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114_1_sol_midterm_2011_Part1 - EE114 Winter Quarter 2011...

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EE114 Introduction to Speech and Image Processing Winter Quarter, 2011 Page 1 of 8 EE114 Midterm Exam Solutions Notes: - For all problems please circle or otherwise clearly indicate your answers! The use of calculators or other electronic devices with calculator-like functionality is not permitted on this test. 1. Signal processing basics (25 points) 1a) Sampling and Reconstruction (7 points) Suppose that a signal consisting of a pure cosine at 4000 Hz is subject to ideal sampling at a sampling frequency of 5000 Hz, and then subject to reconstruction using an ideal low pass filter with a cutoff of 2500 Hz. What is the frequency of the resulting cosine? Answer: 1000 Hz. The spectrum of the original cosine is a pair of delta functions at plus and minus 4 KHz. After sampling, these delta functions will be replicated at intervals of 5 KHz. The delta function locations in KHz will then be . . . .-11,-9,-6, -4, -1,1, 4, 6, 9, 11, 14, 16 . . . After application of the low pass filter, only those delta functions at plus and minus 1KHz will remain. 1b) 1D Convolution (12 points) Figure 1 below shows a triangle function that starts on the horizontal axis at position 4, rises to a maximum height of 1 at 5, and then declines again until it reaches a value of zero again horizontal axis position 6.
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