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Today, I would like to talk to you about the principle and formula of Fast Fourier transform (FFT). Many people may not understand it very well. in order to make you understand better, the editor has summarized the following contents for you. I hope you can get something according to this article.

The principle and Formula of Fast Fourier transform (FFT)

The Fourier transform of aperiodic continuous time signal x (t) can be expressed as

The continuous spectrum of the signal x (t) is calculated in the formula. However, in the actual control system, the discrete sampling value x (nT) of the continuous signal x (t) can be obtained. Therefore, it is necessary to use the discrete signal x (nT) to calculate the spectrum of the signal x (t).

Finite length discrete signal x (n), nympho 1, … The DFT of Nmur1 is defined as:

It can be seen that DFT needs to calculate about N2 times multiplication and N2 times addition. When N is large, the amount of calculation is very large. By using the symmetry and periodicity of WN, the N-point DFT is decomposed into two Nmax 2-point DFT, so that the total amount of computation of the two Nmax 2-point DFT is only half of that of the original, that is, (Nmax 2) 2 + (N-2=N2/2 2) 2=N2/2, which can be further decomposed, and the Nash 2 can be further decomposed into N-DFT and so on. The DFT of 2m points can be decomposed into 2-point DFT, so the amount of computation can be reduced to (Nppam2) log2N times multiplication and Nlog2N times addition. Figure 1 shows the curve of the relationship between the amount of computation required by FFT and DFT- and the number of calculated points. The superiority of FFT algorithm can be clearly seen from the diagram.

Decompose x (n) into the sum of two sequences of even and odd numbers, namely

If the lengths of x1 (n) and x2 (n) are both even and odd, the length of x1 (n) is even and x2 (n) is odd, then

Where X1 (k) and X2 (k) are 2-point DFT of x1 (n) and x2 (n), respectively. Since both X1 (k) and X2 (k) have Nmax 2 as the period and WN k+N/2=-WN k, X (k) can be expressed as:

The operation of the above formula can be represented by figure 2, which is called butterfly operation according to its shape. And so on, the N-point DFT is decomposed into two two-point DFT after one decomposition of M-Mel. Figure 3 shows the decomposition process of 8-point FFT.

The principle of FFT algorithm is to realize large-scale transformation through many small and easier transformations, which reduces the operation requirements and improves the operation speed. FFT is not an approximate operation of DFT, they are completely equivalent.

Description of FFT accuracy:

Because this transformation uses floating-point operations, sufficient precision is required so that the exact integer value of each component of the result is still recognizable in the event of a rounding error. For the rounding error of the FFT, the binary of the log2 (log2N) bit should be allowed to be increased several times. A number with a base of 256 and a length of N bytes can produce convolution components of order (256) 2N, so in order to store correctly, 16+log2N bit precision is required. If the number I is the binary number of the floating point Mantissa, there are conditions:

For any N value of interest (N > 256), single precision is not appropriate for iTunes 24, while double precision for iFix 53 allows N greater than 106, equivalent to millions of decimal digits. Therefore, when using FFT for large number multiplication, the vector array chooses the double precision type.

After reading the above, do you have any further understanding of the principle and formula of fast Fourier transform FFT? If you want to know more knowledge or related content, please follow the industry information channel, thank you for your support.

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