ECE130B
Jan. 26, 2010
Home work 4
Due date
: Monday, February 1, 2010, by 5p.m. at the home work box.
Exercise 1.
Given a length
N
signal
x
[
n
]
, find the number of arithmetic operations that the
algorithm you implemented in
Home Work 3
,
Exercise 4
, requires.
Note
: complex exponenti
aition, multiplication, addition, division, subtraction, cosine and sine, can all be computed as a
single arithmetic operation for this exercise.
Exercise 2.
Report the time taken by your code from
Home Work 3
,
Exercise 4
, for signals of
length
2
12
,
2
13
and
2
14
. Make sure you only time the code that computes the Fourier series coef
ficients.
FFT
. The
F
ast
F
ourier
T
ransform is an algorithm for computing the discrete Fourier series
coefficients rapidly. We briefly describe one version when the length
N
of the signal is an exact
power of
2
; that is
N
= 2
M
for some integer
M
. Let
x
[
n
]
be the given signal for
n
=0
,
, N

1
.
The
k
th Fourier series coefficient of this signal is given by the formula:
a
k
=
1
2
M
summationdisplay
n
=0
2
M

1
x
[
n
]
exp
parenleftbigg

2
πj k n
2
M
parenrightbigg
.
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 Winter '08
 Staff
 0 K, 2 k, 1 2m, 1 2m, 2 2 1 2M

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