[PDF] [PDF] Fourier Series

[PDF] 3-4 Fourier Series

8 sept 2012 · This is the mathematical theory of Fourier series which uses the periodic waveforms, such as square waves, triangle waves, rectified sinusoids, and so on simple property of the complex exponential signal—the integral of a The mathematical formula for the full-wave rectified sine signal is just the 

[PDF] Fourier Series

Fourier Series: Half-wave Rectifier • Ex A sinusoidal voltage Esinωt, is passed through a half-wave rectifier that clips the negative portion of the wave Find the

[PDF] Chapter 16 The Fourier Series

It is also useful to know the values of the cosine, sine, and exponential functions for Determine the Fourier series for the half-wave rectified cosine function

fourier series - gulden kokturks home page

4 jui 2010 · What is the conditions of Fourier series expansion for a signal? Answer: Exponential Fourier series DEU, Electrical Half-wave Symmetry; If a signal has an half wave symmetry, only Full-wave rectifier DEU, Electrical 

[PDF] Fourier series coefficients for a rectified sine wave

The period of the sinusoid (inside the absolute value symbols) is T1 = 2π/ω1 The period of the rectified sinusoid is one half of this, or T = T1/2 = π/ω1 Therefore,

[PDF] Half-Wave Rectifier - Zenon Bank

Vm / , so the rms value of a half-wave rectified sine wave is Vrms If periodic voltage and current waveforms represented by the Fourier series case), the exponential term in equation 3 1 becomes very small compared to unity and the diode

[PDF] EE − 210Signals and Systems Solutions of - CYPHYNETS

Q2 Fourier series of the output voltage of an ideal full-wave Diode Rectifier The following circuit is an ideal full-wave rectifier It is often used as a first stage to

[PDF] 1 Experiment 1 Spectral Analysis 1 Objectives: • To investigate the

1 5 cos(1000 )t Figure (1) 2 (a) Find the exponential Fourier series expansion for the time-shifted version of the halfwave rectified sinusoidal signal: y (t) = x2(t 

[PDF] π π π π - Chemical Engineering - UC Santa Barbara

Problem 10: Show that the exponential form of a Fourier series representation of a function ( ) f x through a half-wave rectifier, which clips the negative portion 

[PDF] exponential fourier series of square wave

[PDF] exponential function to log calculator

[PDF] exponential to log conversion calculator

[PDF] exponential to natural log calculator

[PDF] exponentielle de 0 7

[PDF] exponentielle de 0 valeur

[PDF] exponentielle de x u003d0

[PDF] exponentielle traduction en arabe

[PDF] exponents and logarithms worksheet

[PDF] exponents surds and logarithms pdf

[PDF] exposition art contemporain paris janvier 2019

[PDF] exposition art paris avril 2019

[PDF] exposition art paris mai 2019

[PDF] exposition art paris mars 2019

[PDF] exposition cite architecture paris

© 2012, Ching-Han Hsu, Ph.D.

Fourier Series

Ching-Han Hsu

© 2012, Ching-Han Hsu, Ph.D.

Periodic Function

A function f(x)is called periodic, if it is

defined for every xin the domain of fand if there is some positive number psuch that f(x+p)= f(x)

The number pis called a periodof f(x).

If a periodic function fhas a smallest

period p, this is often called the fundamental periodof f(x).

© 2012, Ching-Han Hsu, Ph.D.

Periodic Function

p f(x) x

© 2012, Ching-Han Hsu, Ph.D.

Periodic Function

For any integer n,

f(x+np) = f(x), for allx

If f(x)and g(x)have period p, then so does

h(x) = af(x) + bg(x), a and b are constants.

© 2012, Ching-Han Hsu, Ph.D.

Trigonometric Functions

Consider the following periodic functions

ʌ1, sin x, cos x, sin2x,

cos2xnx, cos nx

© 2012, Ching-Han Hsu, Ph.D.

Trigonometric Series

The trigonometric series is of the form

a0, a1, a2, a3b1, b2, b3are real constants and are called the coefficientsof the series.

If the series converges, its sum will be a

1 0 22110
sincos1

2sin2cossincos1

n nnnxbnxaa xbxaxbxaa3

© 2012, Ching-Han Hsu, Ph.D.

Fourier Series

Assume f(x)is a periodic function of

trigonometric series:

That is, we assume that the series

converges and has f(x)as its sum.

Question: how to compute the coefficients?

1

0sincos1

n nnnxbnxaaxf

© 2012, Ching-Han Hsu, Ph.D.

Properties of Trigonometric

Functions

Recall some basic properties of

trigonometric functions

0sin1sin

0cos1cos

S S S S S S S S dxnxnxdx dxnxnxdx 0sin2 1sin2 1 sincos S S S S S S xdxmnxdxmn nxdxmx

© 2012, Ching-Han Hsu, Ph.D.

Properties of Trigonometric

Functions

z mn mnxdxmnxdxmn nxdxmx 0cos2 1cos2 1 sinsin S S S S S z mn mnxdxmnxdxmn nxdxmx 0cos2 1cos2 1 coscos S S S S S

© 2012, Ching-Han Hsu, Ph.D.

Determination of the Constant

Term a0

Integrating both sides from to :

0 1 0 2 sincos a dxnxbnxaadxxf n nn S S S f

Sdxxfa2

1 0

© 2012, Ching-Han Hsu, Ph.D.

Determination of the Coefficients anof the Cosine Term

Multiply by cosmxwith fixed positive integer

and then integrate both sides from to : m n nn a mxdxnxbnxaa mxdxxf S S S f cossincos cos 1 0

Smxdxxfamcos1

© 2012, Ching-Han Hsu, Ph.D.

Determination of the Coefficients bnof the Sine Term

Multiply by sinmxwith fixed positive integer

and then integrate both sides from to : m n nn b mxdxnxbnxaa mxdxxf S S S f sinsincos sin 1 0

Smxdxxfbmsin1

© 2012, Ching-Han Hsu, Ph.D.

Summary

1

0sincos

n nnnxbnxaaxf

Sdxxfa2

1 0

Snxdxxfbnsin1

Snxdxxfancos1

© 2012, Ching-Han Hsu, Ph.D.

Fourier Series: Square Wave

Ex. Find the Fourier coefficients of the

periodic function f(x) xfxfxk xkxf SS S2,0 0

© 2012, Ching-Han Hsu, Ph.D.

Fourier Series: Square Wave

@02 1 2 1 2 1 0 0 0 SS SS S S S kk kdxkdxdxxfa

0sinsin1

coscos1 cos1 0 0 0 0 S S S S S S S nxn knxn k nxdxknxdxk nxdxxfan

Q: Is f(x)even or odd?!

© 2012, Ching-Han Hsu, Ph.D.

S SSS S S S S S S S S nn k nn knn k nxn knxn k nxdxknxdxk nxdxxfbn cos12

1coscos11

coscos1 sinsin1 sin1 0 0 0 0

Snncoscosquotesdbs_dbs10.pdfusesText_16