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Continuous-Time Fourier Series (CTFS)

Module by: Richard Baraniuk. E-mail the author

Summary: Details the Continuous-Time Fourier Series.

Fourier Series Coefficients

c n =1T⁢∫0Tf⁢t⁢e−(i⁢2⁢πT⁢n⁢t)d t c n 1 T t 0 T f t 2 T n t
(1)

Fourier Series Synthesis

f⁢t=∑x∈R c n ⁢ei⁢2⁢πT f t n x c n 2 T n t
(2)

Relevant Spaces

The Continuous-Time Fourier Series maps finite-length (or TT-periodic), continuous-time signals in L2 L2 to infinite-length, discrete-frequency signals in l2 l2.

Figure 1: Mapping L 2 ⁢ 0 T L 2 0 T in the time domain to l 2 ⁢Z l 2 in the frequency domain.
Figure 1 (CTFS1.png)

Periodic Extension of the Fourier Series

The Fourier Series is defined for finite-length signals. However, the complex exponential basis functions used to reconstruct the original signal are infinite-length. What happens is that the reconstructed signal is actually the original signal repeated in both the positive and negative directions. This is what is called periodic extension. This is also what necessitates the use of circular convolution with the Fourier Series.

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