Sampling theorem in theory

Given an analog signal $x(t)$ defined in the continuous time $t$. It has the spectrum $X(\omega)$ defined by the Fourier transform
$$
X(\omega) = \int_{-\infty}^{\infty} x(t) \cdot e^{-j\omega t} dt.
$$
Now, the signal $x(t)$ is sampled in order to be stored, processed, or transmitted digitally. The sampling with the **sampling rate** $f_\textrm{s} = \frac{1}{T_\textrm{s}}$ where $T_\textrm{s}$ is the sampling interval can be described by multiplying the signal in time domain with the sampling function $d(t)$:
$$
d(t) = \sum_{k=-\infty}^{\infty} \delta(t - k \cdot T_\textrm{s}),
$$
where $\delta(t)$ is the Dirac function. The spectrum $D(\omega)$ of the function $d(t)$ is known to be
$$
D(\omega) = 2\pi f_\textrm{s} \cdot \sum_{n=-\infty}^{\infty} \delta(\omega - n \cdot 2 \pi f_\textrm{s})
$$
Thus, the sampled signal has the form
$$
x_1(t) = K \cdot x(t) \cdot d(t) = K \cdot x(t) \cdot \sum_{k=-\infty}^{\infty} \delta(t - k \cdot T_\textrm{s}) = K \cdot \sum_{k=-\infty}^{\infty}x(k T_\textrm{s}) \cdot \delta(t - k \cdot T_\textrm{s}).
$$
We can simplify the term $x(k T_\textrm{s})$ by the sequence $x_n$ in the discrete time $n$. When working with digital discrete-time signals, only this notation is used.
Applying the correspondence of the multiplication in the time domain and the convolution in the frequency domain we obtain the Fourier spectrum $X_1(\omega)$ of the sampled signal $x_1(t)$ as
$$
X_1(\omega) = K \cdot f_\textrm{s} \cdot \sum_{n=-\infty}^{\infty} X(\omega - n \cdot 2 \pi f_\textrm{s}).
$$
We see that the spectrum $X_1(\omega)$ is a periodic extension of the original spectrum $X(\omega)$, from which we can derive two conditions for non-destructive sampling:
**Nyquist frequency** $f_\textrm{N}$ is introduced, which is the highest frequency component that can be reconstructed with a given sampling frequency, in this context called the **Nyquist rate**
$$
f_\textrm{s} = 2 \cdot f_\textrm{N}.
$$
To reconstruct the original analog signal from the sampled one, if the sampling conditions were fulfilled, one has to remove the non-baseband copies (images) of the original spectrum. This can be achieved by low pass filtering the sampled signal (anti-imaging filtering) using a reconstruction filter (RLP) with the cut-off frequency $f_\textrm{RLP}$ from the range $f_\textrm{max} < f_\textrm{RLP} < f_\textrm{s} - f_\textrm{max}$. The wider this range is (i.e. high sampling frequency $f_\textrm{s}$), the less steep (less complex) the filter has to be.

The webdemo on the following two slides will visualise these relationships.

Implementation detail for interested readers: To obtain a signal with the defined value of $f_\textrm{max} = 1 \textrm{Hz}$, a 4-PAM signal is generated as for the transmission of digital data with the symbol duration $T_\textrm{sym}=0.5\textrm{s}$ and with a Raised-Cosine pulse shaping filter with the roll-off factor $\alpha = 0$ (see also the webdemo "Pulse shaping" for more insight into this topic).

- The signal $x(t)$ has to be band-limited such that its spectrum is zero for frequencies larger than a value $f_\textrm{max}$ specific for the signal
- The sampling rate $f_\textrm{s}$ has to be at least twice the frequency $f_\textrm{max}$ in order to prevent overlapping of the periodical subspectra of $X_1(\omega)$

The webdemo on the following two slides will visualise these relationships.

Implementation detail for interested readers: To obtain a signal with the defined value of $f_\textrm{max} = 1 \textrm{Hz}$, a 4-PAM signal is generated as for the transmission of digital data with the symbol duration $T_\textrm{sym}=0.5\textrm{s}$ and with a Raised-Cosine pulse shaping filter with the roll-off factor $\alpha = 0$ (see also the webdemo "Pulse shaping" for more insight into this topic).