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ecen314-fall11-hw4.5

ecen314-fall11-hw4.5 - a k for the following...

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ECEN 314: Signals and Systems, Fall ‘11 Homework #4.5 Homework Assignment #4.5 Due date – None. Just for practice. Problem 1. Fourier Series Representation Consider a continuous-time periodic signal x ( t ) , which is real valued and has a fundamental period of T = 8 . The nonzero Fourier series coefficients for x ( t ) are as follows: a 1 = a - 1 = 2 , a 3 = a * - 3 = 4 j Express x ( t ) in the form x ( t ) = k a k cos( ω k t + φ k ) . Problem 2. Fourier Series Representation Consider the following continuous-time periodic signal x ( t ) : x ( t ) = 2 + cos 2 π 3 t + sin 5 π 3 t . (a) What is the fundamental frequency w 0 of x ( t ) ? (b) Find the Fourier series coefficients a k such that x ( t ) = k a k e jkω 0 t . Problem 3. Fourier Series Representation Compute the Fourier series coefficients
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Unformatted text preview: a k for the following continuous-time periodic signal x ( t ) : x ( t ) = ³ 1 . 5 , ≤ t < 1-1 . 5 , 1 ≤ t < 2 with fundamental frequency ω = π . Problem 4. Fourier Series Representation Let x 1 ( t ) be a continuous-time periodic signal with fundamental frequency ω 1 and Fourier series coefficients a k . Now, define x 2 ( t ) as follows: x 2 ( t ) = x 1 (1-t ) + x 1 ( t-1) . (a) What is the fundamental frequency ω 2 of the signal x 2 ( t ) ? (b) Find the relationship between the Fourier coefficients b k of x 2 ( t ) and the coefficients a k ....
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