Fourier%20Analysis.

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
  writtenwhere,
• 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 k2pn/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.
• ExampleTop hat
• Similar to Fourier series but now a continuous
  function of k.

7
Fourier transform of a Gaussian

• Gaussain with r.m.s. deviation Dxs.
• Note
• Fourier transform
• Integration can be performed by completing the
  square of the exponent -(x2/2s2ikx).
• where,

Öp
8
Transforms

• The Fourier transform of a Gaussian is a
  Gaussian.
• Note Dk1/s. i.e. DxDk1
• 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/kconstant (independent of k) then the
  disturbance is unchanged i.e. f(x-vt).
• We have a non-dispersive wave.
• When w/kf(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.
• ConverselyThe 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.