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I'm having a hard time wrapping my head around this. Whenever you edit an audio file and pause it at a specific moment in time, the audio spectrum shows different levels for each frequency, all in that one moment in time. Indeed, in the real world, at any moment in time there are numerous frequencies of sound flying around. If you could "pause" your reality I would imagine there would be numerous frequencies happening, all at a different intensity. A bird chirping, the wind rushing past, people talking, etc.

Say that a microphone generates a single voltage of like +1.12 mV at a single moment in time. The audio spectrum at T=0 is composed of numerous values though.

How is a single voltage reading at a single moment in time able to correlate to so many different frequencies occurring at so many different levels happening at that single moment in time?

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  • \$\begingroup\$ Bad analogy time: If I showed you a picture of a moving car you couldn't tell me its speed or acceleration, but if I showed you a given frame with a video of a moving car, you could probably give me some information using the other frames. \$\endgroup\$ – Wesley Lee Oct 17 '17 at 23:07
  • \$\begingroup\$ @hobbs - actually frequencies do exist at a single point in time. But you can't determine them from a single amplitude measurement. You can determine them from other type of readings however - consider light shining through a prism. \$\endgroup\$ – Chris Stratton Oct 18 '17 at 3:23
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When you pause an audio file in an editor, the spectrum display is that of the audio surrounding that point in time, not the spectrum of one instantaneous sample (because as above, one sample is not enough information to contain harmonics). Of course this varies from one program to another, and only the heavy pro programs come with tech support that will tell you how their software actually "thinks". But in roughly the same way your ears and mind hold a snapshot of a complex sound when you hit the pause button, the software does sorta the same thing.

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How is a single voltage reading at a single moment in time able to correlate to so many different frequencies occurring at so many different levels happening at that single moment in time?

It doesn't. A single sample contains no frequency information.

It appears you are confusing a time domain measurement with a frequency domain measurement. That is to say, in the time domain, you make a measurement at a particular point in time. Your voltage level for example. But in the frequency domain, you are, for instance, taking a measurement of all the different frequencies which comprise the current sound. For example, the sustained periodic sound of a piano note. The sound between the attack and the release. For this, you would need several time domain samples. Enough so that the different frequency components can be discerned.

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  • \$\begingroup\$ In the time domain, a single voltage level coming from a microphone corresponds to what then? \$\endgroup\$ – fuzzybabybunny Oct 17 '17 at 22:45
  • \$\begingroup\$ It corresponds to the instantaneous air pressure at that time (or velocity depending upon microphone design). \$\endgroup\$ – Kevin White Oct 17 '17 at 22:51
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    \$\begingroup\$ Look, @fuzzybabybunny, think of it this way: If someone took a picture of you in a car (sampled where you were for a given instant in time), you could not tell if you were waiting at a stop light or rolling down the road. You need some more pictures (samples) to discern if you were at a stop or moving. It's like that for sound. In order to know the frequency spectrum at a given time, you need to know something about the past sound samples. \$\endgroup\$ – st2000 Oct 17 '17 at 23:03
  • \$\begingroup\$ What the audio software is actually doing it more akin to artificially creating a picture of the speedometer. Although a single point in time can't let you see the car move, it can show an instantaneous value of a metric that measures how the car is moving in the region of that point in time. \$\endgroup\$ – Chris Stratton Oct 18 '17 at 3:19
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The microphone gives you a signal analagous to the instantaneous sound pressure. If we plot this over time (such as on an oscilloscope) you get familiar sine waves and all sorts of other waveforms.

Fourier showed that any periodic waveform - one that repeats - could be made up of sine waves of the fundamental frequency and its harmonics.

There is a Super animation by LucasVB explaining the Fourier decomposition of a square wave:

[][6]

Figure 1. Fourier transform of the signal from the time domain into the frequency domain.

Note how the same information can be viewed in two radically different ways.

How is a single voltage reading at a single moment in time able to correlate to so many different frequencies occurring at so many different levels happening at that single moment in time?

It doesn't. It is simply the sum of the sound pressures (positive and negative) reaching the microphone at that instant. You can't have frequencies without time so it's only as we analyse the signal with respect to time (or listen long enough) that we can extract the frequencies (or hear the tones).

You should be able to demonstrate this aurally using a program such as WavePad. Start with a very short clip - only one cycle of a low frequency note or tone and see how many cycles you need to hear before you can identify it as a note.

For more see What exactly are harmonics and how do they "appear"? (from whence I copied the animation).

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  • \$\begingroup\$ Thanks for pointing to LukasVB; there's some good stuff there! \$\endgroup\$ – bitsmack Oct 18 '17 at 0:06
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You're thinking of the frequency domain, the actual waves travel in the time domain. You're looking at the different frequencies because somewhere in software somebody has used a transformation (probably a Fourier transform or FFT) to break the components down into their respective frequencies.

However, the microphone was moving back and forth as the sound waves hit it so if you were to measure the position of the mic, it would have been in one position at any given time, but all ways moving continuously as each wave hit it. The voltages would be singular for any moment of time, and the ADC in your sound card only records one sample per one instance of time. (for a sound card sampling happens more often than not at 24kHz, 48kHz or 96kHz)

One point can be represented as a frequency, but the only thing we know about it is it has zero frequency or DC. If you want to actually sample a sine wave, you need more than one point, or the sine wave couldn't be re-created, there would not be enough information:

Sampling theorem

The sampling theorem (often called "Shannons Sampling Theorem") states that a continuous signal must be discretely sampled at least twice the frequency of the highest frequency in the signal.

More precisely, a continuous function f(t) is completely defined by samples every 1/fs (fs is the sample frequency) if the frequency spectrum F(f) is zero for f > fs/2. fs/2 is called the Nyquist frequency and places the limit on the minimum sampling frequency when digitizing a continuous signal.

Source: FFT\DFT enter image description here

So to answer your question:

How is a single voltage reading at a single moment in time able to correlate to so many different frequencies occurring at so many different levels happening at that single moment in time?

It can't, the sound processing software is probably using a trick to cut the waveform off before you see it or filling in information to get frequencies at T=0.

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