Spectral Analysis
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Basic Definitionsand
The Spectral Estimation Problem
Lecture 1
Lecture notes to accompany Introduction to Spectral Analysis Slide L11by P. Stoica and R. Moses, Prentice Hall, 1997

Informal Definition of Spectral Estimation
Given: A finite record of a signal.
Determine: The distribution of signal power overfrequency.
t
signal
t=1, 2, . . . +
spectral density
! = (angular) frequency in radians/(sampling interval)f = !=2 = frequency in cycles/(sampling interval)
Lecture notes to accompany Introduction to Spectral Analysis Slide L12by P. Stoica and R. Moses, Prentice Hall, 1997

Applications
Temporal Spectral Analysis
Vibration monitoring and fault detection
Hidden periodicity finding
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 accompany Introduction to Spectral Analysis Slide L13by P. Stoica and R. Moses, Prentice Hall, 1997

Deterministic Signals
fy(t)g
1
t=1
= discretetime deterministic datasequence
If:1
X
t=1
jy(t)j
2

Energy Spectral Density
Parseval's Equality:
1
X
t=1
jy(t)j
2
=
1
2
Z
S(!)d!
where
S(!)
4
= jY (!)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 accompany Introduction to Spectral Analysis Slide L15by P. Stoica and R. Moses, Prentice Hall, 1997

Random Signals
Random Signal
t
random signal probabilistic statements about future variations
current observation time
Here:1
X
t=1
jy(t)j
2
=1
But: En
jy(t)j
2
o

First Definition of PSD
(!) =
1
X
k=1
r(k)e
i!k
where r(k) is the autocovariance sequence (ACS)r(k) = E fy(t)y
(t k)g
r(k) = r
(k); r(0) jr(k)j
Note that
r(k) =
1
2
Z
(!)e
i!k
d! (Inverse DTFT)
Interpretation:
r(0) = E
n
jy(t)j
2
o
=
1
2
Z
(!)d!
so
(!)d! = infinitesimal signal power in the band!
d!
2
Lecture notes to accompany Introduction to Spectral Analysis Slide L17by P. Stoica and R. Moses, Prentice Hall, 1997

Second Definition of PSD
(!) = lim
N!1
E
8
>
:
1
N
N
X
t=1
y(t)e
i!t
2
9
>
=
>
;
Note that
(!) = lim
N!1
E
1
N
jY
N
(!)j
2
where
Y
N
(!) =
N
X
t=1
y(t)e
i!t
is the finite DTFT of fy(t)g.
Lecture notes to accompany Introduction to Spectral Analysis Slide L18by P. Stoica and R. Moses, Prentice Hall, 1997

Properties of the PSD
P1: (!) = (!+2) for all !.
Thus, we can restrict attention to
! 2 [; ] () f 2 [1=2;1=2]
P2: (!) 0
P3: If y(t) is real,
Then: (!) = (!)
Otherwise: (!) 6= (!)
Lecture notes to accompany Introduction to Spectral Analysis Slide L19by P. Stoica and R. Moses, Prentice Hall, 1997

Transfer of PSD Through Linear Systems
System Function: H(q) =1
X
k=0
h
k
q
k
where q1 = unit delay operator: q1y(t) = y(t 1)
e(t)
e
(!)
y(t)
y
(!) = jH(!)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
(!) = jH(!)j2
e
(!)
Lecture notes to accompany Introduction to Spectral Analysis Slide L110by P. Stoica and R. Moses, Prentice Hall, 1997

The Spectral Estimation Problem
The Problem:
From a sample fy(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 thePSD
Lecture notes to accompany Introduction to Spectral Analysis Slide L111by P. Stoica and R. Moses, Prentice Hall, 1997

Periodogramand
CorrelogramMethods
Lecture 2
Lecture notes to accompany Introduction to Spectral Analysis Slide L21by P. Stoica and R. Moses, Prentice Hall, 1997

Periodogram
Recall 2nd definition of (!):
(!) = lim
N!1
E
8
>
:
1
N
N
X
t=1
y(t)e
i!t
2
9
>
=
>
;
Given : fy(t)gNt=1
Drop limN!1
and E fg to get
^
p
(!) =
1
N
N
X
t=1
y(t)e
i!t
2
Natural estimator
Used by Schuster (1900) to determine hiddenperiodicities (hence the name).
Lecture notes to accompany Introduction to Spectral Analysis Slide L22by P. Stoica and R. Moses, Prentice Hall, 1997

Correlogram
Recall 1st definition of (!):
(!) =
1
X
k=1
r(k)e
i!k
Truncate the P and replace r(k) by r^(k):
^
c
(!) =
N1
X
k=(N1)
r^(k)e
i!k
Lecture notes to accompany Introduction to Spectral Analysis Slide L23by P. Stoica and R. Moses, Prentice Hall, 1997

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 < 0
Lecture notes to accompany Introduction to Spectral Analysis Slide L24by P. Stoica and R. Moses, Prentice Hall, 1997

