# Fourier Series and Filters

#### Mazaag

Joined Oct 23, 2004
255
Hi guys...

I have a few questions about Fourier series and filters....

Here's what I understand about Fourier Series:

Any signal (periodic or aperiodic) can be REPRESENTED (key word) as a sum of sinusoidal components (of various frequencies and amplitudes)

Lets consider an arbitrary periodic signal...

Such a signal can be REPRESENTED as a sum of sinusoids having frequencies which are multiples of its fundamental frequency, with various amplitudes ( or "intensity" )

So far so good ?

Here's the part that confuses me..

Let us consider an ideal low pass filter in which this signal is passed through...

The low pass filter will only pass the SINUSOIDS components of low frequency (harmonics) below the cut off frequency.. The result is a NEW signal with only the low frequency remaining components....

This all makes sense...but what I don't understand is the following :

We started off saying that fourier series could be used to REPRESENT an arbitrary signal...but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....

so how does the filtering work by removing the "sinusoidal components" if inherently there are none ?

I hope my question makes sense Thanks guys

#### Papabravo

Joined Feb 24, 2006
16,840
If two things are equivalent then they must produce the same result. If the filter could somehow distinguish between a "sum of sinusoids" and its equivalent then it would choose which output to produce based on one of two equivalent inputs according to some process. We know from the characteristics of passive devices such as resistors, capacitors, and inductors that such behavior is not possible. In particular resistors have an impedance which does not depend on frequency. Capacitors and inductors have an impedance which depends on frequency. The property of impedance in a capacitor or an inductor does not require a sinusoidal input. It works the same way with edges(step functions), ramps, and impulses.

#### GS3

Joined Sep 21, 2007
408
but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
That's like saying that the number obtained by adding 5+2 is not the same as the number 7. The number 7 is always the same no matter how you arrived at it. A signal, like a number, can be arrived at by many different ways and can be dissected many different ways no matter how it was obtained.

A signal can be split into a (Fourier) series of harmonics and the sum of thos harmonics IS that signal. Just like when you split 7 into 5 + 2, when you add up 5+2 you still have the original 7 and not a 5+2 "7-like" thing.

#### Dave

Joined Nov 17, 2003
6,970
We started off saying that fourier series could be used to REPRESENT an arbitrary signal...but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
In brief, this assumption is not correct. For a periodic signal, the Fourier Series (or more accurately the modes) of an arbitrary signal is an injective function of the original (arbitrary) signal.

Dave

#### Dragon

Joined Sep 25, 2007
42
Ever wondered what a square wave is really made of? It is a combination of sinusoids continuously overlapping each other and distorting into sqaure shapes. Add all these so called harmonics together and you end up with a square wave. Mathematically, only a Fourier series or its analysis gets us there.

More specifically, this finds applications is DSP, where information has to be incoded in the sprectrum of a signal, and hence the concept of fourier transform.

#### Dave

Joined Nov 17, 2003
6,970

#### studiot

Joined Nov 9, 2007
4,998
I think this is a perfectly reasonable question, which has been asked many times by many students but is rarely satisfactorily answered.

Any signal (periodic or aperiodic) can be REPRESENTED (key word) as a sum of sinusoidal components (of various frequencies and amplitudes)
You are quite correct in saying that fourier analysis can be used to represent or approximate other functions.

For some functions the approximation is numerically exact, for others less so.
It does not mean to say that because something is numerically exact it is identical, only that our representation allows us to make useful predictions about what happens.

It is not true to say that (5 + 2) is the same as 7. For example 5 apples plus 2 pears makes 7 fruit, but they are not interchangeable.

In this and all the higher maths you may come to study it is very important to take note of the conditions under which a statement applies or holds true.
Step outside those conditions at your peril.

However splitting an entity into more easily handled components should be familiar. You will have studied vectors for either forces or complex number representation of currents and voltages. Splitting a waveform into suitable 'components' is very similar.
The whole point is that the resulting arithmetic is easier by taking one component at a time than by trying to work with the whole

A fourier representation is only exact if the signal is fully continuous in the first place. For all other waveforms there is an error term. This term varies with the waveform and is infinite at every vertical in a square wave.

http://www.sosmath.com/fourier/fourier3/gibbs.html

You also need to distinguish between digital and analog filters which work in different ways.

A filter is designed to have a non linear response (in the frequency domain). The output of any non linear device will not be a faithful reproduction of the input it will look different. Sometimes this is unwanted and is called distortion.

#### GS3

Joined Sep 21, 2007
408
You are quite correct in saying that fourier analysis can be used to represent or approximate other functions.

