Mathematical methods in communication June 16th, Lecture 12
|
|
- Reynold White
- 5 years ago
- Views:
Transcription
1 2- Mathematical methods in communication June 6th, 20 Lecture 2 Lecturer: Haim Permuter Scribe: Eynan Maydan and Asaf Aharon I. MIMO - MULTIPLE INPUT MULTIPLE OUTPUT MIMO is the use of multiple antennas at both the transmitter and reviver to improve communication performance. It is one of serveral forms of smart antenna technology. MIMO technology has attracted attention in wireless communication, because it offers significant increases in data throughput and link range without additional bandwidth or transmit power. T X R X h h 2 h 3 h 2 2 h 22 2 h 23 h 3 h h 33 Fig.. MIMO illustration We have seen in last lectures that for a Gaussian channel with power constraint, n n i= X2 i P, and Gaussian noise,z N(0,σ 2 ), the capacity is defined by P(X),E(X 2 ) P I(X,Y) = 2 log(+ P σ 2). Let s check the case of parallel several channels for independent noise vectors Z,Z 2,..,Z m. In the parallel case, the power constraint is given by P i = n n j= X2 ij, where, m i= P i P with probability one, and the capacity is defined by f(x,x 2,...,X K),E( k i= X2 i ) P I(X,X 2,...,X K ;Y,Y 2,...,Y K )
2 2-2 Fig. 2. Parallel Gaussian channels. After some calculations (described in lecture 8), we get that m 2 log(+ P i ). i= According to Water-filling algorithm, P i can be found by P i = [ ν σ2 i ]+ (0, ν σ2 i ), where P = P i. In MIMO systems, each output is affected by all channel inputs. The power constraint for this model is given by r t i= r j= X2 i,j following: σ 2 i P, with probability. The channel model can be presented as G Z X Y G 2 G 2 X 2 G 22 Z 2 Y 2 Fig. 3. MIMO Note: In this section we suppose that G is deterministic, therefore we can assume that G is known at the decoder and encoder. Mathematically, Y r is defined by Y r = G r t X t +Z r, where X t is the channel input, G r t is a constant channel gain matrix and Z N(0,K z ) is the Gaussian noise. The element G jk representing the gain of the channel from the transmitter antenna j to the receiver antenna k. Lets assume, without loss of generality, that Z N(0,) ( We will see later that this assumption do not
3 2-3 lead to any kind of losses). Indeed, the channel Y = GX +Z with a general K z 0 can be transformed into the channel Ỹ = K z 2 Y = Kz 2 GX + Z Z = K z 2 Z N(0, r ). Since K z is a covariance matrix, it is a symmetric matrix, i.e K z = Kz T. Therefore, we can write K z as: K z = QΛQ T Q is unitary matrix ( QQ T = r ) and Λ is the eigen values matrix which can be described as: λ λ 2 Λ = λ n Note that since K z is a symmetric matrix, the transpose matrix is equal to the inverted matrix, i.e. λ λ2 Q T = Q. Similarly, K 2 z = QΛ 2Q T, where, Λ 2 = λn As we defined above: Ỹ = K z 2 Y = Kz 2 GX +Kz 2 Z Now, let s calculate the covariance matrix of Z = K 2 Z Z K z = E[ Z Z T ] = E[K 2 z ZZ T K 2 z ] = k 2 z K z K (a) 2 z = k 2 z K 2 z K 2 z K 2 z = r r (a) follows from K z = QΛQ T = QΛ 2Q T QΛ 2Q T = K 2 z K 2 z. The notation r r is the identity matrix of dimension r r. Accordingly, we can conclude that the noise covariance can be assumed, without loss of generality, to be the unit matrix. Now, let s find the capacity of the model: I(X;Y) Pi P (a) tr(k x) P I(X;Y) tr(k x P h(y) h(y X)
4 2-4 tr(k x) P h(y) h(z) (b) 2 log[(2πe)r K Y ] 2 log[(2πe)r ] = 2 log K Y (a) follows from P i = E[Xi 2 ], where, X is a vector of dimension t. (b) follows from h(y) 2 log[(2πe)r K Y ], where, r is the dimension of Y, and from the fact that Z has Noraml distribution. In order to find the capacity, we need to imize the expression 2 log K Y,or actually, to imize K Y : K Y = E[(GX +Z)(GX +Z) T ] = GK X G T + For that reason tr(k x)) P 2 log K Y tr(k x) P 2 log GK XG T + Note: SVD (Single Value Decomposition) - We know that every squared matrix can be decomposed as A n n = QΛQ. The SVD allows us to decompose unsquared matrix i.e. G r t = U r r Σ r t V T t t where U,V are unitary matrices (UU T =, VV T = ), and Σ is a diagonal matrix. Therefore, GG T = UΣV T VΣ T U T = UΣΣ T U T G T G = VΣ T U T UΣV T = VΣ T ΣV T Note that U and V can be calculated from the eigen vectors of GG T and G T G, respectively. Now, using the SVD, we can finally imize K Y : tr(k x)) P 2 log K Y tr(k x) P 2 log UΣV T K X VΣ T U T + (a) tr(k x) P 2 log UT UΣV T K X VΣ T U T U +U T U (b) tr( K x) P 2 log Σ K X Σ T + tr( K X) P 2 log Σ K X Σ T + (a) follows from the property from linear algebra A = UAU T = U A U T for square matrices. (b) follows from the fact that U is unitary matrix, therefore, U T U =. In addition, Kx is defined by
