WHY Fourier Transform? If a function f (t) is not a periodic and is defined on an infinite interval, we cannot represent it by Fourier series. It may be possible, however, to consider the function to be periodic with an infinite period. In this section we shall consider this case …

Read More »## 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 …

Read More »## 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, …

Read More »## 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 …

Read More »## 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 …

Read More »## Laplace Transform Properties in Signal and Systems

The Laplace transform fulfills a number of properties that are quite valuable in various applications. In particular, by using these properties, it is possible to derive many new transform pairs from a basic set of pairs. In this tutorial, we state most fundamental properties of the Laplace transform. Linearity If …

Read More »## 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, …

Read More »## 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 …

Read More »## Continuous Time Graphical Convolution Example

This is the continuation of the PREVIOUS TUTORIAL. Steps for Graphical Convolution First of all re-write the signals as functions of τ: x(τ) and h(τ) Flip one of the signals around t = 0 to get either x(-τ) or h(-τ) Best practice is to flip the signal with shorter interval …

Read More »## 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( …

Read More »## Discrete Time Graphical Convolution Example

This is the continuation of the PREVIOUS TUTORIAL. This example is provided in collaboration with Prof. Mark L. Fowler, Binghamton University.

Read More »## 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[ …

Read More »## 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 …

Read More »## 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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