# 6: Continuous Time Fourier Series (CTFS)

- Page ID
- 22878

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

- 6.1: Continuous Time Periodic Signals
- This module defines a periodic function and describes the two common ways of thinking about a periodic signal.

- 6.2: Continuous Time Fourier Series (CTFS)
- This module describes the continuous time Fourier Series (CTFS). It is based on the following modules: Fourier Series: Eigenfunction Approach at http://cnx.org/content/m10496/latest/ by Justin Romberg, Derivation of Fourier Coefficients Equation at http://cnx.org/content/m10733/latest/ by Michael Haag, Fourier Series and LTI Systems at http://cnx.org/content/m10752/latest/ by Justin Romberg, and Fourier Series Wrap-Up at http://cnx.org/content/m10749/latest/ by Michael Haag and Justin Romberg.

- 6.3: Common Fourier Series
- Constant, Sinusoid, Square, Triangle, and sawtooth waveforms, in depth and summarized.

- 6.4: Properties of the CTFS
- An introduction to the general properties of the Fourier series

- 6.5: Continuous Time Circular Convolution and the CTFS
- This module looks at the basic circular convolution relationship between two sets of Fourier coefficients.

- 6.6: Convergence of Fourier Series
- This module discusses the existence and convergence of the Fourier Series to show that it can be a very good approximation for all signals. The Dirichlet conditions, which are the sufficient conditions to guarantee existence and convergence of the Fourier series, are also discussed.

- 6.7: Gibbs Phenomena
- The Fourier Series is the representation of continuous-time, periodic signals in terms of complex exponentials. The Dirichlet conditions suggest that discontinuous signals may have a Fourier Series representation so long as there are a finite number of discontinuities. This seems counter-intuitive, however, as complex exponentials are continuous functions. It does not seem possible to exactly reconstruct a discontinuous function from a set of continuous ones. In fact, it is not. However, it can