5 2-5 K x = V T K x V. Note that according to the trace properties tr(k X ) = tr(vv T K X ) = tr(v T K X V) = tr( K X ) Remark - Assuming K is a nonnegative definite symmetric n n matrix and K denote the determinant of K, then it follows K Π n i= k ii Using the fact that X has a Noraml distribution ( X n N(0,K) ) and the remark above, we can continue imize 2 log K Y as following: Where: tr(k X) P 2 log Σ K X Σ T + (a) min(r,t) tr( K X) P i= log(γ 2 i P i +) (b) min(r,t) log(γi( K 2 X ) ii +) tr( K X) P i= min(r,t) (c) log(γ 2 P i i +) Pi P (a) follows from the fact that X has Normal distribution X n N(0,K) and from Remark. In addition γ i = Σ ii. (b) follows from P i = ( K X ) ii (c) follows from P i = ( K X ) ii which follows that Σ P i P Note that knowing the γ i s let us solve the problem using Water-filling algorithm with the given constraints. Example - Let s assume SinceG = UΣV T the G matrix can be written by: G 4x2 = K 0 0 K G 4x2 = U i= V T In this example, the SVD decomposition gives us tow eigenvalues K and K 2, which can be easily found by Matlab (K = ,K 2 = ). These tow eigenvalues let us find P i using Water-filling algorithm, and then the diagonal matrix K X. Then it s easy to extract K X by K X = V K X V T.
6 2-6 A. Alternative proof of capacity There is another way to calculate the capacity by transforming the MIMO channel into a parallel channel. The MIMO channel is given by Y = UΣV T X +Z Now, define X = V T X Note that E[ X T X] = E[X T X]. In addition, let s define (a) Y = UΣV T X +Z. (b) Since U is unitary, U T U =. Ỹ = U T Y (a) = U T UΣV T X +U T Z (b) = ΣV T X +U T Z = Σ X + Z Remark - Since U and V are unitar matrices, multiplying in U T or V T does not add any power to channel. Lemma The MIMO channel in figure 4 has the same capacity as in figure 5 Z X G Y Fig. 4. MIMO Channel Z X V X G Y U T Ỹ Fig. 5. Parallel Channel Solution to the MIMO channel can be done by the following:
7 2-7 ) Solve the parallel channel in figure 5 using the water-filling. 2) The input to the MIMO is obtain by X = V X. Having this definitions let us analyze the channel from X to Ỹ (parallel channel), where Ỹ = UT Y, instead of analize the MIMO channel. II. MIMO WITH FADING Now, we assume that G is random. This assumption follows 3 cases: ) G is not known to the transmitter and the receiver. 2) G is known only to the receiver. 3) G is known to transmitter and receiver. Case I : In this case, it s not necessarily that G will get the imum capacity because it doesn t depend on G s distribution. Case II : C tr(k X) P I(X;Y) Case III : I(X;Y G) tr(k X)) P tr(k X)) P (h(y G) h(y G,X)) ΣP(g)h(Y G = g) h(z) tr(k X)) P ΣP(g)h(Xg +Z) tr(k X)) P I(X;Y G) P(x g):tr(x)) P
ELEC E7210: Communication Theory. Lecture 10: MIMO systems
ELEC E7210: Communication Theory Lecture 10: MIMO systems Matrix Definitions, Operations, and Properties (1) NxM matrix a rectangular array of elements a A. an 11 1....... a a 1M. NM B D C E ermitian transpose
More informationMaths for Signals and Systems Linear Algebra in Engineering
Maths for Signals and Systems Linear Algebra in Engineering Lectures 13 15, Tuesday 8 th and Friday 11 th November 016 DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE
More informationInformation Theory for Wireless Communications, Part II:
Information Theory for Wireless Communications, Part II: Lecture 5: Multiuser Gaussian MIMO Multiple-Access Channel Instructor: Dr Saif K Mohammed Scribe: Johannes Lindblom In this lecture, we give the
More informationParallel Additive Gaussian Channels
Parallel Additive Gaussian Channels Let us assume that we have N parallel one-dimensional channels disturbed by noise sources with variances σ 2,,σ 2 N. N 0,σ 2 x x N N 0,σ 2 N y y N Energy Constraint:
More informationMultiple Antennas for MIMO Communications - Basic Theory
Multiple Antennas for MIMO Communications - Basic Theory 1 Introduction The multiple-input multiple-output (MIMO) technology (Fig. 1) is a breakthrough in wireless communication system design. It uses
More informationELEC546 MIMO Channel Capacity
ELEC546 MIMO Channel Capacity Vincent Lau Simplified Version.0 //2004 MIMO System Model Transmitter with t antennas & receiver with r antennas. X Transmitted Symbol, received symbol Channel Matrix (Flat
More informationGaussian channel. Information theory 2013, lecture 6. Jens Sjölund. 8 May Jens Sjölund (IMT, LiU) Gaussian channel 1 / 26
