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Why does a periodic signal only has integer frequency values while a non-periodic one has all real numbers as frequency values?

The frequency is just the number of times one full cycle has happened in one single second of time. Then on a periodic signal, why can't there be like for example, 5.3 or 2.486 cycles happening in one second? And for non-periodic, the whole signal comes in a mess(not of a properly sine wave form), there is no one standard cycle to be considered then how could the frequency be calculated in the first place?

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    \$\begingroup\$ this sounds like many things I have heard someone whom does not understand signals teach someone when trying to teach signals. I am sorry to say the easiest way I know to rectify this problem is to get a white board out and discuss the nature of what frequency is. \$\endgroup\$ – Kortuk Aug 24 '11 at 7:33
  • \$\begingroup\$ This was what I was told during lecture but I don't understand how is this so. \$\endgroup\$ – xenon Aug 24 '11 at 7:46
  • \$\begingroup\$ xEnOn, I hate to hear that. I hope that people wring an excellent answer and clear it up. \$\endgroup\$ – Kortuk Aug 24 '11 at 9:14
  • \$\begingroup\$ Two periodic signals whose frequencies are rational multiples of each other (e.g. 222.2hz and 333.3hz, since 222.2 is 2/3*333.3 and 333.3 is 3/2*222.2) will sum to produce another periodic signal, whose frequency subdivides both (e.g. 111.1Hz). Two periodic signals whose frequencies are not rational multiples of each other will sum to a non-periodic signal. Some, though not all, non-periodic signals may be expressed as the sum of a finite number of periodic signals, some as the sum of a countably-infinite number of such signals, and some can't be expressed as any such sum at all. \$\endgroup\$ – supercat Aug 24 '11 at 22:18
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Say you have a 100Hz fundamental frequency. That means it repeats every 10ms, the blue curve in the graph.

Harmonics

This signal may have a 3rd harmonic (purple curve), so that's at 300Hz and hence repeats every 3.3ms. It has to be an exact multiple of the ground frequency so that it also starts a new cycle when the fundamental starts a new cycle, namely after 10ms. At that moment 1 cycle of the fundamental has passed, and 3 of the 3rd harmonic. Same with the fifth harmonic (brown curve): after 10 ms five cycle of it have passed and it starts a new cycle at the same time the other harmonics start a new cycle. So the situation after 10ms is exactly the same for all the harmonics as at time 0. And that's what frequency is, that everything repeats after a given period of time.

Suppose I have a fundamental frequency of 200Hz (blue curve below), so you see the same wave again every 5ms. And that you have a second component at 500Hz (pink), so that's not an exact multiple of 200Hz. What happens? The 200Hz starts a new cycle at 0ms, 5ms, 10ms, 15ms, 20ms, etc. The 500Hz component starts a new cycle at 0ms, 2ms, 4ms, 6ms, 8ms, 10ms, 12ms, etc. At 5ms the first starts a new cycle, but the second doesn't, therefore we won't see a repeat of the 0ms at that point. We do at 10ms, because then both components start a new cycle. And they also will at 20ms, not sooner.

Harmonics graph

We see that the fundamental frequency is 100Hz (brown curve), even if there's no wave at that frequency! But that's the highest frequency which has 200Hz and 500Hz as exact multiples.

So don't we need that fundamental frequency then? It appears that sometimes we don't. Our brain can reconstruct the missing fundamental in audio. Take an audio system where the speakers can't reproduce say 50Hz because too low. If your signal is composed of a 50Hz, a 100Hz and a 150Hz wave only the 100Hz and 150Hz will be reproduced. Yet we perceive this as a 50Hz signal!

The fundamental frequency is the greatest common divider (GCD) of all its components. The more frequencies are present which don't have simple ratios the lower this fundamental frequency will be, until you get to noise, which theoretically has all frequencies in it and has a fundamental frequency of zero, i.e. that it doesn't repeat at all!
Digital noise generators try to create a good quality noise by making the fundamental frequency as low as possible, for instance 0.001Hz, which means the same sequence repeats once every 18 minutes.

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  • \$\begingroup\$ @stephenvh - Nice answer. @ all - Regarding audio perceptions, I found this link very interesting: en.wikipedia.org/wiki/Binaural_beats \$\endgroup\$ – Oli Glaser Aug 24 '11 at 9:35
  • \$\begingroup\$ Thanks for having such a clear explanation. So is the reason why periodic signals have only integer frequency values is because there will definitely be an integer value for its harmonics? Like 3f, 5f, all of it are integers. Are the frequency harmonics allowed to be in decimals like 2.5f? \$\endgroup\$ – xenon Aug 24 '11 at 10:41
  • \$\begingroup\$ @xEnOn - they have to be integers. If you would have 2.5f the fundamental will actually be 0.5f. Otherwise the first "period" will look different from the second. \$\endgroup\$ – stevenvh Aug 24 '11 at 10:56
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    \$\begingroup\$ @xEnOn: the harmonics are integer multiples of the fundamental. The fundamental (or harmonics, for that matter) does NOT have to be an integer value... whoever said that is flat wrong. A second is basically a made up unit. If we defined a new time unit called say a wibble, with a conversion ratio of 2.333_ seconds to a wibble, and use that to specify frequency then what is and isn't an integer frequency changes. The wave itself hasn't changed, and the property of it being periodic hasn't changed either. Periodic is as simple as "does it repeat?". \$\endgroup\$ – darron Aug 24 '11 at 17:36
  • \$\begingroup\$ @stevenvh: Yea, if I had 2.5f, the fundamental frequency also has to change. But be it at 2.5f or 5f, the curves would look the same, is this right? Is the brown curve in the graph the composite signal of the others? How was the composite signal curve derived from the harmonic curves? \$\endgroup\$ – xenon Aug 24 '11 at 17:57
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There can be 5.3 or 2.486 cycles per second. This would be 5.3Hz, or 2.483Hz. There does not have to be an integer number of cycles in one second.

With non periodic signals, there is no one fundamental frequency associated with them, no repeating pattern - they can maybe be thought of as a periodic function with an infinite period. However any signal (including periodic) may have multiple frequency components, and we can look at a spectrum analysis of such a signal to determine frequency components. With a non periodic waveform the components can be continuous, a periodic waveform (ideally) will remain static with harmonically related compnents.

For instance a square wave has power at odd harmonics of the fundamental - so a square wave of 1Hz would have a frequency 1/3 the amplitude at 3Hz, and another 1/5 the amplitude at 5Hz, and so on to infinity (in practice infinity is never reached as signals cannot change instantly in the real world) These will be static as long as the square wave does not change in frequency or amplitude. See here:

Square wave wiki page

For further reading you might want to look into Fourier and Laplace transforms. Most good books on signal processing have plenty of good information on this and much more.

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