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Lecture 8

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

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Lecture 8

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

  2. Fourier Theory • It is possible to represent (almost) any function as a superposition of harmonic functions. • Periodic functions: • Fourier series • Non-periodic functions: • Fourier transforms • Mathematical formalism • Function f(x), which is periodic in x, can be written:where, • Expressions for An and Bn follow from the “orthogonality” of the “basis functions”, sin and cos.

  3. Complex notation • Example: simple case of 3 terms • Exponential representation: • with k=2pn/l.

  4. Example • Periodic top-hat: • N.B. Fourier transform f(x) Zero when n is a multiple of 4

  5. Fourier transform variables • x and k are conjugate variables. • Analysis applies to a periodic function in any variable. • t and w are conjugate. • 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 nwf is R(nwf), then the total response is Linear in both response and driving amplitude Linear in both response and driving amplitude

  6. Fourier Transforms • 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) • This is called Fourier Analysis. • f(x) and g(k) are Fourier Transforms of each other. • Example:Top hat • Similar to Fourier series but now a continuous function of k.

  7. Fourier transform of a Gaussian • Gaussain with r.m.s. deviation Dx=s. • Note • Fourier transform • Integration can be performed by completing the square of the exponent -(x2/2s2+ikx). • where, =Öp

  8. Transforms • The Fourier transform of a Gaussian is a Gaussian. • Note: Dk=1/s. i.e. DxDk=1 • Important general result: • “Width” in Fourier space is inversely related to “width” in real space. (same for top hat) • Common functions (Physicists crib-sheet) • d-function Û constant cosine Û 2 d-functions sine Û 2 d-functions infinite lattice Û infinite lattice of d-functions of d-functions top-hat Û sinc function Gaussian Û Gaussian • In pictures………... d-function

  9. Pictorial transforms • Common transforms

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

  11. Convolution • Convolution: a central concept in Physics. • It is the “smearing” or “blurring” of one function by the other. • Examples occur in all experimental situations where the limited resolution of the apparatus results in a measurement “broader” than the original. • In this case, f1 (say) represents the true signal and f2 is the effect of the measurement. f2 is the point spread function. Convolution symbol Convolution integral h is the convolution of f1 and f2 h is the convolution of f1 and f2 h is the convolution of f1 and f2

  12. Convolution theorem • Convolution and Fourier transforms • Convolution theorem: • The Fourier transform of a PRODUCT of two functions is the CONVOLUTION of their Fourier transforms. • Conversely:The Fourier transform of the CONVOLUTION of two functions is a PRODUCT of their Fourier transforms. • Proof: F.T. of f1.f2 Convolution of g1 and g2

  13. Convolution…………. • Summary: • If,thenand • Examples: • Optical 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. • Crystallography. Far field diffraction pattern is a Fourier transform. A perfect crystal is a convolution of “the lattice” and “the basis”.

  14. Convolution Summary • Must know…. • Convolution theorem • How to convolute the following functions. • d-function and any other function. • Two top-hats • Two Gaussians.

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