Lecture 8
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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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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

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.


  • Complex notation

    Complex notation

    • Example: simple case of 3 terms

    • Exponential representation:

      • with k=2pn/l.


    Example

    Example

    • Periodic top-hat:

      • N.B.

    Fourier transform

    f(x)

    Zero when n

    is a multiple of 4


    Fourier transform variables

    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


    Fourier transforms

    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.


    Fourier transform of a gaussian

    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


    Transforms

    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-functionÛconstantcosineÛ2 d-functions sineÛ2 d-functions infinite lattice Û infinite lattice of d-functions of d-functions top-hatÛsinc functionGaussianÛGaussian

    • In pictures………...

  • d-function


    Pictorial transforms

    Pictorial transforms

    • Common transforms


    Wave packets

    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.


    Convolution

    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


    Convolution theorem

    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


    Convolution1

    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”.


    Convolution summary

    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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