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
=
N1
X
k=(N1)
r^(k)e
i!k
=
^
c
(!)
^
p
(!) =
^
c
(!)
Consequence:Both ^
p
(!) and ^c
(!) can be analyzed simultaneously.
Lecture notes to accompany Introduction to Spectral Analysis Slide L25by P. Stoica and R. Moses, Prentice Hall, 1997

Statistical Performance of ^p
(!) and ^c
(!)
Summary:
Both are asymptotically (for large N ) unbiased:
E
n
^
p
(!)
o
! (!) as N !1
Both have large variance, even for large N .
Thus, ^p
(!) and ^c
(!) have poor performance.
Intuitive explanation:
r^(k) r(k) may be large for large jkj
Even if the errors fr^(k) r(k)gN1jkj=0
are small,there are so many that when summed in[
^
p
(!) (!)], the PSD error is large.
Lecture notes to accompany Introduction to Spectral Analysis Slide L26by P. Stoica and R. Moses, Prentice Hall, 1997

Bias Analysis of the Periodogram
E
n
^
p
(!)
o
= E
n
^
c
(!)
o
=
N1
X
k=(N1)
E fr^(k)g e
i!k
=
N1
X
k=(N1)
1
jkj
N
!
r(k)e
i!k
=
1
X
k=1
w
B
(k)r(k)e
i!k
w
B
(k) =
8
n overdetermined YW system of equations; leastsquares solution for fa^
i
g.
Note: For narrowband ARMA signals, the accuracy offa^
i
g is often better for M > n
Lecture notes to accompany Introduction to Spectral Analysis Slide L417by P. Stoica and R. Moses, Prentice Hall, 1997

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)
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Intro
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ectra
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1997

Parametric Methodsfor
Line Spectra Part 1
Lecture 5
Lecture notes to accompany Introduction to Spectral Analysis Slide L51by P. Stoica and R. Moses, Prentice Hall, 1997

Line Spectra
Many applications have signals with (near) sinusoidalcomponents. Examples:
communications
radar, sonar
geophysical seismology
ARMA model is a poor approximation
Better approximation by Discrete/Line Spectrum Models

6
6
6
6
2
!
1
!
2
!
3
(2
2
1
)
(2
2
2
)
(2
2
3
)
!
(!)
An Ideal line spectrumLecture notes to accompany Introduction to Spectral Analysis Slide L52by P. Stoica and R. Moses, Prentice Hall, 1997

Line Spectral Signal Model
Signal Model: Sinusoidal components of frequenciesf!
k
g and powers f2k
g, superimposed in white noise ofpower 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, uniformlydistributed on [; ]
(realistic and mathematically convenient)
A3: e(t) = circular white noise with variance 2
E fe(t)e
(s)g =
2
t;s
E fe(t)e(s)g = 0
(can be achieved by slow sampling)
Lecture notes to accompany Introduction to Spectral Analysis Slide L53by P. Stoica and R. Moses, Prentice Hall, 1997

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 fy(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 accompany Introduction to Spectral Analysis Slide L54by P. Stoica and R. Moses, Prentice Hall, 1997

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 leastsquares method from
y(t) =
n
X
k=1
k
e
i!^
k
t + residuals
with k
=
k
e
i'
k
.
Lecture notes to accompany Introduction to Spectral Analysis Slide L55by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares (NLS) Method
min
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
iN!
1
e
iN!
n
3
7
5
Lecture notes to accompany Introduction to Spectral Analysis Slide L56by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares (NLS) Method, con't
Then:
F = (Y B)
(Y B) = kY Bk
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
!^ = arg max
!
Y
B(B
B)
1
B
Y
Lecture notes to accompany Introduction to Spectral Analysis Slide L57by P. Stoica and R. Moses, Prentice Hall, 1997

NLS Properties
Excellent Accuracy:
var (!^k
) =
6
2
N
3
2
k
(for N 1)
Example: N = 300SNR
k
=
2
k
=
2
= 30 dB
Thenq
var(!^k
) 10
5
.
Difficult Implementation:
The NLS cost function F is multimodal; it is difficult toavoid convergence to local minima.
Lecture notes to accompany Introduction to Spectral Analysis Slide L58by P. Stoica and R. Moses, Prentice Hall, 1997

Unwindowed Periodogram as anApproximate NLS Method
For a single (complex) sinusoid, the maximum of theunwindowed periodogram is the NLS frequency estimate:
Assume: n = 1
Then: BB = N
B
Y =
N
X
t=1
y(t)e
i!t
= Y (!) (finite DTFT)
Y
B(B
B)
1
B
Y =
1
N
jY (!)j
2
=
^
p
(!)
= (Unwindowed Periodogram)So, with no approximation,
!^ = argmax
!
^
p
(!)
Lecture notes to accompany Introduction to Spectral Analysis Slide L59by P. Stoica and R. Moses, Prentice Hall, 1997

Unwindowed Periodogram as anApproximate NLS Method, con't
Assume: n > 1
Then:
f!^
k
g
n
k=1
' the locations of the n largestpeaks of ^
p
(!)
provided that
inf j!
k
!
p
j > 2=N
which is the periodogram resolution limit.
If better resolution desired then use a High/SuperResolution method.
Lecture notes to accompany Introduction to Spectral Analysis Slide L510by P. Stoica and R. Moses, Prentice Hall, 1997

