dft - DISCRETE FOURIER TRANSFORM JOHN RANDALL Contents 1 2...

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
.