DISCRETE FOURIER TRANSFORM
JOHN RANDALL
Contents
1.
Representations of polynomials
1
2.
Comparison of the representations
2
3.
Roots of unity
2
4.
Discrete Fourier Transform
2
5.
Fast Fourier Transform
2
1.
Representations of polynomials
A polynomial
f
(
x
) of degree less than
n
can be represented in
coefficient form
as
f
(
x
) =
a
0
+
a
1
x
+
a
2
x
2
+
· · ·
+
a
n
-
1
x
n
-
1
,
or, more succinctly, as the sequence of
n
numbers
a
= (
a
0
, . . . , a
n
-
1
).
This is not the only representation. If we choose
n
distinct points
x
=
x
0
, . . . , x
n
,
we can calculate the values
y
0
=
f
(
x
0
)
, . . . , y
n
-
1
=
f
(
x
n
-
1
)
.
The sequence of
n
numbers
y
= (
y
0
, . . . , y
n
-
1
) is called the
point-value form
of
f
with respect to the points
x
.
The coefficient form and the point-value form are equivalent. To recover
a
from
y
, we find the the unique polynomial of degree less than
n
that interpolates the
points (
x
0
, y
0
)
, . . . ,
(
x
n
-
1
, y
n
-
1
). Note that
a
and
y
are related by
x
0
0
x
1
0
. . .
x
n
-
1
0
.
.
.
.
.
.
x
0
n
-
1
x
1
n
-
1
. . .
x
n
-
1
n
-
1
a
0
.
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- Spring '10
- CAREY
- JOHN RANDALL
-
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