Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa.

Apresentação em tema: "Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa."— Transcrição da apresentação:

1 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa da Costa University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code 70.919-970, Brasília - DF Homepage: http://www.pgea.unb.br/~lasp

2 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (1)   Stochastic (or random) process   Evolution of a statistical phenomenon according to probabilistic laws   Before starting the process, it is not possible to define the exactly way it evolves.   Infinite number of realizations of the process   Strictly stationary Statistical properties are invariant to the time shift   If the Probability Density Function (PDF) f(x) is known 2   All the moments can be computed. In practice, the PDF is not known. Therefore, in most cases, only the first and the second moments can be estimated with samples.

3 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (2)   Statistical functions   The first moment also known as mean-value function 3 E{ } stands for the expected-value operator (or statistical value operator) and u(n) is the sample at the n-th instant.   The autocorrelation function   The autocovariance function   Note that all the three functions are assumed constant with time. Therefore, they do not depend on n.

4 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (3)   Relation between the statistical functions 4   If mean is zero the autocovariance and the correlation functions are equal.   The relation between mean-value, autocorrelation and autocovariance functions is given by   Proof of the relation:

5 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (4)   Wide-sense Stationary and mean estimate   Note that if 5   If these equations are satisfied, then the process is wide-sense stationary or stationary to the second-order.   In practice, only a limited number of samples are available. Therefore, the mean, the autocovariance and the autocorrelation are estimated.   Estimate of the mean

6 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (5)   Mean estimate 6   Applying the expected-value operator:

7 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (6)   Mean ergodicity 7   A process is mean ergodic in the mean square-sense error sense if   Another way to represent The order doesn’t matter due to the modulus.

8 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (7)   Mean ergodicity 8 Replacing l = n – k, and after some algebra:

9 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (8)   Mean ergodicity 9 If the process is asymptotically uncorrelated, i.e. c(l) → 0 when l increases, then the process is mean ergodic.

10 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (9)   Correlation ergodicity 10   Similarly a process can also be correlation ergodic

11 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (10)   Correlation matrix 11   L by 1 observation vector   Correlation matrix The main diagonal contains always real-valued elements.

12 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (11)   Correlation matrix 12   The correlation matrix is Hermitian, i.e. Proof:

13 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (12)   Correlation matrix 13   The correlation matrix is Hermitian, i.e. As a consequence of the Hermitian property

14 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (13)   Correlation matrix 14   The correlation matrix is Hermitian, i.e. The correlation matrix is Toeplitz, i.e. the elements of the main diagonal are equal as well as the elements of each diagonal parallel to the main diagonal. Important: Wide sense stationary  R is Toeplitz R is Toeplitz  Wide sense stationary

15 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (14)   Correlation matrix 15   The correlation matrix is always nonnegative definite and almost always positive definite.   We define the scalar, where x is constant, then   We know that Nonnegative definite or positive semidefinite   If positive definite Also if

16 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (15)   Example of correlation matrix 16   We consider the following data model where u(n) is the received signal, is the signal of interest and v(n) is the zero mean i.i.d. noise component with variance. Note: independent and identically distributed (i.i.d.)   Computing the autocorrelation function

17 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (16)   Example of correlation matrix 17

18 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stationary Processes (17)   Example of correlation matrix 18   Parameter estimation: Given the noise variance and computing r(0), estimate |  | 2. Given the estimate of |  | 2 and computing r(l), estimate .   Assuming the case where - All lines and all columns are linearly dependent. - R is rank 1. - Only one eigenvalue is not zero. - The model order is one.

19 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (1)   Model: Hypothesis to explain or to describe the hidden laws that governs or constraints the generation of some physical data. 19   We consider three models   Autoregressive (AR): no past values of the input model;   Moving average (MA): no past values of the output model;   Mixed autoregressive-moving average (ARMA): include both cases.   Data model   If the data is completely random, then no prediction is possible. However, if there is some dependence of the previous data (AR or MA), then the prediction becomes possible.

20 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (2)   Autoregressive model 20   Applying the z transform

21 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (3)   Autoregressive model 21

22 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (4)   Autoregressive model 22   Multiplying by and applying   In the matrix representation (called Yule-Walker equation)

23 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos   Moving average   Moving average of M input samples   Pure FIR (Finite Response Filter) – All zero   ARMA is a mix of a FIR and IIR generator   Systematics Generator whitecolored Analyser white Processo ARIIR (all pole)FIR (all zero) Processo MAFIR (all zero)IIR (all pole) Processo ARMA Mix Mathematical Background: Stochastic Models (5)

24 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (6)   Example   Power Spectral Density 24

25 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (7)   Example   Pole zero diagram 25

26 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (8)   Example   Frequency response 26

27 Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Mathematical Background: Stochastic Models (9)   Example   Impulse response 27