# Estimate number of sinusoids in periodic signal?

#### blah2222

Joined May 3, 2010
582
Hello, just wondering if there is a simple way to estimate the number of sinusoids (and their frequencies) from a plot of a periodic signal if they are all integer multiples a certain fundamental frequency that takes part in the signal. Without using a DFT of course.

Thanks!

#### Wendy

Joined Mar 24, 2008
23,397
It is called Fourier Transform. It is a specific branch of calculus.

http://en.wikipedia.org/wiki/Fourier_transform

I'm not sure you would call it simple. There are several formulas for different waveforms.

Don't know if this was what you were asking, hope it helps.

#### thatoneguy

Joined Feb 19, 2009
6,359
Do you have an example?

A pure sine with an extremely lower or higher pure sine is usually easy to see. The closer they are in frequency, the more impossible it becomes.

#### blah2222

Joined May 3, 2010
582
It is called Fourier Transform. It is a specific branch of calculus.

http://en.wikipedia.org/wiki/Fourier_transform

I'm not sure you would call it simple. There are several formulas for different waveforms.

Don't know if this was what you were asking, hope it helps.
lol, I know that but the question was regarding estimating the components visually without taking the DFT.

Thanks!

#### MrChips

Joined Oct 2, 2009
30,520
Create band pass filters and use zero-crossings.

#### joeyd999

Joined Jun 6, 2011
5,186
If you know the fundamental frequency, and you are sure the signal is really periodic, you are half way to your solution, if you don't mind a little bit of math.

You can be sure that each component sinusoid will be an integer multiple of the fundamental. Therefore, you can throw away all the analysis for any frequencies that are *not* integer multiples of the fundamental. This is opposed to a true Fourier Transform that computes all frequencies up to the Nyquist frequency.

Preforming a quadrature analysis of the original waveform against a unit sinusoid at the integer multiple frequencies will give you both amplitude and phase information for each component. You can carry this computation out to the limit of the Nyquist frequency.

Other than that, I don't know of a way to do it without *some* math!

Of course, this all assumes that you have a digital representation of the signal with an appropriate sampling rate. If so, this can easily (and quickly!) be done on an excel spreadsheet.

#### blah2222

Joined May 3, 2010
582
Cool, thanks for the input everyone! I actually figured out a pretty barbaric approach to estimating it. You can figure out the fundamental frequency pretty easily, and as previously mentioned, you can assure yourself that all other components will be integer multiples of that. Another thing is that you will notice that the peaks of that curve will have roughly the same period between each other as the highest frequency component. This also assumes that the components involved have comparable amplitudes, or else they would be hidden as noise. From there you can develop a range between f0 and fmax where the frequencies in between contribute to shifting the peaks away from the fmax frequency. Hard to explain, but it does give a great approximation and works for every test case I've had. Finding the mid frequencies as well as the number of mid frequencies is a chore though. I can only say that there are at least three frequency components but probably more, depending on the size of the range and offset of the peaks. This is also assuming the sinusoids have zero phase difference.

#### joeyd999

Joined Jun 6, 2011
5,186
Cool, thanks for the input everyone! I actually figured out a pretty barbaric approach to estimating it. You can figure out the fundamental frequency pretty easily, and as previously mentioned, you can assure yourself that all other components will be integer multiples of that. Another thing is that you will notice that the peaks of that curve will have roughly the same period between each other as the highest frequency component. This also assumes that the components involved have comparable amplitudes, or else they would be hidden as noise. From there you can develop a range between f0 and fmax where the frequencies in between contribute to shifting the peaks away from the fmax frequency. Hard to explain, but it does give a great approximation and works for every test case I've had. Finding the mid frequencies as well as the number of mid frequencies is a chore though. I can only say that there are at least three frequency components but probably more, depending on the size of the range and offset of the peaks. This is also assuming the sinusoids have zero phase difference.
Rest assured that pretty much any real-world signal is going to have harmonics from here to infinity* due to distortion and noise. The question is how much of a signal is considered a signal? Yes, you can identify large amplitude components by inspection, but this gets much harder at higher harmonics and as the amplitudes decreases dramatically.

*probably not in reality -- but at least theoretically!

#### joeyd999

Joined Jun 6, 2011
5,186
After thinking about your problem further, I realized that inspection is probably a very difficult way to determine harmonics except for a very small class of signals where there are only a few harmonics of very high amplitude.

For instance, think of a (perfect) square wave or triangle wave: we know precisely what harmonics exist and their amplitudes and phases, but only by definition. It would be *impossible* to deduce the harmonics just by looking for periodic waveforms at a multiple of the fundamental.

It gets even harder when things are not perfect. In the square wave case, think of edges that aren't sharp or bounce and ringing at the transitions...these are all contributed by various combinations of harmonics.

Further, think of a clipped sine wave (an overdriven amplifier, for instance). Here you'll just see a flat peak at the top of each phase. How can you determine harmonics by inspection in this case?

Since this is in the 'homework help' forum, I assume your prof gave you a relatively simple waveform and ask you to identify its components. If that is the case, your way may work. But in real-world practice, I wouldn't depend on it!

#### blah2222

Joined May 3, 2010
582
After thinking about your problem further, I realized that inspection is probably a very difficult way to determine harmonics except for a very small class of signals where there are only a few harmonics of very high amplitude.

For instance, think of a (perfect) square wave or triangle wave: we know precisely what harmonics exist and their amplitudes and phases, but only by definition. It would be *impossible* to deduce the harmonics just by looking for periodic waveforms at a multiple of the fundamental.

It gets even harder when things are not perfect. In the square wave case, think of edges that aren't sharp or bounce and ringing at the transitions...these are all contributed by various combinations of harmonics.

Further, think of a clipped sine wave (an overdriven amplifier, for instance). Here you'll just see a flat peak at the top of each phase. How can you determine harmonics by inspection in this case?

Since this is in the 'homework help' forum, I assume your prof gave you a relatively simple waveform and ask you to identify its components. If that is the case, your way may work. But in real-world practice, I wouldn't depend on it!
Yep, the question was given signals with very few components so it could be easy to estimate. Thank you for you help!