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8/12/2019 Basic Definitions and Spectral Estimation
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Basic Definitions
and
The Spectral Estimation Problem
Lecture 1
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L11
by P. Stoica and R. Moses, Prentice Hall, 1997
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Informal Definition of Spectral Estimation
Given:A finite record of a signal.
Determine:The distribution of signal power over
frequency.
t
signal
t=1, 2, ... +
spectral density
! = (angular) frequency in radians/(sampling interval)
f =! =
2 = frequency in cycles/(sampling interval)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L12
by P. Stoica and R. Moses, Prentice Hall, 1997
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Applications
Temporal Spectral Analysis
Vibration monitoring and fault detection
Hidden periodicityfinding
Speech processing and audio devices
Medical diagnosis
Seismology and ground movement study
Control systems design
Radar, Sonar
Spatial Spectral Analysis
Source location using sensor arrays
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L13
by P. Stoica and R. Moses, Prentice Hall, 1997
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Deterministic Signals
f y ( t ) g
1
t =; 1
= discrete-time deterministic data
sequence
If:1
X
t =; 1
j y ( t ) j
2
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Energy Spectral Density
Parseval's Equality:
1
X
t =; 1
j y ( t ) j
2 =1
2
Z
;
S ( ! )d !
where
S ( ! )
4
= j Y ( ! ) j
2
= Energy Spectral Density
We can write
S ( !) =
1
X
k =; 1
( k ) e
;i ! k
where
( k) =
1
X
t =; 1
y ( t ) y
( t ; k )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L15
by P. Stoica and R. Moses, Prentice Hall, 1997
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Random Signals
Random Signal
t
random signal probabilistic statements about
future variations
current observation time
Here:1
X
t =; 1
j y ( t ) j
2
= 1
But:E
n
j y ( t ) j
2
o
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First Definition of PSD
( ! )
=
1
X
k =; 1
r ( k ) e
;i ! k
where r ( k ) is theautocovariance sequence (ACS)
r ( k) =
E f y ( t ) y
( t ; k ) g
r ( k) =
r
( ; k ) r ( 0 ) j
r ( k ) j
Note that
r ( k) =
1
2
Z
;
( ! ) e
i ! k
d !(Inverse DTFT)
Interpretation:
r( 0 ) =
E
n
j y ( t ) j
2
o
=
1
2
Z
;
( ! )d !
so
( ! )d !
= infinitesimal signal power in the band
!
d !
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L17
by P. Stoica and R. Moses, Prentice Hall, 1997
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Second Definition of PSD
( ! ) =l i m
N! 1
E
8
>
:
1
N
N
X
t= 1
y ( t ) e
;i ! t
2
9
>
=
>
Note that
( !) = l i m
N! 1
E
1
N
j Y
N
( ! ) j
2
where
Y
N
( !) =
N
X
t= 1
y ( t ) e
;i ! t
is thefinite DTFT off y ( t ) g .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L18
by P. Stoica and R. Moses, Prentice Hall, 1997
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Properties of the PSD
P1: ( !) =
( !+ 2
) for all ! .
Thus, we can restrict attention to
! 2 ;
]( )
f 2 ; 1 = 2 1 =2 ]
P2: ( ! ) 0
P3: Ify ( t ) is real,
Then: ( ! ) = ( ; ! )
Otherwise: ( ! ) 6= ( ; ! )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L19
by P. Stoica and R. Moses, Prentice Hall, 1997
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Transfer of PSD Through Linear Systems
System Function: H ( q) =
1
X
k= 0
h
k
q
; k
where q ; 1 = unit delay operator: q ; 1 y ( t) =
y ( t ;1 )
e ( t )
e
( ! )
y ( t )
y
( !) =
j H ( ! ) j
2
e
( ! )
H ( q )
--
Then
y ( t ) =1
X
k= 0
h
k
e ( t ; k )
H ( ! ) =1
X
k= 0
h
k
e
;i ! k
y
( ! ) = j H ( ! ) j 2 e
( ! )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L110
by P. Stoica and R. Moses, Prentice Hall, 1997
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The Spectral Estimation Problem
The Problem:
From a sample f y( 1 )
: : : y ( N ) g
Find an estimate of ( ! ) : f ( ! ) !
2 ;
] g
Two Main Approaches :
Nonparametric:
Derived from the PSD definitions.
Parametric:
Assumes a parameterized functional form of the
PSD
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L111
by P. Stoica and R. Moses, Prentice Hall, 1997
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Periodogram
and
Correlogram
Methods
Lecture 2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L21
by P. Stoica and R. Moses, Prentice Hall, 1997
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Correlogram
Recall 1st definition of ( ! ) :
( !) =
1
X
k =; 1
r ( k ) e
;i ! k
Truncate theP
and replace r ( k ) by r ( k ) :
c
( ! ) =N ; 1
X
k = ; ( N ;1 )
r ( k ) e
;i ! k
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L23
by P. Stoica and R. Moses, Prentice Hall, 1997
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Covariance Estimators
(or Sample Covariances)
Standard unbiased estimate:
r ( k) =
1
N ; k
N
X
t = k+ 1
y ( t ) y
( t ; k ) k
0
Standard biased estimate:
r ( k) =
1
N
N
X
t = k+ 1
y ( t ) y
( t ; k ) k
0
For both estimators:
r ( k) =
r
( ; k ) k
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Relationship Between p
( ! ) and c
( ! )
If: the biased ACS estimator r ( k ) is used in c
( ! ) ,
Then:
p
( !) =
1
N
N
X
t= 1
y ( t ) e
;i ! t
2
=
N ; 1
X
k = ; ( N ;1 )
r ( k ) e
;i ! k
=
c
( ! )
p
( !) =
c
( ! )
Consequence:
Both p
( ! ) and c
( ! ) can be analyzed simultaneously.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L25
by P. Stoica and R. Moses, Prentice Hall, 1997
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Statistical Performance of p
( ! ) and c
( ! )
Summary:
Both are asymptotically (for large N ) unbiased:
E
n
p
( ! )
o
! ( ! ) as N! 1
Both havelargevariance, even for large N .
Thus, p
( ! ) and c
( ! ) havepoor performance.
Intuitive explanation:
r ( k ) ; r ( k ) may be large for large j k j
Even if the errorsf r ( k ) ; r ( k ) g
N ; 1
j k j= 0 are small,
there areso manythat when summed in
p
( ! ) ; ( !) ]
, the PSD error is large.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L26
by P. Stoica and R. Moses, Prentice Hall, 1997
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Bias Analysis of the Periodogram
E
n
p
( ! )
o
= E
n
c
( ! )
o
=
N ; 1
X
k = ; ( N ;1 )
E f r ( k ) g e
;i !
=
N ; 1
X
k = ; ( N ;1 )
1 ;
j k j
N
!
r ( k ) e
;i ! k
=
1
X
k =; 1
w
B
( k ) r ( k ) e
;i ! k
w
B
( k) =
8
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Bartlett Window WB
( ! )
W
B
( !) =
1
N
"
s i n ( ! N = 2 )
s i n ( ! = 2 )
#
2
W
B
( ! )= W
B
( 0 )
, forN
= 2 5
3 2 1 0 1 2 360
50
40
30
20
10
0
dB
ANGULAR FREQUENCY
Main lobe 3dB width 1= N
.
Forsmall N , WB
( ! ) may differ quite a bit from ( ! ) .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L28
by P. Stoica and R. Moses, Prentice Hall, 1997
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Smearing and Leakage
Main Lobe Width:smearingorsmoothing
Details in ( ! ) separated in f by less than 1= N are not
resolvable.
()
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Periodogram Bias Properties
Summary of Periodogram Bias Properties:
Forsmall N , severe bias
AsN
! 1
,W
B
( ! ) ! ( ! )
,so ( ! ) is asymptotically unbiased.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L210
by P. Stoica and R. Moses, Prentice Hall, 1997
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Periodogram Variance
As N ! 1
E
n h
p
( !
1
) ; ( !
1
)
i h
p
( !
2
) ; ( !
2
)
i o
=
(
2
( !
1
) !
1
= !
2
0 !
1
6= !
2
Inconsistent estimate
Erratic behavior
()^
1 st. dev = () too+-
asymptotic mean = ()
()^
Resolvability properties depend onbothbias and variance.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L211
by P. Stoica and R. Moses, Prentice Hall, 1997
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Discrete Fourier Transform (DFT)
Finite DTFT: YN
( !) =
N
X
t= 1
y ( t ) e
;i ! t
Let ! = 2 N
k and W = e ; i2
N .
Then YN
(
2
N
k ) is the Discrete Fourier Transform (DFT):
Y ( k) =
N
X
t= 1
y ( t ) W
t k
k = 0 : : : N ; 1
Direct computation off Y ( k ) g N ; 1k
= 0
from f y ( t ) g Nt
= 1
:
O ( N
2
) flops
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L212
by P. Stoica and R. Moses, Prentice Hall, 1997
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Radix2 Fast Fourier Transform (FFT)
Assume: N= 2
m
Y ( k) =
N =2
X
t= 1
y ( t ) W
t k
+
N
X
t =N = 2 + 1
y ( t ) W
t k
=
N =2
X
t= 1
y ( t) +
y ( t +N = 2 )
W
N k
2
] W
t k
with WN k
2
=
+ 1 for even k
; 1 for odd k
Let ~N =N =
2 and ~W = W 2 = e ; i 2 =~
N .
For k = 0 2 4 : : : N ; 24
= 2p :
Y( 2
p) =
~
N
X
t= 1
y ( t) +
y ( t +
~
N) ]
~
W
t p
For k= 1
3 5 : : : N ;1 = 2
p+ 1
:
Y( 2
p+ 1 ) =
~
N
X
t= 1
f y ( t ) ; y ( t +
~
N) ]
W
t
g
~
W
t p
Each is a ~N =N =
2 -point DFT computation.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L213
by P. Stoica and R. Moses, Prentice Hall, 1997
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FFT Computation Count
Let ck
= number offlops for N = 2 k point FFT.
Then
c
k
=
2
k
2
+ 2c
k ; 1
) c
k
=
k 2
k
2
Thus,
c
k
=
1
2
Nl o g
2
N
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L214
by P. Stoica and R. Moses, Prentice Hall, 1997
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Zero Padding
Append the given data by zeros prior to computing DFT
(or FFT):
f y( 1 ) : : : y
( N ) 0 : : :
0
| { z }
N
g
Goals:
Apply a radix-2 FFT (so N = power of 2)
Finer sampling of ( ! ) :
2
N
k
N ; 1
k= 0
!
2
N
k
N ; 1
k= 0
()^continuous curve
sampled, N=8
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L215
by P. Stoica and R. Moses, Prentice Hall, 1997
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Improved Periodogram-Based
Methods
Lecture 3
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L31
by P. Stoica and R. Moses, Prentice Hall, 1997
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Blackman-Tukey Method
Basic Idea:Weighted correlogram, with small weight
applied to covariances r ( k ) withlarge j k j .
B T
( !) =
M ; 1
X
k = ; ( M ;1 )
w ( k)
r ( k ) e
;i ! k
f w ( k ) g = Lag Window
1
-M M
w(k)
k
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L32
by P. Stoica and R. Moses, Prentice Hall, 1997
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Blackman-Tukey Method, con't
B T
( !) =
1
2
Z
;
p
( ) W ( ! ; )d
W ( !) = DTFT f w ( k ) g
= Spectral Window
Conclusion: B T
( !) = locallysmoothed periodogram
Effect:
Variance decreases substantially
Bias increases slightly
By proper choice of M :
MSE = var + bias2 ! 0 as N! 1
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L33
by P. Stoica and R. Moses, Prentice Hall, 1997
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Window Design Considerations
Nonnegativeness:
B T
( !) =
1
2
Z
;
p
( )
| { z }
0
W ( ! ; )d
IfW ( ! ) 0 (
, w ( k ) is a psd sequence)
Then: B T
( ! ) 0 (which is desirable)
Time-Bandwidth Product
N
e
=
M ; 1
X
k = ; ( M ;1 )
w ( k )
w( 0 )
= equiv time width
e
=
1
2
Z
;
W ( ! )d !
W( 0 )
=
equiv bandwidth
N
e
e
= 1
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L34
by P. Stoica and R. Moses, Prentice Hall, 1997
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Window Design, con't
e
= 1 = N
e
= 0 ( 1 = M)
is the BT resolution threshold.
As M increases, bias decreases and variance
increases.
) Choose M as a tradeoff betweenvarianceand
bias.
Once M is given, Ne
(and hence e
) is essentially
fixed.
) Choose window shape to compromise between
smearing(main lobe width) andleakage(sidelobe
level).
The energy in the main lobe and in the sidelobescannot be reducedsimultaneously, once M is given.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L35
by P. Stoica and R. Moses, Prentice Hall, 1997
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Window Examples
Triangular Window,M
= 2 5
3 2 1 0 1 2 360
50
40
30
20
10
0
dB
ANGULAR FREQUENCY
Rectangular Window, M = 2 5
3 2 1 0 1 2 360
50
40
30
20
10
0
dB
ANGULAR FREQUENCY
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L36
by P. Stoica and R. Moses, Prentice Hall, 1997
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Bartlett Method
Basic Idea:
1 2 . . .
average
. . . ()^
1 ()^
2 ()^
L
()^B
Mathematically:
y
j
( t) =
y
(
( j ;1 )
M + t
)
t= 1
: : : M
= the j th subsequence
( j= 1
: : : L
4
= N = M ] )
j
( !) =
1
M
M
X
t= 1
y
j
( t ) e
;i ! t
2
B
( !) =
1
L
L
X
j= 1
j
( ! )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L37
by P. Stoica and R. Moses, Prentice Hall, 1997
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Comparison of Bartlett
and Blackman-Tukey Estimates
B
( !) =
1
L
L
X
j= 1
8
>
:
M ; 1
X
k = ; ( M ;1 )
r
j
( k ) e
;i ! k
9
>
=
>
=
M ; 1
X
k = ; ( M ;1 )
8
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Welch Method
Similar to Bartlett method, but
allow overlap of subsequences (gives more
subsequences, and thusbetteraveraging)
use data window for each periodogram; gives
mainlobe-sidelobe tradeoff capability
1 2 N
subseq#1
. . .
subseq#2
subseq#S
Let S = # of subsequences of length M .(Overlapping means
S >
N = M] ) better
averaging.)
Additional flexibility:
The data in each subsequence are weighted by atemporal
window
Welch is approximately equal to B T
( ! ) with a
non-rectangular lag window.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L39
by P. Stoica and R. Moses, Prentice Hall, 1997
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Daniell Method
By a previous result, for N 1 ,
f
p
( !
j
) g are (nearly) uncorrelated random variables for
!
j
=
2
N
j
N ; 1
j= 0
Idea: Local averagingof( 2 J + 1 ) samples in the
frequency domain should reduce the variance by about
( 2J
+ 1 ) .
D
( !
k
) =
1
2 J+ 1
k + J
X
j = k ; J
p
( !
j
)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L310
by P. Stoica and R. Moses, Prentice Hall, 1997
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Daniell Method, con't
As J increases:
Bias increases (more smoothing)
Variance decreases (more averaging)
Let = 2 J = N
. Then, for N 1 ,
D
( ! ) '
1
2
Z
;
p
( ! )d !
Hence: D
( ! ) '
B T
( ! ) with arectangular spectral
window.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L311
by P. Stoica and R. Moses, Prentice Hall, 1997
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Summary of Periodogram Methods
Unwindowed periodogram
reasonable bias unacceptable variance
Modified periodograms
Attempt to reduce the variance at the expense of (slightly)increasing the bias.
BT periodogram
Local smoothing/averaging of p
( ! ) by a suitably selectedspectralwindow.
