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Fourier Analysis


Fourier Analysis

Hi !

Today i begin to share a new area , a genius theory from Joseph Fourier. Fourier says that any engineering signals can be represented as a series sum of sine , even square and triangle waves .

Sine waves have many interesting properties , many natural operations deal with a set of differing frequency sine waves as if they were processed individually .

We will learn the technique of expressing waves in terms of sine,this will allow us to use them as we want them to be.

Suppose your signal has noise at a particular frequency (may be 30-40 HZ) , we can pick apart the data to its constituent parts,remove the noise frequency , then stictch the rest back together to get a signal without the noise . This kind of notch filtering is very useful in audio signals .

In image processing,it is possible to look at an image and distil it down to small blocks , then look at each of the line;

now we can treat each line as a 1D signal,it’s then easy to see how you would apply the same techniques to the data that you would with any other signal . Let’s begin our theoric expansions.

square wave     

sine wave

Trigonometric Fourier Series

What it means to be periodic ? When we use periodic for a signal , we say that you get the same samples in the same intervals ;

f (t) = f (t+nT) means that for every period T .

Any periodic function of frequency wo can be expressed as an infinite sum of sine and cosine functions that are integral multiples of wo.

 

f(t) = a0 + a1cos(w0) + b1sin (w0)+…..

in this equation,we can take (a0) as DC part of our signal,which is linear and not wavy .(a0) is also called the average value of f(t) (remember the average value of sinusoids are zero)

Also ;

w0 = 2*pi/T; which is called “fundamental frequency in radians per second”

Another important term is harmonic; The sin(nwot) or cos(nwot) is called nth harmonic of f(t).

 

(an) and (bn) are called “Fourier Coeficcients .” Our work is to find these coefficients to transform our time domain signal through frequency domain.

so ,let’s ask; Why we need Fourier Series of a function,why function itself is not enough for us ?

The fourier series of a periodic function  f(t) is a representation that analyzes f(t) into a DC component and AC component comprising an infinite series of harmonic sinusoids.

 

Your fourier series must contain these conditions ;

1).f(t) is single-valued everywhere in the domain

2).f(t) has a finite number of finite discontinuities in any one period .

3).f(t) has a finite number of maxima and minima in any one period .

As i say earlier , the process of determining the coefficients is called “Fourier Analysis” .

Let’s translate what we say into math !

I will share some trigonometric properties in the chart below , It tells you how to calculate ao,an and bn values for analysis .You will see that average values of sinusoids are zero in an interval.

 


Here is a useful Fourier Analysis video , that tells what it means to transform functions , A must see 3 video  :

Fourier transform-1

Fourier tranform-2

Fourier Transform-3

 

See al videos after reading my paper , See you next lesson

Take Care!

 

Reklamlar

Ekim 31, 2010 Posted by | Circuit Analysis | , , , | Yorum bırakın

Circuit analysis


Hi,

Today i introduce Circuit Analysis principles . i will tell you about systems of units first ,because they play a big critical role in physics,and then how to get and solve problems.

FUNDAMENTALS OF CİRCUİT ANALYSİS

1) System of units :

Length : meter  /  mass kilogram  / time : second  /  electric current : amper  /

Thermodynamic temperature : Kelvin / luminous intensity : candela

Electric current is the time rate change in charge   : i =  dq / dt

*a direct current is a current that remains constant with time

*an alernative current is a current that varies sinusoidally with time

Voltage : to move the electron in a conductor in a particular direction requires some work or energy transfer,Potential difference is also known the energy to move a unit charge from a to b.

Vab = dW/dq

Vab = – Vba

Power: power is the time rate of expanding or absorbing energy ,measured in watts.

P= dw /dt,p=vi

Power absorbed = power supplied

Energy : Energy is the capacity to do work ,measured in joules .

W=p x dt = i.v.t

Circuit Elements

Circuit analysis  is the process of determining voltages across  (or the current) the elements of circuit.

Two types of elements found in electric circuits : passive and active elements .

Active element is capable of generating energy while a Passive element is not.

Passive elements;resistors capacitors inductors,Active elements; batteries operational amplifiers.

#The most important active elements are voltage or current sources ,1)dependent 2 ) independent

İdeal independent source : active element that provides a specified voltage or current that is completely independent of other circuit variables,batteries generators …

Dependent source : (controlled) is an active element in which the source quantity is controlled by another voltage or current .

*İndependent source symbolized by  circle , dependent source is symbolized by diamond

Dependent sources are useful in modelling elements such as  transistors operational amplifiers and integrated circuits.

PROBLEM SOLVİNG


1.Carefully define the Problem

The most important part of solving process.Do all you can to make sure that you understand the problem clearly,it it important to develop questons that need to be addressed before continuing the solving process,with those answers now you can refine the problem and use the refinement as the problem statement

2.Present everything you know about the problem

Ready to write down everything you know about the problem and its possible solutions

3.Establish a set of “alternative solutions” and determine the one that promises the greatest likelihood of success

Almost every problem will have a number of possible paths that can lead to a solution.Evaluating alternative solutions can save you time,Document this process well since you will want to come back to it if the first approach does not work.

4.Attempt a problem solution

The process you follow must be well documented in order  to present a detailed solution if succesful,Many times  it is wise to fully set up a solution before putting numbers into equations .This will help in checking your result.

5.Evaluate the solution and chech for accuracy

6.Has the problem solved Satisfactorily ? If so,present the solution : if not; then return to step 3 and continue through the process again.

Ağustos 28, 2010 Posted by | Circuit Analysis | , , , | Yorum bırakın