Linear System Response to Random Inputs. M. Sami Fadali Professor of Electrical Engineering University of Nevada
|
|
|
- Άριστόδημος Φραγκούδης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Linear System Response to Random Inputs M. Sami Fadali Professor of Electrical Engineering University of Nevada 1
2 Outline Linear System Response to deterministic input. Response to stochastic input. Characterize the stochastic output using: mean, mean square, autocorrelation, spectral density. Continuous-time systems: stationary, nonstationary. Discrete-time systems. 2
3 Response to Deterministic Input Analysis: Given the initial conditions, input, and system dynamics, characterize the system response. Use time domain or s-domain methods to solve for the system response. Can completely determine the output. 3
4 Response to Random Input Response to a given realization is useless. Characterize statistical properties of the response in terms of: Moments. Autocorrelation. Power spectral density. 4
5 Analysis of Random Response 1. Stationary steady-state analysis: Stationary input, stable LTI system After a sufficiently long period Stationary response, can solve in s-domain. 2. Nonstationary transient analysis: Time domain analysis. Can consider unstable or time-varying systems. 5
6 Calculus for Random Signals Dynamic systems: integration, differentiation. Integral and derivative are defined as limits. Random Signal: Limit may not exist for all realizations. Convergence to a limit for random signals (law, probability, q th mean, almost sure). Mean-square Calculus: using mean-square convergence. 6
7 Continuity Deterministic continuous function x t at t 0 Lim t t 0 x t = x t 0 Mean-square continuous random function X t at t 0 Lim E X t X t 2 0 = 0 t t 0 Can exchange limit and expectation for finite variance. 7
8 Interchange Limit & E{.} (Shanmugan & Breipohl, p. 162) For a mean square continuous process with finite variance and any continuous function g. Lim t t 0 E g X t = E g X t 0 8
9 Continuity Theorem (Shanmugan & Breipohl, Stark & Woods) WSS X(t) is mean-square continuous if its autocorrelation function R XX τ is continuous at τ = 0. Proof: E X t + τ X t 2 = R XX 0 + R XX 0 2R XX τ Lim E X t + τ X t 2 τ 0 = 2 R XX 0 Lim τ 0 R XX τ = 0 if R XX τ is continuous at τ = 0. 9
10 Mean-square Derivative Ordinary Derivative xሶ t = Lim ε 0 x t+ε x t For a finite variance stationary process. M.S. Derivative: Limit exists in a mean-square sense. ε ሶ X t = l. i. m. ε 0 X t+ε X t ε Lim E ε 0 ሶ X t X t + ε X t ε 2 = 0 10
11 Mean-square Stochastic Integral T 2X I = න t dt = l. i. m. In T 1 n I n = i=1 X t i Δt i Integral exists when lim n E I I n 2 = 0 Integral is a linear operator 11
12 Mean-square Calculus Provides definitions of integrals and derivatives that are valid for random processes. Allows us to use calculus for stochastic systems. Allows us to write integro-differential equations to describe stochastic systems. 12
