By J. F. James
This re-creation of a profitable textbook for undergraduate scholars in physics, desktop technology and electric engineering describes very important modern rules in sensible technology and data expertise at an comprehensible point, illustrated with labored examples and copious diagrams. the sphere is roofed widely instead of extensive, and contains references to extra prolonged works on a number of themes. This new version is a bit of improved, and comprises extra new fabric within the functions sections.
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Extra resources for A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering
The double-sawtooth waveform. The double-sawtooth waveform: This can not be regarded as the convolution of two rectangular waveforms of equal mark-space4 ratio, since the etlect of integration is to give an embarrassing infinity. Instead it is the convolution of a top-hat of width a with another identical top-hat and with a Dirac comb of period 2a. Thus: I1 a (t) * I1 a (t) * 1lI2a (t) ~ (a/2) sinc2 rrva . 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 , ...
9: 1 h 1 x_ Fig. 9. A saw-tooth waveform, antisymmetrical about the origin. 8 Worked examples By choosing the origin half way up one of the teeth, the function is clearly made antisymmetrical, so that there are no cosine amplitudes. (2rrnx) B n = -2jP/2 2xh - SIll - - dx P -P/2 P P h [ -x cos (2rrnx) =4-p2 P = (- 2h / rr n ) cos rr n 2 P p --+ s i(2rrnx)] n - - P/2 2 2 2rrn 4rr n P -P/2 since sin rr n =0 so that Bo = 0 Bn = (-It+ I (2h/rrn), n -=I- 0 As a matter of interest, it is worth while calculating the sine-amplitudes when the origin is taken at the tip of a tooth, to see how changing the position of the origin changes the amplitudes.
5 Other theorems 31 Transforming both sides: 8(x + a12) - 8(x - a12) ~ e- Jfipa = - eJfipa = -2i sinrrpa -2rrip[a sinc(rrpa)] The theorem extends to further derivatives: and much use is made of this in mathematics. Example 2: if the moment of inertia about the y-axis of a symmetrical curve is infinite, its Fourier transform has a cusp at the origin. Because: 00 100 j(x)dx = ¢(O) and then if there is a discontinuity in (a j I ax) at the origin. Example 3: the differential equation of simple harmonic motion is: 2 md F(t)ldt 2 + kF(t) = 0 where F(t) is the displacement ofthe oscillator from equilibrium at time t.
A Student's Guide to Fourier Transforms: With Applications in Physics and Engineering by J. F. James