We know that \ (FFT \) is a circular convolution.
A discrete Fourier transform on the nature of this nature is satisfied:
\ [C_K = \ SUM \ limits_ {I, J} [K = I + J (MOD \ n-)] a_ib_j \]
However, because we do sufficient length ( \ (n-\) is sufficiently large) volume is not so this circular convolution back.
This leads us to only do a certain length \ (FFT, n = 2 ^
w \) if we need to do any length circular convolution it?
In fact, a little pushing and expression on it. . .
Just say \ (DFT \) , \ (IDFT \) is similar.
In fact, we require is a polynomial:
\ [the DFT (A, X) = \ SUM \ limits_ {I = 0} ^ {n-} X ^ I \ SUM \ limits_ {J = 0} ^ {n-} a_iw_n ^ { ij} \]
so-called point value expression.
Which is required then:
\ [C_K = \ SUM \ limits_ I = {0}} ^ {n-a_iw_n IK ^ {} \]
we can use our small \ (Trick \) th convolution into a form.
\ [c_k = \ sum \ limits_
{i = 0} ^ {n} a_iw_n ^ {\ frac {i ^ 2 + k ^ 2- (ki) ^ 2} {2}} \] This removed a:
\ [ c_k = w_n ^ {\ frac { k ^ 2} {2}} \ sum \ limits_ {i = 0} ^ {n} \ left (a_iw_n ^ {\ frac {i ^ 2} {2}} \ right) w_n ^ {- \ frac {(ki
) ^ 2} {2}} \] convolutional it!
That
we do a circular convolution of any length required \ (the FFT \) 9 times over. . .
Constant is still quite large.
There is a god way is to go in with the imaginary part of the press \ (DFT \) , I do not quite understand.
At night to go and see.