HighOrder YuleWalker Method
Recall:
y(t) = x(t) + e(t) =
n
X
k=1
k
e
i(!
k
t+'
k
)
 {z }
x
k
(t)
+e(t)
Degenerate ARMA 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
(t1)+'
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 accompany Introduction to Spectral Analysis Slide L511by P. Stoica and R. Moses, Prentice Hall, 1997

HighOrder YuleWalker Method, con't
Estimation Procedure:
Estimate f^bi
g
L
i=1
using an ARMA MYW technique
Roots of ^B(q) give f!^k
g
n
k=1
, along with L nspurious roots.
Lecture notes to accompany Introduction to Spectral Analysis Slide L512by P. Stoica and R. Moses, Prentice Hall, 1997

HighOrder 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 highorder (if L > n) and overdetermined(if M > L) system of YW equations.
Lecture notes to accompany Introduction to Spectral Analysis Slide L513by P. Stoica and R. Moses, Prentice Hall, 1997

HighOrder and Overdetermined YW Equations,con't
Fact: rank() = n
SVD of : = UV
U = (M n) with UU = In
V
= (n L) with V V = In
= (n n), diagonal and nonsingular
Thus,
(UV
)b =
The MinimumNorm solution is
b =
y
= V
1
U
Important property: The additional (L n) spuriouszeros of B(q) are located strictly inside the unit circle, ifthe MinimumNorm solution b is used.
Lecture notes to accompany Introduction to Spectral Analysis Slide L514by P. Stoica and R. Moses, Prentice Hall, 1997

HOYW Equations, Practical Solution
Let ^ = but made from fr^(k)g instead of fr(k)g.
Let ^U , ^, ^V be defined similarly to U , , V from theSVD of ^.
Compute ^b = ^V ^1^U^
Then f!^k
g
n
k=1
are found from the n zeroes of ^B(q) thatare closest to the unit circle.
When the SNR is low, this approach may give spuriousfrequency estimates when L > n; this is the price paid forincreased accuracy when L > n.
Lecture notes to accompany Introduction to Spectral Analysis Slide L515by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methodsfor
Line Spectra Part 2
Lecture 6
Lecture notes to accompany Introduction to Spectral Analysis Slide L61by P. Stoica and R. Moses, Prentice Hall, 1997

The Covariance Matrix Equation
Let:
a(!) = [1 e
i!
: : : e
i(m1)!
]
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(tm+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(tm+1)]
T
Then
R
4
= E f~y(t)~y
(t)g = APA
+
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 accompany Introduction to Spectral Analysis Slide L62by P. Stoica and R. Moses, Prentice Hall, 1997

Eigendecomposition of R and Its Properties
R = APA
+
2
I (m > n)
Let:
1
2
: : :
m
: eigenvalues of R
fs
1
; : : : s
n
g: orthonormal eigenvectors associatedwith f
1
; : : : ;
n
g
fg
1
; : : : ; g
mn
g: orthonormal eigenvectors associatedwith f
n+1
; : : : ;
m
g
S = [s
1
: : : s
n
] (m n)
G = [g
1
: : : g
mn
] (m (m n))
Thus,
R = [S G]
2
6
4
1
...
m
3
7
5
"
S
G
#
Lecture notes to accompany Introduction to Spectral Analysis Slide L63by P. Stoica and R. Moses, Prentice Hall, 1997

Eigendecomposition of R and Its Properties,con't
As rank(APA) = 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:
RS = APA
S+
2
S = S
2
6
4
1
0
...
0
n
3
7
5
S = A(PA
S
1
)
4
= AC
with jCj 6= 0 (since rank(S) = rank(A) = n).Therefore, since SG= 0,
A
G = 0
Lecture notes to accompany Introduction to Spectral Analysis Slide L64by P. Stoica and R. Moses, Prentice Hall, 1997

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 the unique solutions ofa
(!)GG
a(!) = 0.
Let:
^
R =
1
N
N
X
t=m
~y(t)~y
(t)
^
S;
^
G = S;G made from theeigenvectors of ^R
Lecture notes to accompany Introduction to Spectral Analysis Slide L65by P. Stoica and R. Moses, Prentice Hall, 1997

Spectral and Root MUSIC Methods
Spectral MUSIC Method:
f!^
k
g
n
k=1
= the locations of the n highest peaks of thepseudospectrum function:
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
(m1)
]
T
Note: Both variants of MUSIC may produce spuriousfrequency estimates.Lecture notes to accompany Introduction to Spectral Analysis Slide L66by P. Stoica and R. Moses, Prentice Hall, 1997

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 accompany Introduction to Spectral Analysis Slide L67by P. Stoica and R. Moses, Prentice Hall, 1997

MinNorm Method
Goals: Reduce computational burden, and reduce risk offalse frequency estimates.
Uses m n (as in MUSIC), but only one vector inR(G) (as in Pisarenko).
Let"
1
g^
#
=
the vector in R( ^G), with first element equalto one, that has minimum Euclidean norm.
Lecture notes to accompany Introduction to Spectral Analysis Slide L68by P. Stoica and R. Moses, Prentice Hall, 1997

