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Signals and Systems

Exponential Fourier Series with Solved Example

Replacing the sinusoidal terms in the trigonometric Fourier series by the exponential equivalents, $\cos (n{{\omega }_{o}}t)=\frac{1}{2}({{e}^{jn{{\omega }_{o}}t}}+{{e}^{-jn{{\omega }_{o}}t}})$ And $\sin (n{{\omega }_{o}}t)=\frac{1}{j2}({{e}^{jn{{\omega }_{o}}t}}-{{e}^{-jn{{\omega }_{o}}t}})$ Now, let us put the above exponential equivalents in the trigonometric Fourier series and get the Exponential Fourier Series expression: The trigonometric Fourier series can be …

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Symmetry Properties of the Fourier series

If a function has symmetry about the vertical axis or the origin, then the computation of the Fourier coefficients may be greatly facilitated. A function f (t) which is symmetrical about the vertical axis is to be an even function and has the property \[f(t)=f(-t)\] For all t. that is, …

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Trigonometric Fourier Series Solved Examples

Why Fourier series? There are many functions that are important in engineering which are not sinusoids or exponentials. A few examples are square waves, saw-tooth waves, and triangular pulses. Indeed, a function may be represented by a set of data points and have no analytical representation given at all. In …

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Classification of Systems in Signals and Systems

Systems can be classified into following different categories in signals and systems because of their inherent properties: Order of the system Causal and non-causal systems Linear and Non-Linear Systems Fixed and Time-Varying Systems Lumped and Distributed parameter Systems Continuous-time and Discrete-time Systems Instantaneous and dynamic systems Before proceeding to more …

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Laplace Transform:Introduction and Example

Laplace Transform Definition The Laplace transform X(s) is a complex-valued function of the complex variable s. In other words, given a complex number s, the value X(s) of the transform at the point s is, in general, a complex number. Given a function x (t) of the continuous-time variable t, …

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Z Transform Introduction | Z Transform Properties

INTRODUCTION TO Z-TRANSFORM  For the sake of analyzing continuous-time linear time-invariant (LTI) system, Laplace transformation is utilized. And z-transform is applied for the analysis of discrete-time LTI system.  Z-transform is fundamentally a numerical tool applied for a transition of a time domain into frequency domain and is a mathematical function …

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Continuous Time Convolution Properties | Continuous Time Signal

Continuous Time Convolution For linear time-invariant (LTI) systems, the convolution is being utilized in order to achieve output response from the knowledge of input and impulse response. Given two continuous-time signals x(t) and h(t), the convolution is defined as $y\left( t \right)=\sum\limits_{\tau =-\infty }^{\infty }{x\left( \tau  \right)h\left( t-\tau  \right)d\tau }~~~~~~~~~~~~~~~~~~~~~~~~\left( …

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Discrete Time Convolution Properties | Discrete Time Signal

Discrete-Time Convolution Convolution is such an effective tool that can be utilized to determine a linear time-invariant (LTI) system’s output from an input and the impulse response knowledge. Given two discrete time signals x[n] and h[n], the convolution is defined by $x\left[ n \right]*h\left[ n \right]=y\left[ n \right]=\sum\limits_{i=-\infty }^{\infty }{{}}x\left[ …

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Basic System Properties

Definition of a System A common way of viewing a system is in terms of a “black box” with terminals, as illustrated in the following figure: In the figure, x1(t), x2(t)… xp(t) are the signals applied to the p input terminals of the system and y1 (t), y2 (t)… yq …

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Linear Difference Equations

Consider the single–input single –output discrete time system given by the input/output difference equation. $y\left( kT+nT \right)+\underset{i=0}{\overset{n-1}{\mathop \sum }}\,{{a}_{i}}y\left( kT+iT \right)=\underset{i=0}{\overset{m}{\mathop \sum }}\,{{b}_{i}}x\left( kT+iT \right)~~~~~~~~~~~~\left( 1 \right)$ In (1), T is a fixed real number, k is a variable that takes its values from the set of integers, and the …

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