Implemented by truncating and weighting r ( k ) using alagwindow in
c
( ! )
Bartlett, Welch periodograms
Approximate interpretation: B T
( ! ) with a suitablelagwindow (rectangular for Bartlett; more general for Welch).
Implemented by averaging subsample periodograms.
Daniell Periodogram
Approximate interpretation: B T
( ! ) with a rectangularspectralwindow.
Implemented by local averaging of periodogram values.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L312
by P. Stoica and R. Moses, Prentice Hall, 1997
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Parametric Methods
for
Rational Spectra
Lecture 4
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L41
by P. Stoica and R. Moses, Prentice Hall, 1997
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Basic Idea of Parametric Spectral Estimation
ObservedData
Assumedfunctional
form of (,)
Estimateparameters
in (,)
Estimate
PSD
() =(,)^ ^
possibly revise assumption on ()
Rational Spectra
( !) =
P
j kj
m
k
e
;i ! k
P
j kj
n
k
e
;i ! k
( ! ) is arational functionin e ; i ! .
ByWeierstrass theorem, ( ! ) can approximate arbitrarily
wellany continuous PSD, provided m and n are chosen
sufficiently large.
Note, however:
choice ofm and n is not simple
some PSDs arenotcontinuous
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L42
by P. Stoica and R. Moses, Prentice Hall, 1997
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AR, MA, and ARMA Models
BySpectral Factorizationtheorem, a rational ( ! ) can befactored as
( !) =
B ( ! )
A ( ! )
2
2
A ( z) = 1 +
a
1
z
; 1
+
+ a
n
z
; n
B ( z) = 1 +
b
1
z
; 1
+
+ b
m
z
; m
and,e.g., A ( ! ) = A ( z ) jz = e
i !
Signal Modeling Interpretation:
e ( t )
e
( !) =
2
w h i t e n o i s e
y ( t )
y
( !) =
B ( ! )
A ( ! )
2
2
B ( q )
A ( q )
l t e r e d w h i t e n o i s e
--
ARMA: A ( q ) y ( t) =
B ( q ) e ( t )
AR:A ( q ) y ( t
) =e ( t )
MA: y ( t) =
B ( q ) e ( t )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L43
by P. Stoica and R. Moses, Prentice Hall, 1997
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ARMA Covariance Structure
ARMA signal model:
y ( t) +
n
X
i
= 1
a
i
y ( t ; i) =
m
X
j
= 0
b
j
e ( t ; j ) ( b
0
= 1 )
Multiply by y ( t ; k ) and take E f g to give:
r ( k) +
n
X
i= 1
a
i
r ( k ; i) =
m
X
j= 0
b
j
E f e ( t ; j ) y
( t ; k ) g
=
2
m
X
j= 0
b
j
h
j ; k
= 0for
k > m
where H ( q) =
B ( q )
A ( q )
=
1
X
k= 0
h
k
q
; k
( h
0
= 1 )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L44
by P. Stoica and R. Moses, Prentice Hall, 1997
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AR Signals: Yule-Walker Equations
AR: m= 0 .
Writing covariance equation in matrix form for
k= 1 : : : n
:2
6
6
6
4
r( 0 )
r ( ;1 ) : : : r
( ; n )
r( 1 )
r( 0 )
...... . . . r ( ;
1 )
r ( n ): : : r ( 0 )
3
7
7
7
5
2
6
6
6
4
1
a
1
...
a
n
3
7
7
7
5
=
2
6
6
6
4
2
0
...
0
3
7
7
7
5
R
"
1
#
=
"
2
0
#
These are theYuleWalker (YW) Equations.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L45
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AR Spectral Estimation: YW Method
Yule-Walker Method:
Replace r ( k ) by r ( k ) and solve for f ai
g and 2 :
2
6
6
6
4
r( 0 ) ^
r ( ;1 ) : : :
r ( ; n )
r( 1 ) ^
r( 0 )
...... . . . r ( ;
1 )
r ( n ): : :
r( 0 )
3
7
7
7
5
2
6
6
6
4
1
a
1
...
a
n
3
7
7
7
5
=
2
6
6
6
4
2
0
...
0
3
7
7
7
5
Then the PSD estimate is
( !) =
2
j
A ( ! ) j
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L46
by P. Stoica and R. Moses, Prentice Hall, 1997
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AR Spectral Estimation: LS Method
Least Squares Method:
e ( t) =
y ( t) +
n
X
i= 1
a
i
y ( t ; i) =
y ( t) +
'
T
( t )
4
= y ( t) +
y ( t )
where ' ( t) =
y ( t ;1 )
: : : y ( t ; n) ]
T .
Find =
a
1
: : : a
n
]
T to minimize
f ( ) =
N
X
t = n+ 1
j e ( t ) j
2
This gives = ; ( Y Y ) ; 1 ( Y y ) where
y =
2
6
4
y ( n + 1 )
y ( n + 2 )
...
y ( N )
3
7
5
Y =
2
6
4
y ( n ) y ( n ; 1 ) y ( 1 )
y ( n + 1 ) y ( n ) y ( 2 )
... ...
y ( N ; 1 ) y ( N ; 2 ) y ( N ; n )
3
7
5
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L47
by P. Stoica and R. Moses, Prentice Hall, 1997
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LevinsonDurbin Algorithm
Fast, order-recursive solution to YW equations
2
6
6
6
4
0
; 1
; n
1
0
. . . ...... . . . . . .
; 1
n
1
0
3
7
7
7
5
| { z }
R
n + 1
2
6
6
6
4
1
n
3
7
7
7
5
=
2
6
6
6
4
2
n
0
...
0
3
7
7
7
5
k
= either r ( k ) or r ( k ) .
Direct Solution:
For one given value of n : O ( n3
) flops
For k= 1
: : : n : O ( n 4 ) flops
LevinsonDurbin Algorithm:
Exploits the Toeplitz form ofRn
+ 1
to obtain the solutions
for k = 1 : : : n in O ( n 2 ) flops!
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L48
by P. Stoica and R. Moses, Prentice Hall, 1997
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Levinson-Durbin Alg, con't
Relevant Properties of R :
R x
= y $ R ~x= ~
y , where ~x=
x
n
: : : x
1
]
T
Nested structure
R
n+ 2
=
2
4
R
n + 1
n + 1
~r
n
n + 1
~r
n
0
3
5
~r
n
=
2
6
4
n
...
1
3
7
5
Thus,
R
n + 2
2
4
1
n
0
3
5
=
2
4
R
n + 1
n + 1
~r
n
n + 1
~r
n
0
3
5
2
4
1
n
0
3
5
=
2
4
2
n
0
n
3
5
where n = n
+ 1
+ ~r
n
n
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L49
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Levinson-Durbin Alg, con't
R
n + 2
2
4
1
n
0
3
5
=
2
4
2
n
0
n
3
5
R
n + 2
2
4
0
~
n
1
3
5
=
2
4
n
0
2
n
3
5
Combining these gives:
R
n + 2
8
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MA Signals
MA: n= 0
y ( t) =
B ( q ) e ( t )
= e ( t) +
b
1
e ( t ;1 ) +
+ b
m
e ( t ; m )
Thus,
r ( k) = 0
for j k j> m
and
( !) =
j B ( ! ) j
2
2
=
m
X
k = ; m
r ( k ) e
;i ! k
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L411
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MA Spectrum Estimation
Two main ways to Estimate ( ! ) :
1. Estimate f bk
g and 2 and insert them in
( !) =
j B ( ! ) j
2
2
nonlinear estimation problem
( ! ) is guaranteed to be 0
2. Insert sample covariances f r ( k ) g in:
( !) =
m
X
k = ; m
r ( k ) e
;i ! k
This is B T
( ! ) with a rectangular lag window of
length 2 m+ 1 .
( ! ) isnotguaranteed to be 0
Both methods are special cases of ARMA methods
described below, with AR model order n= 0 .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L412
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ARMA Signals
ARMA models can represent spectra with both peaks
(AR part) and valleys (MA part).
