# Rom fourier transfom to fourier series

In summary, the student is seeking help in finding a relationship between the coefficients of the Fourier transform and those for the Fourier series. They are looking for a formula to find the Fourier series coefficients from the Fourier transform coefficients. The teacher has assigned this task to be done independently. The desired formula is to be in the form of an and bn, which correspond to cosine and sine coefficients, respectively, and Fn represents the Fourier transform coefficients. The student has not been able to derive the formula on their own and has researched online and in books without success. They are seeking help from others.
HI,
I really need to find a relationship between the coefficients of the Fourier transform coefficients and those for Fourier series, especially how to find Fourier series coefficients from the Fourier transform coefficients.

The formula should be like this:
an= X1*(Fn)+X2*(F-n)
bn= X3*(Fn)+X4*(F-n)

Where an and bn are the Fourier series coefficients for cosine and sine respectively. Fn are the Fourier transform coefficients.
Finally X1,X2,X3,X4 are some coefficients( I don't know if they are constant or not)

I could not derive it by myself and did some research with books and online but I have not found something good.

Thank you
B

What do you MEAN by "coefficients" of a Fourier transform? A Fourier transform of a function is a function, not a series or polynomial and does not have "coefficients".

Also I don't see how you can associate a specific Fourier series with a specific Fourier Transform. A Fourier series is always calculated for a function over a finite interval (and periodic with that interval as period) while a Fourier transform is of a function defined for all x (and not, in general, periodic).

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

The Fourier transform is a mathematical operation that decomposes a signal into its constituent frequencies, while the Fourier series is a way to represent a periodic signal as a sum of sinusoidal functions.

## 2. How is the Fourier transform related to the Fourier series?

The Fourier transform is a continuous version of the Fourier series, where the frequency spectrum is continuous rather than discrete. The Fourier series can be obtained from the Fourier transform by sampling the continuous spectrum at regular intervals.

## 3. What is the significance of the Fourier transform and series in signal processing?

The Fourier transform and series are fundamental tools in signal processing, used for analyzing and manipulating signals in both time and frequency domains. They are essential in a wide range of applications, such as image and audio processing, communication systems, and data compression.

## 4. What are the main properties of the Fourier transform and series?

The Fourier transform and series have many important properties, including linearity, time and frequency shifting, convolution, and Parseval's theorem. These properties make them powerful and versatile tools for signal processing and analysis.

## 5. Are there any limitations or drawbacks of the Fourier transform and series?

While the Fourier transform and series have many useful properties, they also have some limitations. For example, they assume that the signal is periodic or has infinite duration, which may not always be the case in real-world applications. Additionally, the Fourier transform and series may not be suitable for non-linear or non-stationary signals.

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