Sampling Theorem

Material: 1.4.2, 1.4.6, 6.1

Idea: How fast must we sample the continuous-time signal without losing information? And then how can we go back to an analog signal from the digital signal? Nyquist Sampling Theorem

Wagon Wheel Effect

The illusion of a wheel spinning in the wrong direction

Imagine a phasor rotating at angular speed $\omega =2\pi F$ rad/s (counter-clockwise), and you take a snapshot every T second (for examples FPS of a camera), $nT=\frac{n}{F_s}$. Can we find values for T that the sampled phasor appears to rotate in clockwise direction rather than counter-clockwise? Example video

Link to original

---

We sample a signal at a regular interval given by:

$$x[n]=x(t_n)=x(nT)=x\left(\frac{n}{F_s}\right)$$

where $F_s$ is the sampling frequency and $T$ is the sampling interval.

---

Sampling of sinusoids and aliasing

Many practical signals can be represented by the Fourier transform:

$$x(t)={\int }_{-\infty }^{\infty }X(F)e^{j\ast 2\ast \pi \ast F\ast t}dx$$

---

Given

$$x(t)=\cos (\Omega t)=\cos (2\pi Ft)$$

Where $\Omega$ is the angular frequency, and $F$ is the frequency

Then

$$f=\frac{F}{F_s}\le \frac{1}{2}$$

Which means that all f that is not within $-\frac{1}{2}\le f\le \frac{1}{2}$ can be represented with this

Where $k$ is an integer

---

Transclude of Nyquist-Sampling-Theorem

Relation between sampled analog spectra

$$\begin{array}{rl}X(F)& =F_s\sum _{k=\infty }^{\infty }{X}_a(F-k{F}_s),\text{ce}whichisthesameas\\ X(f)& =F_s\sum _{k=-\infty }^{\infty }{X}_a([f-k]F_s)\end{array}$$