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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 - )


( 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


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


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


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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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
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