13 Expectation of Output E x t E x t = E න = න = න G t, τ f τ dτ G t, τ E f τ G t, τ dτ m f The integral is bounded if the linear system is BIBO stable. dτ General result for MIMO time-varying case. 13
14 Stationary Steady-state Analysis Expectation of Output Assume Stable LTI system in steady-state. Stationary random input process. න E x t = න G t τ dτ = න G t τ dτ m f G τ e jωτ dτ m x = G jω ሿ ω 0 m f Mean is scaled by the DC gain ω 0 14
15 Stationary Steady-state Analysis Autocorrelation of Output R xx τ = E x t x T t + τ = E න G u f t u du න G v f t + τ v dv T = න න R xx τ = න G u E f t u f T t + τ v න G T v du dv G u R ff τ + u v G T v du dv 15
16 Change of variable R xx τ = න න Change of variable ξ = u + τ R xx τ = න G u R ff τ + u v G T v du dv න Change order of integration G ξ τ R ff ξ v G T v dξ dv 16
17 Autocorrelation & Power Spectral Density of Output R xx τ = න G ξ τ න R ff ξ v G T v dv dξ = න G τ ξ න = G τ R ff τ G T τ R ff ξ v G T v dv R ff ξ G T ξ dξ Laplace transform: S xx s = G s S ff s G T s 17
18 Stationary Steady-state Analysis Stable LTI system with transfer function G s X s = G s F s, S xx s = G s S ff s G T (s) For SISO case: S XX s = G s G s S FF s X jω = G jω F jω S XX jω = G jω 2 S FF jω 18
19 Example: Gauss-Markov Process Used frequently to model random signals R FF τ = σ 2 e βτ S FF s = 2σ2 β s 2 + β 2 = 2σ2 β s + β 2σ2 β s + β 19
20 Example: SDF of Output Spectral factorization G s = 1 Ts+1 S XX s = G s G s S FF s = 1 Ts Ts + 1 2σ2 β s + β 2σ2 β s + β = 2σ 2 β Ts + 1 s + β 2σ 2 β Ts + 1 s + β 20
21 Mean-square Value of Output j R XX 0 = 1 2πj න S XX s ds j = 1 j 2πj න S XX s S + XX s ds j S + XX s = 2σ 2 β s + β Ts + 1 = 2σ 2 β Ts 2 + βt + 1 s + β Use integration table for 2-sided LT I 2 = c R 2d 0 d XX 0 = σ2 1 βt
22 System with White Noise Input S FF jω = A, G s = 1 Ts + 1 S XX jω = G jω 2 A S FF jω = ωt E X 2 = 1 2π න A ωt dω A = 2πT tan 1 ωt = A 2T 22
23 Bandlimited White Noise Input S FF jω = ቊ A, ω ω c, G s = 1 0, ω > ω c Ts + 1 S XX jω = G(jω 2 S FF jω A = ቐ ωt 2 + 1, ω ω c 0, ω > ω c E X 2 = 1 ω c 2π න A ω c ωt dω = A πt tan 1 ω c T AΤ 2T for ω c T 1 (ω c ω BW = 1/T) Approximately the same as white noise. 23
24 Response of Physical filter E X 2 ൧ physical filter j = 1 2πj න AG s G s ds j 24
25 Noise Equivalent BW Approximate physical filter with ideal filter E X 2 ൧ = 1 ideal filter 2πβ 2π න Adω = 2Aβ 2πβ j E X 2 ൧ = 1 physical filter 2πj න j 1 AG s G s ds BW of ideal filter =noise equivalent BW β = j 1 2 2πj න G s G s ds j Gain 2 Ideal filter: BW 25
26 Example Find the noise equivalent bandwidth for the filter 1 G s = T 1 s + 1 T 2 s + 1 = 1 T 1 T 2 s 2 + T 1 + T 2 s + 1 j 1 β = noise equivalent BW = 2 2πj න G s G s ds j I 2 = c = 2d 0 d 1 2 T 1 + T 2 1 β = 4 T 1 + T 2 26
27 Shaping Filter Use spectral factorization to obtain filter TF S XX s = S XX s S + XX s = G s G s 1 F(s) Unity White Noise G s = S + XX s G(s) Shaping Filter X(s) Colored Noise 27
28 Example: Gauss-Markov S XX s = 2σ2 β s 2 + β 2 = = S XX s S + XX s 2σ2 β s + β 2σ 2 β s + β Spectral Factorization gives the shaping filter G s = 2σ2 β s + β 28
