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Full Name: ECE 6534 (Spring 2015) { Homework #4 Due Date: Feb. 17, 2015 Question #1: (2.5 pts) How many hours did you spend on this homework? Question #2: (15 pts) Linear operator systems (a) Consider the system B = A−1 where A is time-invariant and BIBO stable. Is B guaranteed to be memoryless? Causal? Time-invariant? BIBO stable? Explain why. (b) Consider the system B defined by elements ( 1 if i = j and i is even bij = ; 0 otherwise Is B memoryless? Causal? Time-invariant? BIBO stable? Explain why. What does B do? (c) Consider the system B defined by elements ( 1 if i = j=2 bij = ; 0 otherwise Is B memoryless? Causal? Time-invariant? BIBO stable? Explain why. What does B do? Question #3: (22.5 pts) Properties of eigenvalue decomposition (a) Show that An has the same eigenvectors as A for n > 0. (b) Show that if all eigenvalues of A consist of 1's or 0's, then A is a projection operator. What space does A project a vector onto? (c) Show that if A is a projection operator, then I − A is a projection operator (use eigendecom- position to show this). (d) Show that the eigenvalues of A∗A are positive and the eigenvectors are orthogonal. (e) Show that A∗A is semi-positive definite. That is, x∗A∗Ax ≥ 0 for any vector x. (f) If the eigenvalues of A are all positive, is A semi-positive definite? Explain why or why not. (g) Show that if A is full rank and λ1; λ2;::: are the eigenvalues of A, then the values 1/λ1; 1/λ2;::: are the eigenvalues of A−1. 1 v v2 6 v4 v8 v1 v3 v v v 7 10 5 v9 Question #4: (22.5 pts) Graph signals and consensus In this problem we consider the properties of graph signals and graph systems. Let v1; v2; v3;::: 2 V be defined as nodes of our graph. The nodes represent the \samples" of the data, each which contains a value at a given time. The set V contains all nodes in the graph. Let eij 2 E be an edge in our graph that directs values in node vi to node vj. The set E contains all edges in the graph. The adjacency matrix of a graph is defined by the matrix A with elements ( 1; if eij 2 E aij = 0; otherwise In a directed graph, each eij is indepedent (signals flow in only one direction). In an undirected graph, eij = eji (signals flow in both directions). The adjacency matrix is the \graph shift" matrix. For an undirected graph, the Laplacian of a graph is defined by the matrix L such that L = D − A: The diagonal matrix D is defined by the elements dii = number of non-zero entries in column i of matrix A: The matrix D is known as the degree matrix of a graph (the degree of node vi is the number of outward edges from that node). In this problem, we will analyze the matrix P = I − L Assume the adjacency matrix A is always undirected. (a) Compute the adjacency and Laplacian matrices for the graph above. (b) Show that the eigenvalues of an arbitrary, undirected adjacency matrix A are real. (c) Show that the normalized ones vector 1=k1k (1 = [1 1 1 ··· ]T ) is an eigenvector of P . (d) Determine the eigenvalue of the normalized 1 eigenvector. (e) Show that the absolute values of the eigenvalues of P are ≤ 1 for a given set of values. Determine those values of . (f) Show that when the absolute values of the eigenvalues are ≤ 1, then P nx = α1 as n ! 1. Determine α. (g) The matrix P is known as the consensus operator. Why do you think it is called this? 2 Question #5: (15 pts) The DFT Matrix Define the N-point unitary DFT matrix F where each element fkn is defined by 1 −j 2π nk fkn = p e N N Use the DFT matrix to answer the following questions. (a) For an arbitrary N × 1 vector x, determine F 2x in terms of x. (b) Show that the absolute value of every eigenvalue of F matrix is jλnj = 1, for 0 ≤ n ≤ N (c) Now specifically determine the eigenvalues for the unitary 4-point DFT matrix. (note that the N-point DFT matrix for N > 4 is not diagonalizable. Instead the matrix has a Jordan form. We will not go into this, but it is important in advanced controls applications. ) Question #6: (25 pts) Graphical Consensus The consensus operator in Question #4 is used in distributed computing to analyze multiple noisy measurements in a sensor network. The method is distributed because each node (i.e., sensor) only communicates with their nearby neighbors. There is no \fusion center" that processes all of the data together. We execute consensus by “filtering” the graph signal xk at every time step k, xk+1 = P xk : On the course website, download the MATLAB `.mat' file called \consensus.mat." Program a consensus procedure in MATLAB to find the mean value of the vector initial values in \con- sensus.mat." Build and use the adjacency matrix from the graph illustrated in question #4. Run consensus for = 1=4; 1=8; and 1=16. For each value, plot the `2 square error 2 e = kxk − xk as a function of k, where x is the mean value of initial values. Question #7: (2.5 pts) Project Identify one paper from the IEEE Signal Processing Magazine's top 50 articles (http://ieeexplore. ieee.org/xpl/topAccessedArticles.jsp?punumber=79) to present material from in class. This article should be related to your project. Please attach the paper or provide a link to the paper. In the next homework, you will further identify 2 or 3 additional (more in-depth) papers on this topic. (Note: if you think a paper outside of the given selection would be better suited for your project, please let me know and we can discuss.) 3.
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