Gaussian channel Information theory 2013, lecture 6 Jens Sjölund 8 May 2013 Jens Sjölund (IMT, LiU) Gaussian channel 1 / 26 Outline 1 Definitions 2 The coding theorem for Gaussian channel 3 Bandlimited
More informationMultiple Antennas in Wireless Communications
Multiple Antennas in Wireless Communications Luca Sanguinetti Department of Information Engineering Pisa University luca.sanguinetti@iet.unipi.it April, 2009 Luca Sanguinetti (IET) MIMO April, 2009 1 /
More information12.4 Known Channel (Water-Filling Solution)
ECEn 665: Antennas and Propagation for Wireless Communications 54 2.4 Known Channel (Water-Filling Solution) The channel scenarios we have looed at above represent special cases for which the capacity
More informationLecture 7 MIMO Communica2ons
Wireless Communications Lecture 7 MIMO Communica2ons Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2014 1 Outline MIMO Communications (Chapter 10
More informationLECTURE 18. Lecture outline Gaussian channels: parallel colored noise inter-symbol interference general case: multiple inputs and outputs
LECTURE 18 Last time: White Gaussian noise Bandlimited WGN Additive White Gaussian Noise (AWGN) channel Capacity of AWGN channel Application: DS-CDMA systems Spreading Coding theorem Lecture outline Gaussian
More informationAdvanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung
Advanced Topics in Digital Communications Spezielle Methoden der digitalen Datenübertragung Dr.-Ing. Carsten Bockelmann Institute for Telecommunications and High-Frequency Techniques Department of Communications
More informationVector Channel Capacity with Quantized Feedback
Vector Channel Capacity with Quantized Feedback Sudhir Srinivasa and Syed Ali Jafar Electrical Engineering and Computer Science University of California Irvine, Irvine, CA 9697-65 Email: syed@ece.uci.edu,
More informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationA Proof of the Converse for the Capacity of Gaussian MIMO Broadcast Channels
A Proof of the Converse for the Capacity of Gaussian MIMO Broadcast Channels Mehdi Mohseni Department of Electrical Engineering Stanford University Stanford, CA 94305, USA Email: mmohseni@stanford.edu
More informationSingular Value Decomposition (SVD)
School of Computing National University of Singapore CS CS524 Theoretical Foundations of Multimedia More Linear Algebra Singular Value Decomposition (SVD) The highpoint of linear algebra Gilbert Strang
More informationLecture 6 Channel Coding over Continuous Channels
Lecture 6 Channel Coding over Continuous Channels I-Hsiang Wang Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw November 9, 015 1 / 59 I-Hsiang Wang IT Lecture 6 We have
More informationErgodic and Outage Capacity of Narrowband MIMO Gaussian Channels
Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels Yang Wen Liang Department of Electrical and Computer Engineering The University of British Columbia April 19th, 005 Outline of Presentation
More informationLecture 5: Antenna Diversity and MIMO Capacity Theoretical Foundations of Wireless Communications 1. Overview. CommTh/EES/KTH
: Antenna Diversity and Theoretical Foundations of Wireless Communications Wednesday, May 4, 206 9:00-2:00, Conference Room SIP Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication
More informationPrincipal Component Analysis
Principal Component Analysis CS5240 Theoretical Foundations in Multimedia Leow Wee Kheng Department of Computer Science School of Computing National University of Singapore Leow Wee Kheng (NUS) Principal
More informationErgodic and Outage Capacity of Narrowband MIMO Gaussian Channels
Ergodic and Outage Capacity of Narrowband MIMO Gaussian Channels Yang Wen Liang Department of Electrical and Computer Engineering The University of British Columbia, Vancouver, British Columbia Email:
More informationLecture 6 Sept Data Visualization STAT 442 / 890, CM 462
Lecture 6 Sept. 25-2006 Data Visualization STAT 442 / 890, CM 462 Lecture: Ali Ghodsi 1 Dual PCA It turns out that the singular value decomposition also allows us to formulate the principle components
More informationA L T O SOLO LOWCLL. MICHIGAN, THURSDAY. DECEMBER 10,1931. ritt. Mich., to T h e Heights. Bos" l u T H I S COMMl'NiTY IN Wilcox
G 093 < 87 G 9 G 4 4 / - G G 3 -!! - # -G G G : 49 q» - 43 8 40 - q - z 4 >» «9 0-9 - - q 00! - - q q!! ) 5 / : \ 0 5 - Z : 9 [ -?! : ) 5 - - > - 8 70 / q - - - X!! - [ 48 - -!
