USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

USC - EE - 483

Introduction to Digital Signalto Digital SignalProcessingRichard M. LeahyDept. Electrical EngineeringUniversity of Southern CaliforniaCopyright: R. Leahy 201121Copyright: R. Leahy 2010Personalcommunications andentertainmentBiomedicalsignal a

USC - EE - 483

Outline (lecture #2) A (very) brief review of continuoustime signals and LTI sytemstime signals and LTI sytems Examples of discrete time signals Discrete time systemstime systems Building blocks Examples How do we characterize these systems?1Co

USC - EE - 483

Black box=importantOutline (lecture #3) Properties of Discrete Time LTI SystemsLinearityTime (shift) invarianceCausalityStability Time Domain Characterization of DiscreteTime LTI SystemsTime LTI Systems Impulse response Parallel and cascade s

USC - EE - 483

Outline (lecture #4) Finite Dimensional LTI Systems Finite order linear difference equationsFilidiff FIR systems and their impulse response IIR systems and their impulse response Multiple solutions for IIR forms of LDEs Importance of initial cond

USC - EE - 483

Outline (lecture #5) The Discrete Time Fourier Transform(DTFT)(DTFT)DefinitionPeriodicity and symmetriesInverse DTFTConvergencePropertiesBlack box=important1Copyright 2005, S. K. Mitra, 2006-2010 R.LeahyJean Baptiste Joseph Fourier 1768-183

USC - EE - 483

Outline (lecture #6) The Discrete Time Fourier Transform(DTFT)(DTFT) - continued DTFT Properties LTI Systems and the Frequency Response Magnitude and Phase ResponseBlack box=important1Copyright 2005, S. K. Mitra, 2006-2010 R.LeahyImportant Pro

USC - EE - 483

OutlineOutline (lecture #7) The Discrete Time Fourier Transform(DTFT)(DTFT) - continued The importance of phase in signals andsystem Phase response and discontinuities Linear phase filters Group delayBlack box All pass filters=important1Cop

USC - EE - 483

OutlineOutline (lecture #8) The Discrete Fourier Transform (DFT)Forward and inverse DFT, definition and proofDFTExamplesDFT in matrix formBasis function of the DFTThe FFT AlgorithmBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010 R.L

USC - EE - 483

OutlineOutline (lecture #9) The DFT and Other Unitary TransformsThThe unitary transformDiscrete Cosine TransformThe Haar Transform2D Transforms Relationship between DFT and DTFTbetween DFT and DTFT (coming soon .)Black box=important1Copyrig

USC - EE - 483

OutlineOutline (lecture #10) Relationship between the Fouriertransforms the Samping Theorem andtransforms, the Samping Theorem andSpectral Analysis I 1. Relationship between DTFT and DFT Relationship between FT and DTFT (samplingtheorem) Analog s

USC - EE - 483

OutlineOutline (lecture #11) Relationship between the Fouriertransforms the Samping Theorem andtransforms, the Samping Theorem andSpectral Analysis IIBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010, R.LeahyPractical Spectral Analysis

USC - EE - 483

Lecture #12 OutlineZ-transform definitiondefinitionRegions of convergenceRational Z-transforms: poles and zerosInverse Z-transformBlack box=important1Copyright 2005, S. K. Mitra; 2006-2010, R.LeahyThe z-Transform A generalization of the DTFT d

USC - EE - 483

Lecture #13 Outline LTI systems: the impulse response and thesystems: the impulse response and thesystem function. Rational z-transforms and LTI systems Causality and stability in LTI systems System functions and the frequencyfunctions and the freq

USC - EE - 483

Lecture #14 OutlineA couple of random examplesGeneralized linear phase (again)Minimum and all pass filtersDifferentiators and Hilbert transformsDesign of general purpose filters .Black box=important1Copyright 2005, S. K. Mitra/R. Leahy 2006-2010

USC - EE - 483

Lecture #15 Outline FIR Filter design develop a numericalFilter designnumericaltechnique for determing FIR filter coefficientsto optimally approximate the desired frequencyresponse. Least squares design Design using windows Equiripple filter desi

