fft Fast Fourier Transform (1)

The fft transformation is not a new algorithm, it is just a method to speed up the dft operation. The expression of dft is as follows:

Obviously, finding X(k) of N points requires $N^2$ times of complex number multiplication and N*(N - 1) times of complex number addition; we all know that realizing a complex number multiplication requires four real numbers multiplied by two real number additions, and realizing a complex number addition requires two real number additions; then when N is large , the amount of calculation will be quite large. As the most commonly used algorithm in signal processing, dft obviously cannot be well applied in practice with such a large amount of calculation, so fft appears; how does fft work? A detailed introduction will be made below.

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1. Time-based extraction (DIT) base 2fft algorithm
For the following formula

, let$N=x^M$，$M$ is a positive integer, and$x\left(n\right)$ is divided into two groups according to parity, base order$n=2r$ and$n=2r+1$ , and$r=0,1,...,N/2-1$ , then we

have

A

$A\left(k\right)andB\left(k\right)$ are both$dft\; of\; N/2$ points,$X\left(k\right)$ is N-point dft, so X ( k ) X(k)in the above formula$X\left(k\right)$ is not in perfect form, but

so$A\left(k\right),B\left(k\right)$ can completely represent$X\left(k\right)$ , when$N=8$ ,$The\; relationship\; between\; A\left(k\right),B\left(k\right)andX\left(k\right)$ can be shown in the figure below.A ( k ) and B ( k )

deduced above$A\left(k\right)andB\left(k\right)arestilldft\; ofhighcompositenumber(N/2)$ , and can be further decomposed according to the above method, respectively let$r=2l，r\_=2l+1，l=0,1，...，N/4-1$ , then$A\left(k\right)andB\left(k\right)$ canbe

respectivelyexpressedas__








$N=8$，那么$C\left(k\right),D\left(k\right),E\left(k\right),F\left(k\right)$ are two-point dft, no need to continue to expand, then if

N$N=16,32orahigherpowerof2$ , you can continue to expand according to the above method; the above method is to expand the time n according to parity, so it is called time extraction DIT, the 8-point DFT of the above method can be expressed as$follows$ Figure:

The basic operation unit is as follows: