# E3004 Signal Processing tutorial Questions 1a - Transtutors

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Nov 24, 2013 (4 years and 5 months ago)

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ICM3004 Signal Processing tutorial Questions 1a

1.

Determine the impulse responses of the following causal systems using the z
transform. Plot the poles and zeros. Which systems are stable?

(a)

y[n] = 0.75 y[n
-
1]

0.125 y[n
-
2] + x[n]

(b)

y[n] = y[n
-
1]

0.5 y[n
-
2] + x[n] + x[n
-
1]

(c)

3
1
1
1
)
1
(
)
1
(
)
(

z
z
z
z
H

(d)

y[n] = 0.6 y[n
-
1]

0.08 y[n
-
2] + x[n]

(e)

y[n] = 0.7 y[n
-
1]

0.1 y[n
-
2] + 2 x[n]

x[n
-
2]

2.

Draw the direct form

I

of the filters for which:

(a)

h[n] = {1,2,3,4,3,2,1} for n = 0,1,2,…,6

(b)

H(z) = 1 + 2.88 z
-
1

+ 3.40 z
-
2

+1.74 z
-
3

+0.40 z
-
4

Which of these is linear phase? For this filter (if either) draw an implementation
minimising the number of coefficient multiplications.

3.

Find the poles and zeros of the filter

)
15
.
0
2
.
0
1
)(
1
(
1
)
(
2
1
2
4

z
z
z
z
z
H

Simplify, if possible, by pole/zero cancellation and find the first four terms of the impulse
response by means of long division.

4.

Describe the steps in designing a low
-
pass FIR filter by the window method.

Note that

3
2
)
1
(
)
1
(
]
[

z
z
z
n
u
n
z