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7: Discrete Time Fourier Series (DTFS)

Last updated

Feb 23, 2021
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- 6.7: Gibbs Phenomena
- 7.1: Discrete Time Periodic Signals

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7.1: Discrete Time Periodic Signals
This page introduces the Discrete Time Fourier Series for finite-length discrete time signals, covering periodic signals that repeat every $N$ and are analyzed over $[0,N]$ , thereby eliminating redundancy. It differentiates aperiodic signals, which do not repeat, and provides examples of periodic sinusoidal signals. The conclusion emphasizes that a discrete periodic signal is fully defined by its values within one period.
7.2: Discrete Time Fourier Series (DTFS)
This page covers the Discrete Time Fourier Series (DTFS), detailing its derivation and properties related to expanding discrete-time periodic functions into harmonic complex sinusoids. It emphasizes the role of complex exponentials as eigenfunctions in linear time-invariant systems and illustrates their orthonormal basis for effective signal representation.
7.3: Common Discrete Fourier Series
This page offers a comprehensive overview of Fourier series analysis, detailing the derivation of Fourier coefficients for common signals such as square, constant, sinusoid, triangle, and sawtooth waveforms. It discusses the signals' properties and classifications as even or odd. A table summarizing discrete Fourier transforms for these signals is included. The conclusion underscores the variety and key characteristics of Fourier transforms.
7.4: Properties of the DTFS
This page explores the Discrete-Time Fourier Series (DTFS) and Discrete Fourier Transform (DFT), detailing definitions of Fourier coefficients and their implications on phase shifts and energy relationships as per Parseval's relation. It also covers finite energy conditions for signals, the effect of differentiation and integration in the Fourier domain, and the equivalence of time-domain multiplication to frequency-domain circular convolution.
7.5: Discrete Time Circular Convolution and the DTFS
This page explores circular convolution of periodic signals and its connection to Fourier domain multiplication. It explains how circular convolution leads to efficient DFT-based multiplication of Fourier coefficients, vital for finite-length sequence convolution. The content includes a mathematical foundation for circular convolution and an algorithm for its computation.

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