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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
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-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

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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