MinNorm Method, con't
Spectral MinNorm
f!^g
n
k=1
= the locations of the n highest peaks in thepseudospectrum
1 =
a
(!)
"
1
g^
#
2
Root MinNorm
f!^g
n
k=1
= the angular positions of the n roots of thepolynomial
a
T
(z
1
)
"
1
g^
#
that are closest to the unit circle.
Lecture notes to accompany Introduction to Spectral Analysis Slide L69by P. Stoica and R. Moses, Prentice Hall, 1997

MinNorm Method: Determining g^
Let ^S ="
S
#
g 1
g m 1
Then:"
1
g^
#
2 R(
^
G) )
^
S
"
1
g^
#
= 0
)
S
g^ =
MinNorm solution: g^ = S(SS)1
As: I = ^S^S = + SS, (SS)1 exists iff
= kk
2
6= 1
(This holds, at least, for N 1.)
Multiplying the above equation by gives:
(1 kk
2
) = (
S
S)
) (
S
S)
1
= =(1 kk
2
)
) g^ =
S=(1 kk
2
)
Lecture notes to accompany Introduction to Spectral Analysis Slide L610by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Method
Let A1
= [I
m1
0]A
A
2
= [0 I
m1
]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
m1
0]S
S
2
= [0 I
m1
]S
Recall S = AC with jCj 6= 0. Then
S
2
= A
2
C = A
1
DC = S
1
C
1
DC
 {z }
So has the same eigenvalues as D. is uniquelydetermined as
= (S
1
S
1
)
1
S
1
S
2
Lecture notes to accompany Introduction to Spectral Analysis Slide L611by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Implementation
From the eigendecomposition of ^R, find ^S, then ^S1
and^
S
2
.
The frequency estimates are found by:
f!^
k
g
n
k=1
= arg(^
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 MUSICor MinNorm)
accurate frequency estimates
Lecture notes to accompany Introduction to Spectral Analysis Slide L612by P. Stoica and R. Moses, Prentice Hall, 1997

Sum
mar
yo
fFre
quen
cyEs
timat
ion
Met
hods
Com
puta
tiona
lA
ccur
acy
/R
iskfo
rFa
lseM
etho
dBu
rden
Res
olut
ion
Freq
Estim
ates
Perio
dogr
amsm
all
med
ium
hig
hm
ediu
mN
onlin
earL
Sver
yhi
ghver
yhi
ghver
yhi
ghYu
leW
alke
rm
ediu
mhi
ghm
ediu
mPi
sare
nko
smal
llo
wn
on
e
MU
SIC
high
high
med
ium
Min
Nor
mm
ediu
mhi
ghsm
all
ESPR
ITm
ediu
mver
yhi
ghn
on
e
Rec
omm
enda
tion:
Use
Peri
odog
ram
form
ediu
mre
solu
tion
appl
icat
ions
Use
ESPR
ITfo
rhig
hre
solu
tion
appl
icat
ions
Lect
ure
note
sto
acco
mpa
ny
Intro
duct
ion
toSp
ectra
lAnal
ysis
Slid
eL6
13by
P.St
oic
aan
dR.
Mose
s,Pr
entic
eH
all,
1997

Filter Bank Methods
Lecture 7
Lecture notes to accompany Introduction to Spectral Analysis Slide L71by P. Stoica and R. Moses, Prentice Hall, 1997

Basic Ideas
Two main PSD estimation approaches:
1. Parametric Approach: Parameterize (!) by afinitedimensional model.
2. Nonparametric Approach: Implicitly smoothf(!)g
!=
by assuming that (!) is nearlyconstant 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 compromiseassociated with all nonparametric methods.
Lecture notes to accompany Introduction to Spectral Analysis Slide L72by P. Stoica and R. Moses, Prentice Hall, 1997

Filte
rBan
kA
ppro
ach
toSp
ectra
lEst
imat
ion




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)co
nsis
tent
pow
erca
lcul
atio
n
(b)Id
ealp
assb
and
filte
rwith
band
wid
th
(c)
(
)
const
anto
n
2
[
!
2
;
!
+
2
]
Not
eth
atas
sum
ptio
ns(a)
and
(b),a
sw
ella
s(b)
and
(c),a
reco
nfli
ctin
g.
Lect
ure
note
sto
acco
mpa
ny
Intro
duct
ion
toSp
ectra
lAnal
ysis
Slid
eL7
3by
P.St
oic
aan
dR.
Mose
s,Pr
entic
eH
all,
1997

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~!(Nt)
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
iN(~!!)
1
e
i(~!!)
1
center frequency of H(!) = ~!
3dB bandwidth of H(!) ' 1=N
Lecture notes to accompany Introduction to Spectral Analysis Slide L74by P. Stoica and R. Moses, Prentice Hall, 1997

