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Cosine and Sine Fourier Transforms

Problem 1: Consider a function f(t) that is zero for t<0 and equals e^-t/2r for t>=0. Find its Cosine and Sine Fourier Transforms A(w) and B(w). Make a nice plot of A(w) and B (w). Find a convenient value of tau.

Problem 2: Find the Cosine and Sine Fourier Transforms A(w) and B(w) for the sinusoidal pulse f(t) given by

f(t) = sin(w_0 * t) for-t_0 / 2 <= t <= t_0 / 2

and f(t) = 0 outside this range. Make plots of B(w) for cases where
(i) T = 2pie/w_0 << t_0
(ii) T = 2pie/w_0 = t_0
(iii) T = 2pie / w_0 >> t_0

Problem 3: Now suppose that a pulse of duration t_0 as in Problem 2 contains two sine waves, ie.

f_2 (t) = sin(w_0 t) + sin(w_1 t) for - t_0 / 2 <= t <= t_0/2

and f(t) = 0 outside this range. Use the result from Problem 2 to write down (no more evaluation of integrals needed) the Cosine and Sine Fourier Transforms A(w) and B(w) for f_2(t).

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Solution Summary

The solution to problem (1) includes three pages of calculations neatly formatted in an attached Word document and graphs for A(w), for tau = 0.5 and B(w) for the same value of tau. The calculations and answers for problem (2) and (3) are formatted and explained. 12 pages.

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