29 Nonstationary Analysis Linear system: superposition Random initial conditions & random input Total response = zero-input response + zerostate response Assume zero cross-correlation to add autocorrelations. 29
30 Natural (Zero-input) Response Zero-input response: response due to initial conditions. Response for state-space model xሶ t = Fx t y t = Cx t y t = Ce F t t 0 x t 0 e Ft = L 1 si F 1 30
31 Example: RC Circuit RC circuit in the steady state. Capacitor charged by unity Gaussian white noise input voltage source. Close switch and discharge capacitor. Random initial condition but deterministic discharge. Unity Gaussian R G s = 1/(Cs) white noise voltage source 2R + 1/(Cs) 1 = 2RCs + 1 = 1 2Ts R C
32 Steady-state Response G s = E v 2 c = 1 j 2πj න j 1 2Ts + 1 G s G s ds Use Table 3.1 (Brown & Hwang, pp. 109) with d 1 = 2T, d 0 = c 0 = 1 E v c 2 = I 1 = c 0 2 2d 0 d 1 = 1 4T 32
33 Close switch at t = 0 Tvሶ c t + v c t = 0 sv c s v c 0 + V c s T = 0 Unity Gaussian white noise voltage source V c s = v c 0 s + 1/T v c t = v c 0 e t/t E v c 2 t = E v c 2 0 e 2t/T = 1 4T e 2t/T R R C 33
34 Plot of Natural Response v c 2 t t 34
35 Forced (Zero-state) Response MIMO Time-Varying Case R XX t 1, t 2 = E x t 1 x T t 2 t x t = න G t, τ f τ dτ 0 = න 0 t 1 න 0 t 2G t1, u E f u f T v G T t 2, v du dv = න 0 t 1 න 0 Mean square with t 1 = t 2 t 2G t1, u R ff u, v G T t 2, v du dv 35
36 Forced (Zero-state) Response SISO Time-invariant Case Autocorrelation t X t = න g u F t u du 0 R XX t 1, t 2 = E X t 1 X t 2 t 2 t 1g = න න t1 u g t 2 v R FF u v dudv = න t 2 න 0 t 1g u g v RFF u v + t 2 t 1 dudv 36
37 Mean Square: SISO Timeinvariant Case R XX t 1, t 2 = න 0 R XX t, t t 2 න 0 t 1g u g v RFF u v + t 2 t 1 dudv = E X 2 t t = න 0 t න g u g v R FF u v dudv 0 37
38 Example: RC Circuit g u = 1 T e Τ u T, u 0 Mean Square t t E X 2 = න 0 න 0 = A T 2 න 0 = A T 2 න 0 t R FF u = Aδ u g u g v R FF u v dudv t න e uτt e vτt δ u v dudv 0 t e Τ 2v T dv = A 2T 1 e Gaussian white noise voltage source f 2t Τ T R C 38
39 Example: Autocorrelation R XX t 1, t 2 = න 0 = t 2 t 1 න 1 0 g u = 1, u > 0, R FF u = δ u t 2 t 1g = න න u g v RFF u v + t 2 t 1 dudv 0 0 t 1du න, t2 t 1 0 න 0 1 δ u v + t2 t 1 dudv t 2dv, t1 > t 2 = min t 1, t 2 t 2 t 2 t 1 F(s) v 1/s 39 v=u+t 2 t 1 t 1 u X(s)
40 Discrete-Time (DT) Analysis Difference equations. z-transform solution. z = time advance operator (z 1 = delay) Similar analysis to continuous time but summations replace integrals. 40
41 Z-Transform (2-sided) G k =, G 1, G 0,, G i, G z = + G 1 z 1 + G G i z i + Linear: Z af k + bg k = af z + bg z Convolution Theorem Z G k i f i = G z F z f k i= =, f 1, f 0,, f i, 41
42 Response of DT System x k = i= X z = G z F z, Convolution Summation G i impulse response G i f k i = Z-transform Z i= G z = G k i f i i= G i z i G(z) transfer function 42
43 Even Function G m = G m G z = G m z m = m= m= = G l z l = G z 1 l= G z = G z 1 G m z m 43
