We think you have liked this presentation. If you wish to download it, please recommend it to your friends in any social system. Share buttons are a little bit lower. Thank you!
Presentation is loading. Please wait.
Published byShania Nelson
Modified about 1 year ago
AGC DSP AGC DSP Professor A G Constantinides©1 Modern Spectral Estimation Modern Spectral Estimation is based on a priori assumptions on the manner, the observed process has been generated Validity of these assumptions is taken to hold over all possible realisations and to be of infinite temporal extent. Thus limitations of FFT-based methods are circumvented These assumptions may be entirely statistical or deterministic model-based or both.
AGC DSP AGC DSP Professor A G Constantinides©2 Modern Spectral Estimation Statistical methods make assumptions on the probabilities pertaining to data generation. Wiener-Hopf, and Bayesian methods are typical examples Model-based deterministic methods assume a linear or a non-linear equation for the input/output process driven by a stochastic or a deterministic signal. Linear Predictive Least SquaresTechniques are typical of this class
AGC DSP AGC DSP Professor A G Constantinides©3 Modern Spectral Estimation Main directions are: Least Squares Maximum Entropy
AGC DSP AGC DSP Professor A G Constantinides©4 Modern Spectral Estimation An optimisation problem: Measurements: Problem: Find the best FIR model to filter to yield a given signal We need a) order of FIR system b) decide on how to measure “best fit”
AGC DSP AGC DSP Professor A G Constantinides©5 Modern Spectral Estimation “Order Estimation” is an area by itself Goodness of fit is another large area Usually: we have some idea beforehand on the order we select an “error criterion” which reasonably reflects reality and is analytically tractable
AGC DSP AGC DSP Professor A G Constantinides©6 Modern Spectral Estimation Formulation: Assume FIR order be and unknown filter weights Output of FIR filter is Instantaneous error is
AGC DSP AGC DSP Professor A G Constantinides©7 Modern Spectral Estimation The best solution would be when all such errors are zero. However, this may not possible because of many reasons e.g. the order is not correct, the actual model is not FIR, or is not linear, the noise present in the data, etc Hence need to be selected to minimise some measure of the error.
AGC DSP AGC DSP Professor A G Constantinides©8 Modern Spectral Estimation Error measure can take many forms We draw a distinction between stochastic and deterministic measures For example (a) Stochastic (b) Deterministic
AGC DSP AGC DSP Professor A G Constantinides©9 Modern Spectral Estimation With Problem (a) is known as the Wiener filtering problem Problem(b) is known as the Least Squares problem These problems are also analytically easily tractable
AGC DSP AGC DSP Professor A G Constantinides©10 Modern Spectral Estimation Extensive work has been done in these problems in their various forms. The absolute value squared error is Or
AGC DSP AGC DSP Professor A G Constantinides©11 Modern Spectral Estimation Where for the stochastic case While for the deterministic case we have the same expressions but Expectations are replaced by Summations.
AGC DSP AGC DSP Professor A G Constantinides©12 Modern Spectral Estimation In both cases we have is the crosscorrelation between the measurements (data) and the desired signal is the autocorrelation matrix of the data
AGC DSP AGC DSP Professor A G Constantinides©13 Modern Spectral Estimation The autocorrelation matrix for real signals is symmetric, positive definite This is seen, for the stochastic case, from Expanding
AGC DSP AGC DSP Professor A G Constantinides©14 Modern Spectral Estimation Differentiating with respect to and setting the result to zero we obtain Or Differentiating again yields the autocorrelation matrix, which is positive definite and hence we have a minimum
AGC DSP AGC DSP Professor A G Constantinides©15 Modern Spectral Estimation Differentiating with respect to and setting the result to zero we obtain However,
AGC DSP AGC DSP Professor A G Constantinides©16 Modern Spectral Estimation On taking expectations we obtain This is known as the orthogonality condition “At the optimum the error vector is orthogonal to the data”
AGC DSP AGC DSP Professor A G Constantinides©17 Modern Spectral Estimation For the stochastic case this solution is known as the Wiener–Hopf solution. For the deterministic case the solution is known as the Yule-Walker solution. The framework of modelling has been FIR or Moving Average (MA). It can be extended to include more involved linear models such as Autoregressive (AR), and ARMA
AGC DSP AGC DSP Professor A G Constantinides©18 AR Spectral Estimation This is also known as the Maximum Entropy Method and the Burg Method. Burg solved the problem of extrapolating a given finite set of autocorrelations to an infinite set while keeping the autocorrelation matrix positive semidefinite. In view of the infinite possibile solutions he postulated selecting that which produces the flattest PSD. Equivalently it maximises uncertainty (entropy) or randomness.
