This is the most basic and common graphical analysis tool. It
relies on the fact, known since the work of the French mathematician
Joseph Fourier (1768–1830), that any periodic signal of a given frequency
can be decomposed into a sum of pure
(or sinusoidal) signals
whose frequencies are multiples of the frequency of the original
signal. For example, the following picture shows the signal from page
5, as well as the two sinusoidal signals (blue and red lines) into
which it can be decomposed.
The Spectrum tool takes a piece of signal of a certain length T (the length in question is equal to twice the number, in samples, selected in the Size field of the dialog that comes up when you choose Spectrum) and decomposes it into a superposition of sine waves of periods T, T=2, T=3, etc. as explained above. It then represents the amplitudes of these pure sine waves as a function of their frequencies.
The location of the piece of signal in question relative to the current selection can be chosen in the Spectrum location field. At this stage, you may wonder about the fact that the period T chosen for the spectrum analysis is just a fixed number chosen by the user (like 4096 samples for example), which does not necessarily correspond to the period of the signal that is being analyzed. Furthermore, it may well happen that the signal in question is just not periodic at all. What Amadeus Pro (and just about any other analysis tool) does is to consider the periodic continuation of the signal. In other words, it just pretends that the signal is periodic of the given period, even though this is probably not the case. Let us see what this means for our favourite signal (still the one from page 5). The following figure shows that signal in black, and it has an interval marked as coresponding to the period T which has nothing to do with the true period of the signal.
The blue line then shows to the periodic continuation of the
signal according to the marked interval. Note that this line is
discontinuous, i.e. it has a jump
at the endpoints of
the marked interval. It is intuitively clear that this discontinuity
is not a desirable feature. One way of getting rid of it is to
multiply the signal by a windowing function which is close to
one inside the marked interval and goes to zero at its endpoints. The
red line in the above figure shows the periodic continuation of the
windowed
signal. It does no longer present a discontinuity but
of course it does no longer perfectly match the original signal inside
the marked interval. Such a windowing function can be selected in the
Windowing function field when you make a spectrum analysis.
A list of the most popular windowing functions can be found at this Wikipedia page.
The spectrum analysis window has several features that facilitate its reading. The Current location text field in the toolbar shows the frequency / intensity corresponding to the current location of the mouse. Intensities are given in decibels. The Nearest note text field shows the note (on a standard tempered scale with A4 at 440Hz) which is nearest to the frequency corresponding to the current location of the mouse. The exact frequency corresponding to that note is shown in brackets.
In order to facilitate the reading, a thin green line is drawn at
the current location of the mouse. Furthermore, the harmonics
(i.e. multiples) of the corresponding frequency are also shown
as green lines. The reason why harmonics are important is because a
note played on a musical instrument does not consist of a single
frequency but of a fundamental frequency and all its multiples, or harmonics. The
ratios between the intensities of the harmonics determine the
color
of the sound and allow us to distinguish between
different musical instruments. The number of harmonics that are
displayed is 5 by default, but can be changed by a hidden preference,
see Section 10.
The scale of the display can be changed by clicking on the Settings icon in the toolbar.
A spectrum analysis can be exported as a text file for processing in another program by choosing Save A Copy As in the File menu. This creates a file with three columns. The first column contains the frequency of a given component, the second and third contain its amplitude in complex notation. (So the intensity of a component is the sum of the squares of the real and the imaginary part.) If the spectrum was created by choosing Average over selection, then all the imaginary parts are 0.