44 Expectation of the Output E x k = E G i f k i = G i E f k i i= i= Stationary G e jωt = frequency response m x = G i m f = G i z i m f i= i= = G 1 m f z=1 The mean is scaled by the DC gain 44
45 Discrete-time Processes R m = E x l x l + m Power spectrum: Discrete-time Fourier Transform (DTFT) of autocorrelation. S z = R m z m, S e jω = R m e jmω m= π m= R m = 1 2π න S e jω e jmω dω π 45
46 Properties of R and S R real, even: S real, even, cosine series R m = R m S z = R m z m = S z 1 m= = R 0 + R m (z m + z m ) m=1 S e jω = R R m cos mω m=1 46
47 Autocorrelation of the Output E x k x T k + n = E G i f k i G j f k + n j i= j= = G i E f k i f T k + n j i= j= = G i R ff n + i j G T j i= j= T G T j 47
48 Autocorrelation of the Output E x k x T k + n = i= j= Substitute m = n + i, i = m n = m= G n m j= G i R ff n + i j G T j R ff m j G T j R ff m G T m R xx n = G n R ff n G T n 48
49 Z-Transform of G n G n =, G 1, G 0,, G i, G z = + G 1 z 1 + G G i z i + G n =, G 1, G 0,, G i, Z G n = + G 1 z 1 + G G i z i + = G z 1 49
50 Spectral Density Function S xx z = R xx n z n = G i R ff n + i j G T j n= n= i= i= m = n + i j n = m + j i S xx z = G i R ff m G T j z m+j i m= i= j= = G i z i R ff m z m G T j z j i= m= j= = G z 1 S ff z G T z Same result using the convolution theorem. 50 z n
51 Mean Square of Output R xx n = G i R ff n + i j G T j i= j= E x k x T k + n ൧ = R n=0 xx 0 R xx 0 = G i R ff i j G T j i= j= 51
52 Cross Correlation E x k f T k + n = E = i= i= G i f k i f T k + n G i E f k i f T k + n R xf n = i= G i R ff n + i 52
53 Cross Correlation (2) E f k x T k + n = E f k = i= i= G i f k + n i E f k f T k + n i R fx n = i= G T i R ff n i G T i T 53
54 Cross Spectral Density S xf z S xf z = S xf z = n= R xf n z n = m= i= = i= G i z i n= i= G i R ff n + i G i R ff m z m i, m = n + i m= R ff m z m S xf z = G z 1 S ff z, G z 1 = S xf z S 1 ff z z n 54
55 Cross Spectral Density S fx z S fx z = n= S fx z = R fx n z n = m= i= = m= n= i= R ff n i G T i R ff m G T i z m+i, m = n i R ff m z m i= S fx z = S ff z G T z G z = S 1 ff z S fx z T = S T fx G T i z i z S T ff z z n 55
56 Expressions for SISO Case Autocorrelation: use R xx n = G n R ff n G T n R xx n = g i g j R ff n + i j i= j= Power Spectral Density S xx z = G z 1 G z S ff z S xx e jωt = G e jωt 2 S ff e jωt Identification: S fx z = S ff z G z G z = S fx z /S ff z = S fx z for unity white noise 56
57 Example For the transfer function G z = bz z a, a < 1 Determine the PSD of the output due to a unity white noise sequence. S xx z = G z 1 G z S ff z S xx z = S ff z = 1 b 2 z a z 1 a, a < 1 57
58 Shaping Filter F(z) G(z) X(z) Colored noise with PSD S XX Spectral factorization of the PSD S xx z = r 2 L z L z 1, L = 1 Transfer function of the shaping filter G z = rl z 58
59 Example S xx z = b 2 z a z 1 a, a < 1 Find the transfer function of the shaping filter for colored noise with PSD S XX. S xx z = b 2 L z L z 1 = bz bz 1 z a z 1 a G z = bz F(z) X(z) z a G(z) 59
