Download A Student's Guide to Fourier Transforms: With Applications by J. F. James PDF

By J. F. James

This re-creation of a winning textbook for undergraduate scholars in physics, desktop technological know-how and electric engineering describes vital modern principles in useful technological know-how and knowledge know-how at an comprehensible point, illustrated with labored examples and copious diagrams. the sector is roofed widely instead of extensive, and comprises references to extra prolonged works on a number of subject matters. This new version is a little elevated, and contains extra new fabric within the purposes sections.

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Extra resources for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering

Example text

The intensity pattern, sinc 2 (na sine/A), from diffraction at a single slit. W - W' is a wavefront (a surface of constant phase) and if we choose a moment when the phase is zero at the origin, the phase at x at that moment is given by (2n / A)x. sin ¢, and the phase factor that must multiply Eo is e(-2rri/A)x sinrjJ. The magnitude at P is then E = Eoe2rriro/A i: A(x)e(-2rri/A)x(sine+sinrjJ) dx and when the Fourier transform is done, the oblique incidence is accounted for by remembering that p = (sine + sin¢)/A.

For example, we can describe the electric field of a wave-packet2 by a function which is continuous, finite and single-valued everywhere, and as the wave-packet contains only a finite amount of energy, the electric field is square-integrable. 1 2 The classical nonconformist example is Weierstrass's function, W(x), which has the property that W (x) = 1 if x is rational and W (x) = 0 if x is irrational. It looks like a straight line but it is not transformable, since it can be shown that between any two rational numbers, however close, there is at least one irrational number, and between any two irrational numbers there is at least one rational number, so that the function is everywhere discontinuous.

1lIl- (v) 2a So that the amplitudes, which occur at v = 1/2a, l/a, 3/2a, ... are: 2a/rr 2 , 0, 2a/9rr 2 , 0, 2a/25rr 2 , ... 4 The term 'equal mark-space ratio' comes from radio jargon, and implies that the signal is zero for the same interval that it is not. 1 Fraunhofer diffraction The application of Fourier theory to Fraunhofer diffraction problems and to interference phenomena generally, was hardly recognized before the late 1950s. Consequently, only textbooks written since then mention the technique.

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