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2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute of Technology Department of Mechanical Engineering 2.161 Signal Processing - Continuous and Discrete Fall Term 2008 1 Lecture 1 Reading: • Class handout: The Dirac Delta and Unit-Step Functions 1 Introduction to Signal Processing In this class we will primarily deal with processing time-based functions, but the methods will also be applicable to spatial functions, for example image processing. We will deal with (a) Signal processing of continuous waveforms f(t), using continuous LTI systems (filters). a LTI dy nam ical system input ou t pu t f(t) Continuous Signal y(t) Processor and (b) Discrete-time (digital) signal processing of data sequences {fn} that might be samples of real continuous experimental data, such as recorded through an analog-digital converter (ADC), or implicitly discrete in nature. a LTI dis crete algorithm inp u t se q u e n c e ou t pu t seq u e n c e f Di screte Si gnal y { n } { n } Pr ocessor Some typical applications that we look at will include (a) Data analysis, for example estimation of spectral characteristics, delay estimation in echolocation systems, extraction of signal statistics. (b) Signal enhancement. Suppose a waveform has been contaminated by additive “noise”, for example 60Hz interference from the ac supply in the laboratory. 1copyright c D.Rowell 2008 1–1 inp u t ou t p u t ft + ( ) Fi lte r y( t) + n( t) ad d it ive no is e The task is to design a filter that will minimize the effect of the interference while not destroying information from the experiment. -
8.5 Least Squares Solutions to Inconsistent Systems
8.5 Least Squares Solutions to Inconsistent Systems Performance Criterion: 8. (f) Find the least-squares approximation to the solution of an inconsistent system of equations. Solve a problem using least-squares approximation. (g) Give the least squares error and least squares error vector for a least squares approximation to a solution to a system of equations. Recall that an inconsistent system is one for which there is no solution. Often we wish to solve inconsistent systems and it is just not acceptable to have no solution. In those cases we can find some vector (whose components are the values we are trying to find when attempting to solve the system) that is “closer to being a solution” than all other vectors. The theory behind this process is part of the second term of this course, but we now have enough knowledge to find such a vector in a “cookbook” manner. Suppose that we have a system of equations Ax = b. Pause for a moment to reflect on what we know and what we are trying to find when solving such a system: We have a system of linear equations, and the entries of A are the coefficients of all the equations. The vector b is the vector whose components are the right sides of all the equations, and the vector x is the vector whose components are the unknown values of the variables we are trying to find. So we know A and b and we are trying to find x. If A is invertible, the solution vector x is given by x = A−1 b. -
Lecture 2 LSI Systems and Convolution in 1D
Lecture 2 LSI systems and convolution in 1D 2.1 Learning Objectives Understand the concept of a linear system. • Understand the concept of a shift-invariant system. • Recognize that systems that are both linear and shift-invariant can be described • simply as a convolution with a certain “impulse response” function. 2.2 Linear Systems Figure 2.1: Example system H, which takes in an input signal f(x)andproducesan output g(x). AsystemH takes an input signal x and produces an output signal y,whichwecan express as H : f(x) g(x). This very general definition encompasses a vast array of di↵erent possible systems.! A subset of systems of particular relevance to signal processing are linear systems, where if f (x) g (x)andf (x) g (x), then: 1 ! 1 2 ! 2 a f + a f a g + a g 1 · 1 2 · 2 ! 1 · 1 2 · 2 for any input signals f1 and f2 and scalars a1 and a2. In other words, for a linear system, if we feed it a linear combination of inputs, we get the corresponding linear combination of outputs. Question: Why do we care about linear systems? 13 14 LECTURE 2. LSI SYSTEMS AND CONVOLUTION IN 1D Question: Can you think of examples of linear systems? Question: Can you think of examples of non-linear systems? 2.3 Shift-Invariant Systems Another subset of systems we care about are shift-invariant systems, where if f1 g1 and f (x)=f (x x )(ie:f is a shifted version of f ), then: ! 2 1 − 0 2 1 f (x) g (x)=g (x x ) 2 ! 2 1 − 0 for any input signals f1 and shift x0. -
