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Unformatted text preview: Homework 9 (ECE220 - Fall 2007) • Due: Wednesday, November 14 by 2PM in the drop box outside 219 Phillips Hall. • Reasoning and work must be shown to gain full/partial credit. • WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN! 1. (15 points) Sampling The sampling theorem derived in class, requires that a signal have a Fourier Transform. In general that is not necessary. Use the following steps to prove that any bandlimited signal can be sampled and perfectly reconstructed. i) Assume that your signal is bandlimited by a bandwidth of B. This means if the signal x ( t ) was filtered by an IDLF (ideal low-pass filter) the output would be x ( t ). Mathematically describe this process in the time domain. Remember that you cannot use the Fourier Transform, as we do not know if one exists for the signal x ( t ). ii) Using the sampling theorem write sinc( πB ( t- τ )) as a reconstruction of its samples. iii) Use the results from the first two parts to show that x ( t ) can be written as a sum of its own samples. (You may have to interchange an integral and a summation, you may assume that it is ok to do so.) Motivation: This extension of the sampling period to a broader group of signals is important in...
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