More informationPithy P o i n t s Picked I ' p and Patljr Put By Our P e r i p a tetic Pencil Pusher VOLUME X X X X. Lee Hi^h School Here Friday Ni^ht
G G QQ K K Z z U K z q Z 22 x z - z 97 Z x z j K K 33 G - 72 92 33 3% 98 K 924 4 G G K 2 G x G K 2 z K j x x 2 G Z 22 j K K x q j - K 72 G 43-2 2 G G z G - -G G U q - z q - G x) z q 3 26 7 x Zz - G U-
More informationTransmit Directions and Optimality of Beamforming in MIMO-MAC with Partial CSI at the Transmitters 1
2005 Conference on Information Sciences and Systems, The Johns Hopkins University, March 6 8, 2005 Transmit Directions and Optimality of Beamforming in MIMO-MAC with Partial CSI at the Transmitters Alkan
More informationAPPENDIX A. Background Mathematics. A.1 Linear Algebra. Vector algebra. Let x denote the n-dimensional column vector with components x 1 x 2.
APPENDIX A Background Mathematics A. Linear Algebra A.. Vector algebra Let x denote the n-dimensional column vector with components 0 x x 2 B C @. A x n Definition 6 (scalar product). The scalar product
More informationCapacity of multiple-input multiple-output (MIMO) systems in wireless communications
15/11/02 Capacity of multiple-input multiple-output (MIMO) systems in wireless communications Bengt Holter Department of Telecommunications Norwegian University of Science and Technology 1 Outline 15/11/02
More informationThe Capacity Region of the Gaussian MIMO Broadcast Channel
0-0 The Capacity Region of the Gaussian MIMO Broadcast Channel Hanan Weingarten, Yossef Steinberg and Shlomo Shamai (Shitz) Outline Problem statement Background and preliminaries Capacity region of the
More informationOn the Use of Division Algebras for Wireless Communication
On the Use of Division Algebras for Wireless Communication frederique@systems.caltech.edu California Institute of Technology AMS meeting, Davidson, March 3rd 2007 Outline A few wireless coding problems
More informationInformation Theory for Wireless Communications. Lecture 10 Discrete Memoryless Multiple Access Channel (DM-MAC): The Converse Theorem
Information Theory for Wireless Communications. Lecture 0 Discrete Memoryless Multiple Access Channel (DM-MAC: The Converse Theorem Instructor: Dr. Saif Khan Mohammed Scribe: Antonios Pitarokoilis I. THE
More information18.06 Problem Set 10 - Solutions Due Thursday, 29 November 2007 at 4 pm in
86 Problem Set - Solutions Due Thursday, 29 November 27 at 4 pm in 2-6 Problem : (5=5+5+5) Take any matrix A of the form A = B H CB, where B has full column rank and C is Hermitian and positive-definite
More informationSolutions to Homework Set #4 Differential Entropy and Gaussian Channel
Solutions to Homework Set #4 Differential Entropy and Gaussian Channel 1. Differential entropy. Evaluate the differential entropy h(x = f lnf for the following: (a Find the entropy of the exponential density
More information5. Random Vectors. probabilities. characteristic function. cross correlation, cross covariance. Gaussian random vectors. functions of random vectors
EE401 (Semester 1) 5. Random Vectors Jitkomut Songsiri probabilities characteristic function cross correlation, cross covariance Gaussian random vectors functions of random vectors 5-1 Random vectors we
More informationThe Singular Value Decomposition
The Singular Value Decomposition An Important topic in NLA Radu Tiberiu Trîmbiţaş Babeş-Bolyai University February 23, 2009 Radu Tiberiu Trîmbiţaş ( Babeş-Bolyai University)The Singular Value Decomposition
More informationLecture 8: MIMO Architectures (II) Theoretical Foundations of Wireless Communications 1. Overview. Ragnar Thobaben CommTh/EES/KTH
MIMO : MIMO Theoretical Foundations of Wireless Communications 1 Wednesday, May 25, 2016 09:15-12:00, SIP 1 Textbook: D. Tse and P. Viswanath, Fundamentals of Wireless Communication 1 / 20 Overview MIMO
More informationSingular Value Decomposition
Singular Value Decomposition Motivatation The diagonalization theorem play a part in many interesting applications. Unfortunately not all matrices can be factored as A = PDP However a factorization A =