A ( q ) y ( t) =
B ( q ) e ( t )
( !) =
2
B ( ! )
A ( ! )
2
=
P
m
k = ; m
k
e
;i ! k
j A ( ! ) j
2
where
k
= E f B ( q ) e ( t) ]
B ( q ) e ( t ; k) ]
g
= E f A ( q ) y ( t) ]
A ( q ) y ( t ; k) ]
g
=
n
X
j= 0
n
X
p= 0
a
j
a
p
r ( k + p ; j )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L413
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ARMA Spectrum Estimation
Two Methods:
1. Estimate f ai
b
j
2
g in ( !) =
2
B ( ! )
A ( ! )
2
nonlinear estimation problem; can use an approximatelineartwo-stage least squaresmethod
( ! ) is guaranteed to be 0
2. Estimate f ai
r( k ) g in ( !
) =
P
m
k = ; m
k
e
; i ! k
j A ( ! ) j
2
linear estimation problem (the Modified Yule-Walker
method).
( ! ) isnotguaranteed to be 0
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L414
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Two-Stage Least-Squares Method
Assumption:The ARMA model is invertible:
e ( t) =
A ( q )
B ( q )
y ( t )
= y ( t) +
1
y ( t ;1 ) +
2
y ( t ;2 ) +
= AR( 1 ) with j k
j !0 as k
! 1
Step 1: Approximate, for some large K
e ( t ) ' y ( t) +
1
y ( t ;1 ) +
+
K
y ( t ; K )
1a) Estimate the coefficients f k
g
K
k= 1
by using AR
modelling techniques.
1b) Estimate the noise sequence
e ( t) =
y ( t) +
1
y ( t ;1 ) + +
K
y ( t ; K )
and its variance
2
=
1
N ; K
N
X
t = K+ 1
j e ( t ) j
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L415
by P. Stoica and R. Moses, Prentice Hall, 1997
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Two-Stage Least-Squares Method, con't
Step 2: Replace f e ( t ) g by e ( t ) in the ARMA equation,
A ( q ) y ( t ) ' B ( q)
e ( t )
and obtain estimates of f ai
b
j
g by applying least squares
techniques.
Note that the ai
and bj
coefficients enter linearly in the
above equation:
y ( t ) ; e ( t ) ' ; y ( t ;1 ) : : :
; y ( t ; n )
e ( t ;1 ) : : :
e ( t ; m) ]
=
a
1
: : : a
n
b
1
: : : b
m
]
T
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L416
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Modified Yule-Walker Method
ARMA Covariance Equation:
r ( k) +
n
X
i= 1
a
i
r ( k ; i) = 0 k > m
In matrix form for k = m+ 1
: : : m + M
2
6
4
r ( m ) : : : r ( m ; n + 1 )
r ( m + 1 ) r ( m ; n + 2 )
... . . .
...
r ( m + M ; 1 ) : : : r ( m ; n + M )
3
7
5
"
a
1
...
a
n
#
= ;
2
6
4
r ( m + 1 )
r ( m + 2 )
...
r ( m + M )
3
7
5
Replace f r ( k ) g by f r ( k ) g and solve for f ai
g .
IfM = n , fast Levinson-type algorithms exist for
obtaining f ai
g .
IfM > n overdetermined Y W system of equations; least
squares solution for f ai
g .
Note:For narrowband ARMA signals, the accuracy of
f a
i
g is often better forM > n
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L417
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Sum
maryofParametricMethodsfor
RationalSpectra
Computational
Guarantee
Method
Burden
Accuracy
(
!
)
0
?
Usefor
AR:YWor
LS
low
medium
Yes
Spectrawith(narrow
)peaksbut
novalley
MA:BT
low
low-medium
No
Broadbandspectrapo
ssiblywith
valleysbutnopeaks
ARMA:MYW
low-medium
medium
No
Spectrawithbothpea
ksand(not
toodeep)valleys
ARMA:2-Sta
geLS
medium-high
medium-high
Yes
Asabove
LecturenotestoaccompanyIntroductiontoSpectralAnalysis
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lideL418
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Parametric Methods
for
Line SpectraPart 1
Lecture 5
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L51
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Line Spectra
Many applications have signals with (near) sinusoidal
components. Examples:
communications
radar, sonar
geophysical seismology
ARMA model is apoor approximation
Better approximation byDiscrete/Line Spectrum Models
-
6
6
6
6
2
;
!
1
!
2
!
3
( 2
2
1
)
( 2
2
2
)
( 2
2
3
)
!
( ! )
AnIdealline spectrum
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L52
by P. Stoica and R. Moses, Prentice Hall, 1997
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Line Spectral Signal Model
Signal Model:Sinusoidal components of frequencies
f !
k
g and powers f 2k
g , superimposed in white noise of
power 2 .
y ( t) =
x ( t) +
e ( t ) t= 1
2 : : :
x ( t) =
n
X
k= 1
k
e
i ( !
k
t +
k
)
| { z }
x
k
( t )
Assumptions:
A1: k
> 0 !
k
2 ;
]
(prevents model ambiguities)
A2: f 'k
g = independent rv's, uniformly
distributed on ;
]
(realistic and mathematically convenient)
A3: e ( t) = circular white noise with variance 2
E f e ( t ) e
( s ) g =
2
t s
E f e ( t ) e ( s ) g= 0
(can be achieved byslowsampling)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L53
by P. Stoica and R. Moses, Prentice Hall, 1997
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Covariance Function and PSD
Note that:
E
n
e
i '
p
e
;i '
j
o
= 1, for p = j
E
n
e
i '
p
e
;i '
j
o
= E
n
e
i '
p
o
E
n
e
;i '
j
o
=
1
2
Z
;
e
i '
d '
2
= 0, for p 6= j
Hence,
E
n
x
p
( t ) x
j
( t ; k )
o
=
2
p
e
i !
p
k
p j
r ( k) =
E f y ( t ) y
( t ; k ) g
=
P
n
p= 1
2
p
e
i !
p
k
+
2
k 0
and
( !) = 2
n
X
p= 1
2
p
( ! ; !
p
) +
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L54
by P. Stoica and R. Moses, Prentice Hall, 1997
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Parameter Estimation
Estimate either:
f!
k
k
'
k
g
n
k= 1
2 (Signal Model)
f!
k
2
k
g
n
k= 1
2 (PSD Model)
Major Estimation Problem: f !k
g
Once f !k
g are determined:
f
2
k
g can be obtained by a least squares method from
r ( k) =
n
X
p= 1
2
p
e
i !
p
k + residuals
OR:
Both f k
g and f 'k
g can be derived by a least
squares method from
y ( t) =
n
X
k= 1
k
e
i !
k
t
+ residuals
with k
=
k
e
i '
k .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L55
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Nonlinear Least Squares (NLS) Method
m i n
f !
k
k
'
k
g
N
X
t= 1
y ( t ) ;
n
X
k= 1
k
e
i ( !
k
t + '
k
)
2
| { z }
F (! '
)
Let:
k
=
k
e
i '
k
=
1
: : :
n
]
T
Y=
y( 1 )
: : : y ( N) ]
T
B =
2
6
4
e
i !
1
e
i !
n
... ...
e
i N !
1
e
i N !
n
3
7
5
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L56
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Nonlinear Least Squares (NLS) Method, con't
Then:
F= (
Y ;B
)
( Y ;B ) =
k Y ;B
k
2
= ; ( B
B )
; 1
B
Y ]
B
B ]
; ( B
B )
; 1
B
Y ]
+ Y
Y ; Y
B ( B
B )
; 1
B
Y
This gives:
= (
B
B )
; 1
B
Y
!=
!
and
!= a r g m a x
!
Y
B ( B
B )
; 1
B
Y
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L57
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NLS Properties
Excellent Accuracy:
var(
!
k
) =
6
2
N
3
2
k
( for N 1 )
Example: N= 3 0 0
SNRk
=
2
k
=
2
= 3 0 dB
Thenq
var(
!
k
) 1 0
; 5 .
