Spectral Estimation 2
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Basic Denitions and The Spectral Estimation Problem
Lecture 1
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L11
Informal Denition of Spectral Estimation
Given: A nite record of a signal. Determine: The distribution of signal power over frequency.signal t 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 by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L12
Applications Temporal Spectral Analysis Vibration monitoring and fault detection Hidden periodicity nding Speech processing and audio devices Medical diagnosis Seismology and ground movement study Control systems design Radar, Sonar
Spatial Spectral Analysis Source location using sensor arraysLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L13
Deterministic Signals
fy(t)g1 ;1 = t=If:
discretetime deterministic data sequence
X
1
t=;1
jy(t)j2 < 1X
Then:
Y (! ) =
1
t=;1
y(t)e;i!t
exists and is called the DiscreteTime Fourier Transform (DTFT)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L14
Energy Spectral Density
Parseval's Equality:
jy(t)j2 = 21 ; S (!)d! t=;1X Z
1
where
4 S (!) = jY (!)j2 = Energy Spectral DensityWe can write
S (!) =where
X
1
k=;1X
(k)e;i!k
(k) =
1
t=;1
y(t)y (t ; k)Slide L15
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Random Signals
Random Signalrandom signal probabilistic statements about future variations t current observation time
Here:
X
1
t=;1n
jy(t)j2 = 1o
But:
E jy(t)j2 < 1
E f g = Expectation over the ensemble of realizations E jy(t)j2n o
= Average power in y
(t)
PSD = (Average) power spectral density
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L16
First Denition of PSD
(! ) =where r
X
1
k=;1
r(k)e;i!k
(k) is the autocovariance sequence (ACS) r(k) = E fy(t)y (t ; k)g r(k) = r (;k) r(0) jr(k)jZ
Note that
r(k) = 21 ; (!)ei!k d!n o
(Inverse DTFT)
Interpretation:
so
r(0) = E jy(t)j2 = 21 ; (!)d!Z
(!)d! =
innitesimal signal power in the band
! d! 2
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L17
Second Denition of PSD
1 N y(t)e;i!t (!) = Nlim E N !1 t=1X > :
8 > = >
Note that
1 jY (!)j2 (!) = Nlim E N N !1YN (!) =X
where
N
is the nite DTFT of
fy(t)g.
t=1
y(t)e;i!t
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L18
Properties of the PSD
P1:
(!) = (! + 2 ) for all !.Thus, we can restrict attention to
!2 ;P2:
] () f 2 ;1=2 1=2]
(!) 0 (t) is real, Then: (! ) = (;! ) Otherwise: (! ) 6= (;! )
P3: If y
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L19
Transfer of PSD Through Linear Systems
System Function: where q ;1
H (q) =
X
1
k=0
hk q;k
= unit delay operator: q;1y(t) = y(t ; 1)H (q ) y (t) 2 y (! ) = jH (! )je(
e(t) e (! )Then
!)
y(t) =
X
1
k=0
hk e(t ; k) hk e;i!k
H (! ) =
X
1
y (!) = jH (!)j2 e(!)
k=0
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L110
The Spectral Estimation Problem
The Problem: From a sample
fy(1) : : : y(N )g(!): f ^(!) ! 2 ; ]g
Find an estimate of
Two Main Approaches :
Nonparametric: Derived from the PSD denitions.
Parametric: Assumes a parameterized functional form of the PSD
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L111
Periodogram and Correlogram Methods
Lecture 2
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L21
Periodogram Recall 2nd denition of (! ):
1 N y(t)e;i!t (!) = Nlim E N !1 t=1X > :
8 > = >
Given : Drop
fy(t)gN t=1lim and E
N !1
f g to getX
^p(!) = 1 y(t)e;i!t N t=1Natural estimator
N
2
Used by Schuster ( 1900) to determine hidden periodicities (hence the name).
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L22
Correlogram
Recall 1st denition of (! ):
(! ) =P
X
1
Truncate the and replace r
k=;1 N ;1 X
r(k)e;i!k
(k) by ^(k): rr ^(k)e;i!k
^c(!) =
k=;(N ;1)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L23
Covariance Estimators (or Sample Covariances)
Standard unbiased estimate:
1 N y(t)y (t ; k) k 0 ^ r(k) = N ; k t=k+1X
Standard biased estimate:
1 N y(t)y (t ; k) k 0 r(k) = N ^ t=k+1X
For both estimators:
r ^(k) = ^ (;k) k < 0 r
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L24
Relationship Between ^p (! ) and ^c(! )
If: the biased ACS estimator r Then:
^(k) is used in ^c(!),N
^p(!) = 1 y(t)e;i!t N t=1X
2
=
N ;1 X
= ^c(!)
