Midterm Exam for EECS 121: Information Theory and Probability, Spring 2000, Exams of Digital Communication Systems

Download Midterm Exam for EECS 121: Information Theory and Probability, Spring 2000 and more Exams Digital Communication Systems in PDF only on Docsity! 1 of 4 Name: _______________________________ UNIVERSITY OF CALIFORNIA College of Engineering Department of Electrical Engineering and Computer Sciences Professor David Tse Spring 2000 EECS 121 — MIDTERM (7:00-9:00 p.m., 8 Wednesday 2000) Please explain your answers carefully. There are 100 total points and Question 4, part c) is a bonus. Problem 1 (30 points) [8 pts.] 1a)Argue that for any binary code satisfying the prefix-free condition, the codeword lengths must satisfy the Kraft’s inequality: . [6 pts.] b) Is it true that for any source, the codeword lengths for the binary Huffman code must satisfy Kraft’s inequality with equality? Explain. [6 pts.] c) Suppose now the coded symbols are from a general alphabet of size . The Kraft inequality becomes: . Is it true that the Huffman code must satisfy Kraft’s inequality with equality? Explain. [10 pts.] d) Consider a source for which the letter probabilities are of the form , where is an integer. Con- struct the Huffman code and give the corresponding codeword lengths. Justify that the code is opti- mal. li{ } 2 li– 1≤ i ∑ D D li– i ∑ 1≤ 2 k– k 2 of 4 Problem 2 (30 points) Let be a zero-mean WSS Gaussian process with autocorrelation function . [6 pts.] a) Find its power spectral density. [8 pts.] b) Suppose we sample this process every seconds. Is the resulting discrete-time process Gaussian? WSS? If so, compute its autocorrelation function. [8 pts.] c) Let be the sampled process. We perform DPCM quantization by LLSE prediction of from . Find the distribution of the residual error . [8 pts.] d) The residual error is quantized by a single bit quantizer to values . Find the optimal choice of as a function of . What happens when ? X t( ){ } Rx τ( ) e τ–= T Yn{ } Yn Yn 1– Yn Ŷn– ∆± ∆ T T 0→