Difficult Implementation:
The NLS cost function F is multimodal; it is difficult to
avoid convergence to local minima.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L58
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Unwindowed Periodogram as an
Approximate NLS Method
For a single (complex) sinusoid, the maximum of the
unwindowed periodogram is the NLS frequency estimate:
Assume: n= 1
Then: B B = N
B
Y =
N
X
t= 1
y ( t ) e
;i ! t
= Y ( ! ) (finite DTFT)
Y
B ( B
B )
; 1
B
Y =
1
N
j Y ( ! ) j
2
=
p
( ! )
= (Unwindowed Periodogram)
So, withno approximation,
!= a r g m a x
!
p
( ! )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L59
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Unwindowed Periodogram as an
Approximate NLS Method, con't
Assume:n >
1
Then:
f !
k
g
n
k= 1
' the locations of the n largest
peaks of p
( ! )
provided that
i n fj !
k
; !
p
j > 2 = N
which is the periodogram resolution limit.
If better resolution desired then use a High/Super
Resolutionmethod.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L510
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High-Order Yule-Walker Method
Recall:
y ( t) =
x ( t) +
e ( t) =
n
X
k= 1
k
e
i ( !
k
t + '
k
)
| { z }
x
k
( t )
+ e ( t )
DegenerateARMA equation for y ( t ) :
( 1; e
i !
k
q
; 1
) x
k
( t )
=
k
n
e
i ( !
k
t + '
k
)
; e
i !
k
e
i !
k
( t ;1 ) +
'
k
]
o
= 0
Let
B ( q) = 1 +
L
X
k= 1
b
k
q
; k
4
= A ( q )
A ( q )
A ( q) = ( 1
; e
i !
1
q
; 1
) ( 1
; e
i !
n
q
; 1
)
A ( q) =
arbitrary
Then B ( q ) x ( t ) 0 )
B ( q ) y ( t) =
B ( q ) e ( t )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L511
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High-Order Yule-Walker Method, con't
Estimation Procedure:
Estimate f bi
g
L
i= 1
using an ARMA MYW technique
Roots of
B ( q )
givef !
k
g
n
k= 1
, along withL ; n
spuriousroots.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L512
by P. Stoica and R. Moses, Prentice Hall, 1997
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High-Order and Overdetermined YW Equations
ARMA covariance:
r ( k) +
L
X
i= 1
b
i
r ( k ; i) = 0 k > L
In matrix form for k = L+ 1
: : : L + M
2
6
6
6
4
r ( L ): : : r ( 1 )
r ( L+ 1 ) : : : r ( 2 )
.
.. .
..r ( L + M ;
1 ) : : : r ( M )
3
7
7
7
5
| { z }
4
=
b = ;
2
6
6
6
4
r ( L+ 1 )
r ( L+ 2 )
...
r ( L + M )
3
7
7
7
5
| { z }
4
=
This is a high-order (ifL > n
) and overdetermined
(ifM > L
) system of YW equations.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L513
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High-Order and Overdetermined YW Equations,
con't
Fact: rank( ) =
n
SVD of : =
U V
U= (
M n ) with U U = In
V
= (n L ) with V V = I
n
= (
n n ) , diagonal and nonsingular
Thus,
( U V
) b = ;
The Minimum-Norm solution is
b = ;
y
= ; V
; 1
U
Important property:The additional ( L ; n ) spurious
zeros ofB ( q ) are located strictlyinsidethe unit circle, if
the Minimum-Norm solution b is used.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L514
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HOYW Equations, Practical Solution
Let =
but made from f r ( k ) g instead off r ( k ) g .
Let U , , V be defined similarly to U , , V from the
SVD of .
Compute b = ; V ; 1 U
Then f !k
g
n
k= 1
are found from the n zeroes of B ( q ) that
are closest to the unit circle.
When the SNR is low, this approach may give spurious
frequency estimates whenL > n
; this is the price paid for
increased accuracy whenL > n
.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L515
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Parametric Methods
for
Line SpectraPart 2
Lecture 6
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L61
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The Covariance Matrix Equation
Let:
a ( !) = 1
e
;i !
: : : e
; i ( m ;1 )
!
]
T
A=
a ( !
1
): : : a
( !
n
) ] (m n )
Note: rank( A) =
n (for m n )
Define
~y ( t )
4
=
2
6
6
6
4
y ( t )
y ( t ;1 )
...
y ( t ; m+ 1 )
3
7
7
7
5
= A ~x ( t) + ~
e ( t )
where
~x ( t) =
x
1
( t ) : : : x
n
( t) ]
T
~ e ( t) =
e ( t ) : : : e ( t ; m+ 1 ) ]
T
Then
R
4
= E f ~y ( t) ~
y
( t ) g =A P A
+
2
I
with
P = E f ~x ( t) ~
x
( t ) g =
2
6
4
2
1
0
. . .
0
2
n
3
7
5
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L62
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Eigendecomposition of R and Its Properties
R =A P A
+
2
I (m > n
)
Let:
1
2
: : :
m
: eigenvalues of R
f s
1
: : : s
n
g : orthonormal eigenvectors associated
with f 1
: : :
n
g
f g
1
: : : g
m ; n
g : orthonormal eigenvectors associated
with f n
+ 1
: : :
m
g
S=
s
1
: : : s
n
] (m n )
G=
g
1
: : : g
m ; n
] (m ( m ; n
) )
Thus,
R= S G
]
2
6
4
1
. . .
m
3
7
5
"
S
G
#
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L63
by P. Stoica and R. Moses, Prentice Hall, 1997
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Eigendecomposition of R and Its Properties,
con't
As rank(A P A
) =n :
k
>
2
k= 1
: : : n
k
=
2
k = n+ 1
: : : m
=
2
6
4
1
;
2
0
. ..
0
n
;
2
3
7
5
=
nonsingular
Note:
R S=
A P A
S +
2
S = S
2
6
4
1
0
. . .
0
n
3
7
5
S = A (P A
S
; 1
)
4
=A C
with j Cj 6= 0
(since rank( S) = rank( A
) =n ).
Therefore, since S G= 0
,
A
G= 0
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L64
by P. Stoica and R. Moses, Prentice Hall, 1997
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MUSIC Method
A
G =
2
6
4
a
( !
1
)
...
a
( !
n
)
3
7
5
G= 0
) fa ( !
k
) g
n
k= 1
? R( G )
Thus,
f !
k
g
n
k= 1
are theunique solutionsof
a
( ! )G G
a ( !) = 0
.
Let:
R =
1
N
N
X
t = m
~y ( t) ~
y
( t )
S
G =S G made from the
eigenvectors of R
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L65
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spectral and Root MUSIC Methods
Spectral MUSIC Method:
f !
k
g
n
k= 1
= the locations of the n highest peaks of the
pseudo-spectrumfunction:
1
a
( ! )
G
G
a ( ! )
!2 ;
]
Root MUSIC Method:
f !
k
g
n
k= 1
= the angular positions of the n roots of:
a
T
( z
; 1
)
G
G
a ( z) = 0
that are closest to the unit circle. Here,
a ( z) = 1 z
; 1
: : : z
; ( m ;1 )
]
T
Note: Both variants of MUSIC may produce spurious
frequency estimates.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L66
by P. Stoica and R. Moses, Prentice Hall, 1997
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Pisarenko Method
Pisarenko is a special case of MUSIC with m = n+ 1
(the minimum possible value).
If: m = n+ 1
Then: G=
g
1
,
) f!
k
g
n
k= 1
can be found from the roots of
a
T
( z
; 1
) g
1
= 0
no problem with spurious frequency estimates
computationally simple
(much) less accurate than MUSIC with m n + 1
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L67
by P. Stoica and R. Moses, Prentice Hall, 1997
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Min-Norm Method
Goals:Reduce computational burden, and reduce risk of
false frequency estimates.
Uses m n (as in MUSIC), but onlyonevector in
R ( G ) (as in Pisarenko).