AGC DSP AGC DSP Professor A G Constantinides©19 Modern Spectral Estimation Thus the problem becomes the constrained optimisation problem Subject to
AGC DSP AGC DSP Professor A G Constantinides©20 Modern Spectral Estimation Thus if the PSD of the observations is taken to be that of the output of an AR system driven by a white Gaussian process the problem reduces to finding the parameters of the following model
AGC DSP AGC DSP Professor A G Constantinides©21 Modern Spectral Estimation Where N is the number or poles. are obtainable in the autocorrelation method from (N+1)X(N+1)
AGC DSP AGC DSP Professor A G Constantinides©22 Modern Spectral Estimation Where the autocorrelation sequence is estimated as The signal above is extended by padding with zeros whever the argument demands more samples.
AGC DSP AGC DSP Professor A G Constantinides©23 Modern Spectral Estimation If we take only the entral part of the autocorrelation matrix containing no zero padding then we have the Covariance Method. The signal vector in both cases may be windowed prior to the computations.
AGC DSP AGC DSP Professor A G Constantinides©24 Modern Spectral Estimation While the Burg method is a decided improvement over the non-parametric methods, it has several disadvantages 1) Exhibits Spectral Line Splitting particularly at high SNR 2) For high order systems introduces spurious spectral peaks 3) In estimating sinusoids in noise it shows a bias dependent on the initial sinusoid phases
AGC DSP AGC DSP Professor A G Constantinides©25 Linear Prediction Assume From the measurements in conjunction with the assumed model we can write
AGC DSP AGC DSP Professor A G Constantinides©26 Linear Prediction
AGC DSP AGC DSP Professor A G Constantinides©27 Linear Prediction The above can be seen as solving an underlying AR prediction problem In a compact form The solution to this can be cast as an optimisation problem
AGC DSP AGC DSP Professor A G Constantinides©28 Modern Spectral Estimation Form the error function to be minimised as the difference between the two sides of the equation Then seek solution as The solution is (normal equations)
AGC DSP AGC DSP Professor A G Constantinides©29 Modern Spectral Estimation The autocorrelation matrix is computed directly from the given signal. Hence we obtain Again in the covariance method, only a subset of the total possible rows used in the autocorrelation method, is taken.
AGC DSP AGC DSP Professor A G Constantinides© Estimation Theory We seek to determine from a set of data, a set of parameters such that their values would.
AGC DSP AGC DSP Professor A G Constantinides©1 A Prediction Problem Problem: Given a sample set of a stationary processes to predict the value of the process.
AGC DSP AGC DSP Professor A G Constantinides©1 Adaptive Signal Processing Problem: Equalise through a FIR filter the distorting effect of a communication.
OPTIMUM FILTERING. This is an optimum filter that is based on the minimization of mean square error between the filter output and the a desired signal.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: AR Spectral Estimation MA Spectral Estimation ARMA Spectral Estimation.
Adv DSP Spring-2015 Lecture#11 Spectrum Estimation Parametric Methods.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: The Linear Prediction Model The Autocorrelation Method Levinson and Durbin.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: Deterministic vs. Random Maximum A Posteriori Maximum Likelihood Minimum.
AGC DSP AGC DSP RAVI KISHORE1 Power Spectral Estimation The purpose of these methods is to obtain an approximate estimation of the power spectral density.
1 Linear Prediction. 2 Linear Prediction (Introduction) : The object of linear prediction is to estimate the output sequence from a linear combination.
1 The Problem: Consider a two class task with ω 1, ω 2 LINEAR CLASSIFIERS.
SUPA Advanced Data Analysis Course, Jan 6th – 7th 2009 Advanced Data Analysis for the Physical Sciences Dr Martin Hendry Dept of Physics and Astronomy.
AGC DSP AGC DSP Professor A G Constantinides©1 Hilbert Spaces Linear Transformations and Least Squares: Hilbert Spaces.
ELG5377 Adaptive Signal Processing Lecture 13: Method of Least Squares.
