信号与系统：习题Ch1(2013)

Exercise Ch. 1
1.21(b)注意先平移再翻转
1/2
1.11 Determine the fundamental period of the signal = 1 +e """式
Solution: The period of e
Ni=7. jl^iS
The period of e is
N ：=5.
So N = N I N2=35 1.25. (a). Periodic. T=-T/2.
Solution: T=2^/4=^/2.
(b) . Periodic. T=2.
Solution: T=2^/^=2.
(c) x(r) = [ 1 + cos(4l-2勿/3)]/2. Periodic, pcri(Kl= 2^/4 = /rZ2
(d) . Peritxlic. T=0.5.
Solution: x(/) = E v {cos(4^r)w(/)J
=-{ COS (4E )M (I ) + COS (4TT (T ))W (T ) }
*
=-cos(4^r) {:/(/) + w(-z)}
—
= -cos(4^r)
2
So. T=2^/4^=0.5
1.26 Determine whether or no each of (he following discrete-time signals is periodic. If (he signals is periodic, determine
its fundamental period.
(a) . Periodic. N=7
(d) (e) x[n]u[n-3]-x[n}
is
2“
Solution: N=———* m =7, m=3.
6Z7
(b). Not periodic.
Solution: N= --------- * m = 16〃m , it s not rational number.
1/8
(c)Periodic. period=8 + J = cos| —(?: + ^V)2] = cos(—+—nN + —N2)
8 8 4 8
若乌N +服2=2Jbr・k为整数,对所有n成立.HP2nN^N2 = \6k对所有n成立，则有 4 8 2N + N?均为16的整故倍.故x[n|为周期的,基波周期为8
(d)= cosG〃)cosg〃)As 饲/2芥=1/4,躲/2芥=1/8, the fundamental period is T= 8.
(e). Periodic. N=16
Solution as follow:
A n] = 2cos(—n) + sm(—〃) 一2cos(— " + —)
4 8 2 6
in (his equation,
sin(—n), it's period is N=2^ #m/(^/8)=16, m=l.
it's period is N=2 龙*m/( n /2)=4. m=l.
So, the fundamental period of.v| n] is N=(8,16,4)=16.
1.49. (f) Solution:
(1 +,)5= (VI)%，=4y[2e
|1 + ,| = + = Z^ = tan l(I) = -
补充：Please represent the Figure R1.2I in P. 60 with the step function w(lo).
Solution:
x(r) = -(2+0[w(/ + 2)-M(r+l)]+[w(r+l)-u(/)]+2[M(n-w(r-l)]+(2-0[w(/-l)-w(/-2)]
= -(2+t)u(t+2)+(3++1)+w(/) T • u(t-1)+(2 2)
1.19 For each of the following inpul-output relationships, determine whether the corresponding system is linear, time
invariant or both.
(c ) The system is linear and time-invariant.
1.22(0 同"一 一 2]=上[0]况〃一 2] = y[n]
vl») ・・・三i I~1 -i .|l -i/2 1.31 In this problem wc illustrate one of the most important consequences of the properties of linearity and time invariance.
Specifically, once we know the response of a linear system or a linear time-invariant (LTI) system to a single input or the responses to several inputs, wc can directly compute the responses (o many other input signals. Much of the remainder of this book deals with a thorough exploitation of 【his fact in order to develop results and techniques for analyzing and synthesizing LTI systems.
(a) Consider an LTI system whose response to the signal xi(r) in Figure Pl.31 (a) is the signal yi(z) illustrated in
Figure Pl.31(b). Determine and sketch carefully the response of the system to the input X2(l) depicted in Figure
1.31(c).
(b) Determine and sketch the response of the system considered in part (a)(o the input xj(/) shown in Figure PI.3I
(d).
.弓(。

=K (I +1) + % (/), so,丹(I) =〉T 。

+1) + 出(0
J I I I I y 2
1/2 x 】(t) x(t)
i
Solution: x 2(/) = A ：l (f)-A l (/-2), so, »(，)=月(，)一)'|(，一 2)
y3(t
)
-10 12。