Chapter 2 Problem Answers
2.1 Two quences and are, respectively, given by
and .
Find the quence of the sample-added sum of two quences.
Solution: The sample-added sum is given by
.
2.2 Find the sample-accumulated quence of a given quence
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Solution: For , the sample-accumulated quence is given by
.
For , we have
.
Thus, the sample-accumulated quence is
.
2.3 Assume that a quence is the same as that in problem 2.2. Determine the first-order forward and backward difference quences of , respectively.
Solution: The first-order forward difference quence is given by
,
and the first-order backward difference quence is given by
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2.4 Determine whether or not each of following signals is periodic. If a signal, in the ca, is periodic, specify its fundamental period.
(1)
(2)
(3)
(4)
Solution: (1) Considering
,
and taking , we e that the quence is periodic with period .
Solution: (2) This quence is periodic. Since
,
we can take and obtain the period of the quence as .
Solution: (3) Considering
,
and taking , we e that the period of the quence is .
Solution: (4) Since
,
is not an integer for any integer value of . Thus, the given quence is nonperiodic.
2.10三九大 For each of the following systems, determine whether the system is (a) stable, (b) causal, (c) linear, (d) time invariant, and (e) memoryless.
下列系统，确定系统是否(a)稳定，（b）因果，（C）线性，（D）时不变，和（E）无记忆。
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Solution: (1) The system is defined by .
(a) Consider an input which is bounded by , the system respon to this input is determined as(考虑一个输入x(n)= u(n)这是有界的x(n)| | < = B = 1,系统响应这个输入是确定 )
.
Becau
the system is unstable. 系统不稳定
(b) Since the system output value at prent time , does not depend on any future value of the input , the system is causal.因为系统输出值y(n)目前时间n,并不依赖于任何未来值的输入x(n),系统是因果。原创t恤
(c) Considering that
the system is linear.该系统是线性的
(d) Assume that a new input is . The system respon to this new signal is given by假设一个新的输入。系统对这个新的信号产生的响应为
.
This implies that . Thus, the system is time-invariant.这意味着。因此,该系统是时不变的
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(e) Becau the system output at prent time depends on the prent and past time values of the input, the system is dynamic.因为系统输出当前时间取决于当前和过去的时间值的输入,系统是动态的。
Solution: (2) The system is defined by .
(a) For any bounded input satisfying , we have
Therefore the system is stable.
(b) Becau the output value at prent time does not depend on any future value of the input, the system is causal.与鸭共舞下载
(c) Considering that
the system is nonlinear.
(d) Let . The system respon to this new signal is given by
which satisfies . Thus, the system is time-invariant.
(e) Becau the prent output value depends only on the prent value of the input, the system is memoryless.
Solution: (3) The system is defined by .
(a) If an input satisfies , then
.
越南战争是哪一年Therefore the system is stable.
(b) Becau the prent output value is unrelated to any future input value, the system is causal.
(c) Considering that
We e that the system is linear.
(d) Let . The system respon to this new signal is given by
.
This means that . Thus, the system is time-varying.
(e) Since the prent output value depends only on the prent value of the input , the system is memoryless.
Solution: (4) Considering that the system is defined by .
(a) Let the input of the system be which is bounded by . For and , the system respon to this input is determined as
.
For both cas, we have
Thus the system is unstable.
(b) As the definition of the system, we always assume that . Therefore the prent output value does not depend on any future value of the input . This means that the system is causal.
(c) Becau
the system is linear.
(d) Let . The system respon to this new signal is given by
.
Becau , the system is time-varying.
(e) Generally, the prent output value is related to the input values at other times. Therefore, the system is dynamic.
Solution: (5) Considering that the system is defined by
(a) For a bounded input which satisfies 安全生产宣传内容, we have
.
Therefore, if , then . This shows that the system is stable.
(b) Becau the prent output value is unrelated to any future value of the input, the system is causal.
(c) Considering that
We e that the system is linear.
(d) Let . The system respon to this new signal is given by
.
Becau , the system is time-varying.
(e) Since the prent output value depends only on the prent input value , the system is memoryless.
Solution: (6) Considering that the system is defined by .