Introduction to Aircraft Aeroelasticity and Loads
JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads i JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 ii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Introduction to Aircraft Aeroelasticity and Loads Jan R. Wright University of Manchester and J2W Consulting Ltd, UK Jonathan E. Cooper University of Liverpool, UK iii JWBK209-FM-I JWBK209-Wright November 14, 2007 2:58 Char Count= 0 Copyright C 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wileyeurope.com or www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to [email protected], or faxed to (+44) 1243 770620. -
Linear System Theory
Chapter 2 Linear System Theory In this course, we will be dealing primarily with linear systems, a special class of sys- tems for which a great deal is known. During the first half of the twentieth century, linear systems were analyzed using frequency domain (e.g., Laplace and z-transform) based approaches in an effort to deal with issues such as noise and bandwidth issues in communication systems. While they provided a great deal of intuition and were suf- ficient to establish some fundamental results, frequency domain approaches presented various drawbacks when control scientists started studying more complicated systems (containing multiple inputs and outputs, nonlinearities, noise, and so forth). Starting in the 1950’s (around the time of the space race), control engineers and scientists started turning to state-space models of control systems in order to address some of these issues. These time-domain approaches are able to effectively represent concepts such as the in- ternal state of the system, and also present a method to introduce optimality conditions into the controller design procedure. We will be using the state-space (or “modern”) approach to control almost exclusively in this course, and the purpose of this chapter is to review some of the essential concepts in this area. 2.1 Discrete-Time Signals Given a field F,asignal is a mapping from a set of numbers to F; in other words, signals are simply functions of the set of numbers that they operate on. More specifically: • A discrete-time signal f is a mapping from the set of integers Z to F, and is denoted by f[k]fork ∈ Z.Eachinstantk is also known as a time-step. -
Digital Image Processing Lectures 3 & 4
2-D Signals and Systems 2-D Fourier Transform Digital Image Processing Lectures 3 & 4 M.R. Azimi, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Definitions and Extensions: 2-D Signals: A continuous image is represented by a function of two variables e.g. x(u; v) where (u; v) are called spatial coordinates and x is the intensity. A sampled image is represented by x(m; n). If pixel intensity is also quantized (digital images) then each pixel is represented by B bits (typically B = 8 bits/pixel). 2-D Delta Functions: They are separable 2-D functions i.e. 1 (u; v) = (0; 0) Dirac: δ(u; v) = δ(u) δ(v) = 0 Otherwise 1 (m; n) = (0; 0) Kronecker: δ(m; n) = δ(m) δ(n) = 0 Otherwise M.R. Azimi Digital Image Processing 2-D Signals and Systems 2-D Fourier Transform Properties: For 2-D Dirac delta: 1 1- R R x(u0; v0)δ(u − u0; v − v0)du0dv0 = x(u; v) −∞ 2- R R δ(u; v)du dv = 1 8 > 0 − For 2-D Kronecker delta: 1 1 1- x(m; n) = P P x(m0; n0)δ(m − m0; n − n0) m0=−∞ n0=−∞ 1 2- P P δ(m; n) = 1 m;n=−∞ Periodic Signals: Consider an image x(m; n) which satisfies x(m; n + N) = x(m; n) x(m + M; n) = x(m; n) This signal is said to be doubly periodic with horizontal and vertical periods M and N, respectively. -
CHAPTER TWO - Static Aeroelasticity – Unswept Wing Structural Loads and Performance 21 2.1 Background
