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1: Introduction to Signals

Last updated

Feb 23, 2021
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- About the Book
- 1.1: Signal Classifications and Properties

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1.1: Signal Classifications and Properties
This page introduces the fundamentals of signal classification in signals and systems, covering types such as continuous vs. discrete, analog vs. digital, and periodic vs. aperiodic signals. It also discusses classification based on causality and symmetry, explaining how signals can be expressed as combinations of even and odd components while distinguishing between deterministic and random signals. An example is included to illustrate these classifications.
1.2: Signal Size and Norms
This page explores norms for quantifying the strength of signals in continuous and discrete time, focusing on $L_p$ and $l_p$ norms that generalize signal energy. It details key norms such as $L_1$ , $L_2$ , and $L_{\infty}$ through examples, emphasizing their role in characterizing signals. The text concludes that nonzero periodic signals have infinite energy but finite power due to their periodic nature, thereby affecting their classification in signal processing.
1.3: Signal Operations
This page explains two critical time operations for signals: time shifting, which adjusts a signal's position in time, and time scaling, which compresses or dilates the signal. It also touches on time reversal, achieved by multiplying the time variable by a negative number. Mastery of these concepts is crucial for analyzing signals and their applications in real-world systems.
1.4: Common Continuous Time Signals
This page introduces essential concepts of continuous time analog signals, focusing on key signal types such as sinusoids, complex exponentials, the unit impulse function, and the unit step function. It defines sinusoids using $A \cos (\omega t+\varphi)$ and complex exponentials as $A e^{s t}$ . The importance of the Dirac delta function in signal representation and the use of the unit step function $u(t)$ for testing and defining other signals is emphasized.
1.5: Common Discrete Time Signals
This page covers key discrete time signals, including sinusoids, complex exponentials, unit impulses, and unit step functions. It explains the representation of sinusoids with parameters such as amplitude, frequency, and phase, and discusses the aliasing property of complex exponentials. These fundamental signals are essential for understanding more complex signals in the field of signals and systems.
1.6: Continuous Time Impulse Function
This page discusses the Dirac delta function, a vital concept in engineering and signal processing. Defined as a function with an infinitesimal width and infinite height that integrates to one, it is represented as $\delta(t)$ . Key properties include scaling behavior, symmetry, and its connection to the unit step function.
1.7: Discrete Time Impulse Function
This page discusses the unit impulse function, or unit sample function, a crucial concept in engineering and signal processing. Defined as $\delta[n]$ , it equals one at $n=0$ and zero elsewhere, and helps quantify actions at specific points. Key properties of the unit impulse include symmetry and its relationship to the unit step function. Understanding these characteristics is essential for analyzing discrete time systems and signal processing responses.
1.8: Continuous Time Complex Exponential
This page discusses the significance of complex exponentials in signals and systems as eigenfunctions of linear time-invariant systems, represented as $Ae^{st}$ . It explains the connection between complex exponentials and trigonometric functions via Euler's formula, enabling the decomposition into real and imaginary parts.
1.9: Discrete Time Complex Exponential
This page discusses the significance of complex exponentials in signals and systems as eigenfunctions of linear time-invariant systems. Discrete complex exponentials, expressed as $z^n$ with $z$ in polar coordinates, can be linked to trigonometric functions via Euler's formula.

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