Filter Bank Interpretation of the Periodogram,con't
jH(!)j as a function of (~! !), for N = 50.
3 2 1 0 1 2 340
35
30
25
20
15
10
5
0
dB
ANGULAR FREQUENCY
Conclusion: The periodogram ^p
(!) is a filter bank PSDestimator with bandpass filter as given above, and:
narrow filter passband,
power calculation from only 1 sample of filter output.
Lecture notes to accompany Introduction to Spectral Analysis Slide L75by P. Stoica and R. Moses, Prentice Hall, 1997

Possible Improvements to the Filter BankApproach
1. Split the available sample, and bandpass filter eachsubsample.
more data points for the power calculation stage.
This approach leads to Bartlett and Welch methods.
2. Use several bandpass filters on the whole sample.Each filter covers a small band centered on ~!.
provides several samples for power calculation.
This multiwindow approach is similar to theDaniell method.
Both approaches compromise bias for variance, and in factare quite related to each other: splitting the data samplecan be interpreted as a special form of windowing orfiltering.
Lecture notes to accompany Introduction to Spectral Analysis Slide L76by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Method
Idea: Datadependent bandpass filter 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(tm)
3
7
5
 {z }
~y(t)
E
n
jy
F
(t)j
2
o
= h
Rh; R = E f~y(t)~y
(t)g
H(!) =
m
X
k=0
h
k
e
i!k
= h
a(!)
where a(!) = [1; ei! : : : eim!]T
Lecture notes to accompany Introduction to Spectral Analysis Slide L77by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Method, con't
Capon Filter Design Problem:
min
h
(h
Rh) subject to ha(!) = 1
Solution: h0
= R
1
a=a
R
1
a
The power at the filter output is:
E
n
jy
F
(t)j
2
o
= h
0
Rh
0
= 1=a
(!)R
1
a(!)
which should be the power of y(t) in a passband centeredon !.
The Bandwidth' 1m+1
=
1
(filter length)
Conclusion Estimate PSD as:
^
(!) =
m+1
a
(!)
^
R
1
a(!)
with
^
R =
1
N m
N
X
t=m+1
~y(t)~y
(t)
Lecture notes to accompany Introduction to Spectral Analysis Slide L78by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Properties
m is the user parameter that controls the compromisebetween bias and variance:
as m increases, bias decreases and varianceincreases.
Capon uses one bandpass filter only, but it splits theN data point sample into (N m) subsequences oflength m with maximum overlap.
Lecture notes to accompany Introduction to Spectral Analysis Slide L79by P. Stoica and R. Moses, Prentice Hall, 1997

Relation between Capon and BlackmanTukeyMethods
Consider ^BT
(!) with Bartlett window:
^
BT
(!) =
m
X
k=m
m+1 jkj
m+1
r^(k)e
i!k
=
1
m+1
m
X
t=0
m
X
s=0
r^(t s)e
i!(ts)
=
a
(!)
^
Ra(!)
m+1
;
^
R = [r^(i j)]
Then we have
^
BT
(!) =
a
(!)
^
Ra(!)
m+1
^
C
(!) =
m+1
a
(!)
^
R
1
a(!)
Lecture notes to accompany Introduction to Spectral Analysis Slide L710by P. Stoica and R. Moses, Prentice Hall, 1997

Relation between Capon and AR Methods
Let
^
ARk
(!) =
^
2
k
j
^
A
k
(!)j
2
be the kth order AR PSD estimate of y(t).
Then
^
C
(!) =
1
1
m+1
m
X
k=0
1=
^
ARk
(!)
Consequences:
Due to the average over k, ^C
(!) generally has lessstatistical variability than the AR PSD estimator.
Due to the loworder AR terms in the average,^
C
(!) generally has worse resolution and biasproperties than the AR method.
Lecture notes to accompany Introduction to Spectral Analysis Slide L711by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Methods Part 1
Lecture 8
Lecture notes to accompany Introduction to Spectral Analysis Slide L81by P. Stoica and R. Moses, Prentice Hall, 1997

The Spatial Spectral Estimation Problem
Source n
Source 2
Source 1
B
B
B
B
BN
@
@
@
@R
v
v
v
Sensor 1
@
@
@
@
@
@
Sensor 2
Sensor m
c
c
c
c
c
c
c
c
c
c
c
c
,
,
,
,
,
,
,
,
,
,
,
,
Problem: Detect and locate n radiating sources by usingan 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 overspace (thus the name spatial spectral analysis)
Lecture notes to accompany Introduction to Spectral Analysis Slide L82by P. Stoica and R. Moses, Prentice Hall, 1997

Simplifying Assumptions
Farfield sources in the same plane as the array ofsensors
Nondispersive wave propagation
Hence: The waves are planar and the only locationparameter is direction of arrival (DOA)(or angle of arrival, AOA).
The number of sources n is known. (We do not treatthe detection problem)
The sensors are linear dynamic elements with knowntransfer characteristics and known locations
(That is, the array is calibrated.)
Lecture notes to accompany Introduction to Spectral Analysis Slide L83by P. Stoica and R. Moses, Prentice Hall, 1997

