hw1 - ECE4250 Fall 2008 Homework Set 1 Your solutions to...

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ECE4250 Fall 2008 Homework Set 1 Your solutions to these problems are due in the ECE425 hand-in box at 5:00 PM on Monday, Sept. 8, 2008. The hand-in box is located outside the south entrance to Phillips Hall 219 (usual place) and we use the upper right hand box. No late homework will be accepted without an official university excuse. Please work on these problems well in advance of the due date as we fully expect you may have questions and/or need suggestions. Problem 1 ORDINARY LOWPASS SAMPLING (a) Consider an analog (continuous-time) signal that is bandlimited to ±17 kHz which is sampled at 40 kHz. (Assume a rectangular-shaped spectrum.) Plot the spectrum of the sampled (discrete-time) signal on the interval -100 kHz to +100 kHz. Do not be concerned with the actual amplitude scale of this spectrum. (b) If you had available ideal low-pass filters, give a range of cutoff frequencies for this filter that could be used to recover the analog signal from its samples. (c) A practical (non-ideal) analog low-pass filter that is very popular is the so-called Butterworth filter. [Later we shall see that this filter "characteristic" is very useful as a "prototype" (model) for designing digital filters.] For this problem, what you need to know is that the magnitude of the Butterworth frequency response is given by: A(f) = { f c 2N / [ f 2N + f c 2N ] } 1/2 Here N is the "order" of the filter (number of poles), f c is the -3db cutoff frequency (response drops to 1/√2 or “half-power” frequency), and f is the variable frequency. What is the magnitude response of the Butterworth low-pass at f=0, f=1, and f=∞?
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