More informationLecture 13: Simple Linear Regression in Matrix Format. 1 Expectations and Variances with Vectors and Matrices
Lecture 3: Simple Linear Regression in Matrix Format To move beyond simple regression we need to use matrix algebra We ll start by re-expressing simple linear regression in matrix form Linear algebra is
More informationThe Singular Value Decomposition
The Singular Value Decomposition Philippe B. Laval KSU Fall 2015 Philippe B. Laval (KSU) SVD Fall 2015 1 / 13 Review of Key Concepts We review some key definitions and results about matrices that will
More informationUnsupervised Machine Learning and Data Mining. DS 5230 / DS Fall Lecture 7. Jan-Willem van de Meent
Unsupervised Machine Learning and Data Mining DS 5230 / DS 4420 - Fall 2018 Lecture 7 Jan-Willem van de Meent DIMENSIONALITY REDUCTION Borrowing from: Percy Liang (Stanford) Dimensionality Reduction Goal:
More informationSingle-User MIMO systems: Introduction, capacity results, and MIMO beamforming
Single-User MIMO systems: Introduction, capacity results, and MIMO beamforming Master Universitario en Ingeniería de Telecomunicación I. Santamaría Universidad de Cantabria Contents Introduction Multiplexing,
More informationCLASSIFICATION OF COMPLETELY POSITIVE MAPS 1. INTRODUCTION
CLASSIFICATION OF COMPLETELY POSITIVE MAPS STEPHAN HOYER ABSTRACT. We define a completely positive map and classify all completely positive linear maps. We further classify all such maps that are trace-preserving
More informationExercise 1. = P(y a 1)P(a 1 )
Chapter 7 Channel Capacity Exercise 1 A source produces independent, equally probable symbols from an alphabet {a 1, a 2 } at a rate of one symbol every 3 seconds. These symbols are transmitted over a
More informationChannel capacity estimation using free probability theory
Channel capacity estimation using free probability theory January 008 Problem at hand The capacity per receiving antenna of a channel with n m channel matrix H and signal to noise ratio ρ = 1 σ is given
More informationELEC633: Graphical Models
ELEC633: Graphical Models Tahira isa Saleem Scribe from 7 October 2008 References: Casella and George Exploring the Gibbs sampler (1992) Chib and Greenberg Understanding the Metropolis-Hastings algorithm
More informationLecture 7 (Weeks 13-14)
Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.
More informationSpace-Time Coding for Multi-Antenna Systems
Space-Time Coding for Multi-Antenna Systems ECE 559VV Class Project Sreekanth Annapureddy vannapu2@uiuc.edu Dec 3rd 2007 MIMO: Diversity vs Multiplexing Multiplexing Diversity Pictures taken from lectures
More informationOptimal Power Allocation for Parallel Gaussian Broadcast Channels with Independent and Common Information
SUBMIED O IEEE INERNAIONAL SYMPOSIUM ON INFORMAION HEORY, DE. 23 1 Optimal Power Allocation for Parallel Gaussian Broadcast hannels with Independent and ommon Information Nihar Jindal and Andrea Goldsmith
More informationMorning Session Capacity-based Power Control. Department of Electrical and Computer Engineering University of Maryland
Morning Session Capacity-based Power Control Şennur Ulukuş Department of Electrical and Computer Engineering University of Maryland So Far, We Learned... Power control with SIR-based QoS guarantees Suitable
More informationbe a Householder matrix. Then prove the followings H = I 2 uut Hu = (I 2 uu u T u )u = u 2 uut u
MATH 434/534 Theoretical Assignment 7 Solution Chapter 7 (71) Let H = I 2uuT Hu = u (ii) Hv = v if = 0 be a Householder matrix Then prove the followings H = I 2 uut Hu = (I 2 uu )u = u 2 uut u = u 2u =
More information10-725/36-725: Convex Optimization Prerequisite Topics
10-725/36-725: Convex Optimization Prerequisite Topics February 3, 2015 This is meant to be a brief, informal refresher of some topics that will form building blocks in this course. The content of the
More informationDesigning Information Devices and Systems II Spring 2017 Murat Arcak and Michel Maharbiz Homework 10
