# Thread: Fourier Series : analysis or synthesis equations ?

1. ## Fourier Series : analysis or synthesis equations ?

Apologies for the textual format (no idea how to write these equations properly in a forum) but I hope those who are familiar with Fourier Series will recognise these equations.

Q: In Fourier Series, our teacher has told us that it should be obvious to us whether to use the synthesis :

x(t) = inf.sum of ak*e^(jkw0t)

or the analysis equation :

ak = 1/T0 * integral of x(t)*e^(-jkw0t) dt

But I'm afraid I haven't got the foggiest myself when to use one and not the other and it is not explained anywhere in the notes we have been given or in numerous textbooks I have checked.

Wiki suggests something about one is to break up terms and the other is the restructure but that is not the context my teacher in the engineering course is putting them in and I suspect it is wiki-garbage.

Please can someone help before I start banging my head against the wall with intent of permanent coma.

2. Well, if you are given x(t) and you are asked to find ak, then you use the first one, otherwise use the second equation.

3. Originally Posted by Sling
Apologies for the textual format (no idea how to write these equations properly in a forum) but I hope those who are familiar with Fourier Series will recognise these equations.

Q: In Fourier Series, our teacher has told us that it should be obvious to us whether to use the synthesis :

x(t) = inf.sum of ak*e^(jkw0t)

or the analysis equation :

ak = 1/T0 * integral of x(t)*e^(-jkw0t) dt

But I'm afraid I haven't got the foggiest myself when to use one and not the other and it is not explained anywhere in the notes we have been given or in numerous textbooks I have checked.

Wiki suggests something about one is to break up terms and the other is the restructure but that is not the context my teacher in the engineering course is putting them in and I suspect it is wiki-garbage.

Please can someone help before I start banging my head against the wall with intent of permanent coma.
When given a function of time, the analysis step decomposes the function to give the coeficients of the frequency components that make up the signal.

The synthesis process reconstructs the time function from the coefficients of the frequency decomposition.

The situation with functions of position is analogous, a function of a spacial variable is decomposed by the analysis process to give the coeficients of the spacial frequencies, ...

CB

4. This is not quite the approach that my teacher is using. We are given a signal and are expected to know ourselves how to calculate the Fourier Series.

Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?

I've attached one example to illustrate what we're expected to be able to do.

5. Originally Posted by Sling
This is not quite the approach that my teacher is using. We are given a signal and are expected to know ourselves how to calculate the Fourier Series.

Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?

I've attached one example to illustrate what we're expected to be able to do.
What is the problem with that, you are asked to find the Fourier series of a given waveform. That is to write it in the form:

$\displaystyle x(t)=\sum_{k=-\infty}^{\infty} a_k e^{jk\omega_0t}$

where you have evaluated the $\displaystyle a_k$ 's using:

$\displaystyle a_k=\frac{1}{T_0}\int_{T_0} x(t)e^{-jk\omega_0t}\;dt$

where here $\displaystyle T_0=6$ so the angular frequency of the fundamental is $\displaystyle \omega_0=2\pi/6$, and so the integral becomes the integral over an interval of length $\displaystyle 6$, lets say:

$\displaystyle a_k=\frac{1}{6}\int_{0}^6 x(t)e^{-jk\pi t/3}\;dt$

CB

6. Originally Posted by Sling
if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?
So, is this correct?

If so then the two equations seem to be two methods of doing the same thing, rather than converting the equation from a focus on time to frequency or vice versa.

(I didn't do the above example btw, it was provided... And the teacher started with giving a definition of x(t) but that's not what he seemed to end up calculating although that could be just me being fussy - if I give a definition to aim for, I then calculate that definition and present it at the end of the calculations.)

7. Originally Posted by Sling
Would it be correct to say that if a signal is symmetrical then the analysis equation is to be used, yet if it is not symmetrical the synthesis one should be?
Sorry I did not address this comment. No it would not be correct.

You are trying to represent a time domain signal as the sum of sinusoids (in this case hidden withing the complex exponentials). The analysis bit tells you what the coeficients of each frequency component are and the synthesis expression tell you how to represent your signal as a sum of the frequency components.

CB

8. Originally Posted by CaptainBlack
You are trying to represent a time domain signal as the sum of sinusoids (in this case hidden withing the complex exponentials). The analysis bit tells you what the coeficients of each frequency component are and the synthesis expression tell you how to represent your signal as a sum of the frequency components.

That makes a lot more sense, in fact that's what I was suspecting but the way my teacher was talking it sounded like we had to choose between one or the other equation which just got me confused (the way it's worded in wiki doesn't help much either tbh).

In fact both equations would hold true for any given signal.

I'm guessing that isolating the components will make the Fourier Transform possible which seems to be the really useful part. (We haven't got to that yet though).

The joys of being a student , kk thanks for taking the time to clear that up (if I've got it right now anyway).

9. Originally Posted by Sling
That makes a lot more sense, in fact that's what I was suspecting but the way my teacher was talking it sounded like we had to choose between one or the other equation which just got me confused (the way it's worded in wiki doesn't help much either tbh).

In fact both equations would hold true for any given signal.

I'm guessing that isolating the components will make the Fourier Transform possible which seems to be the really useful part. (We haven't got to that yet though).

The joys of being a student , kk thanks for taking the time to clear that up (if I've got it right now anyway).
Yes, you have it now.

CB