Transcript FFT-based Power Spectrum Estimation

AGC
Eigenvector-based Methods
DSP
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A very common problem in spectral
estimation is concerned with the extraction of
uncorrelated sinusoids from noise
The so-called eigen-decomposition methods
are amongst the best at high SNR
They are used also extensively in array signal
processing eg for the estimation of the
Direction of Arrival (DoA).(Measurements of
spatial frequencies is equivalent to direction
finding)
Professor A G Constantinides©
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AGC
Eigenvector-based Methods
DSP
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These techniques can resolve
frequencies that are closely spaced and
hence are often referred to as “superresolution” methods
Professor A G Constantinides©
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AGC
Mathematical Background
DSP
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The signal model: Assume that the
signal is given by
p
x[n]   ai e


j ( ni i )
i 1
Then
p
jni
rxx [k ]  E{x[n]x[n  k ]}   pi e
i 1
where
2
pi  ai
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AGC
Mathematical Background
DSP

Set
 x[n] 
 x[n  1] 


.


x[n]  

.


 x[n  M  2]
 x[n  M  1] 


rxx [1]
 rxx [0]
 r [1]
rxx [0]
xx
R xx  
.
.

r [ M  1]
.
 xx
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. rxx [( M  1)] 
. rxx [( M  2)]

.
.

.
rxx [0] 
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AGC
Mathematical Background
DSP
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Then we can write the autocorrelation
p
matrix as
H
R xx   pk sk sk
k 1

where
1


 e j1



j 21
 e

si  

.
 j ( M  2 ) 
1
e

 e j ( M 1)1 


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AGC
Mathematical Background
DSP
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And hence
Where
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And
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R xx  SPS H
s  s1 s 2 . . s p 
P  diag ( p1 ,
p2 , . .
p p )
The vector space
S  span{s1 , s 2 , . . s p }
is the signal subspace of {x[ n]}
Professor A G Constantinides©
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AGC
Mathematical Background
DSP
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If M  p then R xx has rank p
Let the eigenvalues of R xx be
1  2  3  ...  M
Corresponding to the normalised
eigenvectors u1 , u 2 , . . u M
Then R xxui  iui
M
and
H
R xx   iuiui
i 1
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AGC
Mathematical Background
DSP

Since R xx is off rank p then
 p 1   p 2   p 3  ...  M  0

And hence
p
R xx   
i 1

H
iuiui
The eigenvectors
u1 , u 2 , . . u p

are the “principal eigenvectors” of
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R xx
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AGC
Mathematical Background
DSP
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An important result is
“The principal eigenvectors span the signal
subspace”
Thus given a sequence of observations we
can determine the autocorrelation matrix and
its eigenvectors.
Knowing the first p eigenvectors we can
determine the space in which the signals
reside even though at this point we do not
know their frequencies.
Professor A G Constantinides©
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AGC
Mathematical Background
DSP

The noise model: We assume that we
observe signal contaminated additively,
by a stationary, zero mean, white noise,
independent of it
y[n]  x[n]  w[n]

Then
ryy [k ]  rxx [k ]    [k ]
2
w
Professor A G Constantinides©
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AGC
Mathematical Background
DSP

From the above we have
p
ryy [k ]   pi e
i 1

Moreover with
y[n]


 y[n  1] 


.


y[n]  

.


 y[n  M  2]
 y[n  M  1] 


jki
   [k ]
2
w
w[ n]


 w[ n  1] 


.


w[ n]  

.


 w[ n  M  2]
 w[ n  M  1] 


Professor A G Constantinides©
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AGC
Mathematical Background
DSP
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We have
R yy  R xx  
Clearly R yy is of rank M
Let its eigenvalues be
2
wI
ie full rank
1  2  3  ...  M
2
p
where the first
are i  i   w
While the rest are all ewual to the variance of
noise
Professor A G Constantinides©
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AGC
Mathematical Background
DSP

Thus we can write
p
M
R yy   (i   )u u    u u
i 1

2
w
H
i i
i  p 1
2
H
w i i
The space
N  span{u p 1 , u p  2 , . . u M }

Is called the noise subspace.
Important result: Any vector in the signal subspace is
orthogonal to the noise subspace
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AGC
Pisarenko
DSP
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The Pisarenko harmonic decomposition
exploits the orthogonality of two
subspaces directly.
Let the number of sinusoids (modes) p
be known
Set M  p  1
so that the noise is
spanned a single vector u M
Professor A G Constantinides©
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AGC
Mathematical Background
DSP
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Note that u M must be orthogonal to all
the signal subspace vectors
H
si u M  0 i  1,2,..., p
With
T
u M  u M ,0 u M ,1 . . u M ,M 1 
We have
M 1
 uM ,k e
k 0
 jki
0
Professor A G Constantinides©
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AGC
Mathematical Background
DSP
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 ji
The last equation is a polynomial in e
and hence its M 1  p which all lie on
the unit circle correspond to the frequencies
of the sinuoidal signal.
The amplitudes are obtained from the
autocorrelation relationships of the
observations as given earlier.
The noise strength is given from the last
eigenvalue of the same autocorrelation
matrix.
Professor A G Constantinides©
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AGC
MUSIC
DSP
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Multiple Signal Classification relies on
the same principle of orthogonality.
T
j
j 2
j ( M 1)
e
. e
Let s ( )  1 e


s ( )x  0
H
Professor A G Constantinides©
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AGC
MUSIC
DSP
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Now define the function
M
M ( )   s ( )u k
2
H
k  p 1
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Clearly M (i )  0 .Hence its reciprocal is infinite
Thus the reciprocal of the above function exhibits
peaks at the input frequencies.
So that whenever
(one of the input
requencies), then for any vector
in the noise
subspace
The signal strengths can be computed as in the
Pisarenko Harmonic Decomposition
Professor A G Constantinides©
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AGC
MUSIC
DSP
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The quantity below is known as the
MUSIC spectrum.
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P( ) 

M ( )
1
M
 s ( )u k
2
H
k  p 1
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