(f) Sketch the polezero diagram of the system.
(g) Is the system a lowpass filter, highpass filter, or neither?
1.9.9 A causal discretetime LTI system is implemented with the di
↵
erence equation
y
(
n
) = 3
x
(
n
) +
3
2
y
(
n

1)
.
(a) Find the output signal when the input signal is
x
(
n
) = 3 (2)
n
u
(
n
)
.
Show your work.
(b) Is the system stable or unstable?
(c) Sketch the polezero diagram of this system.
1.9.10 A causal discretetime LTI system is described by the equation
y
(
n
) =
1
3
x
(
n
) +
1
3
x
(
n

1) +
1
3
x
(
n

2)
where
x
is the input signal, and
y
the output signal.
63
(a) Sketch the impulse response of the system.
(b) What is the dc gain of the system? (Find
H
f
(0).)
(c) Sketch the output of the system when the input
x
(
n
) is the constant unity signal,
x
(
n
) = 1.
(d) Sketch the output of the system when the input
x
(
n
) is the unit step signal,
x
(
n
) =
u
(
n
).
(e) Find the value of the frequency response at
!
=
⇡
. (Find
H
f
(
⇡
).)
(f) Find the output of the system produced by the input
x
(
n
) = (

1)
n
.
(g) How many zeros does the transfer function
H
(
z
) have?
(h) Find the value of the frequency response at
!
=
2
3
⇡
. (Find
H
f
(2
⇡
/
3).)
(i) Find the poles and zeros of
H
(
z
); and sketch the pole/zero diagram.
(j) Find the output of the system produced by the input
x
(
n
) = cos
(
2
3
⇡
n
)
.
1.9.11
TwoPoint Moving Average.
A discretetime LTI system has impulse response
h
(
n
) = 0
.
5
δ
(
n
) + 0
.
5
δ
(
n

1)
.
(a) Sketch the impulse response
h
(
n
).
(b) What di
↵
erence equation implements this system?
(c) Sketch the polezero diagram of this system.
(d) Find the frequency resposnse
H
f
(
!
). Find simple expressions for

H
f
(
!
)

and
\
H
f
(
!
) and sketch them.
(e) Is this a lowpass, highpass, or bandpass filter?
(f) Find the output signal
y
(
n
) when the input signal is
x
(
n
) =
u
(
n
).
Also,
x
(
n
) = cos(
!
o
n
)
u
(
n
) for what value of
!
o
?
(g) Find the output signal
y
(
n
) when the input signal is
x
(
n
) = (

1)
n
u
(
n
).
Also,
x
(
n
) = cos(
!
o
n
)
u
(
n
) for what value of
!
o
?
1.9.12 A causal discretetime LTI system is described by the equation
y
(
n
) =
1
4
3
X
k
=0
x
(
n

k
)
where
x
is the input signal, and
y
the output signal.
(a) Sketch the impulse response of the system.
(b) How many zeros does the transfer function
H
(
z
) have?
(c) What is the dc gain of the system? (Find
H
f
(0).)
(d) Find the value of the frequency response at
!
= 0
.
5
⇡
. (Find
H
f
(0
.
5
⇡
).)
(e) Find the value of the frequency response at
!
=
⇡
. (Find
H
f
(
⇡
).)
(f) Based on (b), (d) and (e), find the zeros of
H
(
z
); and sketch the pole/zero diagram.
(g) Based on the pole/zero diagram, sketch the frequency response magnitude

H
f
(
!
)

.
1.9.13
FourPoint Moving Average.
A discretetime LTI system has impulse response
h
(
n
) = 0
.
25
δ
(
n
) + 0
.
25
δ
(
n

1) + 0
.
25
δ
(
n

2) + 0
.
25
δ
(
n

3)
(a) Sketch
h
(
n
).
(b) What di
↵
erence equation implements this system?
(c) Sketch the polezero diagram of this system.
(d) Find the frequency resposnse
H
f
(
!
). Find simple expressions for

H
f
(
!
)

and
\
H
f
(
!
) and sketch them.
(e) Is this a lowpass, highpass, or bandpass filter?
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 Spring '16
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Inverse Systems