It is often the case that the spectrum of a signal can indicate aspects of the signal that would otherwise not be obvious by looking only at its time-domain representation.
For example, from the time-domain plot below it is clear that the signal is periodic, with a period of about 0.02 seconds. However, few other features of the signal are immediately clear.
Figure 2:
An arbitrary time-domain signal.
The plot below shows the magnitude spectrum of the signal from Fig. 2. It is now obvious that the signal is comprised of only 5 frequency components, the magnitudes of which are inversely proportional to frequency. As implied by the time-domain periodicity, the spectral components are harmonically aligned. From further analysis, we see the 5 components fall only at odd integer multiples of the fundamental frequency.
Figure 3:
The magnitude spectrum of the time-domain signal plotted above.
Frequency-domain representations are also commonly used in evaluating digital filters.
As an example, a simple digital filter has a time-domain representation of the form
. It is certainly possible to gain intuition on the behavior of this filter by considering its output given various input signals in time. However, a relatively easy sequence of steps allows us to realize the filter response for all sinusoidal frequency inputs from as plotted below (where is the digital sampling rate).
Figure 4:
The magnitude frequency response of the filter
.