60 Conclusion Mean, autocorrelation, spectral density. Using white noise in analysis is acceptable. Model colored noise using white noise. Use to extend KF to cases where the noise is not white. Discrete case. 60
61 References 1. R. G. Brown & P. Y. C. Hwang, Introduction to Random Signals and Applied Kalman Filtering, J. Wiley, NY, A. Papoulis & S. U. Pillai, Probability, Random Variables and Stochastic Processes, McGraw Hill, NY, K. S. Shanmugan & A. M. Breipohl, Detection, Estimation & Data Analysis, J. Wiley, NY, T. Söderström, Discrete-time Stochastic Systems : Estimation and Control, Prentice Hall, NY,
Probability and Random Processes (Part II)
Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation
LTI Systems (1A) Young Won Lim 3/21/15
LTI Systems (1A) Copyright (c) 214 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
ΗΜΥ 220: ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ Ι Ακαδημαϊκό έτος Εαρινό Εξάμηνο Κατ οίκον εργασία αρ. 2
ΤΜΗΜΑ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΥΠΡΟΥ ΗΜΥ 220: ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ Ι Ακαδημαϊκό έτος 2007-08 -- Εαρινό Εξάμηνο Κατ οίκον εργασία αρ. 2 Ημερομηνία Παραδόσεως: Παρασκευή
Fundamentals of Signals, Systems and Filtering
Fundamentals of Signals, Systems and Filtering Brett Ninness c 2000-2005, Brett Ninness, School of Electrical Engineering and Computer Science The University of Newcastle, Australia. 2 c Brett Ninness
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Second Order RLC Filters
ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Ψηφιακή Επεξεργασία Φωνής
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Επεξεργασία Φωνής Ενότητα 1η: Ψηφιακή Επεξεργασία Σήματος Στυλιανού Ιωάννης Τμήμα Επιστήμης Υπολογιστών CS578- Speech Signal Processing Lecture 1: Discrete-Time
Nachrichtentechnik I WS 2005/2006
Nachrichtentechnik I WS 2005/2006 1 Signals & Systems wt 10/2005 1 Overview (Signals & Systems) Signals: definition & classification properties basic signals Signal transformations Fourier transformation
CT Correlation (2B) Young Won Lim 8/15/14
CT Correlation (2B) 8/5/4 Copyright (c) 2-24 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C
DERIVATION OF MILES EQUATION FOR AN APPLIED FORCE Revision C By Tom Irvine Email: tomirvine@aol.com August 6, 8 Introduction The obective is to derive a Miles equation which gives the overall response
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE : Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem Solution:. ( 5 ) + (5 6 ) + ( ) cos(5 ) + 5cos( 6 ) + cos(
Numerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
Spectrum Representation (5A) Young Won Lim 11/3/16
Spectrum (5A) Copyright (c) 2009-2016 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
Module 5. February 14, h 0min
Module 5 Stationary Time Series Models Part 2 AR and ARMA Models and Their Properties Class notes for Statistics 451: Applied Time Series Iowa State University Copyright 2015 W. Q. Meeker. February 14,
Forced Pendulum Numerical approach
Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.
