1. Discrete Time Fourier Transform DTFT (Discrete Time Fourier Transform)
Code:
result:
Analysis: It can be seen that the dtft transform of the discrete sequence is periodic, which also conforms to the description of the Nyquist sampling theorem. After the continuous time signal is sampled periodically, the spectrum of the resulting discrete signal is the periodic continuation of the original continuous signal spectrum.
2. Discrete Fourier Transform (DFT)
Unlike DTFT in 1, DTFT's summation interval is the entire frequency domain, which is not achievable for computer calculations. DFT is the finite Fourier transform of the sequence. In fact, the code I gave in 1 is only summing the 1601 points in the frequency domain between -800----+800, and it is not an infinite number of sums.
Implementation code:
Result graph:
Analysis: DFT is just a realistic version of DTFT, because DTFT requires the summation interval to be infinite, and DFT sums only within a limited number of points.
3, Fast Fourier Transform FFT (Fast Fourier Transform)
Although DFT reduces the complexity compared to DTFT, it still has a considerable amount of calculation, which is not conducive to effective processing of information in real time. The DFT discovered in 1965 solved this problem.
Implementation code:
Effect chart:
Analysis: As can be seen from the figure, the frequency center of the fft transformation is not at 0 o'clock. This is caused by the fft algorithm. Changing fft to fftshift can shift the frequency center to 0 o'clock.
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