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solution5 - ECE 101 Linear Systems Fundamentals Problem Set...

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Unformatted text preview: ECE 101 Linear Systems Fundamentals Problem Set # 5 Solutions November 19, 2009 (solutions by Nick Spiliakos. send questions/comments to psiegel@ucsd.edu) Problem 1. (a) x(t) is bandlimited to 20 kHz, so its spectrum is zero for |f| > 20000 Hz. In radial frequency, that just means the spectrum is zero for | | > 20000*2 = 40000 rad/s. In Fourier Transform problems, its often helpful to draw a picture to see whats going on, so well do that here. Suppose X(j ) looked like the shape shown below (A): (A) (B) The signal then gets corrupted with noise in the range 15kHz < |f| < 20kHz 30000 < | | < 40000 (plot (B) above). Recall that when we sample the signal at rate s , the sampled signals FT is just the original spectrum, scaled and copied every s rad/s. To avoid overlap (i.e. aliasing), we need to ensure, as the Sampling Theorem stipulates, that s is fast enough. The only part of the spectrum were interested in is the usable (uncorrupted) part, from -30000 to 30000 rad/s (-15 to 15 kHz). When we sample the signal, as long as we dont get overlapping onto the usable part, were fine. We can permit aliasing on the noise part, because we dont care about it and are eventually going to filter it out anyway. The smallest sampling rate s that gives us copies of the spectrum which dont overlap with the usable part from -15 to 15 kHz is s = 70000 rad/s , or in Hz, f s = 35kHz . To see this, draw the copies of the spectrum you get, centered every s = 70000 rad/s (every 35 kHz): The copies of the noise overlap each other, which is fine. But the uncorrupted spectrum from -30000...
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solution5 - ECE 101 Linear Systems Fundamentals Problem Set...

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