EECS 16B Designing Information Devices and Systems II Spring 2017 Murat Arcak and Michel Maharbiz Homework 10 This homework is due April 12, 2017, at 17:00. 1. MIMO wireless signals Ever wonder why newer
More informationLecture 8: Linear Algebra Background
CSE 521: Design and Analysis of Algorithms I Winter 2017 Lecture 8: Linear Algebra Background Lecturer: Shayan Oveis Gharan 2/1/2017 Scribe: Swati Padmanabhan Disclaimer: These notes have not been subjected
More informationEigenvalues and diagonalization
Eigenvalues and diagonalization Patrick Breheny November 15 Patrick Breheny BST 764: Applied Statistical Modeling 1/20 Introduction The next topic in our course, principal components analysis, revolves
More informationDSP Applications for Wireless Communications: Linear Equalisation of MIMO Channels
DSP Applications for Wireless Communications: Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407) Lecture no. 5 August
More informationData Mining Lecture 4: Covariance, EVD, PCA & SVD
Data Mining Lecture 4: Covariance, EVD, PCA & SVD Jo Houghton ECS Southampton February 25, 2019 1 / 28 Variance and Covariance - Expectation A random variable takes on different values due to chance The
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Numerical Linear Algebra Background Cho-Jui Hsieh UC Davis May 15, 2018 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationDegrees of Freedom Region of the Gaussian MIMO Broadcast Channel with Common and Private Messages
Degrees of Freedom Region of the Gaussian MIMO Broadcast hannel with ommon and Private Messages Ersen Ekrem Sennur Ulukus Department of Electrical and omputer Engineering University of Maryland, ollege
More informationDuality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels
2658 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 49, NO 10, OCTOBER 2003 Duality, Achievable Rates, and Sum-Rate Capacity of Gaussian MIMO Broadcast Channels Sriram Vishwanath, Student Member, IEEE, Nihar
More informationSTA141C: Big Data & High Performance Statistical Computing
STA141C: Big Data & High Performance Statistical Computing Lecture 5: Numerical Linear Algebra Cho-Jui Hsieh UC Davis April 20, 2017 Linear Algebra Background Vectors A vector has a direction and a magnitude
More informationECE Information theory Final (Fall 2008)
ECE 776 - Information theory Final (Fall 2008) Q.1. (1 point) Consider the following bursty transmission scheme for a Gaussian channel with noise power N and average power constraint P (i.e., 1/n X n i=1
More informationMIMO Capacities : Eigenvalue Computation through Representation Theory
MIMO Capacities : Eigenvalue Computation through Representation Theory Jayanta Kumar Pal, Donald Richards SAMSI Multivariate distributions working group Outline 1 Introduction 2 MIMO working model 3 Eigenvalue
More informationLecture 7 Spectral methods
CSE 291: Unsupervised learning Spring 2008 Lecture 7 Spectral methods 7.1 Linear algebra review 7.1.1 Eigenvalues and eigenvectors Definition 1. A d d matrix M has eigenvalue λ if there is a d-dimensional
More informationThe QR Decomposition
The QR Decomposition We have seen one major decomposition of a matrix which is A = LU (and its variants) or more generally PA = LU for a permutation matrix P. This was valid for a square matrix and aided
More informationDimensionality Reduction: PCA. Nicholas Ruozzi University of Texas at Dallas
Dimensionality Reduction: PCA Nicholas Ruozzi University of Texas at Dallas Eigenvalues λ is an eigenvalue of a matrix A R n n if the linear system Ax = λx has at least one non-zero solution If Ax = λx
More informationNote that the new channel is noisier than the original two : and H(A I +A2-2A1A2) > H(A2) (why?). min(c,, C2 ) = min(1 - H(a t ), 1 - H(A 2 )).
l I ~-16 / (a) (5 points) What is the capacity Cr of the channel X -> Y? What is C of the channel Y - Z? (b) (5 points) What is the capacity C 3 of the cascaded channel X -3 Z? (c) (5 points) A ow let.