Array Model Single Emitter Case
x(t) = the signal waveform as measured at a referencepoint (e.g., at the first sensor)
k
= the delay between the reference point and thekth sensor
h
k
(t) = the impulse response (weighting function) ofsensor k
e
k
(t) = noise at the kth sensor (e.g., thermal noise insensor electronics; background noise, etc.)
Note: t 2 R (continuoustime 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 the time delays fk
g with hk
(t)
known but x(t) unknown.
This is a timedelay estimation problem in the unknowninput case.Lecture notes to accompany Introduction to Spectral Analysis Slide L84by P. Stoica and R. Moses, Prentice Hall, 1997

Narrowband Assumption
Assume: The emitted signals are narrowband with knowncarrier frequency !
c
.
Then: x(t) = (t) cos[!c
t+ '(t)]
where (t); '(t) vary slowly enough so that
(t
k
) ' (t); '(t
k
) ' '(t)
Time delay is now' to a phase shift !c
k
:
x(t
k
) ' (t) cos[!
c
t+ '(t) !
c
k
]
h
k
(t) x(t
k
)
' jH
k
(!
c
)j(t) cos[!
c
t+ '(t) !
c
k
+argfH
k
(!
c
)g]
where Hk
(!) = Ffh
k
(t)g is the kth sensor's transferfunction
Hence, the kth sensor output is
y
k
(t) = jH
k
(!
c
)j(t)
cos[!
c
t+ '(t) !
c
k
+ argH
k
(!
c
)] + e
k
(t)
Lecture notes to accompany Introduction to Spectral Analysis Slide L85by P. Stoica and R. Moses, Prentice Hall, 1997

Complex Signal Representation
The noisefree output has the form:
z(t) = (t) cos [!
c
t+ (t)] =
=
(t)
2
n
e
i[!
c
t+ (t)]
+ e
i[!
c
t+ (t)]
o
Demodulate z(t) (translate to baseband):
2z(t)e
!
c
t
= (t)f e
i (t)
 {z }
lowpass+ e
i[2!
c
t+ (t)]
 {z }
highpassg
Lowpass filter 2z(t)ei!ct to obtain (t)ei (t)
Hence, by lowpass filtering and sampling the signal
~y
k
(t)=2 = y
k
(t)e
i!
c
t
= y
k
(t) cos(!
c
t) iy
k
(t) sin(!
c
t)
we get the complex representation: (for t 2 Z)y
k
(t) = (t) e
i'(t)
 {z }
s(t)
jH
k
(!
c
)j e
iarg[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 the complex envelope of x(t).
Lecture notes to accompany Introduction to Spectral Analysis Slide L86by P. Stoica and R. Moses, Prentice Hall, 1997

Vector Representation for a Narrowband Source
Let
= the emitter DOAm = 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 fk
g and fHk
(!
c
)g.
For omnidirectional sensors the fHk
(!
c
)g do not dependon .
Lecture notes to accompany Introduction to Spectral Analysis Slide L87by P. Stoica and R. Moses, Prentice Hall, 1997

Ana
log
Pro
cess
ing
Blo
ckDi
agra
m
Ana
log
proc
essin
gfo
reac
hre
ceiv
ing
arra
yel
emen
t
X
X
X
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Xz
:
E
x
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r
n
a
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o
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I
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i
n
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S
i
g
n
a
l
x
(
t
k
)
S
E
N
S
O
R
,
,
,
l
l
l

C
a
b
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i
n
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a
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r
a
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d
u
c
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r

y
k
(
t
)
D
E
M
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D
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L
A
T
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O
N
O
s
c
i
l
l
a
t
o
r
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o
w
p
a
s
s
F
i
l
t
e
r
L
o
w
p
a
s
s
F
i
l
t
e
r
 
6 ?

s
i
n
(
!
c
t
)
c
o
s
(
!
c
t
)
 
I
m
[
y
k
(
t
)
]
R
e
[
y
k
(
t
)
]
 
I
m
[
~
y
k
(
t
)
]
R
e
[
~
y
k
(
t
)
]
A
/
D
C
O
N
V
E
R
S
I
O
N
!
!
!
!
 
A
/
D
A
/
D
Lect
ure
note
sto
acco
mpa
ny
Intro
duct
ion
toSp
ectra
lAnal
ysis
Slid
eL8
8by
P.St
oic
aan
dR.
Mose
s,Pr
entic
eH
all,
1997

Multiple Emitter Case
Given n emitters with
received signals: fsk
(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, the array equation is:
y(t) = As(t) + e(t)
Use the planar wave assumption to find the dependence of
k
on .
Lecture notes to accompany Introduction to Spectral Analysis Slide L89by P. Stoica and R. Moses, Prentice Hall, 1997

