# 6.5 Convergence of fourier series

 Page 1 / 1
This module discusses the existence and covergence of the Fourier Series to show that it can be a very good approximation for all signals. The Dirichlet conditions, which are the sufficient conditions to guarantee existence and convergence of the Fourier series, are also discussed.

MODULE ID:
m10089

## Introduction

Before looking at this module, hopefully you have become fully convinced of the fact that any periodic function, $f(t)$ , can be represented as a sum of complex sinusoids . If you are not, then try looking back at eigen-stuff in a nutshell or eigenfunctions of LTI systems . We have shown that we can represent a signal as the sum of exponentials through the Fourier Series equations below:

$f(t)=\sum {c}_{n}e^{i{\omega }_{0}nt}$
${c}_{n}=\frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega }_{0}nt)}\,d t$
Joseph Fourier insisted that these equations were true, but could not prove it. Lagrange publicly ridiculedFourier, and said that only continuous functions can be represented by [link] (indeed he proved that [link] holds for continuous-time functions). However, we know now thatthe real truth lies in between Fourier and Lagrange's positions.

## Understanding the truth

Formulating our question mathematically, let $\frac{d {f}_{N}(t)}{d }}=\sum_{n=-N}^{N} {c}_{n}e^{i{\omega }_{0}nt}$ where ${c}_{n}$ equals the Fourier coefficients of $f(t)$ (see [link] ).

$\frac{d {f}_{N}(t)}{d }}$ is a "partial reconstruction" of $f(t)$ using the first $2N+1$ Fourier coefficients. $\frac{d {f}_{N}(t)}{d }}$ approximates $f(t)$ , with the approximation getting better and better as $N$ gets large. Therefore, we can think of the set $\{\forall N, N=\{0, 1, \dots \}()\colon \frac{d {f}_{N}(t)}{d }}\}()$ as a sequence of functions , each one approximating $f(t)$ better than the one before.

The question is, does this sequence converge to $f(t)$ ? Does $\frac{d {f}_{N}(t)}{d }}\to f(t)$ as $N\to$ ? We will try to answer this question by thinking about convergence in two different ways:

1. Looking at the energy of the error signal: ${e}_{N}(t)=f(t)-\frac{d {f}_{N}(t)}{d }}$
2. Looking at $\lim_{N\to }N\to$ f N t at each point and comparing to $f(t)$ .

## Approach #1

Let ${e}_{N}(t)$ be the difference ( i.e. error) between the signal $f(t)$ and its partial reconstruction $\frac{d {f}_{N}(t)}{d }}$

${e}_{N}(t)=f(t)-\frac{d {f}_{N}(t)}{d }}$
If $f(t)\in {L}^{2}(\left[0 , T\right]())$ (finite energy), then the energy of ${e}_{N}(t)\to 0$ as $N\to$ is
$\int_{0}^{T} \left|{e}_{N}(t)\right|^{2}\,d t=\int_{0}^{T} (f(t)-\frac{d {f}_{N}(t)}{d }})^{2}\,d t\to 0$
We can prove this equation using Parseval's relation: $\lim_{N\to }N\to$ t T 0 f t f N t 2 N N n f t n f N t 2 N n n N c n 2 0 where the last equation before zero is the tail sum of theFourier Series, which approaches zero because $f(t)\in {L}^{2}(\left[0 , T\right]())$ .Since physical systems respond to energy, the Fourier Series provides an adequate representation for all $f(t)\in {L}^{2}(\left[0 , T\right]())$ equaling finite energy over one period.

## Approach #2

The fact that ${e}_{N}\to 0$ says nothing about $f(t)$ and $\lim_{N\to }N\to$ f N t being equal at a given point. Take the two functions graphed below for example:

Given these two functions, $f(t)$ and $g(t)$ , then we can see that for all $t$ , $f(t)\neq g(t)$ , but $\int_{0}^{T} \left|f(t)-g(t)\right|^{2}\,d t=0$ From this we can see the following relationships: $\mathrm{energy convergence}\neq \mathrm{pointwise convergence}$ $\mathrm{pointwise convergence}\implies {\mathrm{convergence in L}}^{2}(\left[0 , T\right]())$ However, the reverse of the above statement does not hold true.

It turns out that if $f(t)$ has a discontinuity (as can be seen in figure of $g(t)$ above) at ${t}_{0}$ , then $f({t}_{0})\neq \lim_{N\to }N\to$ f N t 0 But as long as $f(t)$ meets some other fairly mild conditions, then $f({t}^{\prime })=\lim_{N\to }N\to$ f N t if $f(t)$ is continuous at $t={t}^{\prime }$ .

These conditions are known as the Dirichlet Conditions .

## Dirichlet conditions

Named after the German mathematician, Peter Dirichlet, the Dirichlet conditions are the sufficient conditions to guarantee existence and energy convergence of the Fourier Series.

## The weak dirichlet condition for the fourier series

For the Fourier Series to exist, the Fourier coefficients must be finite. The Weak Dirichlet Condition guarantees this. It essentially says that the integral of the absolute value of the signal must befinite.

The coefficients of the Fourier Series are finite if

## Weak dirichlet condition for the fourier series

$\int_{0}^{T} \left|f(t)\right|\,d t$

This can be shown from the magnitude of the Fourier Series coefficients:

$\left|{c}_{n}()\right|=\left|\frac{1}{T}\int_{0}^{T} f(t)e^{-(i{\omega }_{0}()nt)}\,d t\right|\le \frac{1}{T}\int_{0}^{T} \left|f(t)\right|\left|e^{-(i{\omega }_{0}()nt)}\right|\,d t$
Remembering our complex exponentials , we know that in the above equation $\left|e^{-(i{\omega }_{0}()nt)}\right|=1$ , which gives us:
$\left|{c}_{n}\right|\le \frac{1}{T}\int_{0}^{T} \left|f(t)\right|\,d t< \infty$
$\implies (\left|{c}_{n}\right|)$

If we have the function: $\forall t, 0< t\le T\colon f(t)=\frac{1}{t}$ then you should note that this function fails the above condition because: $\int_{0}^{T} \left|\frac{1}{t}\right|\,d t$

## The strong dirichlet conditions for the fourier series

For the Fourier Series to exist, the following two conditions must be satisfied (along with the WeakDirichlet Condition):

1. In one period, $f(t)$ has only a finite number of minima and maxima.
2. In one period, $f(t)$ has only a finite number of discontinuities and each one is finite.
These are what we refer to as the Strong Dirichlet Conditions . In theory we can think of signals that violate these conditions, $\sin \lg t$ for instance. However, it is not possible to create a signal that violates these conditions in a lab. Therefore, anyreal-world signal will have a Fourier representation.

Let us assume we have the following function and equality:

$\frac{d f(t)}{d }}=\lim_{N\to }N\to$ f N t
If $f(t)$ meets all three conditions of the Strong Dirichlet Conditions, then $f(\tau )=\frac{d f(\tau )}{d }}$ at every $\tau$ at which $f(t)$ is continuous. And where $f(t)$ is discontinuous, $\frac{d f(t)}{d }}$ is the average of the values on the right and left.

The functions that fail the strong Dirchlet conditions are pretty pathological - as engineers, we are not too interested inthem.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Rhodes By OpenStax By By Brooke Delaney By Brooke Delaney By Anh Dao By Jazzycazz Jackson By Maureen Miller By Stephen Voron By OpenStax