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Nicolas
2006-Feb-23, 10:52 PM
OK so I've chosen a Master which includes all sorts of courses carrying fancy names, but it all boils down to stochastics. ugh. ANyway, I'm trying to get "to the bottom" of a course reader here, finding proofs for all formulas. There's just one page where I'm stuck with my proofs. I understand everything, but I can't prove it.

We start from the a/b definition of discrete Fourrier series:

http://upload.talk2.nl/files/168703temp.GIF

Some time after that, we're going to the complex form of this Fourrier Transform. I link to the 2 relevant sheets.

http://upload.talk2.nl/files/368140temp.GIF (http://upload.talk2.nl/files/368140temp.GIF)

http://upload.talk2.nl/files/419750temp.GIF (http://upload.talk2.nl/files/419750temp.GIF)


OK so in sheet 1:
*how do you prove that indeed that "c" notation is the same as the a/b notation? I'm stuck with an imaginary part that shouldn't be there...
*related to that, how do you "invent" the "c" complex form in the first place?

On sheet 2:
*you guessed it: prove the formula in the rectangle, below "it is easy to prove that..."
*How can I prove that 3.9 and 3.12 are each other's inverse?


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This is not homework, I just want to understand the basics to the bottom. I've searched on these 4 questions for a long time, but I'm unable to find them. It's strange, asit doesn't seem difficult compared to other proofs that I did find...

Any help is appreciated! I don't need explanations as to the Fourrier series itself, only how to (mathematically proof) the equality of the notations and the validity of the claims.

Edit: "Fourier"...

TheBlackCat
2006-Feb-23, 11:52 PM
The important thing to remember is that cos(x)=(e^jx+e^-jx)/2 and sin(x)=(e^kx-e^-jx)/(2j). Plug those in to the sin and cos in your formula,, expand, and factor out e^jkwt and e^-jkwt. That leaves you with:

1/2(ak-jbk)*e^jkwt+1/2(ak+jbk)*e^-jkwt

Now since your series is from 0 to n-1, you can split it into two series, each from 0 to n-1:

1/2(ak-jbk)*e^jkwt and 1/2(ak+jbk)*e^-jkwt

However, the second series has a negative k term, so by using change of variables you can make that series go from 0 to -(n-1)=-n+1:

1/2(ak+jbk)*e^jkwt

That leaves you with two series, one from 0 to n-1 and the other from 0 to -n+1. The only place these equations overlap is at zero. At this point the e terms reduce to 1 and the jbk terms cancel, so you are left with just ak. For all other points the e terms are the same and you use 1/2(ak-jbk) for k>0 and 1/2(ak+jbk) for k<0. This allows you to combine the two series into a single series. And that gives the answer they have.

To solve the sheet two thing you probably need the definitions of the ak and bk terms. Combine them in the way listed in the first sheet.

Proving they are the inverse just involves solving 3.9 by plugging 3.12 into the ck term and working it out. Don't forget when you have a series and an integral you can switch which is solved first.

papageno
2006-Feb-23, 11:54 PM
OK so in sheet 1:
*how do you prove that indeed that "c" notation is the same as the a/b notation? I'm stuck with an imaginary part that shouldn't be there...

Have you considered that ejwt is also a complex number?
Maybe if you write explicitly a couple of examples it will be easier.




*related to that, how do you "invent" the "c" complex form in the first place?

It boils down to ex = (sin(x) + j cos(x))/2, which can be seen using the series defining the functions.



On sheet 2:
*you guessed it: prove the formula in the rectangle, below "it is easy to prove that..."

Have you tried to insert (3.12) into (3.9)?



*How can I prove that 3.9 and 3.12 are each other's inverse?

(3.9) writes x(t) in terms of ck;
(3.12) writes ck in terms of x(t).

Have they explained Fourier transformation in terms of vectorial spaces?

Nicolas
2006-Feb-24, 12:15 AM
TheBlackCat: thanks, that solved sheet n&#176; 17!!!

Over to sheet 18:
Proving that they are each other's inverse by plugging 3.12 into the ck of 3.9:
I think I tried that, but I didn't really succeed. I'll try another time when I find the time. I "see" that they are each other's inverse, but I can't show it with correct maths...

That leaves me with the "easy" proof. (I hate it when they say that :)). I have the definitions of a and b, but I did not manage to prove it.

Again, thanks THeBlackCat and Papageno for helping me out this far!

Disinfo Agent
2006-Feb-24, 12:29 AM
It boils down to ex = (sin(x) + j cos(x))/2That isn't right...

Surely, you meant ejx = sin(x) + j cos(x).

peter eldergill
2006-Feb-24, 01:46 AM
Is there a reason you're using "j" instead of "i"?

The only time I've seen "j" is for quaternions..

Pete

TheBlackCat
2006-Feb-24, 04:07 AM
j is used in electrical circuits because i is already used for current. Lots of engineering-oriented classes, especially digital and analog signalling processing, (which both have large circuit component), also use j as a result. It just a different convention, another mathematicians vs. scientists/engineers disagreement.

Nicolas
2006-Feb-24, 09:54 AM
Matlab also uses j, as i is usually used as a running integer. You get problems when you define a variable j yourself though :).

Anyway, I did not manage to "break" sheet 18 yet...but I haven't given up yet!

papageno
2006-Feb-24, 02:11 PM
That isn't right...

Surely, you meant ejx = sin(x) + j cos(x).
:doh:
Yes, I meant that...
It was late... :shifty:




Anyway, I did not manage to "break" sheet 18 yet...but I haven't given up yet!

Don't forget when you have a series and an integral you can switch which is solved first.
And in the integral x(t) can be approximated with its average between t0 and t0 + T, if T is small enough (there is a theorem in calculus, somewhere).

TheBlackCat
2006-Feb-24, 06:01 PM
Matlab also uses j, as i is usually used as a running integer. You get problems when you define a variable j yourself though :).

Matlab supports use of both i and j for complex numbers, and allows both to be overwritten as iteration variables (I use them both for iterations pretty routinely).

Nicolas
2006-Feb-27, 10:25 AM
:doh:
Yes, I meant that...
It was late... :shifty:


And in the integral x(t) can be approximated with its average between t0 and t0 + T, if T is small enough (there is a theorem in calculus, somewhere).

Wouldn't that be average*T? Anyway I still haven't finished sheet 18 yet, but I did work up to sheet 64 with only 1 small extra problem at sheet 60 :)

Wolverine
2006-Feb-27, 10:43 AM
Moved from BABBling to General Science.

Nicolas
2006-Feb-27, 10:44 AM
No problem :)

papageno
2006-Feb-27, 11:11 AM
And in the integral x(t) can be approximated with its average between t0 and t0 + T, if T is small enough (there is a theorem in calculus, somewhere).
Wouldn't that be average*T?

Indeed.
That's why in the definition of ck, the integral is divided by T.