k=;(N ;1)
r ^(k)e;i!k
^p(!) = ^c(!)Consequence: Both p ! and
^( )
^c(!) can be analyzed simultaneously.Slide L25
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Statistical Performance of ^p(! ) and ^c(! ) Summary: Both are asymptotically (for large N ) unbiased:
E ^p(!) ! (!) as N ! 1n o
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 r k r k Nj;1 are small, jk =0 there are so many that when summed in ! , the PSD error is large. p!
f^( ) ; ( )g
^ ( ) ; ( )]
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L26
Bias Analysis of the Periodogram
E ^p(!) = E ^c(!) = E f^(k)g e;i!k r k=;(N ;1) N ;1 jkj r(k)e;i!k = 1; N k=;(N ;1) 1 = wB (k)r(k)e;i!k k=;1n o n o X ! X
N ;1 X
wB (k) =
8 < :
=Thus,n o
0
k 1 ; jNj
jkj N ; 1 jkj N
Bartlett, or triangular, window
E ^p(!) = 21 ; ( )WB (! ; ) dZ
Ideally:
WB (!) = Dirac impulse (!).Slide L27
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Bartlett Window WB (! )
1 sin(!N=2) 2 WB (!) = N sin(!=2)" #0
WB (!)=WB (0), for N = 25
10
20
dB
30
40
50
60
3
2
1 0 1 ANGULAR FREQUENCY
2
3
Main lobe 3dB width For small N , WB
1=N .
(!) may differ quite a bit from (!).Slide L28
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Smearing and Leakage
Main Lobe Width: smearing or smoothing Details in resolvable.() smearing
(!) separated in f by less than 1=N are not^ ()
< > :
M ;1 X
=
'Thus:
k=;(M ;1) M ;1 X k=;(M ;1)
k=;(M ;1) 8 M ;1 < 1 X L X:
r ^j (k)e;i!k9 =
9 > = >
L j =1
r ^j (k) e;i!k
r ^(k)e;i!k
^B (!) ' ^BT (!)Since B ! implicitly uses method has
with a rectangular lag window wR k
()
^( )
fwR(k)g, the Bartlett
High resolution (little smearing) Large leakage and relatively large varianceLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L38
Welch Method Similar to Bartlett method, but allow overlap of subsequences (gives more subsequences, and thus better averaging) use data window for each periodogram; gives mainlobesidelobe tradeoff capability
1
2 . . . subseq subseq #1 #2 subseq #S
N
Let S # of subsequences of length M . (Overlapping means S > N=M better averaging.)
=
])
Additional exibility: The data in each subsequence are weighted by a temporal window Welch is approximately equal to nonrectangular lag window.
^BT (!) with aSlide L39
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Daniell Method
By a previous result, for N
1,
f ^p(!j )g are (nearly) uncorrelated random variables for 2 j N ;1 ! =N j =0 Idea: Local averaging of (2J + 1) samples in thej
(2 + 1)
frequency domain should reduce the variance by about J .
k +J 1 ^D(!k ) = ^p (!j ) 2J + 1 j=k;JX
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L310
Daniell Method, con't
As J increases: Bias increases (more smoothing) Variance decreases (more averaging)
= 2J=N . Then, for N 1, ^D(!) ' 1 ^p(!)d! 2 ; Hence: ^D (! ) ' ^BT (! ) with a rectangular spectralLetZ
window.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L311
Summary of Periodogram Methods Unwindowed periodogram reasonable bias unacceptable variance
Modied periodograms Attempt to reduce the variance at the expense of (slightly) increasing the bias.
BT periodogram Local smoothing/averaging of spectral window.
^p(!) by a suitably selected ^(k) using a lag
Implemented by truncating and weighting r window in c !
^( )
Bartlett, Welch periodograms
Approximate interpretation: BT ! with a suitable lag window (rectangular for Bartlett; more general for Welch).
^ ()
Implemented by averaging subsample periodograms.
Daniell Periodogram Approximate interpretation: spectral window.
