17
x
(
n
)

SYS 1

SYS 2

y
(
n
)
The impulse response of SYS 1 is
h
1
(
n
) =
δ
(
n
) + 0
.
5
δ
(
n

1)

0
.
5
δ
(
n

2)
and the transfer function of SYS 2 is
H
2
(
z
) =
z

1
+ 2
z

2
+ 2
z

3
.
(a) Sketch the impulse response of the total system.
(b) What is the transfer function of the total system?
1.5
Inverse Systems
1.5.1 The impulse response of a discretetime LTI system is
h
(
n
) =

δ
(
n
) + 2
1
2
n
u
(
n
)
.
(a) Find the impulse response of the stable inverse of this system.
(b) Use MATLAB to numerically verify the correctness of your answer by computing the convolution of
h
(
n
)
and the impulse response of the inverse system. You should get
δ
(
n
). Include your program and plots with
your solution.
1.5.2 A discretetime LTI system
x
(
n
)

h
(
n
)

y
(
n
)
has the impulse response
h
(
n
) =
δ
(
n
) + 3
.
5
δ
(
n

1) + 1
.
5
δ
(
n

2)
.
(a) Find the transfer function of the system
h
(
n
).
(b) Find the impulse response of the
stable inverse
of this system.
(c) Use MATLAB to numerically verify the correctness of your answer by computing the convolution of
h
(
n
)
and the impulse response of the inverse system. You should get
δ
(
n
). Include your program and plots with
your solution.
1.5.3 Consider a discretetime LTI system with the impulse response
h
(
n
) =
δ
(
n
+ 1)

10
3
δ
(
n
) +
δ
(
n

1)
.
(a) Find the impulse response
g
(
n
) of the
stable
inverse of this system.
(b) Use MATLAB to numerically verify the correctness of your answer by computing the convolution of
h
(
n
)
and the impulse response of the inverse system. You should get
δ
(
n
). Include your program and plots with
your solution.
1.5.4 A
causal
discretetime LTI system
x
(
n
)

H
(
z
)

y
(
n
)
18
is described by the difference equation
y
(
n
)

1
3
y
(
n

1) =
x
(
n
)

2
x
(
n

1)
.
What is the impulse response of the
stable
inverse of this system?
1.6
Difference Equations
1.6.1 A causal discretetime system is described by the difference equation,
y
(
n
) =
x
(
n
) + 3
x
(
n

1) + 2
x
(
n

4)
(a) What is the transfer function of the system?
(b) Sketch the impulse response of the system.
1.6.2 Given the impulse response . . .
1.6.3 A causal discretetime LTI system is implemented using the difference equation
y
(
n
) =
x
(
n
) +
x
(
n

1) + 0
.
5
y
(
n

1)
where
x
is the input signal, and
y
the output signal. Find and sketch the impulse response of the system.
1.6.4 Given the impulse response . . .
19
1.6.5 Given two discretetime LTI systems described by the difference equations
T
1
:
y
(
n
) +
1
3
y
(
n

1) =
x
(
n
) + 2
x
(
n

1)
T
2
:
y
(
n
) +
1
3
y
(
n

1) =
x
(
n
)

2
x
(
n

1)
let
T
be the cascade of
T
1
and
T
2
in series. What is the difference equation describing
T
? Suppose
T
1
and
T
2
are causal systems. Is
T
causal? Is
T
stable?
1.6.6 Consider the parallel combination of two causal discretetime LTI systems.

SYS 1

SYS 2
x
(
n
)
?
6
l
+

r
(
n
)
You are told that System 1 is described by the difference equation
y
(
n
)

0
.
1
y
(
n

1) =
x
(
n
) +
x
(
n

2)
and that System 2 is described by the difference equation
y
(
n
)

0
.
1
y
(
n

1) =
x
(
n
) +
x
(
n

1)
.
Find the difference equation of the total system.
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 Fall '13
 Ivan
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Inverse Systems