure 81.6 Therefore, it is periodic with a fundamental period of 4. (a)
5mm} = émn] + $1[—n]) = %(u[n] — u[n — 41+ u[—n] — u[—n — 4]) Therefore, £v{:r:1[n]} is zero for In] > 3.
(b) Since 12(t) is an odd signal, £U{$2(t)} is zero for all values of t. (c)
1 1 1 ,. l 7.
€v{za[nl} = 5(z1lnl + Illn1) = 5K5) uln — 31—(5) uln — 311
Therefore, £u{:r3[n]} is zero when n < 3 and when n —+ 00.
((1) 5mm} = $0124“) + x4(—t)) = —;—[e5tu(t + 2) — e5‘u(—t + 2) Therefore, £v{:r4(t)} is zero only when t —) oo. 1.8. 1.9. 1.10. 1.11. 1.12. (a) Re{:r1(t)} = 2 = 260‘ cos(0t + 7r) _ (b) Re{:c2(t)} = ﬁcosﬁ) cos(3t + 2n) = cos(3t) = eo‘ cos(3t + ) (c) ’Re{:ra(t)} = e“ sin(3t + 1r) = e“ cos(3t + 3%) (d) 'Re{$4 (t)} = —e'2' sin(100t) = e‘” sin(100t + 1r) = 6‘” cos(100f. + %) (a) :rl(t) is a periodic complex exponential. $1“) =Jej101 ___ ej(10H%) The fundamental period of 2:1(t) is % = (b) 12(t) is a complex exponential multiplied by a decaying exponential. Therefore, 2:2(t)
is not periodic. (c) “[12] is a periodic signal. . .
I3[n] = 6171": = 63an 2:3[n] is a complex exponential with a fundamental period of a} = 2. (d) z4[n] is a periodic signal. The fundamental period is given by N = m(3:75) By choosing m = 3, we obtain the fundamental period to be 10. = meg). (e) 15[n] is not periodic. 2:5[n] is a complex exponential with we = 3/5. We cannot ﬁnd
any integer m such that mtg—7;) is also an integer. Therefore, 2:5[n] is not periodic. x(t) = 2cos(10t + 1) —— sin(4t  1) Period of ﬁrst term in RHS = 2—3 = g Period of second term in RHS = 2T" = % Therefore, the overall signal is periodic with a period which is the least common multiple
of the periods of the ﬁrst and second terms. This is equal to 1r. $[n] = l +9171" —ejzsl" Period of the first term in the RHS = 1
Period of the second term in the HHS = m(;%7) = 7 (when m = 2)
Period of the third term in the RHS = m(%§§) = 5 (when m = 1) Therefore, the overall signal ﬂu] is periodic with a period which is the least common
multiple of the periods of the three terms in a:[n]. This is equal to 35. The signal a:[n] is as shown in Figure $1.12. :1:[n] can be obtained by ﬂipping u[n] and then
shifting the flipped signal by 3 to the right. Therefore, :r[n] = u[—n + 3]. This implies that
M = —1 and no = —3. 10'] —LI o I 2. 3 11
Figure $1.12 1.13.
t t 0, t < —2
W) =/ 3(T)dt =/ (6(1 + 2) — 6(1 — 2))dt = 1, —2 g t g 2
"°° '°° 0, t, > 2
Therefore, 2
EOO=/ dt=4
—2 1.14. The signal z(t) and its derivative g(t) are shown in Figure $1.14. 1m
1 Figure 31.14 Therefore,
00 a)
g(t)=3 Z 6(t—2k)—3 Z 6(t—2k—1) k:—oo [CZ00
This implies that A1 = 3, t1 = 0, A2 = —3, and t2 = l.
1.15. (a) The signal zﬂn], which is the input to $2, is the same as y1[n]. Therefore, y2[n] = 1:2]71 — 2] + éxﬂn  3] ll y1[n  2] + éydn  3]
= 22:1[11 — 2] +4xl[n — 3] + %(221[n — 3] + 4z1[n — 4])
= 21:1[71 — 2] + 51:1[71 — 3] + 2x1]n — 4] The inputoutput relationship for S is y[n] = 2:1:[n — 2] + 5:1:[n — 3] + 2:1:[n  4] 1.20. (a) Given _
:1:(t) = cm —> y(t) = e7“ x(t) = e‘jzt ———> y(t) = e”j3‘ Since the system is linear, (ej3t + e—j3t) . . 1
mm = gem + arm) —» y1(t) = 5 Therefore,
z1(t) = cos(2t) —+ y1(t) = cos(3t) (b) We know that
——j ‘2: ej j2t
12(t) = cos (2(t  %)) = _._.__—_e e] : 6 Using the linearity property, we may once again write
z1(t) = $4519” + ejeﬂ‘) ——» 3/1 (t) = é(e‘jejat + evei3!) = cos(3t  1) Therefore,
:n(t) = cos(2(t — 1/2)) ——+ y1(t) = cos(3t1) 1.21. The signals are sketched in Figure $1.21. 16: I) “(1‘3 1. 2. 1..(Zt+l) 2. z (rt/2) 3 rzmuwjum
2 04’ 0.6
1.
