Show that ∫ dx |f(x)|^2 = ∑ |Cn|^2

[Poirot]

Homework Statement

This is a question regarding Fourier series.
∫ dx |f(x)|^2 = ∑ |Cn|² (note the integral is between -π and π, and the sum is from n= -∞ to ∞)

Homework Equations

Complex Fourier series: f(x) = ∑Cn e^inx (again between n = -∞ and ∞)

The Attempt at a Solution

So I figured the complex conjugate of the Fourier transform would be:
f^*(x)= ∑ C^*n e^-inx

so |f(x)|² = ∑ |Cn|² (as the exponentials cancel)

But I don't understand how the integral comes into things? I think I'm just not overly good at manipulating summations etc, so any help would be greatly appreciated!

Thanks.

 

[ShayanJ]

$|f(x)|^2=f(x) f^*(x)=[\sum_n C_n e^{-inx} ][\sum_m C^*_m e^{imx}]=\sum_n \sum_m C_n C^*_m e^{i(m-n)x}$
Now integrate the above equation!

 

[Poirot]

$|f(x)|^2=f(x) f^*(x)=[\sum_n C_n e^{-inx} ][\sum_m C^*_m e^{imx}]=\sum_n \sum_m C_n C^*_m e^{i(m-n)x}$
Now integrate the above equation!

Hi there, thanks for the help, I see what I did now, forgot to use another dummy variable for the summation!

I ran through the integral, and found that for the condition where n=m (what we want), wouldn't the exponential go to e⁰= 1 so the integral between -π and π be 2pi?

 

[Charles Link]

Hi there, thanks for the help, I see what I did now, forgot to use another dummy variable for the summation!

I ran through the integral, and found that for the condition where n=m (what we want), wouldn't the exponential go to e⁰= 1 so the integral between -π and π be 2pi?

What you are proving is Parseval's theorem for the discrete case. Depending upon how you define the Fourier transform, (with a factor of 1, or $1/\sqrt{2\pi}$, or $1/2\pi$), you will pick up a different factor on the right side of your equation. When the function f(x) becomes a voltage as a function of time, it basically says that the energy that is found from an integral of V^2 ,(without an R, Power P=V^2/R), over time is the same as that of adding up the energy components in the frequency spectrum.

 

[Charles Link]

Addition to the above (post #4): The $1$,$1/\sqrt{2\pi}$, or $1/(2\pi)$ factor is for the continuous case. For the discrete case, there is no such factor, and I think you do need a $1/(2\pi)$ factor on the right side of the equation.

 

[ShayanJ]

I ran through the integral, and found that for the condition where n=m (what we want), wouldn't the exponential go to e0= 1 so the integral between -π and π be 2pi?

Its not that you set n=m! All the terms where n and m differ, actually integrate to zero and for the terms where n=m, you get 2π.

 

[Poirot]

Thanks for all the replies,

But what I am trying to prove is that ∫ dx |f(x)|^2 = ∑ |Cn|^2, which i believe means only the case where m=n matters? And that's where I can't seem to see how to get rid of this extra factor of 2pi? I should also say that a lot of this deep math chat is a little out of my depth.

Thanks again!

 

[ShayanJ]

Thanks for all the replies,

But what I am trying to prove is that ∫ dx |f(x)|^2 = ∑ |Cn|^2, which i believe means only the case where m=n matters? And that's where I can't seem to see how to get rid of this extra factor of 2pi? I should also say that a lot of this deep math chat is a little out of my depth.

Thanks again!

Do you know how to solve the integral $\int_{-\pi}^\pi e^{i(m-n)x} dx$? Are you familiar with limits and L'Hôpital's rule?

That 2π depends on how you define the Fourier series. With your definition, its correct to have it. But you can always normalize your Fourier series by defining it with a $\frac{1}{\sqrt{2\pi}}$ in front of the summation.

 

[Poirot]

Um, I would integrate how you would normally integrate an exponential giving the same function but over i(m-n), but I suppose that doesn't give the nice answer or 2πδ_m,n. And as for L'hopital, from what I remember the Lim f/g = f'/g' or something like that.

We usually use the definitions of Fourier transform with 1/root(2π), but I still can't see where that comes in this proof

 

[ShayanJ]

Um, I would integrate how you would normally integrate an exponential giving the same function but over i(m-n), but I suppose that doesn't give the nice answer or 2πδm,n. And as for L'hopital, from what I remember the Lim f/g = f'/g' or something like that.

So you know that the solution to the integral is $\frac{e^{{\large i(m-n)x}}}{i(m-n)} |_{-\pi}^{\pi}$. Expand this and see what you can learn about the case where $n\neq m$ and the limit $m \to n \Rightarrow (m-n)\to 0 \overset{s=m-n}{\Rightarrow} s\to 0$.

We usually use the definitions of Fourier transform with 1/root(2π), but I still can't see where that comes in this proof

Well...you started with a definition of the Fourier series without $\frac{1}{\sqrt{2\pi}}$!

 

[pasmith]

Homework Statement

This is a question regarding Fourier series.
∫ dx |f(x)|^2 = ∑ |Cn|² (note the integral is between -π and π, and the sum is from n= -∞ to ∞)

Are you sure the statement is not $$\frac{1}{2\pi} \int_{-\pi}^{\pi} |f(x)|^2\,dx = \sum_{n=-\infty}^\infty |c_n|^2$$ as appears on the wikipedia page for https://en.wikipedia.org/wiki/Parseval's_identity]Parseval's[/PLAIN] identity?

 

[Charles Link]

I agree with post #11. My previous statement in post #5 was almost correct. It needs the $1/(2\pi)$ on the left side of the equation.

 

[Poirot]

I'm sure, my lecturer could have made a mistake when writing the question i guess. But yeah thanks anyway guys!