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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 - )

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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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
im not good at math so would this help me
Rachael Reply
yes
Asali
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Samantha
what is the problem that i will help you to self with?
Asali
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
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
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Yasmin
what is the function of carbon nanotubes?
Cesar
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
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
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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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