Transcript FFT-based Power Spectrum Estimation

AGC
Modern Spectral Estimation
DSP
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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.
Professor A G Constantinides©
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AGC
Modern Spectral Estimation
DSP
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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
Professor A G Constantinides©
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AGC
Modern Spectral Estimation
DSP
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Main directions are:
Least Squares
Maximum Entropy
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AGC
Modern Spectral Estimation
DSP
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An optimisation problem:
Measurements: { x [ n ]}
Problem:
Find the best FIR model to filter { x [ n ]} to
yield a given signal { d [ n ]}
We need
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a) order of FIR system
b) decide on how to measure “best fit”
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AGC
Modern Spectral Estimation
DSP
“Order Estimation” is an area by itself
Goodness of fit is another large area
Usually:
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we have some idea beforehand on the
order
we select an “error criterion” which
reasonably reflects reality and is
analytically tractable
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AGC
Modern Spectral Estimation
DSP
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Formulation: Assume FIR order be N
and unknown filter weights { h [ n ]}
Output of FIR filter is
N
y [ n ]   h[ r ] x[ n  r ]  h x
T
r 0
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Instantaneous error is
e[ n ]  d [ n ]  y [ n ]  d [ n ]  h x
T
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AGC
Modern Spectral Estimation
DSP
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The best solution would be when all such
errors are zero. However, this may not
possible because of many reasons
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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 { h [ n ]} need to be selected to minimise
some measure of the error.
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AGC
Modern Spectral Estimation
DSP
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Error measure can take many forms
We draw a distinction between
stochastic and deterministic measures
For example
p
J  min E { e[ n ] }
(a) Stochastic
h
(b) Deterministic J  min  e[ n ] p
h
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n
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AGC
Modern Spectral Estimation
DSP
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With p  2
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
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AGC
Modern Spectral Estimation
DSP
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Extensive work has been done in these
problems in their various forms.
The absolute value squared error is
2
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
e[ n ]  d [ n ]  h x d [ n ]  h x
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T
*
H
*
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Or
2
2
e[ n ]  d [ n ]  g h  h g  h  h
H
H
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H
T
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AGC
Modern Spectral Estimation
DSP
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Where for the stochastic case
g  E { d ( n ) x ( n  j )}
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  E { x ( n  k ) x ( n  j )}  { kj }
*
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While for the deterministic case we
have the same expressions but
Expectations are replaced by
Summations.
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AGC
Modern Spectral Estimation
DSP
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In both cases we have
g is the crosscorrelation between
the measurements (data) and the
desired signal
 is the autocorrelation matrix of the
data
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AGC
Modern Spectral Estimation
DSP
The autocorrelation matrix for real
signals is symmetric, positive definite
 This is seen, for the stochastic case,
from
*
*
E {  x [ n  k ]   x [ n  j ]  x [ n  k ]   x [ n  j ] }  0
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Expanding
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AGC
Modern Spectral Estimation
DSP
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Differentiating e[n ] with respect to h
and setting the result to zero we obtain
0g h
T
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Or
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h
T 1
g
Differentiating again yields the
autocorrelation matrix, which is positive
definite and hence we have a minimum
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AGC
Modern Spectral Estimation
DSP
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Differentiating e[n ] with respect to h
and setting the result
to zero we obtain
T
0g h
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 
h
T 1
g
However,
e[ n ] x
H
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 d [n]  h x x
T
 d [ n ] x  h xx
H
T
H
H
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AGC
Modern Spectral Estimation
DSP
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On taking expectations we obtain
E { e[ n ] x }  E { d [ n ] x  h xx }  g  h   0
H
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H
T
H
T
This is known as the orthogonality
condition
“At the optimum the error vector is
orthogonal to the data”
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AGC
Modern Spectral Estimation
DSP
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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
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AGC
AR Spectral Estimation
DSP
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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.
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AGC
Modern Spectral Estimation
DSP
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Thus the problem becomes the
constrained optimisation problem
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max  ln  P ( e
j
) d 
m p
r [ m ] 
Subject to
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 P (e

j m
) d   rxx [ m ]
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0m p
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AGC
Modern Spectral Estimation
DSP
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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
PMEM ( e
j
G
)
1  a1 e
 j
 a2e
2
 j 2
 ...   a N e
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 jN  2
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AGC
Modern Spectral Estimation
DSP
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Where N is the number or poles.
[ G , a1 , a 2 ,..., a N ] are obtainable in the
autocorrelation method from (N+1)X(N+1)
 r[0]

r [1]

 .
 .

 r[ N ]
*
r [1]
...
r[0]
r [1]
...
r [1]
*
*
r [ N ]  1

a
 1
 .
*
r [1]


r [ 0 ]   a N
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G 2 



0



.





 0 

 0 

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AGC
Modern Spectral Estimation
DSP
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Where the autocorrelation sequence is
estimated as
r [l ] 
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1
L 1
L
n0
 x (n) x(n  l )
*
The signal above is extended by
padding with zeros whever the
argument demands more samples.
Professor A G Constantinides©
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AGC
Modern Spectral Estimation
DSP
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If we take only the entral part of the
autocorrelation matrix containing no
zero padding then we have the
Covariance Method.
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The signal vector in both cases may be
windowed prior to the computations.
Professor A G Constantinides©
23
AGC
Modern Spectral Estimation
DSP
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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
Professor A G Constantinides©
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AGC
Linear Prediction
DSP
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Assume
x [ n ]   ( a1 x [ n  1]  a 2 x [ n  2 ]  ...  a L 1 x [ n  L  1])
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From the measurements in conjunction
with the assumed model we can write
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AGC
Linear Prediction
DSP

 x [1]

 x[ 2 ]

.

 x [ N  1]

.

 x[ L  2 ]

.


0

0


0
x[ 0 ]
0
0
x[ 0 ]
0
x [1]
x[ 0 ]
.
.
.
x[ N  2 ]
x[ N  3 ]
.
.
.
.
x[ L  3 ]
x[ L  4 ]
.
.
.
0
x [ L  1]
x[ L  2 ]
.
0
x [ L  1]
0
.
0
 x [1] 

 x[ 2 ] 

0



 x[ 3 ] 
0




.
.
a 


 1
 
 x[ N ] 
x[ 0 ]
 a2
 



.
.
 a3   



 
x [ L  N  1]  .
 x [ L  1] 
 



.
 a N 
.



x[ L  3 ] 


0



x[ L  2 ] 
0


x [ L  1] 


0
0
Professor A G Constantinides©
26
AGC
Linear Prediction
DSP
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The above can be seen as solving an
underlying AR prediction problem
In a compact form
Xa   x
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The solution to this can be cast as an
optimisation problem
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27
AGC
Modern Spectral Estimation
DSP
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Form the error function to be minimised
as the difference between the two sides
of the equation
e  Xa  x
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Then seek solution as
min e e  min ( Xa  x ) ( Xa  x )
H
a
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H
a
The solution is
(normal equations)
H
H
( X X )a   X x
Professor A G Constantinides©
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AGC
Modern Spectral Estimation
DSP
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The autocorrelation matrix X X  
is computed directly from the given
signal.
Hence we obtain
H
1
a  (X X ) X x
H

H
Again in the covariance method, only a
subset of the total possible rows used in
the autocorrelation method, is taken .
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