Static aeroelasticity – structural loads and performance CHAPTER TWO - Static Aeroelasticity – Unswept wing structural loads and performance 21 2.1 Background ........................................................................................................................... 21 2.1.2 Scope and purpose ....................................................................................................................... 21 2.1.2 The structures enterprise and its relation to aeroelasticity ............................................................ 22 2.1.3 The evolution of aircraft wing structures-form follows function ................................................ 24 2.2 Analytical modeling............................................................................................................... 30 2.2.1 The typical section, the flying door and Rayleigh-Ritz idealizations ................................................ 31 2.2.2 – Functional diagrams and operators – modeling the aeroelastic feedback process ....................... 33 2.3 Matrix structural analysis – stiffness matrices and strain energy .......................................... 34 2.4 An example - Construction of a structural stiffness matrix – the shear center concept ........ 38 2.5 Subsonic aerodynamics - fundamentals ................................................................................ 40 2.5.1 Reference points – the center of pressure..................................................................................... 44 2.5.2 A different -
Linear Algebra I
Linear Algebra I Gregg Waterman Oregon Institute of Technology c 2016 Gregg Waterman This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. The essence of the license is that You are free: to Share to copy, distribute and transmit the work • to Remix to adapt the work • Under the following conditions: Attribution You must attribute the work in the manner specified by the author (but not in • any way that suggests that they endorse you or your use of the work). Please contact the author at [email protected] to determine how best to make any attribution. Noncommercial You may not use this work for commercial purposes. • Share Alike If you alter, transform, or build upon this work, you may distribute the resulting • work only under the same or similar license to this one. With the understanding that: Waiver Any of the above conditions can be waived if you get permission from the copyright • holder. Public Domain Where the work or any of its elements is in the public domain under applicable • law, that status is in no way affected by the license. Other Rights In no way are any of the following rights affected by the license: • Your fair dealing or fair use rights, or other applicable copyright exceptions and limitations; ⋄ The author’s moral rights; ⋄ Rights other persons may have either in the work itself or in how the work is used, such as ⋄ publicity or privacy rights. Notice For any reuse or distribution, you must make clear to others the license terms of this • work. -
Calculus and Differential Equations II
Calculus and Differential Equations II MATH 250 B Linear systems of differential equations Linear systems of differential equations Calculus and Differential Equations II Second order autonomous linear systems We are mostly interested with2 × 2 first order autonomous systems of the form x0 = a x + b y y 0 = c x + d y where x and y are functions of t and a, b, c, and d are real constants. Such a system may be re-written in matrix form as d x x a b = M ; M = : dt y y c d The purpose of this section is to classify the dynamics of the solutions of the above system, in terms of the properties of the matrix M. Linear systems of differential equations Calculus and Differential Equations II Existence and uniqueness (general statement) Consider a linear system of the form dY = M(t)Y + F (t); dt where Y and F (t) are n × 1 column vectors, and M(t) is an n × n matrix whose entries may depend on t. Existence and uniqueness theorem: If the entries of the matrix M(t) and of the vector F (t) are continuous on some open interval I containing t0, then the initial value problem dY = M(t)Y + F (t); Y (t ) = Y dt 0 0 has a unique solution on I . In particular, this means that trajectories in the phase space do not cross. Linear systems of differential equations Calculus and Differential Equations II General solution The general solution to Y 0 = M(t)Y + F (t) reads Y (t) = C1 Y1(t) + C2 Y2(t) + ··· + Cn Yn(t) + Yp(t); = U(t) C + Yp(t); where 0 Yp(t) is a particular solution to Y = M(t)Y + F (t). -
A New Mathematical Model for Tiling Finite Regions of the Plane with Polyominoes
