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Signals and Systems
This module covers the Z-Transform, a key mathematical tool for analyzing discrete-time signals and systems. It details how the Z-Transform simplifies the analysis of difference equations and system behavior in the Z-domain, explaining concepts like the Region of Convergence (ROC), inverse Z-Transform, and various properties of the Z-Transform. The relationship between the Z-Transform, the system function, and the Discrete-Time Fourier Transform (DTFT) is also explored, highlighting their significance in signal processing and system analysis.
Course Chapters
Introduction to Signals and Systems
The course module covers foundational concepts in Signals and Systems, including signal classification, manipulation techniques, and system properties. It establishes key distinctions between continuous-time and discrete-time signals, analog and digital signals, and periodic versus aperiodic signals. By the end of the module, students will be equipped to analyze signals and systems using fundamental operations and understand the behavior of various signal types in engineering contexts.
Time Domain Analysis of Continuous-Time Systems
This module explores the analysis of continuous-time linear time-invariant (LTI) systems in the time domain, building from fundamental principles to advanced concepts like impulse and step responses, convolution integrals, and properties of LTI systems. Practical importance is placed on system behavior understanding through mathematical frameworks, as well as real-world applications such as feedback control systems and differential equations.
Fourier Series Analysis of Continuous-Time Periodic Signals
The chapter delves into Fourier Series Analysis, covering the decomposition of continuous-time periodic signals into harmonically related sinusoidal components. It introduces concepts of orthogonality, establishes the mathematical foundation for calculating Fourier coefficients, and discusses properties of these series. Additionally, key applications including filtering, circuit analysis, and the implications of the Gibbs phenomenon are also highlighted.
Fourier Transform Analysis of Continuous-Time Aperiodic Signals
The comprehensive treatment of Fourier Transform analysis provides critical insights into continuous-time aperiodic signals. It establishes a framework to connect the Fourier Series with the Fourier Transform, focusing on their application in analyzing signal behaviors and system responses in the frequency domain. The chapter emphasizes key properties of the Fourier Transform, its implications for system frequency responses, and the importance of sampling methods in digital signal processing.
Laplace Transform Analysis of Continuous-Time Systems
The chapter explores the Laplace Transform, an essential mathematical tool for analyzing continuous-time systems, providing a transition from the time domain to the frequency domain. It details the benefits over the Fourier Transform, defines integral pairs, and clarifies the Region of Convergence (ROC) alongside essential properties and methods for inversion. Moreover, it elucidates solving differential equations and system characterization via transfer functions, emphasizing the importance of poles, zeros, and stability in system analysis.
Time Domain Analysis of Discrete-Time Systems
This module covers the analysis of Discrete-Time Linear Time-Invariant (DT-LTI) systems, focusing on their behavior in the time domain. Understanding these systems is essential for various engineering fields such as digital signal processing and control systems. The module introduces core concepts such as impulse response, convolution, and the representation of DT-LTI systems via difference equations and block diagrams.
Z-Transform Analysis of Discrete-Time Systems
This module covers the Z-Transform, a key mathematical tool for analyzing discrete-time signals and systems. It details how the Z-Transform simplifies the analysis of difference equations and system behavior in the Z-domain, explaining concepts like the Region of Convergence (ROC), inverse Z-Transform, and various properties of the Z-Transform. The relationship between the Z-Transform, the system function, and the Discrete-Time Fourier Transform (DTFT) is also explored, highlighting their significance in signal processing and system analysis.