USC - EE - 483

Lecture #16 Outline An extended, complete FIR Filter designextended complete FIR Filter designexampleBlack box=important#Copyright 2005, S. K. Mitra/ RM Leahy 2008-2010FIR Design Example Well conclude our study of FIR filter design withconclude

USC - EE - 483

Lecture #17 Outline IIR filter Designfilter Design Analog prototype filters and their designBlack box=important#Copyright 2005, S. K. Mitra/ RM Leahy 2008Analog Prototypes The set of analog filters that are mostset of analog filters that are mo

USC - EE - 483

Lecture #18 Outline IIR filter Designfilter Design The Bilinear transform design method Spectral transformations IIR design by numerical optimizationBlack box=importantCopyright 2005, S. K. Mitra/ RM Leahy 2009Bilinear Transformation Method Ide

USC - EE - 483

Lecture #19An Introduction to AdaptiveFilteringMotivationThe ideal transfer functionFiltering by power minimizationSteepest descentdescentLMS algorithmApplications#Copyright 2010 R. LeahyMotivationMotivation+-G(z)#Copyright 2010 R. Leahy

USC - EE - 483

Lecture #20 Outline: Quanztizationand Digital Filter Structures Quantization: Oversampling DACs Representation of binary signals and quantization noise Sigma-delta converters. Issues in implementation of digital filters: Efficiency (minimize number

USC - EE - 483

EE483: Digital Signal ProcessingInstructor: Professor LeahySAMPLE MIDTERM, FALL 2006Time Allowed: 80 MinutesPlease answer all questions. For partial credit you must show how you reach your solutions.Make sure that any sketches you draw are labelled a

USC - EE - 483

Texas A&M - MATH - 410

Math 410.500Exam 1Solutions1. (15 pts.)(a) What is the Uniform Cauchy Criterion for uniform convergence of asequence of functions fn (x) on a set E ? Answer: For every > 0there exists N so that m, n N implies that |fn (x) fm (x)| < holds for all x

Texas A&M - MATH - 410

Math 410.500Exam 2, version ASolutions1. (20 pts.) Determine the interval of convergence of the power seriesn=1(1)n n (x 3)3n .n8Solution: Its probably easiest to use the ratio test:(1)n+1 (x 3)3(n+1) / n + 18n+1|x 3|3lim= limnn8(1)n (x 3)

Texas A&M - MATH - 410

Math 410.500Exam 2, version BSolutions1. (20 pts.) Determine the interval of convergence of the power seriesn=1(1)n n (x 1)3n .n6Solution: Its probably easiest to use the ratio test:(1)n+1 (x 1)3(n+1) / n + 16n+1|x 1|3lim= limnnn6(1)n (x

Texas A&M - MATH - 410

Math 410.500: answers to exam 11. (a) f is analytic on (a, b) i for each x0 (a, b) there exists a power serieskak (x x0 ) and a number R > 0 so that f (x) =k=0ak (x x0 )kk=0on (x0 R, x0 + R). (It's too much to ask for that f equals a single1powe

Texas A&M - MATH - 410

Math 410.500: answers to exam 21. (a) U Rn is open if and only if for every a U there exists r > 0 sothat Br (a) U .(b) The boundary of U is the set of all points x Rn so that r > 0,both Br (x) U = and Br (x) U c = .(c) E Rn is connected if there doe

Texas A&M - MATH - 410

Math 410.500: answers to exam 31. (a) True. (This is Theorem 11.13.)(b) False. (Problem 2, section 11.2 has rst order partials, but is notdierentiable.)(c) False. (See example 11.11.)(d) True. (This is Theorem 11.15.)(e) False. (See example 11.18.)

