In digital signal processing, sampling refers to the process of converting an time-continuous signal into a time-discrete signal.
Mathematically, a continuous signal $x(t)$ is sampled through a multiplication with an impulse train $\delta_T(t)$ - also referred to as Dirac comb - of infitite length:
$$x[n] = x(t) \delta_T (t) = \sum\limits_{n=-\infty}^{\infty} x(n T) \delta (t-nT)$$
Dirac Comb¶
The Dirac comb is a periodic function:
$$ \delta(t-nT) = \delta(t-nT+T) $$
The following plot shows a Dirac comb for $ T = 0.01$:
Fourier Series of the Dirac Comb¶
This impulse train can be expressed as a Fourier series with the Dirichlet-kernel (which is related to the sinc function):
$$\delta_T = \frac{1}{T} \sum_{n=-\infty}^{\infty} e^{j 2 \pi n \frac{t}{T}}$$
The figure below shows how this series converges towards an impulse train:
Fourier Transform of the Dirac Comb¶
The Fourier transform of a time-domain impulse train is a frequency-domain impulse train:
$$\begin{eqnarray} \mathcal{F}(\delta_T)&= &\int_{-\infty}^{\infty} \delta_T e^{-j 2 \pi f t}dt \\ &=& \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} \delta(t-nT) e^{-j 2 \pi f t}dt \\ \end{eqnarray}$$
The Fourier transform of the shifted impulse is a complex exponential:
$$\begin{eqnarray} \mathcal{F}(\delta_T)&=& {\sum\limits_{n = -\infty}^{\infty} e^{-j 2 \pi n T f}} \\ \delta_F&=& \delta(f - \frac{n}{T}) \end{eqnarray}$$
With the above equation represents the Fourier sieries of a Dirac comb. The Fourier transform of the Dirac comb in the time domain is a Dirac comb in the frequency domain with:
$$ \Delta f = \frac{1}{T} $$
Fourier Transform of the Sampled Signal¶
The convolution with the Dirac comb in the frequency domain results in a periodic spectrum.
$$X[i] = \frac{1}{T} + \sum\limits_{n=-\infty}^{\infty} X(\omega -n \omega_s)$$
From the DFT section we know that our DFT $X[n]$ is defined between $-f_s/2$ and $f_s/2$. According to the convolution theorem, a multiplication of two signals in the time domain is equivalent to a convolution in the frequency domain:
