Jean-Baptiste-Joseph Fourier (French mathematician, 1768-1830)factorization
This is the general way to get periodic signals, and we want to get discretespectrogramSignals can be used like this
As shown in the figure, theFn=1TF0(w)Bringing in the equation shown by the arrow, 2π/T becomes w, which gives the equation on the right. Since the unit period signal is intermittent, the discrete signal is easily obtained.
convolution
It can also be described in terms of convolution
The first formula can be seen as the result of convolving a signal with a periodic shock sequence signal. After they are convolved the periodic shock sequence signal undergoes a period delay.Any signal can be viewed as a signal within its single cycle that has been period-delayed. Mathematically, this can be expressed as the result of convolving a single periodic signal with a sequence of periodic pulses.
We'll start next with a few simple Fourier transforms
Fourier transforms of sine, cosine, and complex exponential signals
General periodic signal Fourier transform
Let's take an example in which the formulae use in places the knowledge related to Fourier decomposition of periodic signals. If you forget, you can find the relevant formulas from the previous section
We see that for the Fourier transform of the unit pulse signal and but for the Fourier transform of the pulse signal after a period delay, it is found. The Fourier transform becomes discrete. Let's be specific about the period-delayed topology below
Periodic extension of signals
The signal is periodically delayed in the time domain it forms a periodic signal, which corresponds to the spectrum of the signal he undergoes discretization. The periodization of a signal can be described by convolution of the signal with a sequence of periodic shocks. The spectrum of the signal can then be analyzed with the help of the convolution of the Fourier transform of the signal. The convolution in the time domain is then multiplied in the frequency domain. The multiplied signal is actually a discretization of the original signal.
We conclude with two examples
