The formula for Fourier transform is
You can also transform the Fourier transform into another form:
It can be seen that the essence of the Fourier transform is the inner product, and the trigonometric function is a complete set of orthogonal functions. The inner product between the trigonometric functions of different frequencies is 0, only when the trigonometric function with the same frequency does the inner product. Not 0.
The following explains the meaning of the Fourier transform from the formula
Because the essence of the Fourier transform is the inner product, when f (t) and the inner product are calculated, only the component with frequency w in f (t) will have the result of the inner product, and the inner product of the remaining components is 0. It can be understood as the projection of f (t) on w, and the integral value is the integral of time from negative infinity to positive infinity, which is to superimpose the components of the signal at w each time, which can be understood as the f (t) on w The superposition of projections, the result of the superposition is the component of frequency w, which forms the frequency spectrum.
The formula for the inverse Fourier transform is
The following analyzes the meaning of the inverse Fourier transform from the formula
The inverse Fourier transform is the inverse process of the Fourier transform. When summing the inner product, only the component inner product at time t will have a result, and the remaining inner component product result is 0, and the integral value is the frequency The integration from infinity to positive infinity is to superimpose the components of the signal at each frequency at time t, and the result of the superposition is the value of f (t) at time t, which returns to the original time domain where we observed the signal.