Math221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Stationary Stochastic Processes Table of Formulas, 2017
Stationary Stochastic Processes, 07 Stationary Stochastic Processes Table of Formulas, 07 Basics of probability theory The following is valid for probabilities: P(Ω), where Ω is all possible outcomes 0
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Sampling Basics (1B) Young Won Lim 9/21/13
Sampling Basics (1B) Copyright (c) 2009-2013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1
Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a
10.7 Performance of Second-Order System (Unit Step Response)
Lecture Notes on Control Systems/D. Ghose/0 57 0.7 Performance of Second-Order System (Unit Step Response) Consider the second order system a ÿ + a ẏ + a 0 y = b 0 r So, Y (s) R(s) = b 0 a s + a s + a
Stationary Stochastic Processes Table of Formulas, 2016
Stationary Stochastic Processes, 06 Stationary Stochastic Processes Table of Formulas, 06 Basics of probability theory The following is valid for probabilities: P(Ω), where Ω is all possible outcomes 0
Αθανάσιος Σκόδρας /
Αθανάσιος Σκόδρας 2610 99 61 67 / 2610 9 97 2 97 skodras@upatras.gr http://www.ece.upatras.gr/gr/personnel/faculty.html?id=672 Ώρες Γραφείου: Τετάρτη Πέµπτη Παρασκευή 11:00-12:00 Γραφείο: 1 ος όροφος Τομέας
Ανάλυση Θορύβου Σε Γραμμικά Κυκλώματα
AO Ηλεκτρονική ΙΙΙ Παύλος - Πέτρος Σωτηριάδης Ανάλυση Θορύβου Σε Γραμμικά Κυκλώματα Θεωρία, Εξαρτήματα και ιατάξεις Εθνικό Μετσόβιο Πολυτεχνείο Σχολή Ηλεκτρολόγων Μηχανικών - Μηχανικών Υπολογιστών Άδεια
ME 365: SYSTEMS, MEASUREMENTS, AND CONTROL (SMAC) I
ME 365: SYSTEMS, MEASUREMENTS, AND CONTROL SMAC) I Dynamicresponseof 2 nd ordersystem Prof.SongZhangMEG088) Solutions to ODEs Forann@thorderLTIsystem a n yn) + a n 1 y n 1) ++ a 1 "y + a 0 y = b m u m)
Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Introduction to Time Series Analysis. Lecture 16.
Introduction to Time Series Analysis. Lecture 16. 1. Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 1 Review: Spectral
ECE 308 SIGNALS AND SYSTEMS FALL 2017 Answers to selected problems on prior years examinations
ECE 308 SIGNALS AND SYSTEMS FALL 07 Answers to selected problems on prior years examinations Answers to problems on Midterm Examination #, Spring 009. x(t) = r(t + ) r(t ) u(t ) r(t ) + r(t 3) + u(t +
EL 625 Lecture 2. State equations of finite dimensional linear systems
EL 625 Lecture 2 EL 625 Lecture 2 State equations of finite dimensional linear systems Continuous-time: ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) Discrete-time: x(t k+ ) = A(t k )x(t k ) +
Πανεπιστήµιο Κύπρου Πολυτεχνική Σχολή
Πανεπιστήµιο Κύπρου Πολυτεχνική Σχολή Τµήµα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών ΗΜΥ 220: ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ Ι Ακαδηµαϊκό έτος 2011-12 Εαρινό Εξάµηνο Ενδιάµεση Εξέταση 1 Παρασκευή 17 Φεβρουαρίου
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
w o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
A Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Assignment 1 Solutions Complex Sinusoids
Assignment Solutions Complex Sinusoids ECE 223 Signals and Systems II Version. Spring 26. Eigenfunctions of LTI systems. Which of the following signals are eigenfunctions of LTI systems? a. x[n] =cos(
Finite difference method for 2-D heat equation
Finite difference method for 2-D heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Abstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
6.003: Signals and Systems. Modulation
6.003: Signals and Systems Modulation May 6, 200 Communications Systems Signals are not always well matched to the media through which we wish to transmit them. signal audio video internet applications
If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
![If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2](/thumbs/53/31933086.jpg "If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2")
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
If we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Lifting Entry (continued)
