Sampling: Theory and Math
In digital signal processing, sampling refers to the process of converting an analog signal into the digital domain.
Mathematically, a continuous signal $x(t)$ is sampled by 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 = 0.1$:
This impulse train can be expressed as a Fourier series:
$$\delta_T = \frac{1}{T} \left[1 +2 \cos(\omega_s t) 2 \cos(2 \omega_s t) + \cdots \right]$$, with $$\omega_s=\frac{2\pi}{T}$$
$$\delta_T = \frac{1}{T} + \sum \left( \frac{2}{T} \cos(n \omega_s t) \right)$$
Fourier Transform of the Dirac Comb¶
The Fourier transform of a time-domain impulse train is a frequency-domain impulse train:
$$\mathcal{F}(\delta_T) = \int_{-\infty}^{\infty} \delta(t-nT) e^{-j k \omega_0 t}dt$$
$$\mathcal{F}(\delta_T) = (\sum C_k e^{j k \omega_0 t})$$
$$\mathcal{F}(\delta_T) = \sum\limits_{m = -\infty}^{\infty} \delta(T f)$$
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¶
According to the convolution theorem, a multiplication of two signals in the time domain is equivalent to a convolution in the frequency domain. The convolution with the Dirac comb in the frequency domain results in a periodic spectrum.
$$X[i] = \frac{1}{T} + \sum\limits_{n=-\infty}^{\infty} X(\omega -n \omega_s)$$