If a transfer function attenuates (makes softer)
certain frequencies, we can say that the function is
filtering out those frequencies. In fact, a system
with a transfer function that passes some frequencies and
attenuates others is called a filter. Just like any
other system, we can characterize a filter by
measuring the output for a range of inputs. A common way to
refer to the transfer function of a filter is as its
frequency response.

Specifically, we can measure the output amplitude (y axis)
for a number of input frequencies (x axis). As long as we
keep the input level of each frequency we put through the
system constant, the shape of the transfer function will
reveal what the system does. Figure 1 shows the brute force
method of measuring the transfer function: input one
frequency at a time, and record the output.

You can image how tedious this can get if you're measuring a
filter's frequency response in real life. Luckily, there is a
shortcut. As long as the filter is linear, we can simply put
all the frequencies through the filter at once. What type of
signal contains a wide range of frequencies that all have the
same amplitude? White noise.

White Noise: a signal containing all
frequencies, where each frequency is the same amplitude.
White noise gets its name from the fact that white light is a
combination of all wavelengths (colors) of light.

Let's look at a Fourier Transform to confirm the spectrum of
white noise is flat (the same amplitude across all frequencies).

Figure 2 shows a white noise is the time and frequency domains (upper
and lower panels, respectively). The time domain shows a jumble of
non-repeating frequencies (about as far from a pure tone as you
can get!), and the frequency domain shows a nice, flat spectrum.
If we pass this noise through a filter, we will be testing every
frequency at once, and the shape of the white noise spectrum
will change to take on the shape of the filter. Figure 3 shows the
time and frequency domains of the white noise after passing it
through a filter that attenuates frequencies above 4 kHz.

The filter in Figure 3 is called a low-pass filter. After
looking at the filter's frequency response, the name makes sense.
The filter in Figure 3 is passing (i.e., not attenuating) low
frequencies, and attenuating high frequencies. There are four main
filter shapes to consider, and each one is named after the shape
of its frequency response. Let's take a look at them.