{"id":1972,"date":"2013-01-01T20:53:00","date_gmt":"2013-01-02T02:53:00","guid":{"rendered":"http:\/\/www.comm.utoronto.ca\/dkundur\/?page_id=1972"},"modified":"2024-07-16T17:52:17","modified_gmt":"2024-07-16T23:52:17","slug":"discrete-time-systems","status":"publish","type":"page","link":"https:\/\/www.comm.utoronto.ca\/dkundur\/course\/discrete-time-systems\/","title":{"rendered":"Discrete-Time Systems"},"content":{"rendered":"<table class=\"CoursePages2\" border=\"0\" width=\"100%\" cellspacing=\"5\" cellpadding=\"5\">\n<tbody>\n<tr>\n<td><img decoding=\"async\" class=\"lazyload alignleft\" title=\"Discrete-Time Systems\" src=\"data:image\/svg+xml,%3Csvg%20xmlns%3D%27http%3A%2F%2Fwww.w3.org%2F2000%2Fsvg%27%20width%3D%27150%27%20height%3D%27150%27%20viewBox%3D%270%200%20150%20150%27%3E%3Crect%20width%3D%27150%27%20height%3D%27150%27%20fill-opacity%3D%220%22%2F%3E%3C%2Fsvg%3E\" data-orig-src=\"\/dkundur\/wp-content\/uploads\/DstTimeSystemsLogo3.png\" alt=\"Discrete-Time Systems\" width=\"150\" height=\"150\" \/>Discrete-time systems theory lays the foundation for digital signal processing algorithm development and design. This course first provides an introduction to discrete-time systems and then elucidates advanced topics in discrete-time signal processing. Topics include: (a) elementary digital signal processing (DSP): discrete-time signals and systems and their properties, z-Transform, frequency analysis, sampling and reconstruction, discrete Fourier transform; (b) multirate digital signal processing: decimation, interpolation, sampling rate conversion, digital filter banks; (c) linear Prediction and Optimum Linear Filters: linear prediction algorithms and properties, Wiener filters; and (d) adaptive filters: LMS algorithm, RLS algorithm.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<hr \/>\n<p>&nbsp;<\/p>\n<h3>Learning Outcomes and Objectives<\/h3>\n<p>The notes and assignments on this resource page are intended to support learning of:<\/p>\n<ul>\n<li>advanced DSP problem solving skills<\/li>\n<li>rigorous thinking and creative visualization<\/li>\n<li>ability to overcome obstacles through ingenuity and resourcefulness<\/li>\n<\/ul>\n<p>and encourage:<\/p>\n<ul>\n<li>an appreciation of the importance of advanced DSP for a broad class of engineering applications<\/li>\n<li>development of technical confidence<\/li>\n<li>a positive learning environment<\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<h3>Text Book and Relevant Sections<\/h3>\n<p>These course resources make use of the following text:<\/p>\n<p style=\"padding-left: 30px;\">John G. Proakis and Dimitris G. Manolakis, <em>Digital Signal Processing: Principles, Algorithms, and Applications<\/em>, 4th edition, 2007.<\/p>\n<p>The following text book sections are covered.<\/p>\n<ul>\n<li>Chapter 1: 1.1, 1.2, 1.3, 1.4<\/li>\n<li>Chapter 2: 2.1, 2.2, 2.3, 2.4, 2.5<\/li>\n<li>Chapter 3: 3.1, 3.2, 3.3, 3.4, 3.5.5<\/li>\n<li>Chapter 4: 4.1, 4.2, 4.3, 4.4<\/li>\n<li>Chapter 5: 5.1, 5.2, 5.4, 5.5<\/li>\n<li>Chapter 6: 6.1, 6.2, 6.4, 6.5<\/li>\n<li>Chapter 7: 7.1, 7.2<\/li>\n<li>Chapter 8: 8.1<\/li>\n<li>Chapter 11: 11.1, 11.2, 11.3, 11.4, 11.5, 11.9, 11.10, 11.11<\/li>\n<li>Chapter 13: 13.1, 13.2, 13.3<\/li>\n<\/ul>\n<div>\n<hr \/>\n<p>&nbsp;<\/p>\n<\/div>\n<h3>Topics and Notes<\/h3>\n<ul>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap1.pdf\" target=\"_blank\" rel=\"noopener\">Introduction<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap2.pdf\" target=\"_blank\" rel=\"noopener\">Discrete-Time Signals and Systems<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap3.pdf\" target=\"_blank\" rel=\"noopener\">z-Transform and Applications<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap4.pdf\" target=\"_blank\" rel=\"noopener\">Frequency Analysis of Signals<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap5.pdf\" target=\"_blank\" rel=\"noopener\">Frequency Domain Analysis of LTI Systems<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap6.pdf\" target=\"_blank\" rel=\"noopener\">Sampling and Reconstruction of Signals<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap7.pdf\" target=\"_blank\" rel=\"noopener\">The Discrete Fourier Transform<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap8.pdf\" target=\"_blank\" rel=\"noopener\">Efficient Computation of the DFT: FFT Algorithms<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap11a.pdf\" target=\"_blank\" rel=\"noopener\">Intro to Decimation and Interpolation<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap11b.pdf\" target=\"_blank\" rel=\"noopener\">Sampling Rate Conversion by a Rational Factor<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap11c.pdf\" target=\"_blank\" rel=\"noopener\">Polyphase Structures, Noble Identities, and Multirate Signal Processing for Phase Shifters and Subband Coding<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap11d.pdf\" target=\"_blank\" rel=\"noopener\">Digital Filter Banks and Quadrature Mirror Filter Banks<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap13a.pdf\" target=\"_blank\" rel=\"noopener\">Introduction to Adaptive Filtering and Background<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap13b.pdf\" target=\"_blank\" rel=\"noopener\">The LMS Algorithm<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/notes\/Kundur_DTS_Chap13c.pdf\" target=\"_blank\" rel=\"noopener\">The RLS Algorithm<\/a><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<h3>Problem Sets and Solutions<\/h3>\n<table class=\"CoursePages2\" border=\"0\" width=\"880px\" cellspacing=\"7\" cellpadding=\"3\">\n<tbody>\n<tr>\n<td width=\"352\">Topic(s)<\/td>\n<td width=\"440\">Problem Set Questions<\/td>\n<td width=\"88\">Solutions<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Introduction to Discrete-Time Systems<\/td>\n<td bgcolor=\"#FFFFFF\">1.2, 1.4, 1.6, 1.7, 2.2, 2.4, 2.5, 2.7 (a,g,j,k)<\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_1.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Discrete-Time Analysis and the z-Transform<\/td>\n<td bgcolor=\"#FFFFFF\"><span class=\"style8\">2.13, 2.23, 3.6, 3.18(d), 3.23, 3.40<\/span><\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_2.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Discrete-Time Frequency Domain Analysis<\/td>\n<td bgcolor=\"#FFFFFF\"><span class=\"style5 style8\">4.13, 4.16, 4.17,4.18,4.23, 5.4 (f,l), 5.11,5.65(a,c)<\/span><\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_3.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Sampling and Reconstruction<\/td>\n<td bgcolor=\"#FFFFFF\">6.1, 6.10, 6.11, 6.13<\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_4.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">DFT and FFT<\/td>\n<td bgcolor=\"#FFFFFF\"><span class=\"style5 style8\">7.1, 7.3, 7.7, 7.13(a), 7.23(a,b,c,h), 7.28, 8.1, 8.3, 8.4, 8.7<\/span><\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_5.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Sampling Rate Conversion<\/td>\n<td bgcolor=\"#FFFFFF\"><span class=\"style5 style8\">11.1, 11.2, 11.3, 11.4, 11.5<\/span><\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_6.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Digital Filter Banks<\/td>\n<td bgcolor=\"#FFFFFF\">11.9, 11.11, 11.12, 11.13, 11.15, 11.25, 11.29<\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_7.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<tr>\n<td bgcolor=\"#FFFFFF\">Adaptive Filtering<\/td>\n<td bgcolor=\"#FFFFFF\">13.1, 13.2, 13.3, 13.12, 13.15<\/td>\n<td bgcolor=\"#FFFFFF\">\n<div align=\"left\"><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/hwsol\/DTS_sol_8.pdf\" target=\"_blank\" rel=\"noopener\">PDF<\/a><\/div>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Note<\/strong>: Problem set solutions are courtesy of Julien Jainsky and Shoeb Mohammed who were former TAs for the course.<\/p>\n<hr \/>\n<p>&nbsp;<\/p>\n<h3>Projects<\/h3>\n<ul>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/project\/Kundur_DTS_Cepstral_Analysis.pdf\" target=\"_blank\" rel=\"noopener\">Cepstral Analysis<\/a><\/li>\n<li><a href=\"http:\/\/www.comm.utoronto.ca\/dkundur\/course_info\/discrete-time-systems\/project\/Kundur_DTS_Rate_Conversion.pdf\" target=\"_blank\" rel=\"noopener\">Sampling Rate Conversion<\/a><\/li>\n<\/ul>\n<hr \/>\n<p>&nbsp;<\/p>\n<table class=\"CoursePages2\" border=\"0\" width=\"100%\" cellspacing=\"5\" cellpadding=\"5\">\n<tbody>\n<tr>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n","protected":false},"excerpt":{"rendered":"<p>Discrete-time systems theory lays the foundation for digital signal processing 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