Fourier Series Transform Proof Help

In summary, the conversation discusses the steps between two equations involving a periodic function and a Fourier series. The first step states that the function is periodic with period T, while the second step provides the general expression for a Fourier series with period 2π/ω. The missing step may involve converting the integral to a sum, with ω being equal to 2π/T.
  • #1
evotunedscc
7
0
Can someone fill in the blank between these two steps? I can't find Fourier series proof anywhere and my professor just left it out.

(1) y(t+nT)=y(t)

(2) y(t)=[tex]A_{0}[/tex] + [tex]\Sigma^{\infty}_{n=1}[/tex][[tex]A_{n}[/tex]cos(n[tex]\omega[/tex]t) + [tex]B_{n}[/tex]sin(n[tex]\omega[/tex]t)]

(The omega is going crazy on me... it's not supposed to be superscripted, just multiplied by n and t)
 
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  • #2
evotunedscc said:
Can someone fill in the blank between these two steps? I can't find Fourier series proof anywhere and my professor just left it out.

(1) y(t+nT)=y(t)

(2) y(t)=[tex]A_{0}[/tex] + [tex]\Sigma^{\infty}_{n=1}[/tex][[tex]A_{n}[/tex]cos(n[tex]\omega[/tex]t) + [tex]B_{n}[/tex]sin(n[tex]\omega[/tex]t)]

(The omega is going crazy on me... it's not supposed to be superscripted, just multiplied by n and t)

What do you mean by "steps between them"? The first just says y is periodic with period T and the second is the general expression of a Fourier series of a function periodic with period [itex]2\pi/\omega[/itex]- there is no mention of "T".

As for the LaTex, I would recommend putting the entire thing in [ t e x] not just individual parts:

[tex]y(t)=A_{0}+ \Sigma^{\infty}_{n=1}[A_{n}cos(n\omega t) + B_{n}sin(n\omega t)][/tex]

It looks better and is easier to type!
 
  • #3
I would say that a general "Fourier expansion" is actually an integral. What (1) implies is that the modes are discrete and thus the integral becomes a sum, and therefore [itex]\omega=2 \pi/T [/itex], as Halls mentioned. Maybe this is the missing step you mean?
 
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  • #4
have u find the gap between those two statements[evotunedscc]?
 
  • #5


I understand the importance of having a complete and accurate proof for any mathematical concept. In this case, the missing step between (1) and (2) in the Fourier Series Transform proof is the application of the Euler's formula. This formula states that e^ix = cos(x) + isin(x), where i is the imaginary unit.

Using this formula, we can rewrite the trigonometric functions in (2) as complex exponential functions. This leads to the final form of the Fourier series:

y(t) = A0 + Σ∞n=1 [An e^inωt + Bn e^-inωt]

I hope this helps in completing the proof and understanding the concept better. Additionally, there are many resources available online and in textbooks that provide a detailed explanation and proof of the Fourier series transform. I encourage you to explore these resources for a more comprehensive understanding.
 

Related to Fourier Series Transform Proof Help

1. What is the Fourier Series Transform?

The Fourier Series Transform is a mathematical tool used to decompose a periodic function into a series of sinusoidal functions. It represents a function as a sum of infinite sine and cosine waves of different frequencies, amplitudes, and phases.

2. How is the Fourier Series Transform used?

The Fourier Series Transform is commonly used in signal processing, image processing, and other fields of science and engineering. It is used to analyze and manipulate periodic signals and to solve differential equations.

3. What is the relationship between the Fourier Series Transform and the Fourier Transform?

The Fourier Series Transform is a special case of the Fourier Transform, which is used to analyze non-periodic signals. The Fourier Transform can be seen as the generalization of the Fourier Series Transform for non-periodic functions.

4. What is the proof for the Fourier Series Transform?

The proof for the Fourier Series Transform involves using the orthogonality of sine and cosine functions to show that any periodic function can be represented as a sum of these functions. This is done by finding the coefficients of the sine and cosine terms using integration.

5. Why is the Fourier Series Transform important?

The Fourier Series Transform is important because it allows us to analyze and manipulate periodic signals using a simple and elegant mathematical tool. It has a wide range of applications in various fields of science and engineering, making it an essential concept for scientists and engineers to understand.

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