Let"
1
g
#
=
the vector in R ( G ) , withfirst element equal
to one, that has minimum Euclidean norm.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L68
by P. Stoica and R. Moses, Prentice Hall, 1997
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Min-Norm Method, con't
Spectral Min-Norm
f ! g
n
k= 1
= the locations of the n highest peaks in the
pseudo-spectrum
1 =
a
( ! )
"
1
g
#
2
Root Min-Norm
f ! g
n
k= 1
= the angular positions of the n roots of the
polynomial
a
T
( z
; 1
)
"
1
g
#
that are closest to the unit circle.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L69
by P. Stoica and R. Moses, Prentice Hall, 1997
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Min-Norm Method: Determining g
Let S =
"
S
#
g 1
g m ; 1
Then:"
1
g
#
2 R(
G ) )
S
"
1
g
#
= 0
)
S
g = ;
Min-Norm solution: g = ; S ( S S ) ; 1
As: I = S S =
+
S
S , ( S S ) ; 1 exists iff
= k k
2
6= 1
(This holds, at least, for N 1 .)
Multiplying the above equation by gives:
( 1 ; k
k
2
) = (
S
S )
) (
S
S )
; 1
= = ( 1 ; k
k
2
)
) g = ;
S = ( 1 ; k k
2
)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L610
by P. Stoica and R. Moses, Prentice Hall, 1997
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ESPRIT Method
Let A1
= I
m ; 1
0 ]A
A
2
= 0I
m ; 1
] A
Then A2
= A
1
D , where
D =
2
6
4
e
;i !
1
0
. . .
0 e
;i !
n
3
7
5
Also, let S1
= I
m ; 1
0 ]S
S
2
= 0I
m ; 1
] S
Recall S = A C with j C j 6= 0 . Then
S
2
= A
2
C = A
1
D C= S
1
C
; 1
D C
| { z }
So has the same eigenvalues as D . is uniquely
determined as
= (
S
1
S
1
)
; 1
S
1
S
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L611
by P. Stoica and R. Moses, Prentice Hall, 1997
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ESPRIT Implementation
From the eigendecomposition of
R
,find
S
, then
S
1 and
S
2
.
The frequency estimates are found by:
f !
k
g
n
k= 1
= ;a r g ( ^
k
)
where f k
g
n
k= 1
are the eigenvalues of
= (
S
1
S
1
)
; 1
S
1
S
2
ESPRIT Advantages:
computationally simple
no extraneous frequency estimates (unlike in MUSIC
or MinNorm)
accurate frequency estimates
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L612
by P. Stoica and R. Moses, Prentice Hall, 1997
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SummaryofFre
quencyEstimationMethods
Computational
Accuracy/
RiskforFalse
Method
Burden
Resolution
FreqEstimates
Periodogr
am
small
medium-high
medium
Nonlinear
LS
veryhigh
veryhigh
veryhigh
Yule-Walker
medium
high
medium
Pisarenko
small
low
none
MUSIC
high
high
medium
Min-Norm
medium
high
small
ESPRIT
medium
veryhigh
none
Recommen
dation:
UsePeriodogramforme
dium-resolutionapplications
UseESPRITforhigh-res
olutionapplications
LecturenotestoaccompanyIntroductiontoSpectralAnalysis
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lideL613
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Filter Bank Methods
Lecture 7
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L71
by P. Stoica and R. Moses, Prentice Hall, 1997
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Basic Ideas
Two main PSD estimation approaches:
1. Parametric Approach:Parameterize ( ! ) by a
finite-dimensional model.
2. Nonparametric Approach:Implicitly smoothf ( ! ) g
! = ;
by assuming that ( ! ) is nearly
constant over the bands
! ; !
+
]
1
2 is more general than 1, but 2 requires
N >1
to ensure that the number of estimated values
(= 2 =
2 = 1 = ) is
< N .
N >1 leads to the variability / resolution compromise
associated with all nonparametric methods.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L72
by P. Stoica and R. Moses, Prentice Hall, 1997
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FilterBankApproachtoSpectra
lEstimation
-
-
-
-
D
i
v
i
s
i
o
n
b
y
l
t
e
r
b
a
n
d
w
i
d
t
h
P
o
w
e
r
C
a
l
c
u
l
a
t
i
o
n
B
a
n
d
p
a
s
s
F
i
l
t
e
r
w
i
t
h
v
a
r
y
i
n
g
!
c
a
n
d
x
e
d
b
a
n
d
w
i
d
t
h
(
!
c
)
P
o
w
e
r
i
n
t
h
e
b
a
n
d
F
i
l
t
e
r
e
d
S
i
g
n
a
l
y
(
t
)
-
6
;
!
c
!
y
(
!
)
@
@
Q
Q
-
6
;
!
c
!
~
y
(
!
)
@
@
-
6
1
!
c
!
H
(
!
)
F
B
(
!
)
(a)
'
1
2
Z
;
j
H
(
)
j
2
(
)
d
=
(b)
'
1
2
Z
!
+
!
;
(
)
d
=
(c)
'
(
!
)
(a)consistentpowercalculation
(b)Idealpassbandfilterwithband
width
(c)
(
)
constanton
2
!
;
2
!
+
2
]
Notethatassumptions(a)and
(b),aswellas(b)a
nd(c),areconflicting.
LecturenotestoaccompanyIntroductiontoSpectralAnalysis
SlideL73
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Filter Bank Interpretation of the Periodogram
p
( ~! )
4
=
1
N
N
X
t= 1
y ( t ) e
; i ~! t
2
=
1
N
N
X
t= 1
y ( t ) e
i ~! ( N ; t )
2
= N
1
X
k= 0
h
k
y ( N ; k )
2
where
h
k
=
(
1
N
e
i ~! k
k = 0 : : : N ; 1
0 otherwise
H ( !) =
1
X
k= 0
h
k
e
;i ! k
=
1
N
e
i N ( ~! ; ! )
; 1
e
i( ~
! ; ! )
; 1
center frequency ofH ( !
) = ~!
3dB bandwidth of H ( ! ) ' 1= N
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L74
by P. Stoica and R. Moses, Prentice Hall, 1997
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Filter Bank Interpretation of the Periodogram,
con't
j H ( ! ) j as a function of( ~
! ; ! ) , for N= 5 0 .
3 2 1 0 1 2 340
35
30
25
20
15
10
5
0
dB
ANGULAR FREQUENCY
Conclusion:The periodogram p
( ! ) is afilter bank PSD
estimator with bandpassfilter as given above, and:
narrowfilter passband,
power calculation from only1sample offilter output.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L75
by P. Stoica and R. Moses, Prentice Hall, 1997
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Possible Improvements to the Filter Bank
Approach
1. Split the available sample, and bandpassfilter each
subsample.
more data points for the power calculation stage.
This approach leads to Bartlett and Welch methods.
2. Useseveral bandpassfilters on the whole sample.
Eachfilter covers a small band centered on ~! .
provides several samples for power calculation.
Thismultiwindow approachis similar to the
Daniell method.
Both approachescompromise bias for variance, and in fact
are quite related to each other: splitting the data sample
can be interpreted as a special form of windowing or
filtering.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L76
by P. Stoica and R. Moses, Prentice Hall, 1997
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Capon Method
Idea:Data-dependent bandpassfilter design.
y
F
( t) =
m
X
k= 0
h
k
y ( t ; k )
= h
0
h
1
: : : h
m
]
| { z }
h
2
6
4
y ( t )
...
y ( t ; m )
3
7
5
| { z }
~y ( t )
E
n
j y
F
( t ) j
2
o
= h
R h R = E f ~y ( t
) ~y
( t ) g
H ( !) =
m
X
k= 0
h
k
e
;i ! k
= h
a ( ! )
where a ( !) = 1 e
;i !
: : : e
;i m !
]
T
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L77
by P. Stoica and R. Moses, Prentice Hall, 1997
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Capon Properties
m is the user parameter that controls the compromise
between bias and variance:
as m increases, bias decreases and variance
increases.
