Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1 iid - PowerPoint PPT Presentation

Feb 14, 2023 •269 likes •366 views

Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1 iid signal x[n], uniform in [-0.5,+0.5] 2 y[.] obtained by passing x[.] through resonant 2 nd -order filter H(z), poles at 0.95e^{j /3} 3 Extracting the portion of x(t) in a

Power Spectral Density (PSD) 6.011, Spring 2018 Lec 18 1
iid signal x[n], uniform in [-0.5,+0.5] 2
y[.] obtained by passing x[.] through resonant 2 nd -order filter H(z), poles at ±0.95e^{j π /3} 3
Extracting the portion of x(t) in a specified frequency band x ( t ) H ( j v ) y ( t ) H ( j v ) ¢ ¢ 1 - v 0 v 0 v 4
Questions (warm-up for Quiz 2!) WSS process x [ · ] with C xx [ m ] = ρδ [ m − 1] + δ [ m ] + ρδ [ m + 1] . δ What is the largest magnitude ρ can have? WSS process x ( · ) with mean µ x and PSD S xx ( j ω ). What is its FSD? Zero-mean WSS process x ( · ) with 1 S xx ( j ω ) = 1 + ω 2 and let y ( t ) = Z + x ( t ), where Z has zero mean, variance σ 2 , and is uncorrelated σ with x ( · ). What are µ y and S yy ( j ω )? 5
Periodograms (e.g., a unit-intensity “white” process) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 v /(2 p ) v /(2 p ) v /(2 p ) v /(2 p ) M = 4, T = 50 M = 4, T = 200 4 4 CT case: X T ( jω ) ↔ x ( t ) windowed to [ − T, T ] − 3 3 2 2 | X T ( j ω ) | 2 1 1 Periodogram = 2 T 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 v /(2 p ) v /(2 p ) M = 16, T = 50 M = 16, T = 200 4 4 X N ( e j Ω ) ↔ x [ n ] windowed to [ − N, N ] − DT case: 3 3 2 2 | X N ( e j Ω ) | 2 Periodogram = 1 1 6 2 N + 1 0 0 v /(2 p ) v /(2 p )
Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 v /(2 p ) v /(2 p ) v /(2 p ) v /(2 p ) M = 4, T = 50 M = 4, T = 200 4 4 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 v /(2 p ) v /(2 p ) M 16, T 50 M 16, T 200 4 4 3 3 2 2 7 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 v /(2 p ) v /(2 p )
Periodogram averaging (illustrating the Einstein-Wiener-Khinchin theorem) M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 M = 1, T = 50 4 4 4 4 3 3 3 3 2 2 2 2 1 1 1 1 0 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 v /(2 p ) v /(2 p ) v /(2 p ) v /(2 p ) M = 4, T = 50 M = 4, T = 200 4 4 3 3 2 2 1 1 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 v /(2 p ) v /(2 p ) M = 16, T = 50 M = 16, T = 200 4 4 3 3 2 2 1 1 8 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 v /(2 p ) v /(2 p )
MIT OpenCourseWare https://ocw.mit.edu 6.011 Signals, Systems and Inference Spring 201 8 For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. 9

Recommend

Power Spectral Density Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical

Power Spectral Density Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay April 10, 2013 1 / 9 Power Spectral Density Fourier transform X ( f ) = x ( t ) exp (

352 views • 9 slides

Lesson 9 Introduction Signal Spectral Analysis: Estimation of the power spectral density

Lesson 9 Introduction Signal Spectral Analysis: Estimation of the power spectral density The problem of spectral estimation is very large and has applications very different from each other Applications: To study the vibrations of

825 views • 79 slides

Power Spectral Density of Digitally Modulated Signals Saravanan Vijayakumaran

Power Spectral Density of Digitally Modulated Signals Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay August 22, 2013 1 / 22 PSD Definition for Digitally Modulated

257 views • 22 slides

THE NEXT GENERATION OF BATTERY FOR HIGH ENERGY DENSITY AND POWER DENSITY Main

THE NEXT GENERATION OF BATTERY FOR HIGH ENERGY DENSITY AND POWER DENSITY Main Principle Investigators Professor Feiyu Kang Department of Material Science, Tsinghua University Beijing 100084, P. R. China

635 views • 7 slides

On the spectral behavior of the Neumann Laplacian under mass density perturbation 9th ISAAC

On the spectral behavior of the Neumann Laplacian under mass density perturbation 9th ISAAC Congress Krakow, August 5 - 9, 2013 Luigi Provenzano, joint work with P. D. Lamberti Department of Mathematics. Doctoral School in Mathematical

1.21k views • 49 slides

Average-Case Acceleration Through Spectral Density Estimation and Universal Asymptotic Optimality

Average-Case Acceleration Through Spectral Density Estimation and Universal Asymptotic Optimality of Polyak Momentum Fabian Pedregosa (Google Brain, Montreal) Damien Scieur (SAIT AI Lab, Montreal) Motivation Consider the convex, quadratic

1.09k views • 59 slides

Average-case Acceleration Through Spectral Density Estimation Fabian Pedregosa (Google Research)

Average-case Acceleration Through Spectral Density Estimation Fabian Pedregosa (Google Research) Damien Scieur (Samsung SAIT AI Lab, Montral) International Conference on Machine Learning 2020 Complexity Analysis in Optimization Worst-case

271 views • 12 slides

1 The appearance of colours Reflected light at each wavelength is the product of illumination

Measurements of Colour relative spectral power of sunlight, made by J. Reading: Chapter 6 Parkkinen and P. Silfsten. Relative spectral power is plotted against wavelength in nm. The visible range is Light is produced in different

466 views • 9 slides

Reducing Power Density through Activity Migration Seongmoo Heo, Kenneth Barr, and Krste Asanovi

