6: Continuous Time Fourier Series (CTFS)

• • Richard Baraniuk et al.
• Victor E. Cameron Professor (Electrical and Computer Engineering) at Rice University

• 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