More informationLinear Algebra, part 3. Going back to least squares. Mathematical Models, Analysis and Simulation = 0. a T 1 e. a T n e. Anna-Karin Tornberg
Linear Algebra, part 3 Anna-Karin Tornberg Mathematical Models, Analysis and Simulation Fall semester, 2010 Going back to least squares (Sections 1.7 and 2.3 from Strang). We know from before: The vector
More informationEigenvectors and SVD 1
Eigenvectors and SVD 1 Definition Eigenvectors of a square matrix Ax=λx, x=0. Intuition: x is unchanged by A (except for scaling) Examples: axis of rotation, stationary distribution of a Markov chain 2
More informationCHANNEL FEEDBACK QUANTIZATION METHODS FOR MISO AND MIMO SYSTEMS
CHANNEL FEEDBACK QUANTIZATION METHODS FOR MISO AND MIMO SYSTEMS June Chul Roh and Bhaskar D Rao Department of Electrical and Computer Engineering University of California, San Diego La Jolla, CA 9293 47,
More informationUpper Bounds on MIMO Channel Capacity with Channel Frobenius Norm Constraints
Upper Bounds on IO Channel Capacity with Channel Frobenius Norm Constraints Zukang Shen, Jeffrey G. Andrews, Brian L. Evans Wireless Networking Communications Group Department of Electrical Computer Engineering
More informationPhysical-Layer MIMO Relaying
Model Gaussian SISO MIMO Gauss.-BC General. Physical-Layer MIMO Relaying Anatoly Khina, Tel Aviv University Joint work with: Yuval Kochman, MIT Uri Erez, Tel Aviv University August 5, 2011 Model Gaussian
More informationA New SLNR-based Linear Precoding for. Downlink Multi-User Multi-Stream MIMO Systems
A New SLNR-based Linear Precoding for 1 Downlin Multi-User Multi-Stream MIMO Systems arxiv:1008.0730v1 [cs.it] 4 Aug 2010 Peng Cheng, Meixia Tao and Wenjun Zhang Abstract Signal-to-leaage-and-noise ratio
More informationTitle. Author(s)Tsai, Shang-Ho. Issue Date Doc URL. Type. Note. File Information. Equal Gain Beamforming in Rayleigh Fading Channels
Title Equal Gain Beamforming in Rayleigh Fading Channels Author(s)Tsai, Shang-Ho Proceedings : APSIPA ASC 29 : Asia-Pacific Signal Citationand Conference: 688-691 Issue Date 29-1-4 Doc URL http://hdl.handle.net/2115/39789
More informationLinear Least Squares. Using SVD Decomposition.
Linear Least Squares. Using SVD Decomposition. Dmitriy Leykekhman Spring 2011 Goals SVD-decomposition. Solving LLS with SVD-decomposition. D. Leykekhman Linear Least Squares 1 SVD Decomposition. For any
More informationSingular Value Decomposition and Principal Component Analysis (PCA) I
Singular Value Decomposition and Principal Component Analysis (PCA) I Prof Ned Wingreen MOL 40/50 Microarray review Data per array: 0000 genes, I (green) i,i (red) i 000 000+ data points! The expression
More informationEE 5407 Part II: Spatial Based Wireless Communications
EE 5407 Part II: Spatial Based Wireless Communications Instructor: Prof. Rui Zhang E-mail: rzhang@i2r.a-star.edu.sg Website: http://www.ece.nus.edu.sg/stfpage/elezhang/ Lecture IV: MIMO Systems March 21,
More informationLecture 2. Capacity of the Gaussian channel
Spring, 207 5237S, Wireless Communications II 2. Lecture 2 Capacity of the Gaussian channel Review on basic concepts in inf. theory ( Cover&Thomas: Elements of Inf. Theory, Tse&Viswanath: Appendix B) AWGN
More informationLecture 13: Simple Linear Regression in Matrix Format
See updates and corrections at http://www.stat.cmu.edu/~cshalizi/mreg/ Lecture 13: Simple Linear Regression in Matrix Format 36-401, Section B, Fall 2015 13 October 2015 Contents 1 Least Squares in Matrix
More informationSingular Value Decomposition
Chapter 5 Singular Value Decomposition We now reach an important Chapter in this course concerned with the Singular Value Decomposition of a matrix A. SVD, as it is commonly referred to, is one of the
More informationOn Gaussian MIMO Broadcast Channels with Common and Private Messages
On Gaussian MIMO Broadcast Channels with Common and Private Messages Ersen Ekrem Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland, College Park, MD 20742 ersen@umd.edu
More information33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM
33AH, WINTER 2018: STUDY GUIDE FOR FINAL EXAM (UPDATED MARCH 17, 2018) The final exam will be cumulative, with a bit more weight on more recent material. This outline covers the what we ve done since the
More informationOn the Optimality of Multiuser Zero-Forcing Precoding in MIMO Broadcast Channels
On the Optimality of Multiuser Zero-Forcing Precoding in MIMO Broadcast Channels Saeed Kaviani and Witold A. Krzymień University of Alberta / TRLabs, Edmonton, Alberta, Canada T6G 2V4 E-mail: {saeed,wa}@ece.ualberta.ca
More informationSingular Value Decomposition
Singular Value Decomposition (Com S 477/577 Notes Yan-Bin Jia Sep, 7 Introduction Now comes a highlight of linear algebra. Any real m n matrix can be factored as A = UΣV T where U is an m m orthogonal
More informationUNIT 6: The singular value decomposition.