^BT (!) with a rectangularSlide L312
Implemented by local averaging of periodogram values.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Parametric Methods for Rational Spectra
Lecture 4
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L41
Basic Idea of Parametric Spectral EstimationObserved Data Assumed functional form of (,) Estimate parameters in (,)
^
Estimate PSD
^ ^ =(,) ()
possibly revise assumption on ()
Rational Spectra
jkj m k e;i!k (!) = P jkj n k e;i!k (!) is a rational function in e;i! .P
By Weierstrass theorem, ! can approximate arbitrarily well any continuous PSD, provided m and n are chosen sufciently large. Note, however: choice of m and n is not simple some PSDs are not continuousLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L42
( )
AR, MA, and ARMA Models
By Spectral Factorization theorem, a rational factored as
(!) can be
B (!) 2 2 (!) = A(!) A(z) = 1 + a1z;1 + + anz;n B (z) = 1 + b1z;1 + + bmz;m and, e.g., A(! ) = A(z )jz =ei!Signal Modeling Interpretation:
e(t) e (! ) =
2

B (q) A(q)
white noise
B (!) 2 2 A(!) ltered white noise
y(t)y (! ) =
ARMA: AR: MA:
A(q)y(t) = B (q)e(t) A(q)y(t) = e(t) y(t) = B (q)e(t)Slide L43
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
ARMA Covariance Structure
ARMA signal model:
y(t)+
X
n
i=1 n
aiy(t ; i) =
X
m
j =0
bj e(t ; j )
(b0 = 1)
Multiply by y
(t ; k) and take E f g to give:X
r (k ) +
X
2 m bj hj ;k = j =0 = 0 for k > m B (q) = 1 h q;k (h = 1) where H (q ) = A(q ) 0 k k=0X X
i=1
air(k ; i) =
m
j =0
bj E fe(t ; j )y (t ; k)g
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L44
AR Signals: YuleWalker Equations
AR: m
= 0.2 1 a1 = 0 . . . . an 03 7 7 7 5 2 6 6 6 4
Writing covariance equation in matrix form for k : : : n:
=1 r(0) r(;1) : : : r(;n) . r(1) r(0) . . . . . r (;1) . r(n) : : : r(0) 1 = 2 R 02 6 6 6 4 7 7 7 5 " # "
32 6 6 6 4
3 7 7 7 5
#
These are the YuleWalker (YW) Equations.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L45
AR Spectral Estimation: YW Method
YuleWalker Method: Replace r2 6 6 6 4
(k) by r(k) and solve for f^ig and ^2: ^ a r ^(0) ^(;1) : : : ^(;n) 1 r r ^2 . . r ^(1) r(0) ^ ^.1 = 0 a . . . . . ^(;1) . . r . r ^(n) : : : r(0) ^ ^n a 032 7 7 7 5 6 6 6 4 3 7 7 7 5 2 6 6 6 4
3 7 7 7 5
Then the PSD estimate is
^2 ^(!) = ^ 2 jA(!)j
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L46
AR Spectral Estimation: LS Method
Least Squares Method:
e(t) = y(t) +
X
n
4 = y(t) + y(t) ^ where '(t) = y (t ; 1) : : : y (t ; n)]T .Find
i=1
aiy(t ; i) = y(t) + 'T (t)
= a1 : : : an]T to minimizef( ) =X
N
This gives2
y=6 4
y(n + 1) y(n + 2) y(N ). . .
^ = ;(Y Y );1(Y y) where3 7 5 2 6
t=n+1
je(t)j2y(1) y(2) y(N ; n). . .3 7 5
Y
=4
y(n) y(n ; 1) y(n + 1) y(n). . .
y(N ; 1) y(N ; 2)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L47
LevinsonDurbin Algorithm
Fast, orderrecursive solution to YW equations2 6 6 6 4 
0 ;1 1 .0 ..
n
..
... ...{z
;n . ;1. .
32 7 7 7 5 } 6 6 6 4
1n
3 7 7 7 5
2 6 6 6 4
^ k = either r(k) or r(k).Direct Solution: For one given value of n: For k
Rn+1
1
0
= 0 . . 0
2 n
3 7 7 7 5
O(n3) ops
= 1 : : : n: O(n4) ops
LevinsonDurbin Algorithm: Exploits the Toeplitz form of Rn+1 to obtain the solutions for k : : : n in O n2 ops!
=1
( )
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L48
LevinsonDurbin Alg, con't
Relevant Properties of R:
Rx = y $ Rx = y, where x = xn : : : x1]T ~ ~ ~Nested structure2
Rn+2 =Thus,2
4
Rn+1n+1
r ~n
n+1 r ~n0
3 5
2
r ~n =
6 4
. .
n
3 7 5
1
Rn+2 4
1 n = 03 5
2 4
Rn+1n+1
r ~n
n+1 r ~n0
32 54
1 n n = 0 0 n3 5 2 4
2
3 5
where
r n = n+1 + ~n n
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L49
LevinsonDurbin Alg, con't2
Rn+2 482 < 4 +2 :
1 n n = 0 0 n3 5 2 4
2
3 5
2
Rn+2 439 = 5 2
0 ~n = 0n 2 1 n3 5 2 4 3 5 2
3 5
Combining these gives:
Rn
0 1 n + kn ~n 0 13 5 2 4
=
4
n + kn n2
0 n + kn
2 n
=
4
n
2 +1
3 5
0 0
Thus,
2 kn = ; n = n )n+1 =
0 + kn 2+1 = n + kn n = n(1 ; jknj2) 2 2 n
"
n
#
"
~n 1
#
Computation count: k ops for the step k
2 )
n2 ops
!k+1 2 to determine f k
k gn=1 kSlide L410
This is O
(n2) times faster than the direct solution.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
MA Signals
MA: n
=0 y(t) = B (q)e(t) = e(t) + b1e(t ; 1) +
+ bme(t ; m)
Thus,
r(k) = 0 for jkj > mand
2 2 = m r(k)e;i!k (!) = jB(!)j k=;mX
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L411
MA Spectrum Estimation Two main ways to Estimate (! ): 1. Estimate
fbkg and 2 and insert them in (!) = jB(!)j2 2
nonlinear estimation problem
^(!) is guaranteed to be 02. Insert sample covariances
(! ) =This is length
X
m
f^(k)g in: rr(k)e;i!k
k=;m
^BT (!) with a rectangular lag window of 2m + 1. ^(!) is not guaranteed to be 0 =0
Both methods are special cases of ARMA methods described below, with AR model order n .Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L412
ARMA Signals
ARMA models can represent spectra with both peaks (AR part) and valleys (MA part).