For some functions the approximation is numerically exact, for others less so.
The summary of a Fourier series is always exactly equal to the function it represents. Not approximately but exatcly.
It does not mean to say that because something is numerically exact it is identical, only that our representation allows us to make useful predictions about what happens.
I have no idea what this means. If two numbers are "numerically exact" then they are "identical".
It is not true to say that (5 + 2) is the same as 7.
Yes it is.
For example 5 apples plus 2 pears makes 7 fruit, but they are not interchangeable.
Well, isn't it good that I never said such a thing?
http://en.wikipedia.org/wiki/Straw_man

A straw man argument can be set up in the following ways, by:

-Presenting a misrepresentation of the opponent's position and then refuting it, thus giving the appearance that the opponent's actual position has been refuted.
- Quoting an opponent's words out of context -- i.e., choosing quotations that are not representative of the opponent's actual intentions
I would have thought it was obvious but let me add a disclaimer:

I state that 5 + 2 = 7

Disclaimer: This statement is only directed at those who have the capacity to abstract numbers from actual objects. Those who have not yet attained the capacity to do this should abstain from interpreting it and keep practising the art of abstraction until they have achieved higher zenhood at which moment they can attempt to interpret the above equation without risk of blowing their minds.

Sheesh.

#### GS3

Joined Sep 21, 2007
408
Have a look at http://en.wikipedia.org/wiki/Fourier_series

Notice the "=" signs.

The "=" sign means what comes before and what comes after are *exactly* the same thing. Not similar, not approximately but the same thing.
but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
It is the same thing. Exactly the same thing. The signal is equal to the sum of the Fourier series and the sum of the Fourier series is equal to the signal. Same thing. No difference.

#### studiot

Joined Nov 9, 2007
4,998
Unfortunately ping pong politics tends to obscure the main point, which in this case was

Remember mathematical methods only 'work' if you apply them within their rules of engagement.

.......................................................................................................
With regard to post#8

I believe it's the Chinese who have a saying

"Beware what you wish for, lest it come true"

I would be interested to see GS3 display a fourier representation of the following perfectly valid mathematical function;

y= sin (1/x) { 0 < x <= 1} and y = 0 {-1 < x <= 0} (the <= stands for less than or equal to)

This is a real world function that 'occurs' when you switch on a sine wave oscillator.

Further I maintain that 5 + 2 = 1, but then I am talking about addition, modulo 6.

This is a real world function that 'occurs' if you take a pulse generator and set the pulse length greater than the period.

#### studiot

Joined Nov 9, 2007
4,998
The "=" sign means what comes before and what comes after are *exactly* the same thing. Not similar, not approximately but the same thing.
The equivalence function is not the same as the identity function - if it were there would be no need for mathematicians to distinguish between the two.

As an example the square root of 4 is equal to 2 but not identical to 2, as it is also equal to -2.

#### GS3

Joined Sep 21, 2007
408
Unfortunately ping pong politics tends to obscure the main point, which in this case was

Remember mathematical methods only 'work' if you apply them within their rules of engagement.
Silly me. I thought the main point of this thread was
but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
I would be interested to see GS3 display a fourier representation of the following perfectly valid mathematical function;

y= sin (1/x) { 0 < x <= 1} and y = 0 {-1 < x <= 0} (the <= stands for less than or equal to)
I have no clue as to how that is related to the OP or to anything I have said. You can go ahead and do it yourself if it insterests you. And, while we are asking for things I'd also like a pony.

My answer to the original question, which I remind us was
but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
is that the addition of any number of functions which add up to the original function is, in fact, the same thing as the original function just the same as 5+2 = 7.

Further I maintain that 5 + 2 = 1, but then I am talking about addition, modulo 6. OK. You win.

#### Distort10n

Joined Dec 25, 2006
429
Never argue with a mathmetician.

Context is key people, context.

#### studiot

Joined Nov 9, 2007
4,998
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.

#### GS3

Joined Sep 21, 2007
408
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.
Ah, more straw men. Sorry but I am not going to waste my time on irrelevant limbs.

The question, no matter how much it is being confused, remains:
but the signal itself, coming from a sensor or any other source, isn't inherently made up of individual sinusoids components summed up together....
and my answer is that the sum of any number of signals (not just sinusoid) which add up to the original signal are in fact the same thing as the original signal. If I am receiving a signal on a wire I have no way of knowing and in fact it does not matter if it originated as a single signal with that shape or as a sum of several signals which sum results in the signal I am receiving. They are for all intents and purposes the same thing. That is my point. The rest is just trying to confuse the question and go off on tangents.

When I pick up a certain sound composed of a fundamental frequency and certain harmonics it really makes not much sense to ask whether the sound was produced like that or whether each harmonic was produced separately. It is really the same thing. A vibrating string produces a number of harmonics but you can have another string for which one of those harmonics is its fundamental resonance.

