# Fourier transform and derivation

• arcTomato
In summary, the derivation of Eq.(2.2) is not at all straightforward and more information is needed to understand it.
arcTomato
Homework Statement: I don't know how can I derivation Eq.(2.2)
Homework Equations: Fourier coefficients

Homework Statement: I don't know how can I derivation Eq.(2.2)
Homework Equations: Fourier coefficients

Dear all.
I don't know how can I derivation Eq.(2.2).
Where Σk is come from??

Moved from a HW section.
The derivation, which is described as capable of being "straightforwardly computed," doesn't seem at all obvious to me.

How is ##t_k## defined? I feel that more information is needed than the image you posted.

Mark44 said:
Moved from a HW section.
The derivation, which is described as capable of being "straightforwardly computed," doesn't seem at all obvious to me.
On that side note, I think words such as ”straightforwardly” or ”simply” simply do not belong in textbooks.

Thank you @Orodruin and I appreciate for your kindness.
This paper doesn’t refer to what ##t_k## is.
That’s is the point I can’t understand.

Can you give a reference to the paper?

##x_k=x(t_k) = \frac{1}{N}\sum_j A_j \cos \omega_j t_k + B_j \sin \omega_j t_k## , plug the solution:
##A_j = \sum_k x_k \cos \omega_j t_k , \ \ \ B_j = \sum_k x_k \sin \omega_j t_k##, and you get:
##x_k=\frac{1}{N} \sum_j \sum_m x_m \cos \omega_j t_m \cos_j t_k + x_m \sin \omega_j t_m \sin \omega_j t_k ##, if ##m\ne k##, then the term on the RHS will vanish, since on the the LHS there are no terms of this form, so the sum over ##m## becomes ##x_k(\cos^2 \omega_j t_k + \sin^2 \omega_j t_k)=x_k##, the sum over ##j## gives you the ##N## factor that cancels with ##N## in the denominator.
That's all there is to it.

arcTomato
I got it! This is the easy-to-understand explanation.
I appreciate for you all.

## 1. What is Fourier transform and why is it important?

The Fourier transform is a mathematical tool used to decompose a complex signal into its individual frequency components. It is important because it allows us to analyze and understand the behavior of a signal in the frequency domain, which can provide valuable insights in various fields such as signal processing, image processing, and physics.

## 2. How is the Fourier transform derived?

The Fourier transform is derived from the Fourier series, which is a representation of a periodic function as a sum of sine and cosine functions. By extending this concept to non-periodic functions, we can obtain the Fourier transform by taking the limit of the Fourier series as the period approaches infinity.

## 3. What is the difference between Fourier transform and Fourier series?

The Fourier transform is used for non-periodic signals, while the Fourier series is used for periodic signals. The Fourier transform gives a continuous representation of a signal in the frequency domain, while the Fourier series gives a discrete representation. Additionally, the Fourier transform is defined for both real and complex signals, while the Fourier series is only defined for real signals.

## 4. How is the Fourier transform used in image processing?

In image processing, the Fourier transform is used to analyze the frequency content of an image. This allows us to apply filters and other operations in the frequency domain, which can enhance certain features of the image. The inverse Fourier transform is then used to convert the processed image back to the spatial domain.

## 5. Can the Fourier transform be applied to any type of signal?

Yes, the Fourier transform can be applied to any signal, as long as it is integrable. This means that the signal must have a finite energy and can be represented by a mathematical function. However, the Fourier transform may not always be the most appropriate tool for analyzing certain types of signals, and other methods such as wavelet transforms may be more suitable.

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