# Complex Fourier transform (represented by Σ)

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• arcTomato
In summary, the speaker is struggling to understand how to derive equation (2.3a) in a paper that describes Fourier coefficients using summations. They question the use of summations instead of integrals and ask for clarification on how to derive the equation in summation form. The responder suggests approximating the integrals with discrete summations and recommends reading on for a better understanding. The speaker thanks the responder for their help and expresses gratitude.
arcTomato
TL;DR Summary
I can't understand how to derive fourier transform(described by Σ).
Dear all.

I can't understand how to derive Eq.(2.3a).

Fourier coefficients, ##A_j## and ##B_j## are described by summation in this paper as (2.2).　I think this is weird.
Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms".
and I understand Fourier coefficients as integral.( I think integral version is more general than ##Σ## version.) so I'm confused.

I would like to ask you how to derive (2.3a) in this paper( in ##Σ## version).
(this is the paper I readFourier_techniques.pdf)

Assume that you are integrating from ##-T## to ##+T##, and consider the evenness or oddness of ##\cos^2\omega_jt##, ##\sin^2\omega_jt##, and ##\sin\omega_jt\cos\omega_jt##. Then take the small step from integration to discrete summation.

I appreciate for you, @tnich.

Can I use ##t_k=kT/N,ω_j=2πj/T##in the process to derive (2.2)??
This paper does not say anything about ##ω_j## in section 2.1.
So you say like this??Can I approximate to Σ from integral?

Yes, I am saying that you can approximate the integrals with discrete summations. Try looking at the definition of an integral as a limit of discrete summations to see how to do the approximation.
I am having trouble reading your derivation, so I don't know if you got it right. I see that you kept ##k## and ##t_k## in the integral. I think you need to integrate over ##t## from ##-T## to ##+T##.
But, that said, on page 2 the author says that this section is a general overview and that he will go into the details later. If you understand the general idea, I would be tempted to just read on.

Ok, I think I got it

tnich

## What is a Complex Fourier transform?

A Complex Fourier transform, represented by Σ, is a mathematical operation that decomposes a complex-valued function into its constituent frequencies. It is an extension of the Fourier transform, which is used for real-valued functions.

## What is the difference between a Complex Fourier transform and a Fourier transform?

The main difference between a Complex Fourier transform and a Fourier transform is that the Complex Fourier transform can handle complex-valued functions, while the Fourier transform is limited to real-valued functions. The Complex Fourier transform also includes information about the phase and amplitude of each frequency component.

## What is the significance of the symbol Σ in the Complex Fourier transform?

The symbol Σ in the Complex Fourier transform represents the summation of the individual frequency components. It is used to denote the addition of all the complex exponential functions that make up the transformed function.

## What are some practical applications of the Complex Fourier transform?

The Complex Fourier transform has many practical applications in various fields such as signal processing, image processing, and quantum mechanics. It is used for analyzing and manipulating signals and images, as well as solving differential equations and studying quantum systems.

## What are the limitations of the Complex Fourier transform?

The Complex Fourier transform has some limitations, such as the assumption that the function being transformed is periodic and the need for the function to be integrable. It also has difficulty handling functions with discontinuities or sharp edges. In addition, the Complex Fourier transform can become computationally expensive for large datasets.

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