# Fourier Series and Filters

#### Papabravo

Joined Feb 24, 2006
15,750
Perhaps it would be good to get back to the original problem which was the suspicion that a "sum of sinusoids" and a "square-wave" are fundamentally different. Just because the shapes are not similar does not allow you to conclude that they are fundamentally different. It is of course correct to say what level of precision you require for equality. Transcendental functions by their very nature are imprecise having both irrational arguments and irrational values. In a sense nothing involving transcendental functions can ever be precise in the sense that a rational result can be.

If there is some difference between the two things would the OP please enlighten us with a convincing argument to support that conjecture.

#### GS3

Joined Sep 21, 2007
408
Since you ask, my question is to look at the simple fourier analysis of a waveform containing a fundamental and second harmonic in phase.

I apologise in advance for the poor quality drawing.

One interesting fact is immediately obvious. The 2nd harmonic reinforces the positive peak and curtails the negative excursion. Thus the positive and negative halves of the waveforma are not mirror images in the resultant.

Now pass this through a circuit which alters the relative phase of the fundamental and the 2nd harmonic.

This will alter the wave shape as the peaks in the 2nd harmonic reinforce different parts of the fundamental.

Is this what you expect of a filter?
I still fail to see how this is related to the matter under discussion so I am not going to address this sidetrack. Again, the matter being discussed is how one simple waveform "generated in one go" can be considered to be formed by the sum of several sinusoidal waveforms and I think I have explained quite clearly that it can be considered to be composed of such a sum because there is no difference on whether it was generated one way or the other and such a waveform will always produce the same results in any given circuit.

If you wish to discuss how filters work and what they do that may be an interesting topic for another thread.

#### GS3

Joined Sep 21, 2007
408
Perhaps it would be good to get back to the original problem which was the suspicion that a "sum of sinusoids" and a "square-wave" are fundamentally different. Just because the shapes are not similar does not allow you to conclude that they are fundamentally different. It is of course correct to say what level of precision you require for equality. Transcendental functions by their very nature are imprecise having both irrational arguments and irrational values. In a sense nothing involving transcendental functions can ever be precise in the sense that a rational result can be.

If there is some difference between the two things would the OP please enlighten us with a convincing argument to support that conjecture.
I think I can understand his gut feeling that intuitively they are different things but given that the problem disappears if we accept that they are in fact the same thing then it makes sense to accept it. There are many things which may not seem intuitive at first but that we learn to accept after rational analysis shows that they are so.

#### Ron H

Joined Apr 14, 2005
7,014
I would also be interested in GS3's comments on the following

If we plot phase change against frequency for a typical filter we see that this varies with frequency so that the phase change for 2f, 3f, 4f etc is different.

So if we pass a bunch of frequencies through the filter each will experience a different phase change.

However if we decompose some signal into those same frequencies using fourier analysis, the phase change is fixed and constant for all components.
I agree that this doesn't seem relevant to our OP's question, but anyhow...
Actually, "a different phase change" is desirable if you want to preserve the overall wave shape.What is needed is constant group delay, which is linear phase shift vs frequency. If you use a non-recursive filter, this can be obtained.

#### thingmaker3

Joined May 16, 2005
5,084
The filter doesn't care about math. (People care about stuff, circuits just conduct electricity.) The filter doesn't do math. The filter simply resists rapid changes in state. We flawed, sensitive, and oh-so-passionate human beings are able to use math to predict how the filter resists steep changes in state.

A fun exercise we had in high-school was to inject Fourier series sine-wave components into a mixer to construct a signal approaching a square wave.

#### Ron H

Joined Apr 14, 2005
7,014
The filter doesn't care about math. (People care about stuff, circuits just conduct electricity.) The filter doesn't do math. The filter simply resists rapid changes in state. We flawed, sensitive, and oh-so-passionate human beings are able to use math to predict how the filter resists steep changes in state.

