1.4.15 Consider the parallel combination of two LTI systems.

h
2
(
n
)

h
1
(
n
)
x
(
n
)
?
6
l
+

y
(
n
)
You are told that the impulse responses of the two systems are
h
1
(
n
) = 3
✓
1
2
◆
n
u
(
n
)
and
h
2
(
n
) = 2
✓
1
3
◆
n
u
(
n
)
(a) Find the impulse response
h
(
n
) of the total system.
(b) You want to implement the total system as a
cascade
of two first order systems
g
1
(
n
) and
g
2
(
n
).
Find
g
1
(
n
) and
g
2
(
n
), each with a single pole, such that when they are connected in cascade, they give the same
system as
h
1
(
n
) and
h
2
(
n
) connected in parallel.
x
(
n
)

g
1
(
n
)

g
2
(
n
)

y
(
n
)
1.4.15)
Solution
h
tot
(
n
) =
h
1
(
n
) +
h
2
(
n
) = 3
✓
1
2
◆
n
u
(
n
) + 2
✓
1
3
◆
n
u
(
n
)
H
tot
(
z
) =
3
z
z

1
/
2
+
2
z
z

1
/
3
=
5
z
2

2
z
(
z

1
/
2)(
z

1
/
3)
=
z
(5
z

2)
(
z

1
/
2)(
z

1
/
3)
so we can set
G
1
(
z
) =
z
z

1
/
2
G
2
(
z
) =
5
z

2
z

1
/
3
so
g
1
(
n
) =
✓
1
2
◆
n
u
(
n
)
To find
g
2
(
n
) we can perform partial fraction expansion on
G
2
(
z
)
/z
.
G
2
(
z
)
z
=
5
z

2
z
(
z

1
/
3)
=
A
z
+
B
z

1
/
3
where we can find that
A
= 6 and
B
=

1, so
G
2
(
z
) = 6

z
z

1
/
3
61
and so
g
2
(
n
) = 6
δ
(
n
)

✓
1
3
◆
n
u
(
n
)
Other answers are also possible. We can check this answer in MATLAB using the following code fragment.
>> n = 0:20;
>> h1 = 3 * (1/2).^n .* (n >= 0);
>> h2 = 2 * (1/3).^n .* (n >= 0);
>> htot = h1 + h2;
>>
>> g1 = (1/2).^n .* (n >= 0);
>> g2 = 6 * (n == 0)  (1/3).^n .* (n >= 0);
>> gtot = conv(g1,g2);
>>
>> % check that htot is the same as gtot for n = 1:10
>> [htot(1:10); gtot(1:10)]’
ans =
5.0000
5.0000
2.1667
2.1667
0.9722
0.9722
0.4491
0.4491
0.2122
0.2122
0.1020
0.1020
0.0496
0.0496
0.0244
0.0244
0.0120
0.0120
0.0060
0.0060
62
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.
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 Fall '14
 Digital Signal Processing, LTI system theory, Impulse response