ISLPED 2003 8/26/2003 Reducing Power Density through Activity Migration Seongmoo Heo, Kenneth Barr, and Krste Asanovi Computer Architecture Group, MIT CSAIL Background Hot Spots Rapid rise of processor power density Uneven

511 views • 27 slides

Poster #190 1 Spectral Clustering of Signed Graphs Poster #190 Our Goal: Extend Spectral

Spectral Clustering of Signed Graphs via Matrix Power Means Pedro Mercado, Francesco Tudisco and Matthias Hein ICML 2019, Long Beach, USA Poster #190 1 Spectral Clustering of Signed Graphs Poster #190 Our Goal: Extend Spectral Clustering to

106 views • 7 slides

6.011: Signals, Systems & Inference Lec 3 Energy spectral density 1 Inner (dot) product of

6.011: Signals, Systems & Inference Lec 3 Energy spectral density 1 Inner (dot) product of signals X < x , v > = x [ k ] v [ k ] for real signals x [ ] , v [ ] k [ n ] X More generally, p [ n ] = x [ k ] v [ k n ]

481 views • 18 slides

Power devices Power BJT - Large devices to reduce current density - Structure is

Power devices Power BJT - Large devices to reduce current density - Structure is similar to the integrated BJT, but there is no need to have the collector contact on the upper face -> collector contact at the bottom - interleaved

512 views • 22 slides

Spectral Clustering Spectral Clustering? Spectral methods Methods using eigenvectors of

Spectral Clustering Spectral Clustering? Spectral methods Methods using eigenvectors of some matrices Involve eigen-decomposition (or spectral decomposition) Seungjin Choi Spectral clustering methods: Algorithms that cluster data

501 views • 10 slides

Reanalyzing the Relationship Between X-ray PSD Breaks and Black Hole Mass Rachael Merritt

Reanalyzing the Relationship Between X-ray PSD Breaks and Black Hole Mass Rachael Merritt Georgia State University AGN Storm Collaboration Meeting Atlanta, Georgia 18 August 2017 Power Spectral Density Functions Radio-quiet AGN X-ray

331 views • 6 slides

An Introduction to Spectral Learning Hanxiao Liu November 8, 2013 An Introduction to Spectral

An Introduction to Spectral Learning An Introduction to Spectral Learning Hanxiao Liu November 8, 2013 An Introduction to Spectral Learning Outline 1 Method of Moments 2 Learning topic models using spectral properties 3 Anchor words An

410 views • 22 slides

Power and Energy Charles Li and Deepak Pallerla Power: A First-Class Architectural Design

Power and Energy Charles Li and Deepak Pallerla Power: A First-Class Architectural Design Constraint Motivations IT was 8% of US electricity usage in 2000 Increasing over time Chip die power density increasing linearly

521 views • 31 slides

Enersine Pro 80A Modular Active Power Filter Ablerex Electronics Co., Ltd.

Enersine Pro 80A Modular Active Power Filter Ablerex Electronics Co., Ltd. http://www.ablerex.com.tw Key Features Modular and easy to extend 3 Level Technology Space-saving high power density design Hot-scalable power modules

150 views • 13 slides

Sparse Robust Regression using Non-concave Penalized Density Power Divergence Subhabrata Majumdar

Sparse Robust Regression using Non-concave Penalized Density Power Divergence Subhabrata Majumdar Joint work with Abhik Ghosh University of Florida Informatics Institute IISA-2018 conference, Gainesville, FL May 19, 2018 Ghosh and Majumdar

810 views • 55 slides

Relative Density Chapters 3.5 Relative Density 1 2/5/2015 Minimum Density Pluviate soil from

2/5/2015 Relative Density Chapters 3.5 Relative Density 1 2/5/2015 Minimum Density Pluviate soil from height of 25 mm Maximum Density Vibrate for 8 minutes 2 2/5/2015 Relative Density Correlations Clean sand (0 F c 5%) Sand w/

372 views • 13 slides

Easyjet A319/A320 vortex generator retrofit April 2016 Retrofit of vortex generators Fuel over

Easyjet A319/A320 vortex generator retrofit April 2016 Retrofit of vortex generators Fuel over pressure check panels , EZY aircraft have one on each wing. Tonal noise is reduced With VG retro-fit Averaged Sound Power Spectral Density at

909 views • 5 slides

Spectral Clustering Lecture 16 David Sontag New York

Spectral Clustering Lecture 16 David Sontag New York University Slides adapted from James Hays, Alan Fern, and Tommi Jaakkola Spectral clustering K-means Spectral clustering twocircles, 2

848 views • 19 slides

Electrode Architectures for High power density Li-ion batteries Electrode Architecture

Electrode Architectures for High power density Li-ion batteries Electrode Architecture Electrochemical Testing 10 2 One, 2.4 mg/cm Discharge Capacity 120 2 1.0 Four, 8.93 mg/cm Active Material Loading 2 ) 2 Active Material Loading

151 views • 14 slides

4-current density . Lets suppose J is the 4-current density, let J =(c , J )

4-current density . Lets suppose J is the 4-current density, let J =(c , J ) So Using the property Smooth function we can rewrite J as P. Piot, PHYS 571 Fall 2007 Charge continuity equation

700 views • 13 slides

$Spectral decimation & its applications to spectral analysis on infinite fractal lattices Joe$

Spectral decimation & its applications to spectral analysis on infinite fractal lattices Joe

Spectral decimation & its applications to spectral analysis on infinite fractal lattices Joe P. Chen Department of Mathematics Colgate University QMath13: Mathematical Results in Quantum Physics Special Session on Quantum Mechanics with

585 views • 19 slides