A fun exercise we had in high-school was to inject Fourier series sine-wave components into a mixer to construct a signal approaching a square wave.
I tried to post a little simulation of this, but the keepers of the forum shrink graphics so much that they were unreadable. Guess I could post them on a free posting site, and provide the URL.

#### thingmaker3

Joined May 16, 2005
5,084
If you e-mail it to me, I'll put it up on my site and link to it.

#### Dave

Joined Nov 17, 2003
6,970
I tried to post a little simulation of this, but the keepers of the forum shrink graphics so much that they were unreadable. Guess I could post them on a free posting site, and provide the URL.
RonH, can you let me know which graphics are posing problems and I will look at addressing the issue.

Dave

#### Ron H

Joined Apr 14, 2005
7,014
RonH, can you let me know which graphics are posing problems and I will look at addressing the issue.

Dave
OK, here they are. The graph was full screen on my computer, and had vivid colors.
It appears that they are shrunk, and .PNG gets converted to .JPG, which I think degrades the resolution.

Ron

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#### Mazaag

Joined Oct 23, 2004
255
The filter doesn't care about math. (People care about stuff, circuits just conduct electricity.) The filter doesn't do math. The filter simply resists rapid changes in state.
So the filter works more on the change in state rather than a periodic characterisitc (frequency) ?

If that is the case then wouldn't it make more sense that fundamentally the filter works on the basis of rapid change in the signal (more of a derivative) rather than "frequency components" which is merely a tool to approximate (with infinite accuracy) a complex signal..?

To add to that, wouldn't saying that the impedance of a capacitor , which is dependant of frequency, also be a way of simplifing the math by working in the "frequency domain"...? so fundamentally, the physics of how a capacitor behaves would NOT be dependant on this thing called "frequency", but could be simplified and represented (with infinite accuracty) to be dependant on frequency...

Am I making sense ? LOL...

#### Dave

Joined Nov 17, 2003
6,970
OK, here they are. The graph was full screen on my computer, and had vivid colors.
It appears that they are shrunk, and .PNG gets converted to .JPG, which I think degrades the resolution.

Ron
Ron, I have tweaked the attachment settings for JPG/JPEGs and PNGs. Can you upload an new version of the attachments in this test thread I have created (I don't wish to derail Mazaags thread further) to see if the problem persists. I'm sure at the moment why the attachments have converted type.

Thanks.

Dave

#### GS3

Joined Sep 21, 2007
408
So the filter works more on the change in state rather than a periodic characterisitc (frequency) ?

If that is the case then wouldn't it make more sense that fundamentally the filter works on the basis of rapid change in the signal (more of a derivative) rather than "frequency components" which is merely a tool to approximate (with infinite accuracy) a complex signal..?

To add to that, wouldn't saying that the impedance of a capacitor , which is dependant of frequency, also be a way of simplifing the math by working in the "frequency domain"...? so fundamentally, the physics of how a capacitor behaves would NOT be dependant on this thing called "frequency", but could be simplified and represented (with infinite accuracty) to be dependant on frequency...
This is like a dentist arguing with a proctologist about whose view of the human body is more "real". They are both just as real and just as valid, and one is better than the other depending on what you are trying to do. If you want to extract a tooth the dentist's view is more useful because, well, extracting a tooth by way of the colon can get a bit complicated.

Analysis in the time domain and in the frequency domain are just as real and one is not any more real than the other. Each one has their uses and some types of analysis would be impossible to do if you restrict yourself to the time domain. Try analyzing a tuned radio receiver without using frequency analysis and see how far you get. You might do better extracting a tooth through the anus.

#### thingmaker3

Joined May 16, 2005
5,084
so fundamentally, the physics of how a capacitor behaves would NOT be dependant on this thing called "frequency", but could be simplified and represented (with infinite accuracty) to be dependant on frequency...
That is exactly the case. Ditto for coils. dV/dt is proportional to f.

Which model should be used depends on one's application, one's background, and one's preference. Either model serves.