Uniform Linear Arrays
Source

+
?
]
p p p
Sensor
m4321
d sin
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)
d sin
c
where c = wave propagation speed
Lecture notes to accompany Introduction to Spectral Analysis Slide L810by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Frequency
Let:
!
s
4
= !
c
d sin
c
= 2
d sin
c=f
c
= 2
d sin
= c=f
c
= signal wavelength
a() = [1; e
i!
s
: : : e
i(m1)!
s
]
T
By direct analogy with the vector a(!) made fromuniform samples of a sinusoidal time series,
!
s
= spatial frequency
The function !s
7! a() is onetoone for
j!
s
j $
dj sin j
=2
1 d =2
As
d = spatial sampling period
d =2 is a spatial Shannon sampling theorem.
Lecture notes to accompany Introduction to Spectral Analysis Slide L811by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Methods Part 2
Lecture 9
Lecture notes to accompany Introduction to Spectral Analysis Slide L91by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Filtering
Spatial filtering useful for
DOA discrimination (similar to frequencydiscrimination of timeseries filtering)
Nonparametric DOA estimation
There is a strong analogy between temporal filtering andspatial filtering.
Lecture notes to accompany Introduction to Spectral Analysis Slide L92by P. Stoica and R. Moses, Prentice Hall, 1997

Analogy between Temporal and Spatial Filtering
Temporal FIR Filter:
y
F
(t) =
m1
X
k=0
h
k
u(t k) = h
y(t)
h = [h
o
: : : h
m1
]
y(t) = [u(t) : : : u(tm+1)]
T
If u(t) = ei!t then
y
F
(t) = [h
a(!)]
 {z }
filter transfer functionu(t)
a(!) = [1; e
i!
: : : e
i(m1)!
]
T
We can select h to enhance or attenuate signals withdifferent frequencies !.
Lecture notes to accompany Introduction to Spectral Analysis Slide L93by P. Stoica and R. Moses, Prentice Hall, 1997

Analogy between Temporal and Spatial Filtering
Spatial Filter:
fy
k
(t)g
m
k=1
= the spatial samples obtained with asensor 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 (noisefree)response to a narrowband ( sinusoidal) wavefront withcomplex envelope s(t) is:
y(t) = a()s(t)
a() = [1; e
i!
c
2
: : : e
i!
c
m
]
T
The corresponding filter output is
y
F
(t) = [h
a()]
 {z }
filter transfer functions(t)
We can select h to enhance or attenuate signals comingfrom different DOAs.Lecture notes to accompany Introduction to Spectral Analysis Slide L94by P. Stoica and R. Moses, Prentice Hall, 1997

Analogy between Temporal and Spatial Filtering
(Temporal sampling)
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(m1)!
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
m1
[h
a(!)]u(t)
(a) Temporal filter
Z
Z
Z
Z
Z~
narrowband source with DOA=
(Spatial sampling)
P
z

q
q
q
?
?
?
@
@
@
@
@R





!
!
a
a

!
!
a
a

!
!
a
a
q
q
q
reference point
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
m1
[h
a()]s(t)
(b) Spatial filterLecture notes to accompany Introduction to Spectral Analysis Slide L95by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Filtering, con't
Example: The response magnitude jha()j of a spatialfilter (or beamformer) for a 10element ULA. Here,h = a(
0
), where 0
= 25
2 4 6 8 100Magnitude
Theta (deg)
90
60
30
0
30
60
90
Lecture notes to accompany Introduction to Spectral Analysis Slide L96by P. Stoica and R. Moses, Prentice Hall, 1997

Spatial Filtering Uses
Spatial Filters can be used
To pass the signal of interest only, hence filtering outinterferences located outside the filter's beam (butpossibly having the same temporal characteristics asthe signal).
To locate an emitter in the field of view, by sweepingthe filter through the DOA range of interest(goniometer).
Lecture notes to accompany Introduction to Spectral Analysis Slide L97by P. Stoica and R. Moses, Prentice Hall, 1997

Nonparametric Spatial Methods
A Filter Bank Approach to DOA estimation.
Basic Ideas
Design a filter h() such that for each
It passes undistorted the signal with DOA =
It attenuates all DOAs 6=
Sweep the filter through the DOA range of interest,and evaluate the powers of the filtered signals:
E
n
jy
F
(t)j
2
o
= E
n
jh
()y(t)j
2
o
= h
()Rh()
with R = E fy(t)y(t)g.
The (dominant) peaks of h()Rh() give theDOAs of the sources.
Lecture notes to accompany Introduction to Spectral Analysis Slide L98by P. Stoica and R. Moses, Prentice Hall, 1997

Beamforming Method
Assume the array output is spatially white:
R = E fy(t)y
(t)g = I
Then: En
jy
F
(t)j
2
o
= h
h
Hence: In direct analogy with the temporally whiteassumption for filter bank methods, y(t) can beconsidered as impinging on the array from all DOAs.
Filter Design:
min
h
(h
h) subject to ha() = 1
Solution:
h = a()=a
()a() = a()=m
E
n
jy
F
(t)j
2
o
= a
()Ra()=m
2
Lecture notes to accompany Introduction to Spectral Analysis Slide L99by P. Stoica and R. Moses, Prentice Hall, 1997

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 the n largest peaks ofa
()
^
Ra().
This is the direct spatial analog of the BlackmanTukeyperiodogram.
Resolution Threshold:
inf j
k
p
j >
wavelengtharray length
= array beamwidth
Inconsistency problem:Beamforming DOA estimates are consistent if n= 1, butinconsistent if n > 1.
Lecture notes to accompany Introduction to Spectral Analysis Slide L910by P. Stoica and R. Moses, Prentice Hall, 1997