Capon uses one bandpassfilter only, but it splits theN -data point sample into ( N ; m ) subsequences of
length m with maximum overlap.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L79
by P. Stoica and R. Moses, Prentice Hall, 1997
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Relation between Capon and Blackman-Tukey
Methods
Consider B T
( ! ) withBartlett window:
B T
( !) =
m
X
k = ; m
m+ 1 ; j
k j
m+ 1
r ( k ) e
;i ! k
=
1
m+ 1
m
X
t= 0
m
X
s= 0
r ( t ; s ) e
;i !
( t ; s )
=
a
( ! )
R a( ! )
m+ 1
R=
r ( i ; j) ]
Then we have
B T
( !) =
a
( ! )
R a( ! )
m+ 1
C
( !) =
m+ 1
a
( ! )
R
; 1
a ( ! )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L710
by P. Stoica and R. Moses, Prentice Hall, 1997
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Relation between Capon and AR Methods
Let
ARk
( !) =
2
k
j
A
k
( ! ) j
2
be the k th order AR PSD estimate ofy ( t ) .
Then
C
( !) =
1
1
m+ 1
m
X
k= 0
1 =
ARk
( ! )
Consequences:
Due to the average over k , C
( ! ) generally hasless
statistical variabilitythan the AR PSD estimator.
Due to the low-order AR terms in the average,
C
( ! ) generally hasworse resolution and bias
propertiesthan the AR method.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L711
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial MethodsPart 1
Lecture 8
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L81
by P. Stoica and R. Moses, Prentice Hall, 1997
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The Spatial Spectral Estimation Problem
S o u r c e n
S o u r c e 2
S o u r c e 1
B
B
B
B
BN
@
@
@
@R
v
v
v
S e n s o r 1
@
@
;
;
@
@
;
;
@
@
;
;
S e n s o r 2
S e n s o r m
c
c
c
c
c
c
c
c
c
c
c
c
,
,
,
,
,
,
,
,
,
,
,
,
Problem:Detectandlocate n radiating sources by using
an array of m passive sensors.
Emitted energy:Acoustic, electromagnetic, mechanical
Receiving sensors:Hydrophones, antennas, seismometers
Applications:Radar, sonar, communications, seismology,
underwater surveillance
Basic Approach:Determine energy distribution over
space(thus the namespatial spectral analysis)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L82
by P. Stoica and R. Moses, Prentice Hall, 1997
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Simplifying Assumptions
Far-field sources in the same plane as the array of
sensors
Non-dispersive wave propagation
Hence:The waves are planar and the only location
parameter isdirection of arrival (DOA)
(or angle of arrival, AOA).
The number of sources n is known. (We do not treat
the detection problem)
The sensors are linear dynamic elements withknown
transfer characteristicsandknown locations
(That is, the array iscalibrated.)
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L83
by P. Stoica and R. Moses, Prentice Hall, 1997
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Array Model Single Emitter Case
x ( t) =
the signal waveform as measured at a reference
point (e.g., at thefirstsensor)
k
= the delay between the reference point and the
k th sensor
h
k
( t) =
the impulse response (weighting function) of
sensork
e
k
( t) = noiseat the k th sensor (e.g., thermal noise in
sensor electronics; background noise, etc.)
Note: t2 R (continuous-time signals).
Then the output of sensor k is
y
k
( t) =
h
k
( t ) x ( t ;
k
) + e
k
( t )
( = convolution operator).
Basic Problem:Estimate thetime delays f k
g with hk
( t )
known but x ( t ) unknown.
This is atime-delay estimation problemin the unknown
input case.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L84
by P. Stoica and R. Moses, Prentice Hall, 1997
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Narrowband Assumption
Assume: The emitted signals are narrowband with knowncarrier frequency !
c
.
Then: x ( t ) = ( t ) c o s !c
t + ' ( t) ]
where ( t ) '
( t ) varyslowly enoughso that
( t ;
k
) ' ( t ) '
( t ;
k
) ' ' ( t )
Time delay is now ' to aphase shift !c
k
:
x ( t ;
k
) ' ( t) c o s
!
c
t + ' ( t ) ; !
c
k
]
h
k
( t ) x ( t ;
k
)
' jH
k
( !
c
) j ( t) c o s
!
c
t + ' ( t ) ; !
c
k
+ a r g f H
k
( !
c
) g ]
where Hk
( !) = F f
h
k
( t ) g is the k th sensor's transfer
function
Hence, the k th sensor output is
y
k
( t) =
j H
k
( !
c
) j ( t )
c o s
!
c
t + ' ( t ) ; !
c
k
+ a r g H
k
( !
c
) ] + e
k
( t )
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L85
by P. Stoica and R. Moses, Prentice Hall, 1997
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Complex Signal Representation
The noise-free output has the form:
z ( t) =
( t) c o s
!
c
t + ( t) ] =
=
( t )
2
n
e
i !
c
t + ( t) ]
+ e
; i !
c
t + ( t) ]
o
Demodulate z ( t ) (translate to baseband):
2 z ( t ) e
; !
c
t
= ( t ) f e
i ( t )
| { z }
lowpass
+ e
; i 2
!
c
t + ( t) ]
| { z }
highpass
g
Lowpassfilter 2 z ( t ) e ; i ! c t to obtain ( t ) e i ( t )
Hence, by low-passfiltering and sampling the signal
~y
k
( t ) =2 =
y
k
( t ) e
;i !
c
t
= y
k
( t) c o s (
!
c
t ) ; i y
k
( t) s i n (
!
c
t )
we get thecomplex representation: (for t2 Z )
y
k
( t) =
( t ) e
i ' ( t )
| { z }
s ( t )
j H
k
( !
c
) j e
i a r g H
k
( !
c
) ]
| { z }
H
k
( !
c
)
e
; i !
c
k
+ e
k
( t )
or
y
k
( t) =
s ( t ) H
k
( !
c
) e
; i !
c
k
+ e
k
( t )
where s ( t ) is thecomplex envelopeofx ( t ) .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L86
by P. Stoica and R. Moses, Prentice Hall, 1997
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Vector Representation for a Narrowband Source
Let
= the emitter DOA
m = the number of sensors
a ( ) =
2
6
4
H
1
( !
c
) e
;i !
c
1
...
H
m
( !
c
) e
;i !
c
m
3
7
5
y ( t) =
2
6
4
y
1
( t )
...
y
m
( t )
3
7
5
e ( t) =
2
6
4
e
1
( t )
...
e
m
( t )
3
7
5
Then
y ( t) =
a ( ) s ( t) +
e ( t )
NOTE: enters a ( ) via both f k
g and f Hk
( !
c
) g .
Foromnidirectionalsensors thef H
k
( !
c
) g
do not dependon .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L87
by P. Stoica and R. Moses, Prentice Hall, 1997
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Multiple Emitter Case
Given n emitters with
received signals: f sk
( t ) g
n
k= 1
DOAs: k
Linear sensors )
y ( t) =
a (
1
) s
1
( t) +
+ a (
n
) s
n
( t) +
e ( t )
Let
A=
a (
1
) : : : a (
n
) ] ( m n )
s ( t) =
s
1
( t ) : : : s
n
( t) ]
T
( n 1 )
Then, thearray equationis:
y ( t) = A s
( t) +
e ( t )
Use theplanar waveassumption tofind the dependence of
k
on .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L89
by P. Stoica and R. Moses, Prentice Hall, 1997
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Uniform Linear Arrays
S o u r c e
-
+
?
