Fourier Transform Proof

In summary, the homework statement says that if F(p) and G(p) are the Fourier transforms of f(x) and g(x) respectively, then∫f(x)g*(x)dx = ∫F(p)G*(p)dp
  • #1
Seda
71
0

Homework Statement



If F(p) and G(p) are the Fourier transforms of f(x) and g(x) respectively, show that

∫f(x)g*(x)dx = ∫ F(p)G*(p)dp

where * indicates a complex conjugate. (The integrals are from -∞ to ∞)

Homework Equations



F(p) = ∫f(x)exp[2∏ipx]dx
G(p) = ∫g(x)exp[2∏ipx]dx
G*(p) = ∫g(x)exp[-2∏ipx]dx
f(x) = ∫F(p)exp[-2∏ipx]dp
g(x) = ∫G(p)exp[-2∏ipx]dp
g*(x) = ∫G(p)exp[2∏ipx]dp

The Attempt at a Solution



Well this question is kind of weird to me since most of the in class examples have been based on knowing the function and then using different methods of integration to find the transforms, but in this proof it's all arbitrary, obviously.

Well simply subbing definitions in I get:

∫f(x)g*(x)dx = ∫[∫F(P)exp[-2∏ipx]dp∫G(p)exp[2∏ipx]dp]dx

and

∫F(p)G*(p)dp = ∫[∫f(x)exp[2∏ipx]dx∫g(x)exp[-2∏ipx]dx]dp

Now I guess if I can show that these two lines simplify to the same thing I have my proof. However, I am not sure how to simplify this. Maybe I am forgetting some basic property of integrals?

Also I have no idea if this approach is even correct and it would be better to start someplace else.
 
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  • #2
You need to be a bit more careful with the integration variables. You have
\begin{align*}
f(x) &= \int F(p)e^{-2\pi ipx}\,dp \\
g^*(x) &= \int G^*(p')e^{2\pi ip'x}\,dp'.
\end{align*} Remember that F(p) and G(p) are complex. So now you have
\begin{align*}
\int f(x)g^*(x)\,dx &= \int\int F(p)e^{-2\pi ipx}\,dp \int G^*(p')e^{2\pi ip'x}\,dp' \,dx \\
&= \iiint F(p) G^*(p') e^{-2\pi ipx} e^{2\pi ip'x}\,dp\,dp'\,dx
\end{align*} Now you want to identify the Dirac delta function that's in there somewhere.
 
  • #3
Do I have my signs backwards? When taking the Fourier transform on a function do I have a negative exponential? My class notes don't match will other things I've found...
 
  • #4
It depends on what convention your instructor has decided to use. There can also be factors of ##2\pi## that move around depending on the convention you choose.

The Wikipedia entry on the Fourier transform lists the common ones.
 
  • #5
vela said:
It depends on what convention your instructor has decided to use. There can also be factors of ##2\pi## that move around depending on the convention you choose.

The Wikipedia entry on the Fourier transform lists the common ones.

My instructor has the 2∏ in the exponential. Yes I've seen cases where it is infront of the integral as a 1/(2∏). I don't fully understand the (non) difference, just sticking with what I've seen.
 
  • #6
Aha, I believe I have to use

FT of g*(x) = G*(-x)
Fixes my sign issue I believe.
 
  • #7
Ugh, still have 1 sign error tripping me up...

going by my definitions..

replacing only g*(x)

∫f(x)g*(x)dx = ∫f(x)[∫G(p)exp[2∏ipx]dp]dx = ∫f(x)[∫G*(p)exp[-2∏ipx]dp]dx

= ∫G*(p)[∫f(x)exp[-2∏ipx]dx]dp

In that last line if the negative wasn't there I could replace it with F(P) from y definitions and be done...
 

What is the Fourier Transform Proof?

The Fourier Transform Proof is a mathematical technique used in signal processing and analysis to decompose a function into its constituent frequencies. It converts a signal from its original domain (often time or space) to a representation in the frequency domain.

What is the purpose of the Fourier Transform Proof?

The purpose of the Fourier Transform Proof is to simplify complex signals and make it easier to analyze them in terms of their frequency components. This allows for a better understanding of the underlying patterns and structures within the signal.

How does the Fourier Transform Proof work?

The Fourier Transform Proof uses a mathematical formula to decompose a signal into its constituent frequencies. The formula involves taking the integral of the signal multiplied by a complex exponential function. This process is repeated for all frequencies of interest, resulting in a representation of the signal in the frequency domain.

What are the applications of the Fourier Transform Proof?

The Fourier Transform Proof has many applications in various fields such as audio and image processing, telecommunications, and physics. It is used to analyze and filter signals, remove noise from data, and compress data without losing important information.

Are there any limitations to the Fourier Transform Proof?

While the Fourier Transform Proof is a powerful tool, it has some limitations. It assumes that the signal is periodic, and it may not accurately represent signals with discontinuities or sharp edges. Additionally, it requires a large amount of computational power for complex signals, and the interpretation of the frequency components may be difficult in some cases.

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