Capon Method
Filter design:
min
h
(h
Rh) subject to ha() = 1
Solution:
h = R
1
a()=a
()R
1
a()
E
n
jy
F
(t)j
2
o
= 1=a
()R
1
a()
Implementation:
f
^
k
g = the locations of the n largest peaks of1=a
()
^
R
1
a():
Performance: Slightly superior to Beamforming.
Both Beamforming and Capon are nonparametricapproaches. They do not make assumptions on thecovariance properties of the data (and hence do notdepend on them).Lecture notes to accompany Introduction to Spectral Analysis Slide L911by P. Stoica and R. Moses, Prentice Hall, 1997

Parametric Methods
Assumptions:
The array is described by the equation:
y(t) = As(t) + e(t)
The noise is spatially white and has the same power inall sensors:
E fe(t)e
(t)g =
2
I
The signal covariance matrix
P = E fs(t)s
(t)g
is nonsingular.
Then:
R = E fy(t)y
(t)g = APA
+
2
I
Thus: The NLS, YW, MUSIC, MINNORM and ESPRITmethods of frequency estimation can be used, almostwithout modification, for DOA estimation.Lecture notes to accompany Introduction to Spectral Analysis Slide L912by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares Method
min
f
k
g; fs(t)g
1
N
N
X
t=1
ky(t)As(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(AA)1A] ^Rg
Thus, f^k
g= argmax
f
k
g
trf[A(AA)1A] ^Rg
For N = 1, this is precisely the form of the NLS methodof frequency estimation.
Lecture notes to accompany Introduction to Spectral Analysis Slide L913by P. Stoica and R. Moses, Prentice Hall, 1997

Nonlinear Least Squares Method
Properties of NLS:
Performance: high
Computational complexity: high
Main drawback: need for multidimensional search.
Lecture notes to accompany Introduction to Spectral Analysis Slide L914by P. Stoica and R. Moses, Prentice Hall, 1997

YuleWalker Method
y(t) =
"
y(t)
~y(t)
#
=
"
A
~
A
#
s(t) +
"
e(t)
~e(t)
#
Assume: E fe(t)~e(t)g = 0
Then:
4
= E fy(t)~y
(t)g =
AP
~
A
(M L)
Also assume:
M > n; L > n () m =M + L > 2n)
rank( A) = rank( ~A) = n
Then: rank() = n, and the SVD of is
= [U
1
{z}
n
U
2
{z}
Mn
]
"
nn
0
0 0
# "
V
1
V
2
#
g n
g Ln
Properties: ~AV2
= 0 V
1
2 R(
~
A)
Lecture notes to accompany Introduction to Spectral Analysis Slide L915by P. Stoica and R. Moses, Prentice Hall, 1997

YWMUSIC DOA Estimator
f
^
k
g= the n largest peaks of1=~a
()
^
V
2
^
V
2
~a()
where
~a(), (L 1), is the array transfer vector for ~y(t)at DOA
^
V
2
is defined similarly to V2
, using
^
=
1
N
N
X
t=1
y(t)~y
(t)
Properties:
Computational complexity: medium
Performance: satisfactory if m 2n
Main advantages: weak assumption on fe(t)g the subarray A need not be calibrated
Lecture notes to accompany Introduction to Spectral Analysis Slide L916by P. Stoica and R. Moses, Prentice Hall, 1997

MUSIC and MinNorm Methods
Both MUSIC and MinNorm methods for frequencyestimation apply with only minor modifications to theDOA estimation problem.
Spectral forms of MUSIC and MinNorm can be usedfor arbitrary arrays
Root forms can be used only with ULAs
MUSIC and MinNorm break down if the sourcesignals are coherent; that is, if
rank(P) = rank(E fs(t)s(t)g) < n
Modifications that apply in the coherent case exist.
Lecture notes to accompany Introduction to Spectral Analysis Slide L917by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Method
Assumption: The array is made from two identicalsubarrays separated by a known displacement vector.
Let
m = # sensors in each subarrayA
1
= [I
m
0]A (transfer matrix of subarray 1)A
2
= [0 I
m
]A (transfer matrix of subarray 2)
Then A2
= A
1
D, where
D =
2
6
4
e
i!
c
(
1
)
0
...
0 e
i!
c
(
n
)
3
7
5
() = the time delay from subarray 1 tosubarray 2 for a signal with DOA = :
() = d sin()=c
where d is the subarray separation and is measured fromthe perpendicular to the subarray displacement vector.
Lecture notes to accompany Introduction to Spectral Analysis Slide L918by P. Stoica and R. Moses, Prentice Hall, 1997

ESPRIT Method, con't
ESPRIT Scenario
subarray 1
subarray 2
source
known displacement vector
Properties:
Requires special array geometry
Computationally efficient
No risk of spurious DOA estimates
Does not require array calibration
Note: For a ULA, the two subarrays are often the firstm1 and last m1 array elements, so m = m1 and
A
1
= [I
m1
0]A; A
2
= [0 I
m1
]A
Lecture notes to accompany Introduction to Spectral Analysis Slide L919by P. Stoica and R. Moses, Prentice Hall, 1997