Aliasing and Leakage

There are two effects that are introduced into the computation of the DFT.

Aliasing

Since when we compute we are necessarily dealing with a time-limited set of data, the signal cannot be bandlimited. The sampling process, with its incumbent spectral duplication, therefore introduces aliasing. This aliasing effect can be reduced by sampling faster.

Leakage

If the function $x(t)$ is not really time limited, then we truncate it in order to obtain a finite set of samples. As viewed above, the mathematics sees the sampled signal as if it were periodic in time. There are two ways of viewing what is going on. First, if we have a function $x(t)$, we can obtain a time-truncated version of it by

$$y(t) = x(t)w(t)$$

where $y(t)$ is the truncated version and $w(t)$ is a windowing function. In the frequency domain, the effect is to smear the spectrum out,

$$Y(\omega) = \frac{1}{2\pi}X(\omega)* W(\omega)$$

This smearing is spectral leakage. Another way of viewing the leakage is this: if we truncate a function then make it periodic, the resulting function is going to have additional frequency components in it that were not in the original function, due to the change from end to end. The only way this does not happen is if the the signal is periodic with respect to the number of samples already.

Leakage can be reduced either by taking more samples (wider windows of data), i.e. increasing $N_0$. It can also be reduced by choosing a different window function. However, it can never be completely eliminated for most functions.