Suppose you have a certain sound from a musical instrument. Now you can synthesize a fundamental and whatever harmonics you like, add them up, send them to a loudspeaker and you have the same sound. The question of whether the sound in one case is a different sound from the other is meaningless. They result in the same waveform and are, in fact, the same sound.

Just like they were obtained differently they can later be split and analyzed differently but they are the same sound.

#### studiot

Joined Nov 9, 2007
4,998
So what do your fourier calculations show for say the phase change at 2f

that the phase difference is unaltered through the filter

or

That the phase difference through the filter is changed by an amount which is the same as measured on the on a phase /frequency plot?

#### Mazaag

Joined Oct 23, 2004
255
and my answer is that the sum of any number of signals (not just sinusoid) which add up to the original signal are in fact the same thing as the original signal. If I am receiving a signal on a wire I have no way of knowing and in fact it does not matter if it originated as a single signal with that shape or as a sum of several signals which sum results in the signal I am receiving. They are for all intents and purposes the same thing.
That all makes sense to me... whether it is made of sinusoids or not the end result is the same.. thats fine..but what I can't get my head round is the fact that the low pass filter (consider a regular RC) is based on the fact that the capactior acts differently at different FREQUENCIES.... true the signal COULD have been made up of seperate sinusoidal components added up, but it could have NOT been made up of them too....but the filter acts on the "assumption" so to speak that the signal IS made up of sinusoidal components with specific frequencies.... "frequency" only makes sense if you're talking about a periodic entity...

If an entity COULD be made up of serveral components, and behaves in a certain way when passed through a filter, why does it guarantee that this entity will always react in the same way to this filter, even if it was created in a different manner.. ?

back to the 5 + 2 = 7 analogy... lets say a black box behaves in a certain way towards this 7 based on the fact that it was made up of 5 and 2 (analogous to periodic sinusoids)...why is it guaranteed that it will always behave the same way as that even if i were to make up this 7 out of a 4 and a 3 (other means of producing final result, analogous to aperiodic) ?

#### GS3

Joined Sep 21, 2007
408
So what do your fourier calculations show for say the phase change at 2f

that the phase difference is unaltered through the filter

or

That the phase difference through the filter is changed by an amount which is the same as measured on the on a phase /frequency plot?
I have not the slightest clue as to the relevance of this question or how it may relate to the OP which is about how a signal originates. Why do you continue to ask these seemingly irrelevant questions? Why don't you make your point if you have one? Or is it that asking vaguely related, obscure questions, makes some kind of point? You continue to ignore what I say and just keep asking these esoteric questions. Why don't you quit this game and make yor point if you are in fact contradicting what I am saying? Or is it just a game of asking questions to leave the others scratching their heads?
How much wood would a woodchuck chuck if Chuck chuck chuck?
Here's what I say: the same signal, however it was produced, will behave the same way when put through the same circuit be it a filter or a meat grinder.

When we are analyzing a signal in the frequency domain we are not looking at something different, we are looking at the same thing from a different angle. It is still the same thing in every respect.

#### GS3

Joined Sep 21, 2007
408
That all makes sense to me... whether it is made of sinusoids or not the end result is the same.. thats fine..but what I can't get my head round is the fact that the low pass filter (consider a regular RC) is based on the fact that the capactior acts differently at different FREQUENCIES.... true the signal COULD have been made up of seperate sinusoidal components added up, but it could have NOT been made up of them too....but the filter acts on the "assumption" so to speak that the signal IS made up of sinusoidal components with specific frequencies.... "frequency" only makes sense if you're talking about a periodic entity...

If an entity COULD be made up of serveral components, and behaves in a certain way when passed through a filter, why does it guarantee that this entity will always react in the same way to this filter, even if it was created in a different manner.. ?
Because, that's the point, that it was created in a different manner does in no way change the fact that the components are there and it is for all purposes the same signal. You are running into a question (problem) based on the assumption that the signals are somehow different. They are not different. They are the same signal in every way. Your problem is based on a faulty assumption. If you accept that they are the same signal the problem disappears.

What is the difference of me giving you 7 or me giving you 5+2? None! If you have to pay me 7 it matters nothing how you obtained the 7. For all purposes it is the same.

Think that sunusoidal analysis is just a convenient tool but you could do other types of analysis and they would prove just as true (although maybe not as useful). You could analyse a waveform as a sum of triangular waveforms or as a sum of sawtooth waveforms but that is not as useful because physical and electrical responses, characteristics and oscillations are of trigonometric nature and trigonometric analysis proves to be the most useful. It's just a point of view of the same phenomenon. You can find other points of view but you tend to pick those that are most useful.

It is not only in electrical signals but also in mechanical and electromechanical analysis that frequency analysis is useful.

#### studiot

Joined Nov 9, 2007
4,998
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?