Fourier series formula is as follows:
among them:
Trigonometric form:
On behalf of the Euler equation:
It can be transformed into:
Will ,
on behalf of the Fourier series obtained:
The (2), (3), (4) into obtain:
Similarly available:
The two substituted into the formula (5) in solution to give:
(Note that when the time:
)
So that equation (6) simplifies to:
Equation (6) for the exponential Fourier series
Then we examine the following equation (6)
Extraction period is defined by the Fourier transform has
( remember it as a positive integer n is not dense integration, such as the denominator is infinitely non-repeating decimals. Riemann must not be so accumulated, but it is integrable Lebesgue, because the points are countable ), so this formula becomes accumulated in the form of differential equations, and we set
it in the
middle because the variable
has been out of points, so the only variable is
, so
are:
We get the Fourier transform:
Then according to (8) we get the inverse Fourier transform
Equation (9), (8) is a well-known Fourier transform, inverse Fourier transform
We take on a formula (7)
among them
Because the Fourier transform order so that a cumulative integration into a formula, and the DFT
will be determined in accordance with a specific value of the input signal points. The calculation formula is:
(NOTE: The calculated because
of a cycle
, N being the number of points sampled your)
Therefore, we can simplify the formula
among them:
Simple Conversion method. After the integration of the two formulas, we can redefine the cycle from two points, so about out get:
Wherein (1) is a discrete Fourier transform, (2) changes as an inverse discrete Fourier.