Volume 15, Number 2, Pages 95{131 ISSN 1715-0868 A NEW MATHEMATICAL MODEL FOR TILING FINITE REGIONS OF THE PLANE WITH POLYOMINOES MARCUS R. GARVIE AND JOHN BURKARDT Abstract. We present a new mathematical model for tiling finite sub- 2 sets of Z using an arbitrary, but finite, collection of polyominoes. Unlike previous approaches that employ backtracking and other refinements of `brute-force' techniques, our method is based on a systematic algebraic approach, leading in most cases to an underdetermined system of linear equations to solve. The resulting linear system is a binary linear pro- gramming problem, which can be solved via direct solution techniques, or using well-known optimization routines. We illustrate our model with some numerical examples computed in MATLAB. Users can download, edit, and run the codes from http://people.sc.fsu.edu/~jburkardt/ m_src/polyominoes/polyominoes.html. For larger problems we solve the resulting binary linear programming problem with an optimization package such as CPLEX, GUROBI, or SCIP, before plotting solutions in MATLAB. 1. Introduction and motivation 2 Consider a planar square lattice Z . We refer to each unit square in the lattice, namely [~j − 1; ~j] × [~i − 1;~i], as a cell.A polyomino is a union of 2 a finite number of edge-connected cells in the lattice Z . We assume that the polyominoes are simply-connected. The order (or area) of a polyomino is the number of cells forming it. The polyominoes of order n are called n-ominoes and the cases for n = 1; 2; 3; 4; 5; 6; 7; 8 are named monominoes, dominoes, triominoes, tetrominoes, pentominoes, hexominoes, heptominoes, and octominoes, respectively. -
Iterative Properties of Birational Rowmotion II: Rectangles and Triangles
Iterative properties of birational rowmotion II: Rectangles and triangles The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Grinberg, Darij, and Tom Roby. "Iterative properties of birational rowmotion II: Rectangles and triangles." Electronic Journal of Combinatorics 22(3) (2015). As Published http://www.combinatorics.org/ojs/index.php/eljc/article/view/ v22i3p40 Publisher European Mathematical Information Service (EMIS) Version Final published version Citable link http://hdl.handle.net/1721.1/100753 Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Iterative properties of birational rowmotion II: rectangles and triangles Darij Grinberg∗ Tom Robyy Department of Mathematics Department of Mathematics Massachusetts Institute of Technology University of Connecticut Massachusetts, U.S.A. Connecticut, U.S.A. [email protected] [email protected] Submitted: May 1, 2014; Accepted: Sep 8, 2015; Published: Sep 20, 2015 Mathematics Subject Classifications: 06A07, 05E99 Abstract Birational rowmotion { a birational map associated to any finite poset P { has been introduced by Einstein and Propp as a far-reaching generalization of the (well- studied) classical rowmotion map on the set of order ideals of P . Continuing our exploration of this birational rowmotion, we prove that it has order p+q on the (p; q)- rectangle poset (i.e., on the product of a p-element chain with a q-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). -
Underdetermined and Overdetermined Linear Algebraic Systems
Underdetermined and Overdetermined Linear Algebraic Systems ES100 March 1, 1999 T.S. Whitten Objectives ● Define underdetermined systems ● Define overdetermined systems ● Least Squares Examples Review B 500 N = − ° = ∑ Fx 0; 500 FBC sin 45 0 = ° − = ∑ Fy 0; FBC cos 45 FBA 0 FBC FBA − sin 45° 0 F − 500 ⋅ BC = ° 1cos42445 4341 1F2BA3 0 coefficients variables Review cont. − sin 45° 0 F − 500 ⋅ BC = ° 1cos42445 4341 1F2BA3 0 coefficients variables The system of matrices above is of the form: Ax = b and can be solved using MATLAB left division thus, x = A\b × results in a 1 2 matrix of values for FBC and FBA Review Summary ● A system of two Equations and two unknowns may yield a unique solution. ● The exception is when the determinant of A is equal to zero. Then the system is said to be singular. ● The left division operator will solve the linear system in one step by combining two matrix operations ●A\B is equivalent to A-1*B Graphical Representation of Unique vs. Singular Systems unique solution y singular system x Underdetermined Systems ● A system of linear equations is may be undetermined if; 1 The determinant of A is equal to zero A = 0 2 The matrix A is not square, i.e. the are more unknowns than there are equations + + = x x 3y 2z 2 1 3 2 2 ⋅ y = x + y + z = 4 1 1 1 4 z Overdetermined Systems ● The converse of an underdetermined system is an overdetermined system where there are more equations than there are variables ● This situation arises frequently in engineering.