1.6.7 Consider a causal discretetime LTI system described by the difference equation
y
(
n
)

5
6
y
(
n

1) +
1
6
y
(
n

2) = 2
x
(
n
) +
2
3
x
(
n

1)
.
20
(a) Find the transfer function
H
(
z
).
(b) Find the impulse response
h
(
n
).
You may use MATLAB to do the partial fraction expansion.
See the
command
residue
. Make a stem plot of
h
(
n
) with MATLAB.
(c) Plot the magnitude of the frequency response

H
(
e
jω
)

of the system. Use the MATLAB command
freqz
.
1.6.8 A room where echos are present can be modeled as an LTI system that has the following rule:
y
(
n
) =
∞
X
k
=0
2

k
x
(
n

10
k
)
The output
y
(
n
) is made up of delayed versions of the input
x
(
n
) of decaying amplitude.
(a) Sketch the impulse response
h
(
n
).
(b) What is transfer function
H
(
z
)?
(c) Write the corresponding finiteorder difference equation.
1.6.9
Echo Canceler.
A recorded discretetime signal
r
(
n
) is distorted due to an echo.
The echo has a lag of 10
samples and an amplitude of 2
/
3. That means
r
(
n
) =
x
(
n
) +
2
3
x
(
n

10)
where
x
(
n
) is the original signal. Design an LTI system with impulse response
g
(
n
) that removes the echo from
the recorded signal. That means, the system you design should recover the original signal
x
(
n
) from the signal
r
(
n
).
(a) Find the impulse response
g
(
n
).
(b) Find a difference equation that can be used to implement the system.
(c) Is the system you designed both causal and stable?
1.6.10 Consider a causal discretetime LTI system with the impulse response
h
(
n
) =
3
2
3
4
n
u
(
n
) + 2
δ
(
n

4)
(a) Make a stem plot of
h
(
n
) with MATLAB.
(b) Find the transfer function
H
(
z
).
(c) Find the difference equation that describes this system.
(d) Plot the magnitude of the frequency response

H
(
e
jω
)

of the system. Use the MATLAB command
freqz
.
1.6.11 Consider a
stable
discretetime LTI system described by the difference equation
y
(
n
) =
x
(
n
)

x
(
n

1)

2
y
(
n

1)
.
(a) Find the transfer function
H
(
z
) and its ROC.
(b) Find the impulse response
h
(
n
).
1.6.12 Two LTI systems are connected in series:

SYS 1

SYS 2

21
The system SYS 1 is described by the difference equation
y
(
n
) =
x
(
n
) + 2
x
(
n

1) +
x
(
n

2)
where
x
(
n
) represents the input into SYS 1 and
y
(
n
) represents the output of SYS 1.
The system SYS 2 is described by the difference equation
y
(
n
) =
x
(
n
) +
x
(
n

1) +
x
(
n

2)
where
x
(
n
) represents the input into SYS 2 and
y
(
n
) represents the output of SYS 2.
(a) What difference equation describes the total system?
(b) Sketch the impulse response of the total system.
1.6.13 Three causal discretetime LTI systems are used to create the a single LTI system.
x
(
n
)
H
1
H
2
H
3
+
y
(
n
)
r
(
n
)
f
(
n
)
g
(
n
)
The difference equations used to implement the systems are:
H
1
:
r
(
n
) = 2
x
(
n
)

1
2
r
(
n

1)
H
2
:
f
(
n
) =
r
(
n
)

1
3
f
(
n

1)
H
3
:
g
(
n
) =
r
(
n
)

1
4
r
(
n

1)
What is the transfer function
H
tot
(
z
) for the total system?
1.6.14 Given the difference equation...
22
1.6.15 Difference equations in MATLAB
Suppose a system is implemented with the difference equation:
y
(
n
) =
x
(
n
) + 2
x
(
n

1)

0
.
95
y
(
n

1)
Write your own MATLAB function,
mydiffeq
, to implement this difference equation using a
for
loop. If the
input signal is
N
samples long (0
≤
n
≤
N

1), your program should find the first
N
samples of the output
y
(
n
) (0
≤
n
≤
N

1). Remember that MATLAB indexing starts with 1, not 0, but don’t let this confuse you.
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 Fall '13
 Ivan
 Digital Signal Processing, Signal Processing, LTI system theory, Impulse response, Inverse Systems