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6: Continuous Time Fourier Series (CTFS)

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    • 6.1: Continuous Time Periodic Signals
      This page covers the Continuous Time Fourier Series, which transforms finite-length continuous-time signals into infinite-length discrete-frequency signals. It defines periodic signals as those that repeat after a period, allowing analysis within a single cycle to avoid redundancy, while aperiodic signals do not repeat. An example of periodic sinusoidal signals is provided, along with additional resources such as a video lecture for further understanding.
    • 6.2: Continuous Time Fourier Series (CTFS)
      This page introduces the Continuous Time Fourier Series (CTFS), illustrating how continuous-time periodic functions can be expanded using harmonic complex sinusoids. It explains the representation of arbitrary functions as a linear combination of complex exponentials, detailing the derivation of Fourier coefficients and their relationship with LTI systems.
    • 6.3: Common Fourier Series
      This page covers the basics of Fourier series analysis, emphasizing common signals like square waves, their properties, and the Gibb's phenomenon. It also discusses other waveforms, including constant, sinusoidal, triangular, and sawtooth, detailing their characteristics and Fourier series representations. A summary table of these signals and their Fourier coefficients is included for easy reference.
    • 6.4: Properties of the CTFS
      This page covers the Continuous-Time Fourier Series (CTFS), detailing properties like linearity, time shifting, and Parseval's Relation. It explains Fourier transformation, highlighting its linearity and effects of time shifting on Fourier coefficients. The module discusses the behavior of real signals, addressing differentiation and integration in the frequency domain, and emphasizes the relationship between multiplication in time and convolution in frequency.
    • 6.5: Continuous Time Circular Convolution and the CTFS
      This page explains the connection between circular convolution of periodic signals and the multiplication of their Fourier coefficients. It introduces a new signal via circular convolution and shows that its Fourier series representation arises from the product of the original signals' coefficients.
    • 6.6: Convergence of Fourier Series
      This page covers the representation of periodic functions via Fourier Series and the historical debate between Fourier and Lagrange. It examines the convergence of these series, error signals, and energy convergence, alongside the essential Dirichlet Conditions for Fourier Series existence. It emphasizes the Strong Dirichlet Conditions, stating that real-world signals ensure valid Fourier representation, while theoretical violations hold little practical importance for engineering applications.
    • 6.7: Gibbs Phenomena
      This page discusses the Fourier Series, which uses complex exponentials to represent continuous, periodic signals, even with finite discontinuities. Despite the impossibility of exact reconstruction at discontinuities, near-exact representations are possible almost everywhere, leading to the Gibbs Phenomenon. This phenomenon, involving persistent ripples at discontinuities, was described by J. Willard Gibbs.

    This page titled 6: Continuous Time Fourier Series (CTFS) is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Richard Baraniuk et al. via source content that was edited to the style and standards of the LibreTexts platform.