UNIT 6: The singular value decomposition. María Barbero Liñán Universidad Carlos III de Madrid Bachelor in Statistics and Business Mathematical methods II 2011-2012 A square matrix is symmetric if A T
More informationAnatoly Khina. Joint work with: Uri Erez, Ayal Hitron, Idan Livni TAU Yuval Kochman HUJI Gregory W. Wornell MIT
Network Modulation: Transmission Technique for MIMO Networks Anatoly Khina Joint work with: Uri Erez, Ayal Hitron, Idan Livni TAU Yuval Kochman HUJI Gregory W. Wornell MIT ACC Workshop, Feder Family Award
More informationLinear Algebra Review. Vectors
Linear Algebra Review 9/4/7 Linear Algebra Review By Tim K. Marks UCSD Borrows heavily from: Jana Kosecka http://cs.gmu.edu/~kosecka/cs682.html Virginia de Sa (UCSD) Cogsci 8F Linear Algebra review Vectors
More informationStat 159/259: Linear Algebra Notes
Stat 159/259: Linear Algebra Notes Jarrod Millman November 16, 2015 Abstract These notes assume you ve taken a semester of undergraduate linear algebra. In particular, I assume you are familiar with the
More informationOptimum Power Allocation in Fading MIMO Multiple Access Channels with Partial CSI at the Transmitters
Optimum Power Allocation in Fading MIMO Multiple Access Channels with Partial CSI at the Transmitters Alkan Soysal Sennur Ulukus Department of Electrical and Computer Engineering University of Maryland,
More informationB553 Lecture 5: Matrix Algebra Review
B553 Lecture 5: Matrix Algebra Review Kris Hauser January 19, 2012 We have seen in prior lectures how vectors represent points in R n and gradients of functions. Matrices represent linear transformations
More informationCOMP 558 lecture 18 Nov. 15, 2010
Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to
More informationSTAT 309: MATHEMATICAL COMPUTATIONS I FALL 2017 LECTURE 5
STAT 39: MATHEMATICAL COMPUTATIONS I FALL 17 LECTURE 5 1 existence of svd Theorem 1 (Existence of SVD) Every matrix has a singular value decomposition (condensed version) Proof Let A C m n and for simplicity
More informationEECS 275 Matrix Computation
EECS 275 Matrix Computation Ming-Hsuan Yang Electrical Engineering and Computer Science University of California at Merced Merced, CA 95344 http://faculty.ucmerced.edu/mhyang Lecture 6 1 / 22 Overview
More informationThe Singular Value Decomposition and Least Squares Problems
The Singular Value Decomposition and Least Squares Problems Tom Lyche Centre of Mathematics for Applications, Department of Informatics, University of Oslo September 27, 2009 Applications of SVD solving
More informationExercise Sheet 1. 1 Probability revision 1: Student-t as an infinite mixture of Gaussians
Exercise Sheet 1 1 Probability revision 1: Student-t as an infinite mixture of Gaussians Show that an infinite mixture of Gaussian distributions, with Gamma distributions as mixing weights in the following
More informationEE263 homework 9 solutions
EE263 Prof S Boyd EE263 homework 9 solutions 1416 Frobenius norm of a matrix The Frobenius norm of a matrix A R n n is defined as A F = TrA T A (Recall Tr is the trace of a matrix, ie, the sum of the diagonal
More informationNotes on Eigenvalues, Singular Values and QR
Notes on Eigenvalues, Singular Values and QR Michael Overton, Numerical Computing, Spring 2017 March 30, 2017 1 Eigenvalues Everyone who has studied linear algebra knows the definition: given a square
More informationIEEE C80216m-09/0079r1
Project IEEE 802.16 Broadband Wireless Access Working Group Title Efficient Demodulators for the DSTTD Scheme Date 2009-01-05 Submitted M. A. Khojastepour Ron Porat Source(s) NEC
More informationOptimal Sequences, Power Control and User Capacity of Synchronous CDMA Systems with Linear MMSE Multiuser Receivers
Optimal Sequences, Power Control and User Capacity of Synchronous CDMA Systems with Linear MMSE Multiuser Receivers Pramod Viswanath, Venkat Anantharam and David.C. Tse {pvi, ananth, dtse}@eecs.berkeley.edu
More informationSum-Power Iterative Watefilling Algorithm
Sum-Power Iterative Watefilling Algorithm Daniel P. Palomar Hong Kong University of Science and Technolgy (HKUST) ELEC547 - Convex Optimization Fall 2009-10, HKUST, Hong Kong November 11, 2009 Outline
More informationCapacity Region of Reversely Degraded Gaussian MIMO Broadcast Channel
Capacity Region of Reversely Degraded Gaussian MIMO Broadcast Channel Jun Chen Dept. of Electrical and Computer Engr. McMaster University Hamilton, Ontario, Canada Chao Tian AT&T Labs-Research 80 Park
More informationRandom Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws. Symeon Chatzinotas February 11, 2013 Luxembourg
Random Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws Symeon Chatzinotas February 11, 2013 Luxembourg Outline 1. Random Matrix Theory 1. Definition 2. Applications 3. Asymptotics 2. Ensembles
More information