signal & system
T.J
ECE124F17HW32.docECE 124 HOMEWORK #3
The following four problems involve DT signals and systems. The remaining four problems involve CT signals and systems. This will make it possible for me avoid starting every problem with a statement about what kind of system we are dealing with. Keep in mind that inputs are labeled
()
xt
or[]
xn
as appropriate, outputs are labeled()
yt
or[]
yn
as appropriate, and impulse responses are labeled(,)
ht
t
,(,0)()
htht
=
or[,]
hnk
,[,0][]
hnhn
=
as appropriate.1. If
[]3[2]2[1][][1]2[2]
hnnnnnn
ddddd
=-+-+-+-+-
and the input to the system is[]2[2]3[1][][1]2[2]
xnnnnnn
ddddd
=+++++---
, determine[]
yn
. Sketch the result.2. Evaluate the convolution
[]([10]2[1][6])[]
n
ynunununun
b
=+--+-Ä
,||1
b
<
. Do NOT assign a numerical value tob
. Use MATLAB to evaluate and plot[]
yn
for the specific case0.9
b
=
.3. Determine the impulse response of the systems described by the following set of input - output relations: a)
[]0.25{[][1][2][3]}
ynxnxnxnxn
=+-+-+-
, b)1
[][]
nN
mnN
ynxm
N
+
=-
=
å
whereN
is a positive integer. Do not assign a numerical value to it, c)[][1][][1][2]
ynxnxnxnxn
=-++--+-
. Sketch each impulse response as a function ofn
, and for b) let4
N
=
for sketching purposes only.4. System impulse responses are given:
1
[,]2[]
nk
hnkunk
-+
=-
,2
[,]2[2]
nk
hnkunk
-+
=-+
,3
[,](1)(0.4)[1]
nk
hnknkunk
-
=-+-+
. a) For each system determine whether or not it is time invariant. b) For each system determine whether or not it is causal. c) For each system determine whether or not it is memoryless. d) For any system that is time invariant, determine also whether or not it is stable.5. If
1
()
2
t
htrect
TT
æö
=
ç÷
èø
and()()
t
xteut
=-
, determine the output()
yt
. Let0
T
>
be a parameter to which no value should be assigned. Plot the output as a function oft
, specifically for the case2
T
=
.6. If
()cos()
xtt
=
and()(/7)
httrit
=
, determine the output()
yt
. Sketch the result.7. Evaluate the convolution specified below and plot the resulting function
()
yt
using MATLAB.2
()[(2.5)(3.7)2(6.1)][(1)((2)/2)]
t
ytttteutrectt
ddd
-
=--++-Ä++-
8. Determine the impulse response
(,0)
ht
of the systems described by the following set of input - output relations: a)()(3)(3)
yttxttxt
=-++-
; b)()(23)
ytxt
=-
; c)1
()(')'
t
ytxtdt
-
-¥
=
ò
; d)1
()(')'
tT
tT
ytxtdt
T
+
-
=
ò
, where0
T
>
is a fixed parameter. Do not assign a value to it. Sketch each impulse response as) a function oft
. This can be done without using MATLAB.9. System impulse responses are given:
1
(,)cos()()
t
htetut
t
tt
-+
=-
,2
(,)(1)
t
hteut
t
tt
-
=-+
,3
(,)()(1)
t
htteut
t
ttt
-
=---
. a) For each system determine whether or not it is time invariant. b) For each system determine whether or not it is causal. c) For each system determine whether or not it is memoryless. d) For any system that is time invariant, determine also whether or not it is stable.Note:
Â
means the set of all real numbers andI
means the set of all integers.1
