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Electrical Engineering 20N - Fall 1998 - Midterm 2

Electrical Engineering 20N - Fall 1998 - Midterm 2 - EECS...

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EECS 20. Midterm 2. November 6, 1998 Solution 1) 24 points . Consider a continuous-time signal x with the following finite Fourier series expansion: for all t Reals , x t k t k ( ) cos( ) = = ϖ 0 0 4 where ϖ 0 = π /4 radians/second. Define Sampler T : ContSignals DiscSignals to be a sampler with sampling interval T (in seconds). Define IdealDiscToCont : DiscSignals ContSignals to be an ideal bandlimited interpolation system. I.e., given a discrete-time signal y ( n ), it constructs the continuous-time signal w where for all t Reals , w t y nT p t nT k ( ) ( ) ( ) = - =-∞ where the pulse p is the sinc function, p t t T t T ( ) sin( / ) / = π π a) Give an upper bound on T (in seconds) such that x = IdealDiscToCont ( Sampler T (x)). b) Suppose that T = 4 seconds. Give a simple expression for y = Sampler T ( x ). c) For the same T = 4 seconds, give a simple expression for w = IdealDiscToCont ( Sampler T ( x )). solution to problem 1: a) The highest frequency term is the k = 4 term in the summation, which has frequency 4 ϖ 0 = π radians/second. By the Nyquist-Shannon sampling theorem, we have to sample at a sampling frequency at least twice this, or 2 π radians/second or 1 Hz. This means that the sampling interval must be at most 1/(1 Hz) = 1 second.
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