EE301HW3 - EE301 Signals and Systems I HW#3 Due 5:00pm 1(a...

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EE301 Signals and Systems I HW#3 Due : November 19, 2009, 5:00pm. 1. (a) Show that if the response of an LTI system to x ( t ) is y ( t ), then the response of the system to x 0 ( t ) = d x ( t ) d t is y 0 ( t ) by: i. directly from the properties of linearity and time-invariance and the definition of derivative, i.e., x 0 ( t ) = lim h 0 x ( t ) - x ( t - h ) h ii. differentiating the convolution integral. (b) Show that if the same input is applied to another LTI system with an impulse response equal to the derivative of the previous system in part (a), i.e., h 0 ( t ), then the output is y 0 ( t ). (c) An LTI system has the response y ( t ) = sin ω 0 t to input x ( t ) = e - 5 t u ( t ). Find its impulse response. 2. Determine the periodic signal x ( t ) with period 4 for each of the Fourier Series (FS) coefficients given below: (a) a k = ( 0 , k = 0 ( j ) k sin kπ/ 4 , k 6 = 0 (b) a k = ( - 1) k sin kπ/ 8 2 (c) a k = ( jk, | k | < 3 0 , otherwise 3. Let x ( t ) be a periodic signal with fundamental period T and spectral coefficients a k . Derive the spectral coefficients of each of the following signals in terms of a k : (a) x ( t - t 0 ) + x ( t + t 0 ) (b) Ev { x ( t
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