Lecture 8

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# Lecture 8 - PowerPoint PPT Presentation

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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Presentation Transcript
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
• 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
• Example: simple case of 3 terms
• Exponential representation:
• with k=2pn/l.
Example
• Periodic top-hat:
• N.B.

Fourier transform

f(x)

Zero when n

is a multiple of 4

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
• 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
• 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
• 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
• Common transforms
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: 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 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

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