4 Sampled Data Systems
Aliasing Lab DemonstrationAll Activities
The block diagram below shows a laboratory setup with an A/D converter, computer program which passes the numbers "straight through," and D/A converter (or ZOH) at the output. Variables $r(t)$, $r^∗(t)$ represent the continuous and sampled inputs, while $r_h(t)$ is the desampled output. The sampling frequency will be 500 Hz, or $T = 0.002$.
A sinusoidal function generator will be used to provide the input, so
$$r(t)=r_o \sin{\omega t}$$and the frequency spectrum $R(j\omega)$ will be a single line at $\omega$. The frequency spectrum of $r_h(t)$ is
$$R_h(j\omega)=R^*(j\omega)H_o(j\omega)$$with magnitude
$$|R_h(j\omega)|=\frac{1}{T}\sum_{n=-\infty}^\infty{|R(j\omega-jn\omega_s|T\bigg|\mathrm{sinc}\frac{\omega T}{2}\bigg|}$$The process of sampling will cause the repeated spectra, but their magnitude will be attenuated by the amplitude ratio of the ZOH (Refer to the textbook Section 4.4).
The spectral amplitudes of the continuous input and the sampled and reconstructed output will be displayed on a spectrum analyzer. The expected amplitudes at several frequencies are shown in the accompanying table. Signals at these frequencies will be attenuated accordingly.
| $f(HZ)$ | $\omega$ | $\omega T/2$ | $\mathrm{sinc}(\omega T/2)$ | db |
|---|---|---|---|---|
| 50 | 314 | 0.314 | 0.984 | -0.14 |
| 100 | 628 | 0.628 | 0.936 | -0.58 |
| 150 | 942 | 0.942 | 0.858 | -1.33 |
| 200 | 1257 | 1.257 | 0.757 | -2.42 |
| 300 | 1885 | 1.885 | 0.505 | -5.94 |
The magnitude of the ZOH frequency response at a sampling frequency of 500 Hz is shown in the plot. If the input is a sinusoid at 100 Hz, there will repeated spectra at 400, 600, 900, 1100... Hz.
The figure below simulates the experiment described above. The simulation assumes a perfect input where data is collected for an infinite amount of time before being analyzed. Of course, you can't do that in an actual experiment and in the actual experiment, you would not see perfect lines on the spectrum analyzer.
- Calculate the aliased frequencies when input frequency is 30Hz. Verify that with the simulation.
- Slowly adjust the input frequency from 50Hz to 240Hz. What happens as the input frequency grows.
- What happens when the input frequency is greater than 250Hz, and why?