The difference and connection of DFT, DTFT, DFS

The difference and connection of DFT, DTFT, DFS/DTFS

Fourier transform can transform discrete sequences and continuous signals.

Fourier transform and Fourier series

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CTFT：$x(t)\to X\left(j\Omega \right)$
transforms the continuous signal from the time domain to the frequency domain, the frequency domain is continuous, in$\Omega (rad/s)$ is the coordinate axis, draw [0,2π], the
computer cannot do this transformation, it can only be derived mathematically, or sample a continuous signal into a discrete sequence, and use FFT to increase the number of points N To approximate CTFT

DTFT(FT): $x(n)\to X\left({and}^{j\omega }\right)$
transforms the discrete sequence from the time domain to the frequency domain. The frequency domain is continuous, with$\omega \left(rad\right)$ is the coordinate axis. Drawing [0,2π]
DTFT uses less, because continuous frequency data cannot be processed by a computer

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Divided into two groups of memory is more convenient, about the frequency coordinate relationship $f,\Omega ,\omega ,k$ can refer tothe relationship between physical frequency, circular frequency, and digital frequency

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DFT：$x(n)\to X\left(k\right)$
transforms the discrete sequence from the time domain to the frequency domain. The frequency domain is discrete, with$k$ is the coordinate axis, draw [0,N]
when using DFT function (or fast Fourier transform FFT), it is necessary to determine the number of transformation points$N$ , generally take$2^n$， k k The maximum value of$k$ is N, which corresponds to 2π in DTFT

%matlab程序
N = 512;
xn = [1 2 3 0 0 0];
Xk = fft(xn,N)

DFS/DTFS：$\tilde{x}(n)\to \tilde{X}\left(k\right)$
Transform the infinitely long sequence from the time domain to the frequency domain. The frequency domain is discrete, with$k$ is the coordinate axis
if$\tilde{x}\left(n\right)$isx (n) x(n)The period extension of$x$$($$n$$)$ , then$\tilde{X}\left(k\right)$is also$X\left(k\right)$ period extension, so DFT and DFS are actuallyTake the main valueversusCycle extensionRelationship

Supplement: DFT is equivalent to sampling DTFT

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