Signal and system 8: Fourier Transformation

The idea of how to transform Fourier series to Fourier transformation is to first extend the non-cyclic signal x (t) to a certain cycle T cycle to obtain a periodic signal x1 (t ), because the periodic signal has Fourier Series

We obtain a series of Fourier Series A (k). When T tends to be infinite, x (t) = x1 (t) the Fourier series at this time is Fourier transformation.

With the definition of Fourier transformation, what is the relationship between the two? We can understand this.

The finite time domain must be infinite in the frequency domain. Assuming that the non-cyclic signal x (t) is finite, its Fourier transformation is infinite, and the signal x1 (t) obtained by periodic Delay) the Fourier series of is one of the samples of x (t) Fourier transformation.

When T is larger, the number of samples increases. When T is infinite, the number of samples is all Fourier transformation.

How is the Fourier transformation of periodic signals defined?

Fourier transformation of periodic signals can be directly constructed using Fourier series,

For verification, it is obtained through Fourier inverse transformation.The definition is the same as that mentioned above, so we should consider that such a definition can make all signals, whether in the cycle or in the non-cycle phaseCorresponding Fourier transformation.

P:

The figure above shows a non-cyclic function with T = 4Periodic delay. The following figure shows a periodic delay of T = 40,

We can see that the number of samples with T = 40 Fourier series is much greater than T = 4.

It can be imagined that when T tends to be infinite, the basic Fourier series are all Fourier Transformations.