a j— JL_JL
4' 6 8 to + o I .1. ﬂy? 3/2. t Figure $1.21 1.22. The signals are sketched in Figure 31.22.
1.23. The even and odd parts are sketched in Figure 81.23. 10 1.24. 1.25. 1.26. 1.27. Figure $1.24 The even and odd parts are sketched in Figure 81.24. (a) Periodic, period = 21r/(4) = 1r/2.
(b) Periodic, period = 21r/(1r) = 2. (c) x(t) = [l + cos(4t — 21r/3)]/2. Periodic, period = 21r/(4) = 1r/2.
(d) :1:(t) = cos(47rt)/2. Periodic, period = 21r/(41r) = 1/2.
(e) z(t) = [sin(47rt)u(t) — sin(47rt)u(——t)]/2. Not periodic. (f) Not periodic. (3) Periodic, period = 7.
(b) Not periodic.
(c) Periodic, period = 8. (d) z[n] = (l/2)[cos(31rn/4) + cos(1m/4)]. Periodic, period = 8. (e) Periodic, period = 16. (3) Linear, stable. (b) Memoryless, linear, causal, stable.
(c) Linear (d) Linear, causal, stable. (e) Time invariant, linear, causal, stable. (f) Linear, stable. (g) Time invariant, linear, causal. l2 1.35. 1.36. 1.37. 1.38. 1.39. We want to ﬁnd the smallest No such that m(21r/N)N0 = Zn]: or N0 = kN / m, where k is an
integer. If No has to be an integer, then N must be a multiple of m/k and m/k must be an
integer. This implies that m/k is a divisor of both m and N. Also, if we want the smallest
possible No, then m/k should be the GCD of m and N. Therefore, N0 = N/gcd(m, N). (a) If :r[n] is periodic ej“°("+N)T = ej“’°"T, where wo = 27r/To. This implies that 2—7:NT = 21rk => T — f = a rational number. To E‘N (b) If T/To = p/q then z[n] = e72"(”/4). The fundamental period is q/gcd(p, q) and the
fundamental frequency is 271' 21r w w T
—qgcd(p,q) = —Egcd(p,q) = flamed = —:Tgcd(p,q) (c) p/gcd(p, q) periods of :c(t) are needed. (a) From the deﬁnition of ¢zy(t), we have foo :c(t + 'r)y(r)dr —oo ¢1y(t) = = [00 y(—t + T)x(T)dT
= ¢y1("t) (b) Note from part (a) that ¢n(t) = ¢ﬂ(—t). This implies that ¢n(t) is even. Therefore,
the odd part of ¢n(t) is zero. (‘3) Here: 451g“) = (1)11“  T) and ¢yy(t) = ¢Iz(t)'
(a) We know that 26A(2t) = 6642“). Therefore, . ' 1
Llano 6A(2t) — 1131310 5613/20) This implies that l (b) The plots are as shown in Figure $1.38. We have AiEOUA(t)5(t) = AiTOuA(O)6(t) = 0. Also, mundane) = gm). l7 We have —00
Therefore,
0, t<0 '.‘6(t—'r)=0
y(t) = 1, t > 0 u(T)6(t — r) = 6(t — T) .
undeﬁned for t = 0 1.40. (a) If a system is additive, then
0 = x(t) — a:(t) —> y(t)  y(t) = 0.
Also, if a. system is homogeneous, then 0 = ow) ——» y(t).0 = 0. (b) y(t) = z2(t) is such a. system.
t
(c) No. For example, consider y(t) = / z(r)d1 with 2(t) = u(t) —u(t— 1). Then :r(t) = 0 —00
for t > 1, but y(t)=1fort>1. 18 ...
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This note was uploaded on 02/06/2012 for the course EE 3610 taught by Professor Liang during the Fall '10 term at LSU.
 Fall '10
 Liang

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