Sound theory: Time-Frequency representation

This is perhaps the most important part in order to fully understand the nature of sound. A mathematical approach to this subject can become extremely complicated , so in our case it will suffice to hint at the main concepts leaving out the kind of detail one would expect perhaps in the scientific field.

The graphs we have up until now seen have been of an Amplitude-Time kind, and described the flow of amplitude in relation to time variations. Let's now consider a different approach to the question and see how its is infact possible to represent amplitude depending on frequency.

In the case of a pure sinusoid with the equation y=A sin(2πft) we can certainly say that both frequency f and amplitude A are constant. In figure we see a diagram representing Amplitude-Frequency , a sinusoid with amplitude A and frequency f whilst the two sinusoids of the previous example shall be represented as shown in the following figure:

Time-Frequency representation

So, a sinusoid can be seen, in an Amplitude-Frequency diagram, as a segment whose length corresponds to the sinusoids amplitude and positioned on its frequency (this last sentence would horrify any physician but as we have already said, we're not interested so much in scientific rigour, but rather we're aiming at gaining an overall understanding of phenomenae). Now, let's put all these things together. Let's imagine a complex sound, in other words, a sound that is composed of all the sinusoids from 20 Hz to 20 KHz (this is more or less the frequency interval that a human ear can hear, so from our point of view the only ones we're interested in). Let's now consider a complex sound signal like the one shown in the following figure:

Time-flow of a complex sound signal

Its frequency spectrum shall vary constantly in time and if we imagine to "photograph" the spectrum in a determined instant in time we will have the following graph from a Amplitude-Frequency diagramme:

A sound never remains still but varies continually in time. This means that every sinusoid's amplitude varies and thus so does the shape of the graph representing the spectrum. This explains what we see when we observe a spectrum analyzer with all its LED's going crazy. What it is actually doing is manifesting the amplitude of its sinusoids. This explains the behaviour of a graphic equalizer (graphic equalizer): it amplifies or attenuates (increases or decreases) the amplitude of the sinusoids. The interval 20Hz-20KHz is a continuous interval, so in an equalizer every cursor controls the amplitude of a single frequency-infact to be more specific of the frequency's sinusoid.