]
S e n s o r
m4321
d s i n
J
J
J
J]
d
-
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
J
h
v v v v v
ULA Geometry
Sensor #1 = time delay reference
Time Delay for sensor k :
k
= (k ;
1 )
ds i n
c
where c = wave propagation speed
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L810
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial Frequency
Let:
!
s
4
= !
c
ds i n
c
= 2
ds i n
c = f
c
= 2
ds i n
=c = f
c
= signal wavelength
a ( ) = 1 e
;i !
s
: : : e
; i ( m ;1 )
!
s
]
T
By direct analogy with the vector a ( ! ) made from
uniform samples of asinusoidal time series,
!
s
= spatial frequency
The function ! s 7! a ( ) isone-to-onefor
j !
s
j $
d js i n
j
=2
1 d =
2
As
d = spatial sampling period
d =
2 is aspatialShannon sampling theorem.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L811
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial MethodsPart 2
Lecture 9
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L91
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial Filtering
Spatialfiltering useful for
DOA discrimination (similar to frequency
discrimination of time-seriesfiltering)
Nonparametric DOA estimation
There is a strong analogy between temporalfiltering and
spatialfiltering.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L92
by P. Stoica and R. Moses, Prentice Hall, 1997
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Analogy between Temporal and Spatial Filtering
Temporal FIR Filter:
y
F
( t) =
m ; 1
X
k= 0
h
k
u ( t ; k) =
h
y ( t )
h=
h
o
: : : h
m ; 1
]
y ( t) =
u ( t ) : : : u ( t ; m+ 1 ) ]
T
Ifu ( t) =
e
i ! t then
y
F
( t) =
h
a ( !) ]
| { z }
filter transfer function
u ( t )
a ( !) = 1 e
;i !
: : : e
; i ( m ;1 )
!
]
T
We can select h to enhance or attenuate signals with
different frequencies ! .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L93
by P. Stoica and R. Moses, Prentice Hall, 1997
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Analogy between Temporal and Spatial Filtering
Spatial Filter:
f y
k
( t ) g
m
k= 1
= thespatial samplesobtained with a
sensor array.
Spatial FIR Filter output:
y
F
( t) =
m
X
k= 1
h
k
y
k
( t) =
h
y ( t )
Narrowband Wavefront: The array's (noise-free)
response to a narrowband ( sinusoidal) wavefront with
complex envelope s ( t ) is:
y ( t) =
a ( ) s ( t )
a ( ) = 1 e
;i !
c
2
: : : e
;i !
c
m
]
T
The correspondingfilter output is
y
F
( t) =
h
a ( ) ]
| { z }
filter transfer function
s ( t )
We can select h to enhance or attenuate signals coming
from different DOAs.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L94
by P. Stoica and R. Moses, Prentice Hall, 1997
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Analogy between Temporal and Spatial Filtering
( T e m p o r a l s a m p l i n g )
z
-
q
q
q
?
?
?
-
-
-
-
@
@
@
@
@R
-
?
?
q
q
q
-
s
-
?
s
P
u ( t ) = e
i ! t
q
; 1
q
; 1
1
m ; 1
0
B
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
e
; i !
.
.
.
e
; i ( m ; 1 ) !
1
C
C
C
C
C
C
C
C
C
C
C
C
C
C
A
| { z }
a ( ! )
u ( t )
h
0
h
1
h
m ; 1
h
a ( ! ) ] u ( t )
(a) Temporalfilter
Z
Z
Z
Z
Z~
n a r r o w b a n d s o u r c e w i t h D O A =
( S p a t i a l s a m p l i n g )
P
z
-
q
q
q
?
?
?
@
@
@
@
@R
-
-
-
-
-
!
!
a
a
-
!
!
a
a
-
!
!
a
a
q
q
q
r e f e r e n c e p o i n t
1
2
m
0
B
B
B
B
B
B
B
B
B
B
B
@
1
e
; i !
2
( )
.
.
.
e
; i !
m
( )
1
C
C
C
C
C
C
C
C
C
C
C
A
| { z }
a ( )
s ( t )
h
0
h
1
h
m ; 1
h
a ( ) ] s ( t )
(b) Spatialfilter
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L95
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial Filtering, con't
Example: The response magnitude j h a ( ) j of a spatial
filter (or beamformer) for a 10-element ULA. Here,
h = a (
0
)
, where
0
= 2 5
2 4 6 8 100
Magnitude
Theta(deg)
90
60
30
0
30
60
90
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L96
by P. Stoica and R. Moses, Prentice Hall, 1997
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Spatial Filtering Uses
Spatial Filters can be used
To pass the signal of interest only, hence filtering out
interferences located outside thefilter's beam (but
possibly having the same temporal characteristics as
the signal).
To locate an emitter in thefield of view, by sweeping
thefilter through the DOA range of interest
(goniometer).
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L97
by P. Stoica and R. Moses, Prentice Hall, 1997
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Nonparametric Spatial Methods
AFilter Bank Approachto DOA estimation.
Basic Ideas
Design afilter h ( ) such that for each
It passes undistorted the signal with DOA =
It attenuates all DOAs 6=
Sweep thefilter through the DOA range of interest,
and evaluate the powers of thefiltered signals:
E
n
j y
F
( t ) j
2
o
= E
n
j h
( ) y ( t ) j
2
o
= h
( )R h
( )
with R = E f y ( t ) y ( t ) g .
The (dominant) peaks ofh ( )R h
( ) give the
DOAs of the sources.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L98
by P. Stoica and R. Moses, Prentice Hall, 1997
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Beamforming Method
Assume the array output isspatially white:
R = E f y ( t ) y
( t ) g = I
Then: En
j y
F
( t ) j
2
o
= h
h
Hence:In direct analogy with the temporally whiteassumption forfilter bank methods, y ( t ) can be
considered as impinging on the array fromallDOAs.
Filter Design:
m i n
h
( h
h ) subject to h a ( ) = 1
Solution:
h = a ( )= a
( ) a ( ) =
a ( )= m
E
n
j y
F
( t ) j
2
o
= a
( )R a
( )= m
2
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L99
by P. Stoica and R. Moses, Prentice Hall, 1997
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Implementation of Beamforming
R =
1
N
N
X
t= 1
y ( t ) y
( t )
The beamforming DOA estimates are:
f
k
g =
the locations of then
largest peaks ofa
( )
R a( ) .
This is the direct spatial analog of the Blackman-Tukey
periodogram.
Resolution Threshold:
i n fj
k
;
p
j >
wavelength
array length
= array beamwidth
Inconsistency problem:Beamforming DOA estimates are consistent if n
= 1 , but
inconsistent ifn >
1 .
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L910
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Parametric Methods
Assumptions:
The array is described by the equation:
y ( t) = A s
( t) +
e ( t )
The noise is spatially white and has the same power in
all sensors:
E f e ( t ) e
( t ) g =
2
I
The signal covariance matrix
P = E f s ( t ) s
( t ) g
is nonsingular.
Then:
R = E f y ( t ) y
( t ) g =A P A
+
2
I
Thus: The NLS, YW, MUSIC, MIN-NORM and ESPRIT
methods of frequency estimation can be used, almost
without modification, for DOA estimation.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L912
by P. Stoica and R. Moses, Prentice Hall, 1997
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Nonlinear Least Squares Method
m i n
f
k
g f s ( t ) g
1
N
N
X
t= 1
k y ( t ) ;A s
( t ) k
2
| { z }
f ( s
)
Minimizing f over s gives
s ( t) = (
A
A )
; 1
A
y ( t ) t = 1
: : : N
Then
f (
s) =
1
N
N
X
t= 1
k I ; A ( A
A )
; 1
A
] y ( t ) k
2
=
1
N
N
X
t= 1
y
( t)
I ; A ( A
A )
; 1
A
] y ( t )
= trf I ; A ( A A ) ; 1 A ] R g
Thus, f k
g= a r g m a x
f
k
g
trf A ( A A ) ; 1 A ] R g
For N= 1 , this is precisely the form of the NLS method
of frequency estimation.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L913
by P. Stoica and R. Moses, Prentice Hall, 1997
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Nonlinear Least Squares Method
Properties of NLS:
Performance: high
Computational complexity: high
Main drawback: need for multidimensional search.
Lecture notes to accompanyIntroduction to Spectral Analysis Slide L914
by P. Stoica and R. Moses, Prentice Hall, 1997
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Yule-Walker Method
y ( t) =
"
y ( t )
~y ( t )
#
=
"
A
~
A
#
s ( t) +
"
e ( t )
~ e ( t )
#
Assume: E f e ( t) ~
e
( t ) g= 0
Then:;
4
= E f y ( t) ~
y
( t ) g =
A P
~
A
( M L )
Also
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