ifting Entry (continued) Basic planar dynamics of motion, again Yet another equilibrium glide Hypersonic phugoid motion Planar state equations MARYAN 1 01 avid. Akin - All rights reserved http://spacecraft.ssl.umd.edu
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Introduction to the ML Estimation of ARMA processes
Introduction to the ML Estimation of ARMA processes Eduardo Rossi University of Pavia October 2013 Rossi ARMA Estimation Financial Econometrics - 2013 1 / 1 We consider the AR(p) model: Y t = c + φ 1 Y
DiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
ECE 468: Digital Image Processing. Lecture 8
ECE 468: Digital Image Processing Lecture 8 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu 1 Image Reconstruction from Projections X-ray computed tomography: X-raying an object from different directions
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
ECE Spring Prof. David R. Jackson ECE Dept. Notes 2
ECE 634 Spring 6 Prof. David R. Jackson ECE Dept. Notes Fields in a Source-Free Region Example: Radiation from an aperture y PEC E t x Aperture Assume the following choice of vector potentials: A F = =
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Θεωρία Πιθανοτήτων & Στοχαστικές Ανελίξεις - 4
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Θεωρία Πιθανοτήτων & Στοχαστικές Ανελίξεις - 4 5.9 Η Στοχαστική Ανέλιξη Gauss (οι διαφάνειες ακολουθούν διαφορετική
Statistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)
HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Outline Analog Communications. Lecture 05 Angle Modulation. Instantaneous Frequency and Frequency Deviation. Angle Modulation. Pierluigi SALVO ROSSI
Outline Analog Communications Lecture 05 Angle Modulation 1 PM and FM Pierluigi SALVO ROSSI Department of Industrial and Information Engineering Second University of Naples Via Roma 9, 81031 Aversa (CE),
HMY 799 1: Αναγνώριση Συστημάτων
HMY 799 : Αναγνώριση Συστημάτων Διάλεξη Στοχαστικές Τυχαίες Μεταβλητές/ Στοχαστικά Σήματα Πειραματικά δεδομένα >Επιλογή τύπου μοντέλου >Επιλογή κριτηρίου >Υπολογισμός >Επικύρωση Προσαρμογή καμπύλης (Curve
1 String with massive end-points
1 String with massive end-points Πρόβλημα 5.11:Θεωρείστε μια χορδή μήκους, τάσης T, με δύο σημειακά σωματίδια στα άκρα της, το ένα μάζας m, και το άλλο μάζας m. α) Μελετώντας την κίνηση των άκρων βρείτε
Class 03 Systems modelling
Class 03 Systems mdelling Systems mdelling input utput spring / mass / damper Systems mdelling spring / mass / damper Systems mdelling spring / mass / damper applied frce displacement input utput Systems
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Appendix to On the stability of a compressible axisymmetric rotating flow in a pipe. By Z. Rusak & J. H. Lee
Appendi to On the stability of a compressible aisymmetric rotating flow in a pipe By Z. Rusak & J. H. Lee Journal of Fluid Mechanics, vol. 5 4, pp. 5 4 This material has not been copy-edited or typeset
Spherical Coordinates
Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical
Magnetically Coupled Circuits
DR. GYURCSEK ISTVÁN Magnetically Coupled Circuits Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 Ch. Alexander,
Iterated trilinear fourier integrals with arbitrary symbols
Cornell University ICM 04, Satellite Conference in Harmonic Analysis, Chosun University, Gwangju, Korea August 6, 04 Motivation the Coifman-Meyer theorem with classical paraproduct(979) B(f, f )(x) :=
EE101: Resonance in RLC circuits
EE11: Resonance in RLC circuits M. B. Patil mbatil@ee.iitb.ac.in www.ee.iitb.ac.in/~sequel Deartment of Electrical Engineering Indian Institute of Technology Bombay I V R V L V C I = I m = R + jωl + 1/jωC
Integrals in cylindrical, spherical coordinates (Sect. 15.7)
Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Pg The perimeter is P = 3x The area of a triangle is. where b is the base, h is the height. In our case b = x, then the area is
Pg. 9. The perimeter is P = The area of a triangle is A = bh where b is the base, h is the height 0 h= btan 60 = b = b In our case b =, then the area is A = = 0. By Pythagorean theorem a + a = d a a =
What happens when two or more waves overlap in a certain region of space at the same time?
