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practice exam - period 1 Find the fourier series of S t 5...

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Math 445 - Spring 2008 Exam 1 Practice Page 1 1. Show that the fourier transform is linear. 2. (a) Find the fourier series expansion for the odd extension of f ( t ) where f ( t ) = πt - t 2 for 0 t π (b) Find the fourier series expansion for the even extension of f ( t ) where f ( t ) = πt - t 2 for 0 t π 3. Let f be a function deﬁned on [ - 1 / 2 , 1 / 2] by f ( t ) = 16 π 4 t 4 - 8 π 4 t 2 and let g ( t ) be the periodization of f ( t ) with period 1. Given the Fourier series for g ( t ) is 7 / 15 π 4 - 48Σ n =1 ( - 1) n n 4 cos(2 πnt ) show that Σ n =1 1 n 4 = π 90 . 4. Let S ( t ) be the sawtooth function, that is, S ( t ) = t for 0 < t < 1 and then periodized to have
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Unformatted text preview: period 1. Find the fourier series of S ( t ). 5. Let f(t) be a signal, s a number, and deﬁne g ( t ) = f ( t ) cos ( s t ) . Assuming that all the transforms exist, show that ˆ g ( w ) = 1 / 2 ˆ f ( w-s ) + 1 / 2 ˆ f ( w + s ). 6. Show that under some conditions on f , the following relation holds F{ f } ( w ) = iw F{ f } ( w ). State these conditions. 7. Show that F{ f * g } ( w ) = √ π F{ f } ( w ) F{ g } ( w )....
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