Why is the spectral view useful?

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 zoomed-in 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.
$\begin{figure}\begin{center} \epsfig{file=figures/timesignal.eps, width=3.5in} \end{center} \end{figure}$
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.
$\begin{figure}\begin{center} \epsfig{file=figures/example-mag.eps, width=3.5in} \end{center} \end{figure}$
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 .
$\begin{figure}\begin{center} \epsfig{file=figures/filter-mag.eps, width=3.5in} \end{center} \end{figure}$