<< Chapter < Page Chapter >> Page >
x ( t ) = a ( 0 ) 2 + k = 1 a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) .

where x k ( t ) = c o s ( 2 π k t / T ) and y k ( t ) = s i n ( 2 π k t / T ) are the basis functions for the expansion. The energy or power in an electrical,mechanical, etc. system is a function of the square of voltage, current, velocity, pressure, etc. For this reason, the natural setting for arepresentation of signals is the Hilbert space of L 2 [ 0 , T ] . This modern formulation of the problem is developed in [link] , [link] . The sinusoidal basis functions in the trigonometric expansion form a completeorthogonal set in L 2 [ 0 , T ] . The orthogonality is easily seen from inner products

( cos ( 2 π T k t ) , cos ( 2 π T t ) ) = 0 T ( cos ( 2 π T k t ) cos ( 2 π T t ) ) d t = δ ( k - )

and

( cos ( 2 π T k t ) , sin ( 2 π T t ) ) = 0 T ( cos ( 2 π T k t ) sin ( 2 π T t ) ) d t = 0

where δ ( t ) is the Kronecker delta function with δ ( 0 ) = 1 and δ ( k 0 ) = 0 . Because of this, the k th coefficients in the series can be foundby taking the inner product of x ( t ) with the k th basis functions. This gives for the coefficients

a ( k ) = 2 T 0 T x ( t ) cos ( 2 π T k t ) d t

and

b ( k ) = 2 T 0 T x ( t ) sin ( 2 π T k t ) d t

where T is the time interval of interest or the period of a periodic signal. Because of the orthogonality of the basis functions, afinite Fourier series formed by truncating the infinite series is an optimal least squared error approximation to x ( t ) . If the finite series is defined by

x ^ ( t ) = a ( 0 ) 2 + k = 1 N a ( k ) cos ( 2 π T k t ) + b ( k ) sin ( 2 π T k t ) ,

the squared error is

ε = 1 T 0 T | x ( t ) - x ^ ( t ) | 2 d t

which is minimized over all a ( k ) and b ( k ) by [link] and [link] . This is an extraordinarily important property.

It follows that if x ( t ) L 2 [ 0 , T ] , then the series converges to x ( t ) in the sense that ε 0 as N [link] , [link] . The question of point-wise convergence is more difficult. A sufficient condition that is adequate for mostapplication states: If f ( x ) is bounded, is piece-wise continuous, and has no more than a finite number of maxima over an interval, the Fourierseries converges point-wise to f ( x ) at all points of continuity and to the arithmetic mean at points of discontinuities. If f ( x ) is continuous, the series converges uniformly at all points [link] , [link] , [link] .

A useful condition [link] , [link] states that if x ( t ) and its derivatives through the q th derivative are defined and have bounded variation, the Fourier coefficients a ( k ) and b ( k ) asymptotically drop off at least as fast as 1 k q + 1 as k . This ties global rates of convergence of the coefficients to local smoothness conditions of the function.

The form of the Fourier series using both sines and cosines makes determination of the peak value or of the location of a particularfrequency term difficult. A different form that explicitly gives the peak value of the sinusoid of that frequency and the location or phase shift ofthat sinusoid is given by

x ( t ) = d ( 0 ) 2 + k = 1 d ( k ) cos ( 2 π T k t + θ ( k ) )

and, using Euler's relation and the usual electrical engineering notation of j = - 1 ,

e j x = cos ( x ) + j sin ( x ) ,

the complex exponential form is obtained as

x ( t ) = k = - c ( k ) e j 2 π T k t

where

c ( k ) = a ( k ) + j b ( k ) .

The coefficient equation is

c ( k ) = 1 T 0 T x ( t ) e - j 2 π T k t d t

The coefficients in these three forms are related by

| d | 2 = | c | 2 = a 2 + b 2

and

θ = a r g { c } = tan - 1 ( b a )

It is easier to evaluate a signal in terms of c ( k ) or d ( k ) and θ ( k ) than in terms of a ( k ) and b ( k ) . The first two are polar representation of a complex value and the last is rectangular. Theexponential form is easier to work with mathematically.

Questions & Answers

how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
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
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
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
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit 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
Good
Berger describes sociologists as concerned with
Mueller Reply
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, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing and digital filter design (draft)' conversation and receive update notifications?

Ask