What is the Difference Between Fourier Series and Fourier Transform?

The main difference between Fourier Series and Fourier Transform lies in the type of signals they are applied to and the way they represent the signals in the frequency domain. Here are the key differences:

• Type of Signal: Fourier Series is applicable to periodic signals, while Fourier Transform can be applied to both periodic and non-periodic signals.
• Representation: Fourier Series represents a periodic function as a sum of cosine and sine terms. On the other hand, Fourier Transform represents a function in the frequency domain using a continuous superposition or integral of complex exponentials.
• Generalization: Fourier Transform is considered a generalization of the Fourier Series, as it allows the Fourier Series to extend to non-periodic functions by converting any function into a sum of simple sinusoids.

In summary, Fourier Series is used for harmonic analysis of arbitrary periodic functions, while Fourier Transform is a mathematical operation that decomposes a signal into its constituent frequency components, applicable to both periodic and non-periodic signals. Fourier Transform can be viewed as a limit of the Fourier Series of a function with a period, and its inverse can be used to transform the frequency domain representation back into the time domain.

Comparative Table: Fourier Series vs Fourier Transform

The main differences between Fourier Series and Fourier Transform are summarized in the following table:

In summary, Fourier series is used for periodic functions, while the Fourier transform is used for both periodic and non-periodic functions. Fourier series decomposes a periodic signal into a sum of sine and cosine terms, whereas the Fourier transform converts a signal from time domain into its frequency domain representation.