The definition of "frequency" in different contexts

Question

I have been doing some research on all kinds of sound-related topics lately and have been a bit confused by the different uses of the term "frequency". Of course, the most general meaning of frequency is just how many times something happens during a certain period of time (a second, usually). However, I've also come across many texts referring to a sound containing multiple frequencies, which seemed weird to me at first. Now, after reading up on stuff like Fourier analysis I've come to understand that "frequencies" in this case refers to the frequencies of the sine waves that make up the sound. So I guess my question comes down to this:

1. Is the above correct, or am I still not understanding it correctly in some way?
2. Assuming it is, is this not pretty ambiguous terminology? You could for example say that a square wave with a frequency of 440 Hz (meaning the waveform repeats 440 times per second) consists of many different frequencies (meaning it consists of many sine waves with different frequencies). It's not hard to see how this could be confusing.
3. And in that case, why can I not easily find a clear explanation of this anywhere? It took me quite some time to piece this together myself and seems like it really should be pretty basic stuff. For example, I first really started thinking about this when I ran into the Nyquist frequency. It seemed logical to me that you needed at least two samples per period to represent any kind of change, so the maximum frequency of the signal could only be half of the sampling rate, but when I started involving different kind of waveforms I realized there would be no way to distinguish between them. It took me a lot of time to figure out that the Nyquist frequency implicitly refers to the maximum frequency that any sine wave present in the signal could have, making it impossible to sample any other waveform at the Nyquist frequency.

Of course, if anyone could explain or point me in the direction of some material explaining the importance of sine waves in a more general fashion, and why these things seem to be supposed to be so obvious, that would be appreciated, but the three questions above are what I'm really wondering about.

Answer 1

1, A pure note consist of a single frequency. It's not really meaningfull or useful to talk about the frequency of a non-repeating sound - like a human voice.

2, A good way of looking at complex sounds is to split them up into a lot of single frequencies that are added together. A square wave consists of a single square frequency but you can also approximate it by a number of different frequency sin waves. A sqaure wave is a very good example of this because an ideal square wave goes from low to high in zero time at the edge, this is impossible in reality - the edges of a real world square wave will always be a little bit sloped. As you add more and more sin waves you can get closer and closer to a perfect square - the same thing happens for any other wave, you can always get closer and closer by adding more sin waves. (see http://www.mathworks.com/products/matlab/demos.html?file=/products/demos/shipping/matlab/xfourier.html)

3, The simple definition of Nyquist frequency makes sense for a sin wave. For a more complex wave the Nyquist limit is the frequency you would have to sample the highest frequency sin wave in the source with to reproduce the source perfectly. Remember the example of the square wave? There isn't a highest frequency to get a perfect square - you have to consider the maximum frequency sin wave you can use (or the maximum frequency you can play back) and as long as you capture that wave with the Nyquist frequency you have at least that level of accuracy.

Answer 2

Yep, it's not good terminology, but when you're familiar with something, you speak loosely. Better to say:

A square wave with a frequency of 440 Hz consists of many different sinusoid components of different frequencies

The square wave has a frequency, the sinusoids have frequencies, and "frequency" has the same meaning for both. The square wave isn't made up of "frequencies", it's made up of sinusoids. The sinusoids have their own frequencies.

And taking a signal and breaking it up into sinusoids is not the only possibility. It's just the most commonly used method (it's the method used by our ears, for instance). You could also take a sine wave and break it up into square waves, or take a musical signal and break it up into wavelets.

Answer 3

To add a bit to the previous answers...

The Frequency (expressed in Hertz unit) designates the number of times your artefact/phenomena occurs in a second. There is an abuse of language consisting in assimilating that 'signal frequency' to the frequency of the 'corresponding' Sinusoidal wave. This stems from the general monopoly of Fourier transform as a way of projecting your signal into singular components (ie. representing your signal on a certain basis of reference functions).

So while a sinewave of frequency $\omega$ is constituted by, well, 1 sine wave of frequency $\omega$, a square wave of the same frequency $\omega$ is constituted by an infinite number of sinewaves, of frequencies $\omega$, $2\omega$, $3\omega$ ... but both the sinewave and the square wave had the same and unique frequency $\omega$. This 'discrepancy' appears because the Fourier transform splits your square wave into sine waves components. As the shapes do not match, you need many (actually an infinity) of sine waves at certain frequencies to approximate your square wave (and it will never be perfect even, because the shapes are inherently incompatible). It is a representation issue.

The Fourier decomposition works well for describing continuous signals. It deals very badly with discontinuities, because its reference base is purely continuous (hence a square wave is described as an infinity of sine waves... that's very non-efficient!). I think this is never emphasized enough in teaching, which (with the prominence of the Fourier transform over other transforms such as Wavelets) leads to the assumption that it's the only way of projecting signals, hence the confusion about actual Frequency and 'closest match' Sine wave Frequency.

Now why is the Fourier transform so prominent? I think we must look at historical reasons, as far as I'm aware it came as the first decomposition/reconstruction method for continuous functions. But maybe in teaching the 'continuous' part is often skipped, which dilutes the fundamental idea that not all references are suitable.

Hope this helps.