Least SquaresELE Adaptive Signal Processing 1 Method of Least Squares.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: Definitions Random Signal Analysis (Review) Discrete Random Signals Random.
Method of Least Squares. Least Squares Method of Least Squares: Deterministic approach The inputs u(1), u(2),..., u(N) are applied to the system The.
AGC DSP AGC DSP Professor A G Constantinides©1 Eigenvector-based Methods A very common problem in spectral estimation is concerned with the extraction.
Adv DSP Spring-2015 Lecture#9 Optimum Filters (Ch:7) Wiener Filters.
Geology 6600/7600 Signal Analysis Last time: Linear Systems The Multiple Coherence Function measures coherence of combined inputs to output: giving it.
LEAST MEAN-SQUARE (LMS) ADAPTIVE FILTERING. Steepest Descent The update rule for SD is where or SD is a deterministic algorithm, in the sense that p and.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: The FIR Adaptive Filter The LMS Adaptive Filter Stability and Convergence.
(Fast) Block LMSELE Adaptive Signal Processing 1 Normalised Least Mean-Square Adaptive Filtering.
RLSELE Adaptive Signal Processing 1 Recursive Least-Squares (RLS) Adaptive Filters.
ECE 8443 – Pattern Recognition ECE 8423 – Adaptive Signal Processing Objectives: Normal Equations The Orthogonality Principle Solution of the Normal Equations.
Real time DSP Professors: Eng. Julian Bruno Eng. Mariano Llamedo Soria.
0 - 1 © 2007 Texas Instruments Inc, Content developed in partnership with Tel-Aviv University From MATLAB ® and Simulink ® to Real Time with TI DSPs Spectrum.
CS 782 – Machine Learning Lecture 4 Linear Models for Classification Probabilistic generative models Probabilistic discriminative models.
Chapter 5ELE Adaptive Signal Processing 1 Least Mean-Square Adaptive Filtering.
Adv DSP Spring-2015 Lecture#10 Spectrum Estimation.
The process has correlation sequence Correlation and Spectral Measure where, the adjoint of is defined by The process has spectral measure where.
Week 2ELE Adaptive Signal Processing 1 STOCHASTIC PROCESSES AND MODELS.
PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 3: LINEAR MODELS FOR REGRESSION.
ECE-7000: Nonlinear Dynamical Systems Overfitting and model costs Overfitting The more free parameters a model has, the better it can be adapted.
280 SYSTEM IDENTIFICATION The System Identification Problem is to estimate a model of a system based on input-output data. Basic Configuration continuous.
T – Biomedical Signal Processing Chapters
PROBABILITY AND STATISTICS FOR ENGINEERING Hossein Sameti Department of Computer Engineering Sharif University of Technology Principles of Parameter Estimation.
CY3A2 System identification Input signals Signals need to be realisable, and excite the typical modes of the system. Ideally the signal should be persistent.
AGC DSP AGC DSP Professor A G Constantinides 1 Digital Filter Specifications Only the magnitude approximation problem Four basic types of ideal filters.
Course AE4-T40 Lecture 5: Control Apllication. The Kalman filter model assumes the true state at time k is evolved from the state at (k − 1) according.
Self tuning regulators ( STR). Introduction to Self Tuning Regulator (STR)
Point estimation, interval estimation Point estimation Desirable properties of point estimations Interval estimations Confidence intervals.
Estimation and the Kalman Filter David Johnson. The Mean of a Discrete Distribution “I have more legs than average”
Chapter Three TWO-VARIABLEREGRESSION MODEL: THE PROBLEM OF ESTIMATION 3.1 THE METHOD OF ORDINARY LEAST SQUARES Under certain assumptions, the method of.
Linear and generalised linear models Purpose of linear models Least-squares solution for linear models Analysis of diagnostics Exponential family and generalised.
EE513 Audio Signals and Systems Wiener Inverse Filter Kevin D. Donohue Electrical and Computer Engineering University of Kentucky.
Dates for term tests 1.Friday, February 07 2.Friday, March 07 3.Friday, March 28.
Chapter 5: Signal Space Analysis Digital Communication Systems 2012 R.Sokullu 1/45 CHAPTER 5 SIGNAL SPACE ANALYSIS.
© 2017 SlidePlayer.com Inc. All rights reserved.