### Comparison table of Fourier transform formulas

The Fourier transform formula is cosω0t=[exp(jω0t)+exp(-jω0t)]/2.

The Fourier transform denotes the ability to express a certain function that satisfies a certain condition as a linear combination of trigonometric functions (sine and/or cosine functions) or their integrals.

The Fourier transform has many different variants in different fields of study, such as the continuous Fourier transform and the discrete Fourier transform. Originally Fourier analysis was proposed as a tool for the analytical analysis of thermal processes.

Related Definitions

1. Fourier transform belongs to harmonic analysis.

2, The inverse transform of the Fourier transform is easy to derive and is very similar in form to the positive transform.

3, The sinusoidal basis function is an eigenfunction of differential operations, thus allowing the solution of linear differential equations to be transformed into the solution of algebraic equations with constant coefficients. Within a linear time-invariant physical system, frequency is an invariant property, so that the response of the system to complex excitations can be obtained by combining its response to sinusoidal signals of different frequencies.

### Formula for the Fourier transform?

According to Euler’s formula, cosω0t=[exp(jω0t)+exp(-jω0t)]/2.

The Fourier transform of a DC signal is 2πδ(ω).

The Fourier transform of exp(jω0t) is 2πδ(ω-ω0) by the frequency shift property.

Again, by the linear property, the Fourier transform of

cosω0t=[exp(jω0t)+exp(-jω0t)]/2 is πδ(ω-ω0)+πδ(ω+ω0).

Expanded Information

Fast methods for calculating the discrete Fourier transform are the FFT algorithm extracted by time and the FFT algorithm extracted by frequency. The former arranges the time-domain signal sequence by even-odd and the latter arranges the frequency-domain signal sequence by even-odd.

They both draw on two characteristics: first, periodicity; second, symmetry, where the symbol * represents its conjugate. In this way, the calculation of the discrete Fourier transform can be divided into a number of steps, the efficiency of the calculation is greatly improved.

Time Extraction Algorithm Let the length of the signal sequence is N=2, where M is a positive integer, the time domain signal sequence x(n) can be decomposed into two parts, one is the even part of x(2n), and the other is the odd part of x(2n+1), and then the discrete Fourier transform of the signal sequence x(n) can be expressed and computed with the discrete Fourier transform of the two N/2 sampling points. Considering and the periodicity of the discrete Fourier transform, Eq. (1) can be written as

(3) where (4a) (4b) It follows that Eq. (4) is two discrete Fourier transforms containing only N/2 points, with G(k) including only the sequence of even points in the original signal sequence, and H(k) including only its sequence of odd points. Although k = 0, 1, 2, …, N-1, both G(k) and H(k) have a period of N/2, and their values repeat with N/2 cycles.