How are Fourier Series and Fourier transform are related?

In summary, Fourier Series and Fourier Transform are two mathematical methods used to analyze periodic and non-periodic functions, respectively. They are related in that Fourier Transform is the continuous version of Fourier Series, but they cannot be used interchangeably as they have different applications and equations. Both methods can only be applied to functions that meet specific criteria and work best for smooth and continuous functions.
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ramdas
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We know that Fourier series is used for periodic sinusoidal signals and Fourier transform is used for aperiodic sinusoidal signals.

But i want to know that
  1. Is there any relation present between Fourier Series and Fourier transform ?
  2. Also,Can we derive mathematical formula of Fourier transform for a aperiodic signal using Fourier series formula or vice versa ?
 
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1. What is the difference between Fourier Series and Fourier Transform?

Fourier Series and Fourier Transform are both mathematical methods used to analyze periodic functions. However, Fourier Series deals with periodic functions that are represented as a sum of sine and cosine waves, while Fourier Transform deals with non-periodic functions that are represented as a continuous spectrum of frequencies.

2. How are Fourier Series and Fourier Transform related?

Fourier Transform can be thought of as the continuous version of Fourier Series. In other words, Fourier Transform is the limit of Fourier Series as the period of the function approaches infinity. This means that Fourier Transform can be used to analyze non-periodic functions by treating them as infinite periodic functions.

3. Can Fourier Series and Fourier Transform be used interchangeably?

No, Fourier Series and Fourier Transform are two distinct mathematical tools that have different applications. While Fourier Series is used to analyze periodic functions, Fourier Transform is used to analyze non-periodic functions. Although they are related, they cannot be used interchangeably.

4. How are the equations for Fourier Series and Fourier Transform different?

The equations for Fourier Series and Fourier Transform are different because they are used for different types of functions. Fourier Series involves a summation of sine and cosine waves, while Fourier Transform involves an integral over a continuous spectrum of frequencies. Additionally, the equations for Fourier Series use discrete values of time, while the equations for Fourier Transform use continuous time.

5. Can Fourier Series and Fourier Transform be applied to any function?

Fourier Series and Fourier Transform can be applied to any function that meets certain criteria. Fourier Series can only be used for periodic functions, while Fourier Transform can only be used for functions that are finite and have a well-defined Fourier Transform. Additionally, both methods work best for functions that are smooth and continuous.

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