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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.

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

• 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-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

• 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.