Many physical systems are assumed to
obey the principle of superposition, which states that the response of some
system to a combination of two inputs is the same as the sum of the responses
to each input acting by itself. That is, Output(A+B)= Output(A) + Output(B).
Bathroom scales, hi-fi amplifiers and many other common systems are assumed to
follow this relationship. Such a system is also called linear.
Although it is not obvious from the
definition, one of the important consequences of the principle of superposition
is that the output of a system that obeys it does not contain frequencies that
were not present at the input. This is especially important for hi-fi
amplifiers, for example, where the output is intended to reproduce the input
signal without additional spurious frequencies. For example, lower-quality amplifiers (which are not completely
linear) can produce harmonics (integer multiples) of the input frequencies in
the output signal. The output of such a
system can also contain frequencies that are the sum or the difference of two
frequencies that are present at the input.
We have already seen that the response
of photographic film is logarithmic, and so photography using film does not satisfy
the principle of superposition. It turns out that neither the eye nor the ear
satisfy this principle either. The ear also has a logarithmic response to
sounds, and the response of the eye is also non-linear in complicated ways. One
of the consequences of these non-linear response functions is that both the eye
and the ear can produce the sensation of frequencies which are not present at
the input. Furthermore, the response of both of them to complex inputs having
many frequencies is not given by the simple sum of the responses to each
frequency individually.
Since light is an electromagnetic
signal, there is a direct relationship between the frequency of a light signal
and its wavelength, and the fact that the eye does not obey the principle of
superposition with respect to frequencies also implies that its response to an
input beam containing several wavelengths will not be the simple sum of the
responses to each of the wavelengths individually. All of our characterization
of color is derived from this fact.
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