# Fourier Series

#### Mazaag

Joined Oct 23, 2004
255
Hey Guys..

I was wondering if someone can give me a hand finding the Fourier Series Coefficients of the function f(t) = Acos(wt) .

I tried using the following definition

Xn = 1/T integral ( f(t) e^-jwnt )

I converted the cos(wt) to its exponential form, then multiplied and combined and integrated.. however it sorta kept going on and on and doesn't seem to end :S

does anyone have a link of the explanation on how to solve this.. ?

thanks guys

#### Dave

Joined Nov 17, 2003
6,970
You need to perform your integral over the time period T:

Say T = π

For f(t) = ACos(wt)

Your even coefficients are given by:

a(n) = 1/π [f(t)Cos(nt)] dt For -π < t < π

Your odd coefficients are given by:

b(n) = 1/π [f(t)Sin(nt)] dt For -π < t < π

Substitute f(t) into the above equation and perform one integral using a suitable method (Parts).

Because you are performing the integral over the period t, you should be able to derive a(n) and b(n) in terms of n. As a pointer, the Fourier Series of ACos(wt) will have and infinite number of even coefficents, a(n), and b(n) = 0. This should be clear because ACos(wt) is an even function.

Clearer?

Dave

#### Mazaag

Joined Oct 23, 2004
255
Okay..

So i got the fourier series of this Cosine wave. Its basically a 2 diracs at w0 and -w0 (fundamental frequency).

now my question is this: what does this tell me ? like... does the fourier series of a signal give me the coefficients Xn of which I can construct that signal ? or am I getting things mixed up... ? If so , then what does the fouier series and fourier transform represent... ?

Thanks guys

#### Dave

Joined Nov 17, 2003
6,970
The Fourier series gives you the frequency representation of a time-domain periodic signal. It decomposes your signal into a series of weighted sinusoidal harmonics that consistutes a mapping of the original signal. When in the frequency domain we can perform signal manipulation, enhancement, filtering etc. You should also be aware of the Discrete Fourier Transform (and FFT) when looking into Fourier analysis in the field of electroninc engineering.

Dave

#### Mazaag

Joined Oct 23, 2004
255
okay so in the case of the cosine function, when I took the transform I got 2 dirac deltas. (Wo and -Wo) what does that mean ? like what do these impulses tell me about the cosine wave.. and what does the y axis of the frequency spectrum represent..

#### Dave

Joined Nov 17, 2003
6,970
You get 2 Dirac deltas which are the real components at frequency Wo and -Wo, and no imaginary components. This tells you that you have a cosine function (because you only have even components) at a single frequency of Wo, and amplitude 2-times the height of the impulse at Wo (to take account of the fact you have an impulse at Wo and -Wo).

Dave

#### alitex

Joined Mar 5, 2007
139
Hey Guys..

I was wondering if someone can give me a hand finding the Fourier Series Coefficients of the function f(t) = Acos(wt) .

I tried using the following definition

Xn = 1/T integral ( f(t) e^-jwnt )

I converted the cos(wt) to its exponential form, then multiplied and combined and integrated.. however it sorta kept going on and on and doesn't seem to end :S

does anyone have a link of the explanation on how to solve this.. ?

thanks guys
this is pdf file about Fourier Series

#### Attachments

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#### Dave

Joined Nov 17, 2003
6,970
this is pdf file about Fourier Series
Its an interesting article, would you happen to have the answers to the problems?

Dave

#### DrNick

Joined Dec 13, 2006
110
if you are finding the exponential form of the fourier series they would be

C_1 = 1/2
C_-1 = 1/2

so your series would be (in exponential form)

f(t) = A [1/2 exp(-jwt) + 1/2 exp(jwt)]

or in cosine form

f(t) = A cos(wt)

or in sine form

f(t) = A sin(wt -pi/2)

in a sense that is kindof a redundent question. Acos(wt) is ALREADY a fourier series representation of f(t).

#### alitex

Joined Mar 5, 2007
139
Its an interesting article, would you happen to have the answers to the problems?

Dave
yes dave if i can solve problem why i don't put solving?

#### Dave

Joined Nov 17, 2003
6,970
yes dave if i can solve problem why i don't put solving?
What I mean is, do you have the solution set for the problems in the pdf article you attached above?

Dave

#### alitex

Joined Mar 5, 2007
139
What I mean is, do you have the solution set for the problems in the pdf article you attached above?

Dave
what do u mean dave?
is the my pdf file useless
actually i have alot of informations about many scopes

#### Dave

Joined Nov 17, 2003
6,970
what do u mean dave?
is the my pdf file useless
actually i have alot of informations about many scopes
No the pdf is useful. However towards the end there are some questions (also referred to as problems), do you have the answers to these questions? It doesn't matter if you haven't I was only wondering, if you have please share them so others can learn from the attachments.

Dave