Wave Superposition What happens when two or more waves overlap in a certain region of space at the same time? To find the resulting wave according to the principle of superposition we should sum the fields
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Matrices and vectors. Matrix and vector. a 11 a 12 a 1n a 21 a 22 a 2n A = b 1 b 2. b m. R m n, b = = ( a ij. a m1 a m2 a mn. def
Matrices and vectors Matrix and vector a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn def = ( a ij ) R m n, b = b 1 b 2 b m Rm Matrix and vectors in linear equations: example E 1 : x 1 + x 2 + 3x 4 =
Solution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Aluminum Electrolytic Capacitors
Aluminum Electrolytic Capacitors Snap-In, Mini., 105 C, High Ripple APS TS-NH ECE-S (G) Series: TS-NH Features Long life: 105 C 2,000 hours; high ripple current handling ability Wide CV value range (47
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Ψηφιακή Επεξεργασία Φωνής
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Επεξεργασία Φωνής Ενότητα 4η: Γραμμική Πρόβλεψη: Ανάλυση και Σύνθεση Στυλιανού Ιωάννης Τμήμα Επιστήμης Υπολογιστών CS578- Speech Signal Processing Lecture
Aluminum Electrolytic Capacitors (Large Can Type)
Aluminum Electrolytic Capacitors (Large Can Type) Snap-In, 85 C TS-U ECE-S (U) Series: TS-U Features General purpose Wide CV value range (33 ~ 47,000 µf/16 4V) Various case sizes Top vent construction
Electronic Analysis of CMOS Logic Gates
Electronic Analysis of CMOS Logic Gates Dae Hyun Kim EECS Washington State University References John P. Uyemura, Introduction to VLSI Circuits and Systems, 2002. Chapter 7 Goal Understand how to perform
Tridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Θεωρία Πιθανοτήτων & Στοχαστικές Ανελίξεις - 3
ΣΤΟΧΑΣΤΙΚΑ ΣΥΣΤΗΜΑΤΑ & ΕΠΙΚΟΙΝΩΝΙΕΣ 1o Τμήμα (Α - Κ): Αμφιθέατρο 4, Νέα Κτίρια ΣΗΜΜΥ Θεωρία Πιθανοτήτων & Στοχαστικές Ανελίξεις - 3 5.6: Μέση Τιμή, Συναρτήσεις Συσχέτισης (Correlation) & Συνδιασποράς (Covariance)
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Balanced Slope Demodulator EEC 112. v o2
Balanced Slope Demodulator EEC 11 The circuit below isabalanced FM slope demodulator. ω 01 i i (t) C 1 L 1 1 Ideal +v o (t) 0 C 0 v o1 v o + + C 0 Ideal 0 ω 0 L C i i (t) It is the same as the circuit
ΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT -
ΔΙΑΚΡΙΤΟΣ ΜΕΤΑΣΧΗΜΑΤΙΣΜΟΣ FOURIER - Discrete Fourier Transform - DFT - Α. ΣΚΟΔΡΑΣ ΣΗΜΑΤΑ ΚΑΙ ΣΥΣΤΗΜΑΤΑ ΙΙ (22Y603) ΕΝΟΤΗΤΑ 4 ΔΙΑΛΕΞΗ 1 ΔΙΑΦΑΝΕΙΑ 1 Διαφορετικοί Τύποι Μετασχηµατισµού Fourier Α. ΣΚΟΔΡΑΣ
HMY 799 1: Αναγνώριση Συστημάτων
HMY 799 : Αναγνώριση Συστημάτων Διαλέξεις 3 4 Στοχαστικά/τυχαία / χ διανύσματα Ντετερμινιστικά και στοχαστικά σήματα στο πεδίο της συχνότητας Στοχαστικά σήματα και γραμμικά συστήματα Deterministic and
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :