Sound theory: Time-Frequency representation

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This is perhaps the most important concept for a full understanding of 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 in a scientific context.

The graphs we have seen so far have illustrated amplitude-time relationships, and have described amplitude rate in relation to time variations. Let's now consider a different approach and see how it is in fact 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 the following figure we see on the left hand side the canonical representation of a sinusoid in an amplitude-time graph. The top right-hand diagram represents this same sinusoid in a graph relating amplitude and frequency, whilst the two sinusoids of the previous example are represented in the bottom right-hand diagram:

Sound theory - Time-frequency representation

Time-frequency representation

So, a sinusoid can be seen, in an amplitude-frequency diagram, as a segment whose length corresponds to the sinusoid's amplitude and positioned on its frequency (this last sentence would horrify any physicist 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 perceive, 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:

Sound theory - Rate of a complex sound signal

Rate of a complex sound signal

Its frequency spectrum shall vary constantly in time and if we were to "photograph" the spectrum at a certain instant in time we would get the following graph relating amplitude and frequency in the moment we took the shot:

Dettagli su: Sound theory - Frequency spectrum of a complex sound signal

Frequency spectrum of a complex sound signal

A sound never remains still but continually changes 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 showing the amplitude of the spectrum's sinusoids. This explains the behaviour of a graphic equalizer [Equalizers and filters ] : it amplifies or attenuates the amplitude of the sinusoids. The interval 20 Hz - 20 KHz is continuous, so in an equalizer every cursor controls the amplitude of one frequency band. The more cursors there are, the narrower the controlled bandwidth will be. In the ideal theoretical case of infinite cursors, each one would control one single sinusoid's amplitude.

  • posted on 28-08-2010 23:14
    Hi SkY! The picture on the left is showing a representation of the signal in time. The pictures on the right represent the frequency representation of the same signal (sinusoid). It is the same signal showed in two different ways. I hope that is clearer now, if not, keep on asking ;-) Cheers
  • posted on 21-08-2010 21:22
    hey, in the pic you say time-frequency represenataion, but in actuality, you represent amplitude against frequency..... do correct it and explain it coz ive purchased your course and the first thing i dont understand is "frequency spectograms" and what do they show.....

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