4 Sampled Data Systems

A square wave can be represented by a Fourier series$$\text{for a square wave of period T, }\optbreak{}f(x)=\frac{4}{\pi}\sum_{n=1,3,5,...}^{\infty}{\sin{\frac{n\pi x}{T}}}$$. As you increase the number of terms in the series, the approximation becomes better, but the corresponding frequency spectrum includes higher frequencies.

The figure below shows the frequency spectrum, $R(j\omega)$, for the Fourier series of square wave and the corresponding time domain signal. Also shown is the sampled spectrum, $R^*(j\omega)$ for that signal and the time-domain signal reconstructed using the sampled spectrum.

Use the controls to change the number of terms in the series and change sampling frequency, $\omega_s$

square wave and spectrum
square wave and spectrum
Figure 1: Square wave approximation and frequency spectrum
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  1. How does time-domain signal and $R(j\omega)$ change as you change the number of terms in the series?
  2. Set the number of terms to 7 and the sampling frequency to 32 Hz. Then change the sampling frequency to 24 Hz. Observe what happens to $T|R^*(j\omega)|$. How and why does this affect the signal reconstructed from $R^*(j\omega)$?
  3. What happens to the reconstructed signal when aliasing occurs?