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- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/13%3A_Capstone_Signal_Processing_Topics/13.02%3A_The_Fast_Fourier_Transform_(FFT)This page explains the Fast Fourier Transform (FFT), an efficient algorithm that computes the Discrete Fourier Transform (DFT) with reduced complexity from O(N^2) to O(N log N) by leveraging symmetrie...This page explains the Fast Fourier Transform (FFT), an efficient algorithm that computes the Discrete Fourier Transform (DFT) with reduced complexity from O(N^2) to O(N log N) by leveraging symmetries and recursive decomposition. The process involves separating even and odd indexed elements and using a "butterfly" structure for output pairing.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/09%3A_Discrete_Time_Fourier_Transform_(DTFT)/9.04%3A_Properties_of_the_DTFTThis page covers the Discrete-Time Fourier Transform (DTFT) properties for aperiodic discrete-time signals, including linearity, symmetry, time-related operations, convolution, and Parseval's relation...This page covers the Discrete-Time Fourier Transform (DTFT) properties for aperiodic discrete-time signals, including linearity, symmetry, time-related operations, convolution, and Parseval's relation. It illustrates how time domain operations translate to the frequency domain, particularly the convolution-multiplication relationship. The module includes a comprehensive table summarizing various DTFT properties to enhance understanding of signal transformations.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/07%3A_Discrete_Time_Fourier_Series_(DTFS)/7.04%3A_Properties_of_the_DTFSThis 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 relatio...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.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/zz%3A_Back_Matter/10%3A_Index
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/03%3A_Time_Domain_Analysis_of_Continuous_Time_SystemsThis page provides an overview of key modules in continuous time systems, including impulse response of LTI systems, convolution properties, eigenfunctions, BIBO stability, and linear constant coeffic...This page provides an overview of key modules in continuous time systems, including impulse response of LTI systems, convolution properties, eigenfunctions, BIBO stability, and linear constant coefficient differential equations. It emphasizes fundamental concepts and techniques for solving related equations within the mathematical framework of these systems.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/09%3A_Discrete_Time_Fourier_Transform_(DTFT)/9.02%3A_Discrete_Time_Fourier_Transform_(DTFT)This page elucidates the Derivation of the Discrete Time Fourier Transform (DTFT) for discrete-time functions, showcasing complex exponentials as eigenfunctions of linear time-invariant systems. It ex...This page elucidates the Derivation of the Discrete Time Fourier Transform (DTFT) for discrete-time functions, showcasing complex exponentials as eigenfunctions of linear time-invariant systems. It explains how discrete time-periodic functions can be expressed via Fourier coefficients and how the DTFT transitions to a continuous frequency representation as signal period increases.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/10%3A_Sampling_and_Reconstruction/10.03%3A_Signal_ReconstructionThis page discusses sampling and reconstruction processes of signals, emphasizing the conversion between continuous and discrete signals. It explains the role of lowpass filters in reconstruction, int...This page discusses sampling and reconstruction processes of signals, emphasizing the conversion between continuous and discrete signals. It explains the role of lowpass filters in reconstruction, introduces cardinal basis splines for smooth signal creation, and details how higher orders of splines yield closer resemblance to the sinc function.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/04%3A_Time_Domain_Analysis_of_Discrete_Time_Systems/4.05%3A_Eigenfunctions_of_Discrete_Time_LTI_SystemsThis page introduces linear time invariant (LTI) systems, explaining how eigenfunctions aid in calculating outputs from complex exponential inputs. It notes that the output is scaled by an eigenvalue ...This page introduces linear time invariant (LTI) systems, explaining how eigenfunctions aid in calculating outputs from complex exponential inputs. It notes that the output is scaled by an eigenvalue derived from the impulse response, highlighting the usefulness of complex exponentials in analyzing discrete time signals through discrete time Fourier transforms and series for both aperiodic and periodic signals.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/03%3A_Time_Domain_Analysis_of_Continuous_Time_Systems/3.06%3A_BIBO_Stability_of_Continuous_Time_SystemsThis page discusses BIBO stability, which is a property of systems where bounded inputs lead to bounded outputs. It emphasizes that a bounded signal stays within a finite limit. For continuous-time LT...This page discusses BIBO stability, which is a property of systems where bounded inputs lead to bounded outputs. It emphasizes that a bounded signal stays within a finite limit. For continuous-time LTI systems, BIBO stability is attained if the impulse response is absolutely integrable. In the Laplace domain, stability is represented by the pole-zero plot, requiring poles to be on the left side of the imaginary axis. BIBO stability is essential for the effectiveness of systems.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/07%3A_Discrete_Time_Fourier_Series_(DTFS)This page covers modules on discrete-time periodic signals and their analysis via the Discrete-Time Fourier Series (DTFS). It includes DTFS derivations, complex sinusoids reviews, and discrete Fourier...This page covers modules on discrete-time periodic signals and their analysis via the Discrete-Time Fourier Series (DTFS). It includes DTFS derivations, complex sinusoids reviews, and discrete Fourier transforms. The text also discusses DTFS properties and circular convolution algorithms pertinent to the DTFS.
- https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/Signals_and_Systems_(Baraniuk_et_al.)/12%3A_Z-Transform_and_Discrete_Time_System_Design/12.03%3A_Properties_of_the_Z-TransformThis page covers essential properties of the Z-Transform for discrete-time signals, including linearity, symmetry, time scaling, time shifting, convolution, and time differentiation. It highlights the...This page covers essential properties of the Z-Transform for discrete-time signals, including linearity, symmetry, time scaling, time shifting, convolution, and time differentiation. It highlights the connection between time-domain and frequency-domain representations, showing how linear combinations and time shifts function in frequency.
