Why Do Spectral Aliases Arise?

Varying sample rates (e.g., $f_s=5, 10, 20$ samples/second) for a continuous-time signal determine the spacing between the samples. The question is how close or far we can sample in time. Here, we explore some interesting phenomena observed when the sample rate is changed beyond certain values.

After sampling, a complex sinusoid has a frequency scaled by $f_s$ because

$$\begin{align} x[n] &= A e^{j \left(2\pi f t + \theta\right)}| _{t=nT_s}\nonumber \\ &= A e^{ j\left(2 \pi \frac{f}{f_s} n + \theta\right)} \end{align}$$

where $f_s=1/T_s$ ...