Signals and Systems HW1
Deadline: 2025/03/07 23:59
李健誌_b12502138_01班
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1. (10%) Let 𝑥[𝑛] be a discrete-time signal, and let
𝑛
y1 [𝑛] = 𝑥[5𝑛]
and
𝑥 [ ] , 𝑛 even
y2 [𝑛] = { 2
0,
𝑛 odd
.
Determine whether the statement “If y2 [𝑛] is periodic, then y1 [𝑛] is periodic” is
true. If the statement is true, find the relationship between the fundamental periods
of these two signals. If the statement is not true, produce a counterexample to it.
2. (30%) Determine the signals as periodic or aperiodic. If periodic, identify the
fundamental period. Justify your answer.
(a) (10%)
𝜋
𝜋 2
7𝜋
𝑥(𝑡) = [𝑐𝑜𝑠 ( 𝑡 − )] + 𝑠𝑖𝑛 ( 𝑡)
4
3
8
(b) (10%)
𝑥[𝑛] =𝑐𝑜𝑠 (𝜋𝑛) +𝑐𝑜𝑠 (2𝜋√7𝑛)
(c) (10%)
𝑛
𝑛
𝑥[𝑛] =𝑐𝑜𝑠 ( − 𝜋) + 2 𝑐𝑜𝑠 ( )
2
4
3. (30%) Consider a system with the input/output relationship given by 𝑦(𝑡) = 𝑥 2 (𝑡).
(a) (10%) Is this system linear? Justify your answer.
(b) (10%) Is this system memoryless? Justify your answer.
(c) (10%) Is this system invertible? Justify your answer.
4. (30%) Consider the following systems:
𝐻: 𝑦(𝑡) = 2𝑥(𝑡 − 8)
−1
𝐺: 𝑦(𝑡) = 𝑥(5𝑡)
(a) (15%) What is the function of 𝐻 , the inverse of 𝐻? What is the function of
𝐺 −1 , the inverse of 𝐺? Justify your answer.
(b) (15%) Consider the system in the following figure. That is, 𝐹 is equivalent to
the cascaded interconnection of 𝐻 and 𝐺. Find 𝐹 −1 , the inverse of 𝐹, and draw
it in block diagram form in terms of 𝐻 −1 and 𝐺 −1 . Justify your answer.
。
…
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n
]
=
x
{π}
since xn is periodio, n
x5
x [ n]
=
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[
]
n+ N .
=
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(
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]
≠
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.
.
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]
.
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(
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.
]
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.
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:
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.
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:
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os
4
:
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,
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: 6
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:
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.
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.
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,
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when
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.
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t)
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aysten
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less
.
x
。
。
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( a)
fr
(t )
y
:
H
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H
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ig
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(
V
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y
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8
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)
>
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-
=
=
β
8
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( t ) →∴ ( t ]
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↑ t)
x(
G
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j
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=
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t
)
+ → Fi y ( t ix ( 5 + )
>
et t
)
he t 5 t
g( )
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↑
,
:
,
zx ( t 8
F ( x (t )]
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x (t)
x t]
= Yzt
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-
( )
②
t
λ ( t 8)
( t 8)
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*
x
x
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y )
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]
8
ee t 5 t 8
=
,
-