Texas A&M - MATH - 410

Math 410.500Exam 12/16/051. (20 pts.) Short answer:(a) Dene: f is analytic on an open interval (a, b).(b) State Dirichlet's test for uniform convergence offk (x) gk (x)k=1on a set E .(c) State the Cauchy criterion for uniform convergence of a seq

Texas A&M - MATH - 410

Math 410.500Exam 23/30/05There are problems on both sides of this sheet!1. (18 pts.) Dene the term in italics :(a) U Rn is an open set.(b) What is the boundary of a set U Rn ?(c) E Rn is a connected set. (Either the denition from the bookor the de

Texas A&M - MATH - 410

Math 410.500Exam 34/29/05There are problems on both sides of this sheet!1. (10 pts.) True or false? (no explanation is needed)(a) If f is dierentiable at a Rn then f is continuous at a.(b) If all rst order partials of f exist at a then f is dierenti

Texas A&M - MATH - 410

Solutions to homework #1 a) False. You can use a divergent p series as a counterexample, or6.1.01k=1 k+ k+1 , which I showed in class was a diverging telescoping series.b) False. As a counterexample, take any convergent series whose termsare nonzer

Texas A&M - MATH - 410

Solutions to homework # 2 6.3.2d. A routine ratio test:limk(1 3 (2k 1) (2k + 1) / (2k + 2)!)(1 3 (2k 1) / (2k )!==limk2k + 1(2k + 1) (2k + 2)0,hence the series converges absolutely and therefore converges. Since everything is obviously positi

Texas A&M - MATH - 410

Solutions # 3x 7.1.1 a) On any interval [a, b], we have |x| max (|a| , |b|), so that n 0 cfw_xmax(|a|,|b|)max(|a|,|b|)x. For any > 0, if< n, then n 0 < , thus nnconverges uniformly to zero on [a, b]. b) There are two things to show.1First, fn

Texas A&M - MATH - 410

Solutions to assignment # 4 7.3.1b: Writing out the series as 1+ x2 +32 x4 + x6 +34 x8 + x10 + , we seethat the sequence of roots of the coecients is 1, 0, 1, 0, 31/2 , 0, 1, 0 , 31/2 ,and so on, so the limit supremum is 3, and the radius of convergenc

Texas A&M - MATH - 410

Solutions to assignment 5 8.1.1: a)x y = x z + z y x z + z y < 2 + 3 = 5 b)|x y x z| = |x (y z)| x y z x y + (z) x (y + z)< 2 (3 + 4) = 14,where I used the Cauchy-Schwartz inequality in the rst step, andthen the triangle inequality. c)|x (

Texas A&M - MATH - 410

Assignment # 6Solutions: 8.3.5 a) The sketch is below: the region in question is the intersection ofthe two disks.b) U is relatively open in E1 , since its the intersection of an open set withE1 . c) U is relatively closed in E2 , since its the inter

Texas A&M - MATH - 410

Solutions to suggested problems for exam 2 9.1.1a: Obviously, we suspect that the limit is (0, 1). To prove thisfrom the denition, we need to produce N () so that k N () implies(1)1k , 1 k2 (0, 1) < . This is equivalent to()1111=,+ 4 < .k

Texas A&M - MATH - 367

Denition 38 Triangular region is the union of a triangle and its interior. Denition 39 A polygonal region is the union of a nite number of triangular regions such that if two triangular regions intersect, their intersection is an edge or vertex of both. D

Texas A&M - MATH - 367

Homework 7 Identify the congruent triangles and show they are congruent. Note a picture can have more than one pair of triangles.1.2.3.4.5.6.7.8.12

Texas A&M - MATH - 367

An axiomatic system example. Assume that a club of two or more students is organized into committees in such a way that each of the following conditions are satised. a) Every committee is a set of one or more students. b) For each pair of students, there

Texas A&M - MATH - 367

Denition 28 Parallel Two lines are parallel in a plane if they dont intersect. Postulate 16 Parallel Postulate Through a point not on a given line there is exactly one parallel to the line. Postulate 17 Lobachevsky and Bolya Postulate Through a point not

Texas A&M - MATH - 367

Theorem 29 If A and B are equidistant from P and Q then every point between A and B has the same property. Theorem 30 If a line L contains the midpoint of P Q and contains another point which is equidistant from P and Q, then L P Q.Theorem 40 Through a g