1
UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410 Digital Signal Processing
Homework 14
Due Wednesday, December 10, 2008
Prof. Bresler / Prof. Jones
Problem 1
(25 points)
Compute the following cyclic convolutions:
(a) {x
n
}
5
n=0
= {1, 2, 3, 4, 5, 6} and {y
n
}
5
n=0
= {1, 0, 0, 1, 0, 0}.
Compute z[n] =
∑
=
−
5
0
6
]
[
]
[
k
k
n
y
k
x
, n=0,1,2,3,4,5.
(b) {x
n
}
8
n=0
={1, 2, 3, 4, 5, 6,0,0,0}; {y
n
}
8
n=0
={1,0,0,1,0, 0, 0,0,0}
Compute p[n] =
∑
=
−
8
0
9
]
[
]
[
k
k
n
y
k
x
n=0,1,2,3,4,5,6,7,8
(c) {x
n
}
7
n=0
={1, 2, 3, 4, 5, 6,0,0}; {y
n
}
7
n=0
={1,0,0,1,0, 0, 0,0}
Compute w[n] =
∑
=
−
7
0
8
]
[
]
[
k
k
n
y
k
x
n=0,1,2,3,4,5,6,7
(d) Compute z
linear
[n], the linear convolution of the sequences in (a).
(e) What is the relationship between the linear convolution of x[n] and y[n] computed in (d) and the
circular convolutions computed in (a), (b) and (c) ?
Problem 2
(15 points)
The nonzero part of the impulse response of a particular lowpass filter is given by
h[n]
4
n=0
= {0.1, 0.5, 1, 0.5, 0.1}
Your goal is to convolve h[n] with the following signal: