harmonics and fundamental frequency

[bhuvanesh]

i need explaination other that guitar string examples(related to electronics).what is harmonics and fundamental frequency. when and where harmonics occurs .thank you in advance

 

[Harald Kapp]

Any periodic signal can be constructed from the addition of sine (or cosine) functions with varying amplitude and phase. Conversely, any periodic signal can be decomposed into a sum of sine frequencies. This is called Fourier sysnthesis or Fourier analysis and results in a Fourier transform of the signal. Any periodic non-sine signal can be treated this way.
The fundamental frequency (f0) of a Fourier series is the inverse of the period of the signal, just as the freqeuncy of a sine wave is the inverse of the period of the sine wave. The harmonics are the sines of higher frequencies (1*f0, 2*f0, 3*f0, ...) with respective amplitudes.

A common example where this technique can be applied is the decomposition of a square wave into component sine waves.

Read the Wikipedia articles I linked, come back if you have further questions.

 

[LvW]

....... when and where harmonics occurs .

Applying a (theoretically) "clean" sinusoidal signal to a non-linear transfer characteristic - and you are producing harmonics of the fundamental frequency.
Remember, each transistor amplifier has such a non-linear characteristic and, therefore, the voltage swing is limited if you need a "good" high-quality output.
It is common practice to use the figure of "Total Harmonic Distortion" (THD) as a measure for the degree of non-linearity. The THD is defined using the square root of the sum of the squared amplitudes for the existing harmonics.

 

[KrisBlueNZ]

Ah, you've gotta love Harald. His idea of a simple explanation is a bunch of formulas! Well, he's German, and they just seem to be born with brilliant, analytical engineering-oriented minds.

The idea of harmonics is a way to analyse waveforms that aren't perfect sinewaves.

A sinewave is a pure tone, sounding like a whistle or a flute. Most sounds that we hear in the real world (e.g. in speech and from musical instruments) aren't sinewaves. Not even close. In fact they're a lot more complicated than even a repetitive waveform, because they include an "envelope" (attack, decay, sustain, release or more complicated than that) during which the timbre (as well as the volume) varies. But certain types of waveforms, sawtooth for example, have distinctive sounds.

The point of studying harmonics is that a simple, continuous waveform, like a triangle, sawtooth, or square wave, can be broken down into multiple sinewaves that are added together. Obviously, there is one sinewave at the fundamental frequency (the frequency of the more complex waveform), then other sinewaves at multiples (harmonics) of that frequency, and with different phase relationships, are added to that sinewave until you end up with the final waveshape.

Another way to look at this is that any regular waveform can be broken down into a sum of sinewaves at the fundamental frequency and multiples thereof, at different relative volume levels and phase relationships.

For example, if you use a filter, you can pick out the individual harmonics of a sound. Listen to the sound of a square wave, for example, through a tube or pipe with variable length. The tube or pipe acts as a filter, and resonates at frequencies that are related to its length. As you adjust the length of the pipe, it will pick out (resonate at) the various harmonics of the fundamental freqency, and you can actually hear them individually. This is a kind of evidence that those frequencies are actually present in the signal, although when you listen to it, it only appears to contain the fundamental frequency.

Harmonics in real world sounds occur because whatever is making the sound - a string, or a reed coupled to some air, or just some air in a pipe, or a resonating drum head - does not move with a perfectly sinusoidal (sinewave-shaped) motion, because of physical factors such as how it is mounted and supported, how the resonating chamber around it is formed, etc.

If every sound in the world was a perfect sinewave, every instrument, every human voice, and so on, would sound like a whistle or a flute. It's the timbre, i.e. the mix of harmonics, that gives sounds their identifiable character. You can tell whether a sound is a guitar or a saxophone or a kettle drum; this partly comes from their harmonics, but mostly from how those harmonics vary over time, during the course of the sound.

You can also distinguish simple continuous waveforms (triangle, sawtooth, square, sine) from each other, once you've heard them all, although they're much less "real-world"-sounding than the complex sounds produced by movement of air in the real world.

I'm not sure if this will help you or not. It's just a bit of a brain dump

Get some samples of real-world sounds and look at them with a waveform editor such as Audacity or Cool Edit Pro. Look at the waveshape, and how it changes over time, and listen to it carefully. Also, view sections of it in the frequency domain, using the FFT (fast Fourier transform) feature of the software. Try to correlate what you hear with what you see.

 

[Harald Kapp]

Kris, I feel being treated a bit unfair
The link to Fourier Transform I gave points to a neat graphical representation. I still believe in th eold adage that 1 image is worth 1000 words. And yes, the link to Fourier analysis points to a lot of math. But: how is one to understand the significance of an FFT (as pointed out by you) if you do not know what a Fourier transform is all about?

 

[KrisBlueNZ]

Sorry Harald. I think you're brilliant, really I do. But sometimes what people need is a simple explanation that they can relate to. In my experience, German folks seem to have a certain mentality, which produces a lot of the best engineering in the world, but those highly mathematical explanations can be hard to grasp for people who don't have that mentality.

I could be totally wrong here, and if so, I apologise. I just think that some real-world explanation would be valuable to the OP.

As for the question of how can someone understand FFT without understanding the mathematics, I would (initially at least) look at it from the other direction. The maths is important to understand HOW the FFT is produced, once you understand WHAT it does!

Please tell me that I haven't insulted you. I think your mindset is great, really I do. But in some cases, I don't think it's the best way to start explaining things!

 

[Harald Kapp]

Kris, don't worry, I'm not feeling the least insulted.

Maybe Bhuvanesh could be our judge and tell us which information was helpful and which information may still be missing?

 

[KrisBlueNZ]

Thank goodness for that.

Yes, good idea.

 

[bhuvanesh]

first i seen harald explaination.he really explained in good way.i understood everything.but the direct link he given to wikipedia was little unfair.
later i seen kris answer which is really really helpfull in all way,rather than answering question krish gave lot more explaination in well grasping way.
finally krish impressed me .it doesnt mean u didnt.i alwasy like your answers(harald)
thank you

 

[KrisBlueNZ]

Very diplomatic, bhuvanesh. I guess that's why I like this site so much. So many different points of view available here.

 

[bhuvanesh]

The point of studying harmonics is that a simple, continuous waveform, like a triangle, sawtooth, or square wave, can be broken down into multiple sinewaves that are added together.

1) you mean that square sawtooth are derived from sine.if yes than how

the mix of harmonics, that gives sounds their identifiable character

how do say that that mix of hormonics is only giving uniq sound to every thing in world .how is that

 

[Harald Kapp]

1) you mean that square sawtooth are derived from sine.if yes than how

Atthe risk of repeating myself, follow the link to Fourier transform. There this is shown graphically how a square wave of a triangular wave is cretated by adding up sines of different frequencies.

how do say that that mix of hormonics is only giving uniq sound to every thing in world .how is that

Almost any sound is a mixture of different frequencies. The human ear is able to discern frequencies from ~20Hz to 20kHz (depending on age etc.) Within th ear different receptros (nerves) are repsonsible for different frequencies. These receptors react to the components of a sound. If two sounds contain diffferent frequency components, the ear detects this and you will hear these as different sounds (or sounds of a difffenret character) even if they have the same fundamental frequency.

This makes the beauty of different instruments playing together harmonically in an orchestra.

 

[LvW]

German folks seem to have a certain mentality, which produces a lot of the best engineering in the world, but those highly mathematical explanations can be hard to grasp for people who don't have that mentality.

I think, A. Einstein (A German who was born in Ulm) can serve as a good counter-example: "Explain everything as simple as possible - but not simpler!
As a good example for his methods I remember how he has explained some of his findings using a running train or an elevator.
And Barkhausens method, for my opinion, to describe his condition of oscillation also was very illustrative.

 

[BobK]

This reminds me of a realization I had recently. I remember studying how early astronomers used epicycles, i.e. smaller cycles in an orbit that could match the orbits better than a circle. I thought at the time that that seemed ridiculous. But recently I realized that they had basically discovered a geometric representation of the Fourier transform. The epicycles are nothing more than the harmonics needed to get from the circle, which produces a pure sine waveform to ellipses.

Bob

 

[KrisBlueNZ]

I think, A. Einstein (A German who was born in Ulm) can serve as a good counter-example: "Explain everything as simple as possible - but not simpler! As a good example for his methods I remember how he has explained some of his findings using a running train or an elevator. And Barkhausens method, for my opinion, to describe his condition of oscillation also was very illustrative.

Fair comments. Every good generalisation has exceptions!

1) you mean that square sawtooth are derived from sine.if yes than how

Not really. First of all, "square sawtooth" is a contradiction. But square and sawtooth waves are (or can be thought of as being) made up from multiple sinewaves; one at the fundamental frequency, and others at harmonics (integer multiples) of it.

Look at http://www.slack.net/~ant/bl-synth/4.harmonics.html and you should be able to see how it works.

[/QUOTE]how do say that that mix of hormonics is only giving uniq sound to every thing in world .how is that[/QUOTE]

Every waveshape has a different sound, and every waveshape is made from a different combination of harmonics in different proportions. That example shows you how a square wave is made from odd harmonics - that is f, 3f, 5f, 7f, etc, at steadily decreasing amplitudes. Different waveforms contain different harmonics, at different amplitudes.

But the sound of any real life noise is not a continuous steady waveshape; it varies in amplitude and shape over time. For example, if you hit a snare drum you get a loud burst of noise mixed with the resonant frequency of the drum head, which decays rapidly; if you blow across the top of a bottle, you get a roughly sinewave-shaped signal that quickly builds from nothing to a constant timbre and volume, then fades out when you stop blowing. So it's not just the mixture of harmonics that makes a sound recognisable; it's the way that mixture, and the overall volume, changes from the start to the end of the sound.

 

[LvW]

Fair comments. Every good generalisation has exceptions!
.

Ohh - I have another exception from this rule: I am sure you have heard about Schroedinger´s cat. Nice explanation.
(I hope you forgive my searching for counter examples - I am german too).

 

[LvW]

So it's not just the mixture of harmonics that makes a sound recognisable;....

Yes - and sometimes even SUBHARMONICS belong to a composite sound signal we appreciate.

 

[KrisBlueNZ]

Ohh - I have another exception from this rule: I am sure you have heard about Schroedinger´s cat. Nice explanation.
(I hope you forgive my searching for counter examples - I am german too).

I guess I shouldn't have tried to make ANY generalisations! In fact, I'm SURE I shouldn't have.

 

[LvW]

No problem. Although I'm not so very young anymore, my sense of humor is not lost.

 

[bhuvanesh]

Fair comments. Every good generalisation has exceptions!


Not really. First of all, "square sawtooth" is a contradiction. But square and sawtooth waves are (or can be thought of as being) made up from multiple sinewaves; one at the fundamental frequency, and others at harmonics (integer multiples) of it.

Look at http://www.slack.net/~ant/bl-synth/4.harmonics.html and you should be able to see how it works.

how do say that that mix of hormonics is only giving uniq sound to every thing in world .how is that

Every waveshape has a different sound, and every waveshape is made from a different combination of harmonics in different proportions. That example shows you how a square wave is made from odd harmonics - that is f, 3f, 5f, 7f, etc, at steadily decreasing amplitudes. Different waveforms contain different harmonics, at different amplitudes.

But the sound of any real life noise is not a continuous steady waveshape; it varies in amplitude and shape over time. For example, if you hit a snare drum you get a loud burst of noise mixed with the resonant frequency of the drum head, which decays rapidly; if you blow across the top of a bottle, you get a roughly sinewave-shaped signal that quickly builds from nothing to a constant timbre and volume, then fades out when you stop blowing. So it's not just the mixture of harmonics that makes a sound recognisable; it's the way that mixture, and the overall volume, changes from the start to the end of the sound.

nice link..why odd harmonics are being choosed to form square wave