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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

- 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.

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.

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