1、What do we hope to achieve with theFourier Transform?We desire a measure of the frequencies present in a wave.This willlead to a definition of the term,the“spectrum.”Plane waves have only one frequency,w.This light wave has many frequencies.And the frequency increases in time(from red to blue).It wi
2、ll be nice if our measure also tells us when each frequency occurs.Light electric fieldTimeLord Kelvin on Fouriers theoremFouriers theorem is not only one of the most beautiful results of modern analysis,but it may be said to furnish an indispensable instrument in the treatment of nearly every recon
3、dite question in modern physics.Lord KelvinJoseph Fourier,our heroFourier was obsessed with the physics of heat and developed the Fourier series and transform to model heat-flow problems.Anharmonic waves are sums of sinusoids.Consider the sum of two sine waves(i.e.,harmonic waves)of different freque
4、ncies:The resulting wave is periodic,but not harmonic.Most waves are anharmonic.Fourier decomposing functionsHere,we write asquare wave as a sum of sine waves.Any function can be written as thesum of an even and an odd function()()()/2()()()/2()()()E xf xfxO xf xfxf xE xO xE(-x)=E(x)O(-x)=-O(x)Fouri
5、er Cosine Series Because cos(mt)is an even function(for all m),we can write an even function,f(t),as:where the set Fm;m=0,1,is a set of coefficients that define the series.And where well only worry about the function f(t)over the interval(,).f(t)1Fmcos(mt)m0 The Kronecker delta function ,1 if 0 if m
6、 nmnmnFinding the coefficients,Fm,in a Fourier Cosine SeriesFourier Cosine Series:To find Fm,multiply each side by cos(mt),where m is another integer,and integrate:But:So:only the m=m term contributesDropping the from the m:yields the coefficients for any f(t)!01()cos()mmf tFmtf(t)cos(m t)dt1m0Fmcos
7、(mt)cos(m t)dt,cos()cos()0m mif mmmtm t dtif mm,01()cos()mm mmf tm t dtF()cos()mFf tmt dtFourier Sine SeriesBecause sin(mt)is an odd function(for all m),we can write any odd function,f(t),as:where the set Fm;m=0,1,is a set of coefficients that define the series.where well only worry about the functi
8、on f(t)over the interval(,).f(t)1F msin(mt)m0Finding the coefficients,Fm,in a Fourier Sine Series Fourier Sine Series:To find Fm,multiply each side by sin(mt),where m is another integer,and integrate:But:So:only the m=m term contributes Dropping the from the m:yields the coefficients for any f(t)!f(
9、t)1F msin(mt)m001()sin()sin()sin()mmf tm t dtFmtm t dt,sin()sin()0m mif mmmtm t dtif mm,01()sin()mm mmf tm t dtF()sin()mFf tmt dtFourier Serieseven component odd component where and0011()cos()sin()mmmmf tFmtFmtFmf(t)cos(mt)dtF mf(t)sin(mt)dtSo if f(t)is a general function,neither even nor odd,it can
10、 be written:We can plot the coefficients of a Fourier SeriesWe really need two such plots,one for the cosine series and another for the sine series.Fm vs.mm525201510301.50Discrete Fourier Series vs.Continuous Fourier Transform Fm vs.mmAgain,we really need two such plots,one for the cosine series and
11、 another for the sine series.Let the integer m become a real number and let the coefficients,Fm,become a function F(m).F(m)The Fourier Transform Consider the Fourier coefficients.Lets define a function F(m)that incorporates both cosine and sine series coefficients,with the sine series distinguished
12、by making it the imaginary component:Lets now allow f(t)to range from to,so well have to integrate from to,and lets redefine m to be the“frequency,”which well now call w:F(w)is called the Fourier Transform of f(t).It contains equivalent information to that in f(t).We say that f(t)lives in the“time d
13、omain,”and F(w)lives in the“frequency domain.”F(w)is just another way of looking at a function or wave.f(t)cos(mt)dt if(t)sin(mt)dtF(m)Fm i Fm=()()exp()Ff ti t dtwwThe FourierTransform The Inverse Fourier Transform The Fourier Transform takes us from f(t)to F(w).How about going back?Recall our formu
14、la for the Fourier Series of f(t):Now transform the sums to integrals from to,and again replace Fm with F(w).Remembering the fact that we introduced a factor of i(and including a factor of 2 that just crops up),we have:0011()cos()sin()mmmmf tFmtFmt1()()exp()2f tFi t dwwwInverse Fourier Transform The
15、 Fourier Transform and its Inverse The Fourier Transform and its Inverse:So we can transform to the frequency domain and back.Interestingly,these functions are very similar.There are different definitions of these transforms.The 2 can occur in several places,but the idea is generally the same.Invers
16、e Fourier TransformFourierTransform ()()exp()Ff ti t dtww1()()exp()2f tFi t dwwwFourier Transform NotationThere are several ways to denote the Fourier transform of a function.If the function is labeled by a lower-case letter,such as f,we can write:f(t)F(w)If the function is labeled by an upper-case
17、letter,such as E,we can write:or:()()E tEw()()E tE tFSometimes,this symbol is used instead of the arrow:The Spectrum We define the spectrum of a wave E(t)to be:2()E tFThis is our measure of the frequencies present in a light wave.Example:the Fourier Transform of arectangle function:rect(t)1/21/21/21
18、/21()exp()exp()1exp(/2)exp(exp(/2)exp(2sin(Fi t dti tiiiiiiiwwwwwwwwwwww (sinc(FwwImaginary Component=0F(w)wSinc(x)and why its importantSinc(x/2)is the Fourier transform of a rectangle function.Sinc2(x/2)is the Fourier transform of a triangle function.Sinc2(ax)is the diffraction pattern from a slit.
19、It just crops up everywhere.The Fourier Transform of the trianglefunction,D D(t),is sinc2(ww)w02sinc(/2)w1t0()tD11/2-1/2The triangle function is just what it sounds like.Well prove this when we learn about convolution.Sometimes people use L(t),too,for the triangle function.Example:the Fourier Transf
20、orm of adecaying exponential:exp(-at)(t 0)0000(exp()exp()exp()exp()11exp()exp()exp(0)10 11Fati t dtati t dtait dtaitaiaiaiaiwwwwwwwww 1(Fiiaww A complex Lorentzian!Example:the Fourier Transform of aGaussian,exp(-at2),is itself!222exp()exp()exp()exp(/4)atati t dtawwFt02exp()atw02exp(/4)awThe details
21、are a HW problem!Some functions dont have Fourier transforms.The condition for the existence of a given F(w)is:Functions that do not asymptote to zero in both the+and directions generally do not have Fourier transforms.So well assume that all functions of interest go to zero at.()f tdt Expanding the
22、 Fourier transform of a function,f(t):Expanding further:()Re()cos()Im()sin()Ff tt dtf tt dtwwwFourier Transform Symmetry Properties ReF(w w)ImF(w w)=0 if Re or Imf(t)is odd =0 if Re or Imf(t)is evenEven functions of w wOdd functions of w w()Re()Im()cos()sin()Ff tif ttit dtwwwIm()cos()Re()sin()if tt dtif tt dtwwFourier Transform Symmetry Examples I Fourier Transform Symmetry Examples II 谢谢！
