Aliasing

The problem with sampling is that you lose information in the process. Two different functions $f(t)$ and $g(t)$ might get sampled and produce the same vector $\mathbf{x}$. For instance, suppose $f$ is a sine wave, $f(t) = \sin(t)$ and $g$ is the zero function, $g(t) = 0$. These are two very different signals. But, with the sampling interval $\Delta t = \pi$, we see that $$\begin{eqnarray*} x_n &=& f(n\pi) = \sin(n\pi) \\ &=& 0 \\ &=& g(n\pi), \end{eqnarray*}$$ so $f$ and $g$ get sampled to appear as the same vector $\mathbf{x}$, which happens to be the zero vector. This is called aliasing: one signal appears the same as another.