<< Chapter < Page Chapter >> Page >
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.


Ricardo Radaelli-Sanchez


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 n c n ω 0 n t
c n 1 T t T 0 f t ω 0 n 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 f N t n N N c n ω 0 n t where c n equals the Fourier coefficients of f t (see [link] ).

f N t is a "partial reconstruction" of f t using the first 2 N 1 Fourier coefficients. f N t approximates f t , with the approximation getting better and better as N gets large. Therefore, we can think of the set N N 0 1 f N t 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 f N t f t as N ? 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 f N t
  2. Looking at N 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 f N t

e N t f t f N t
If f t L 2 0 T (finite energy), then the energy of e N t 0 as N is
t T 0 e N t 2 t T 0 f t f N t 2 0
We can prove this equation using Parseval's relation: N 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 L 2 0 T .Since physical systems respond to energy, the Fourier Series provides an adequate representation for all f t L 2 0 T equaling finite energy over one period.

Approach #2

The fact that e N 0 says nothing about f t and N 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 g t , but t T 0 f t g t 2 0 From this we can see the following relationships: energy convergence pointwise convergence pointwise convergence convergence in L 2 0 T 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 N f N t 0 But as long as f t meets some other fairly mild conditions, then f t N f N t if f t is continuous at t t .

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

t 0 T f t

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

c n 1 T t 0 T f t ω 0 n t 1 T t 0 T f t ω 0 n t
Remembering our complex exponentials , we know that in the above equation ω 0 n t 1 , which gives us:
c n 1 T t 0 T f t
c n

If we have the function: t 0 t T f t 1 t then you should note that this function fails the above condition because: t 0 T 1 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, 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:

f t N f N t
If f t meets all three conditions of the Strong Dirichlet Conditions, then f τ f τ at every τ at which f t is continuous. And where f t is discontinuous, f t is the average of the values on the right and left.

Discontinuous functions, f t .
Got questions? Get instant answers now!
The functions that fail the strong Dirchlet conditions are pretty pathological - as engineers, we are not too interested inthem.

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
anybody can imagine what will be happen after 100 years from now in nano tech world
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?