1 Waves 8 Lecture 8 Fourier Analysis. D Aims: ëFourier Theory: > Description of waveforms in terms of a superposition of harmonic waves. > Fourier series (periodic functions); > Fourier transforms (aperiodic functions). > Wavepackets ëConvolution > convolution theorem.

2 Waves 8 Fourier Theory ëIt is possible to represent (almost) any function as a superposition of harmonic functions. D Periodic functions: ëFourier series D Non-periodic functions: ëFourier transforms D Mathematical formalism  Function f(x), which is periodic in x, can be written: where,  Expressions for A n and B n follow from the “orthogonality” of the “basis functions”, sin and cos.

3 Waves 8 Complex notation D Example: simple case of 3 terms D Exponential representation:  with k=2  n/l.

4 Waves 8 Example D Periodic top-hat: ëN.B. Fourier transform f(x)f(x) f(x)f(x) Zero when n is a multiple of 4 Zero when n is a multiple of 4

5 Waves 8 Fourier transform variables  x and k are conjugate variables. ëAnalysis applies to a periodic function in any variable.  t and  are conjugate. D Example: Forced oscillator  Response to an arbitrary, periodic, forcing function F ( t ). We can represent F ( t ) using [6.1].  If the response at frequency n  f is R(n  f ), then the total response is Linear in both response and driving amplitude

6 Waves 8 Fourier Transforms D Non-periodic functions:  limiting case of periodic function as period . The component wavenumbers get closer and merge to form a continuum. (Sum becomes an integral) ëThis is called Fourier Analysis.  f(x) and g(k ) are Fourier Transforms of each other. D Example: Top hat  Similar to Fourier series but now a continuous function of k.

7 Waves 8 Fourier transform of a Gaussian  Gaussain with r.m.s. deviation  x= . ëNote ëFourier transform  Integration can be performed by completing the square of the exponent -(x 2 /2  2 +ikx). ëwhere, ë = 

8 Waves 8 Transforms ëThe Fourier transform of a Gaussian is a Gaussian.  Note:  k=1/ . i.e.  x  k=1 ëImportant general result: > “Width” in Fourier space is inversely related to “width” in real space. (same for top hat) D Common functions D Common functions (Physicists crib-sheet)   -function  constant cosine  2  -functions sine  2  -functions infinite lattice  infinite lattice of  -functions of  -functions top-hat  sinc function Gaussian  Gaussian ëIn pictures………...  -function

9 Waves 8 Pictorial transforms D Common transforms

10 Waves 8 Wave packets D Localised waves ëA wave localised in space can be created by superposing harmonic waves with a narrow range of k values. ëThe component harmonic waves have amplitude  At time t later, the phase of component k will be kx-  t, so  Provided  /k =constant (independent of k ) then the disturbance is unchanged i.e. f(x-vt). ëWe have a non-dispersive wave.  When  /k=f(k) the wave packet changes shape as it propagates. ëWe have a dispersive wave.

11 Waves 8 Convolution D Convolution: a central concept in Physics. ëIt is the “smearing” or “blurring” of one function by the other. ëExamples occur in all experimental situations where the limited resolution of the apparatus results in a measurement “broader” than the original.  In this case, f 1 (say) represents the true signal and f 2 is the effect of the measurement. f 2 is the point spread function. h is the convolution of f 1 and f 2 Convolution symbol Convolution integral

12 Waves 8 Convolution theorem ëConvolution and Fourier transforms D Convolution theorem: ëThe Fourier transform of a PRODUCT of two functions is the CONVOLUTION of their Fourier transforms. ëConversely: The Fourier transform of the CONVOLUTION of two functions is a PRODUCT of their Fourier transforms. ëProof: F.T. of f 1.f 2 F.T. of f 1.f 2 Convolution of g 1 and g 2 Convolution of g 1 and g 2

13 Waves 8 Convolution…………. D Summary: ëIf, then and D Examples: ëOptical instruments and resolution > 1-D idealised spectrum of “lines” broadened to give measured spectrum > 2-D: Response of camera, telescope. Each point in the object is broadened in the image. ëCrystallography. Far field diffraction pattern is a Fourier transform. A perfect crystal is a convolution of “the lattice” and “the basis”.

14 Waves 8 Convolution Summary D Must know…. ëConvolution theorem ëHow to convolute the following functions.   -function and any other function. > Two top-hats > Two Gaussians.