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As title says, what is the relationship between DTFT and continuous fourier transform? Let's say there is continious signal $f(t)$. Continuous Fourier transform convert this into $F(\omega)$. Now let us create infinite number of uniformly-distanced samples following Nyquist-Shannon sampling criterion, and DTFT these samples into $F_d (\Omega)$. What exactly is the relationship between $F(\omega)$ and $F_d (\Omega)$?

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If $T$ is the sampling period then the relationship can be written as

$$F_d(\Omega)=\frac{1}{T}\sum_{n=-\infty}^{\infty}F\left(\frac{\Omega-2\pi n}{T}\right)\tag{1}$$

So in general $F_d(\Omega)$ is an aliased version of the original spectrum. If $F(\omega)$ is ideally band-limited and the sampling theorem is satisfied, then you have

$$F_d(\Omega)=\frac{1}{T}F\left(\frac{\Omega}{T}\right),\quad\Omega\in [-\pi,\pi]\tag{2}$$

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