数字信号处理习题
第一章
1-4
今对三个正弦信号
12 3
() cos2 , () cos6 , () cos10
aa a
x
ttxt txt t
π
ππ
==−=
进行理想采
样,采样频率为
8
s
π
Ω=
,求着三个采样输出序列,比较其结果.画出
123
(), (), ()
aa a
x
tx tx t
的波形及采
样点位置并解释频谱混迭现象.
解答:由于
1
8
2
2
w
π
π
=<
,没有混迭;
2
8
6
2
w
π
π
=>
,混迭;
3
8
10
2
w
π
π
=>
,混迭.
1-13 下列系统中, 表示输出,
()yn ()
x
n
表示输入,试确定是否是线性系统?是否是时不变系统?
(1) ( ) 2 ( ) 5yn xn=+
(3) ( ) ( )
n
m
yn xm
=−∞
=
∑
解答:
12 12
12
12 12
(1) [ ( ) ( )] 2[ ( ) ( )] 5
[2 () 5] [2 () 5] 5 5 5
[ ()] [ ()] 5 5 5 [ ()] [ ()]
T ax n bx n ax n bx n
axn bxn a b
aT x n bT x n a b aT x n bT x n
+= ++
=+++−−+
=+−−+≠+
所以,非线性;
假设输入为
0
()
x
nn−
,则有
00
[( )] 2( ) 5 ( )Txnn xnn ynn−= −+=−
0
2
0
所以,时不变.
12 1 2
121
(2) [ ( ) ( )] [ ( ) ( )]
() () () ()
n
m
nn
mm
Tax n bx n ax m bx m
axmbxmaynbyn
=−∞
=−∞ =−∞
+= +
=+=+
∑
∑∑
所以,线性;
0
00
[( )] ( ) ( ) ( )
nn
n
mm
Txnn xmn xm ynn
−
=−∞ =−∞
−= −= =−
∑∑
时不变.
1-16 确定下列系统的因果性和稳定性:
0
0
(1)() ()(), ()
(2) ( ) ( ),
(4) ( ) 0.5 ( )
n
kn
n
yn gnxn gn
yn xk n n
hn un
=
=
=>
=
∑
有界
解答:
1