Universidade de Brasília

Transcrição

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
de Brasília
Homepage:Universidade
http://www.pgea.unb.br/~lasp
Laboratório de Processamento de Sinais em Arranjos
1
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
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.
2
Statistical functions
The first moment also known as mean-value function
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.
3
Relation between the statistical functions
The relation between mean-value, autocorrelation and autocovariance
functions is given by
Proof of the relation:
If mean is zero the autocovariance and the correlation functions are
equal.
4
Wide-sense Stationary and mean estimate
Note that if
If these equations are satisfied, then the process is wide-sense
stationary or stationary to the second-order.
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
5
Mean estimate
Applying the expected-value operator:
6
Mean ergodicity
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.
7
Mean ergodicity
Replacing l = n – k, and after some algebra:
8
Mean ergodicity
If the process is asymptotically
uncorrelated, i.e. c(l) → 0
when l increases, then the
process is mean ergodic.
9
Correlation ergodicity
Similarly a process can also be correlation ergodic
10
Correlation matrix
L by 1 observation vector
Correlation matrix
The main diagonal contains always real-valued elements.
11
Correlation matrix
The correlation matrix is Hermitian, i.e.
Proof:
12
Correlation matrix
As a consequence of the Hermitian property
13
Correlation matrix
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
14
Correlation matrix
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
Also if
If
positive definite
Also if
15
Example of correlation matrix
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
16
17
Parameter estimation:
Given the noise variance and computing r(0), estimate |α|2.
Given the estimate of |α|2 and computing r(l), estimate ω.
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.
18
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.
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.
19
Autoregressive model
Applying the z transform
20
21
Multiplying by
and applying
In the matrix representation (called Yule-Walker equation)
22
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
Analyser
Generator
white
colored
white
colored
Processo AR
IIR (all pole)
FIR (all zero)
Processo MA
FIR (all zero)
IIR (all pole)
Processo
ARMA
Mix
Mix
Example
Power Spectral Density
10
3
Φ XX(f)
Φ YY(f)
Φ (f)
Φ ZZ(f)
10
10
2
1
0
0.2
0.4
0.6
0.8
1
f/f p
24
Example
Pole zero diagram
0.8
0.6
0.4
Imag
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-0.5
0
Real
0.5
1
25
Example
Frequency response
10
|G(f)|
10
0.2
0.1
10
10
10
H(f)
A(f)
H(f)*A(f)
0
-0.1
-0.2
0
0.1
0.2
0.3
0.4
0.5
f/f p
26
Example
Impulse response
1
gh(τ)
ga(τ)
gh,a(τ
g(τ)
0.5
0
-0.5
0
2
4
6
8
10
τ
27

Universidade de Brasília

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