A(q)y(t) = B (q)e(t) B (!) 2 = m=;m k e;i!k k (!) = 2 A(!) jA(!)j2P
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 =X
n
X
n
j =0 p=0
aj ap r(k + p ; j )
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L413
ARMA Spectrum Estimation
Two Methods:
2g in (!) = 2 B(!) 2 1. Estimate fai bj A(!)nonlinear estimation problem; can use an approximate linear twostage least squares method
^(!) is guaranteed to be 02. Estimate
fai r(k)g in (!) =
P
m ;i!k k=;m k e
jA(!)j2
linear estimation problem (the Modied YuleWalker method).
^(!) is not guaranteed to be 0Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L414
TwoStage LeastSquares Method Assumption: The ARMA model is invertible:
A e(t) = B(q) y(t) (q ) = y(t) + 1y(t ; 1) + 2y(t ; 2) + = AR(1) with j kj ! 0 as k ! 1 Step 1: Approximate, for some large K e(t) ' y(t) + 1y(t ; 1) + + K y(t ; K )1a) Estimate the coefcients modelling techniques.
f kgK=1 by using AR k
1b) Estimate the noise sequence
^(t) = y(t) + ^1y(t ; 1) + eand its variance
+ ^K y(t ; K )
^2 =
N 1 j^(t)j2 e N ; K t=K +1X
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L415
TwoStage LeastSquares Method, con't
fe(t)g by ^(t) in the ARMA equation, e A(q)y(t) ' B (q)^(t) e and obtain estimates of fai bj g by applying least squaresStep 2: Replace techniques. Note that the ai and bj coefcients enter linearly in the above equation:
y(t) ; ^(t) ' ;y(t ; 1) : : : ; y(t ; n) e ^(t ; 1) : : : ^(t ; m)] e e = a1 : : : an b1 : : : bm]T
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L416
Modied YuleWalker Method
ARMA Covariance Equation:
r(k) +2 6 4
X
n
In matrix form for k
i=1
air(k ; i) = 0 k > m3 7 5 "
r(m) r(m + 1). . .
: : : r(m ; n + 1) r(m ; n + 2)... . . .
= m + 1 ::: m + Ma1 an. . .#
2
=
r(m + M ; 1) : : : r(m ; n + M )
;6 4
Replace
fr(k)g by f^(k)g and solve for faig. r f^ g
. 5 . . r(m + M )
r(m + 1) r(m + 2)
3 7
If M n, fast Levinsontype algorithms exist for obtaining ai . If M > n overdetermined squares solution for ai .
=
f^ g
Y W system of equations; least
f^ g
Note: For narrowband ARMA signals, the accuracy of ai is often better for M > n
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L417
Summary of Parametric Methods for Rational Spectra
Method AR: YW or LS low lowmedium mediumhigh mediumhigh medium lowmedium No No Yes
Computational Burden low Accuracy medium
Guarantee ^(! ) 0 ? Yes
MA: BT
ARMA: MYW
ARMA: 2Stage LS
Use for Spectra with (narrow) peaks but no valley Broadband spectra possibly with valleys but no peaks Spectra with both peaks and (not too deep) valleys As above
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L418
Parametric Methods for Line Spectra Part 1
Lecture 5
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L51
Line Spectra Many applications have signals with (near) sinusoidal components. Examples: communications radar, sonar geophysical seismology ARMA model is a poor approximation Better approximation by Discrete/Line Spectrum Models
2 6(2 1) 2
6
(!)6
(2 2) 2
6
(2 2) 3!Slide L52
; !1
!2
!3

An Ideal line spectrumLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Line Spectral Signal Model Signal Model: Sinusoidal components of frequencies !k and powers 2 , superimposed in white noise of k 2. power
f g
f g
y(t) = x(t) + e(t) t = 1 2 : : : n x(t) = k ei(!k t+ k ) k=1 xk (t)X  {z }
Assumptions: A1: A2:
k>0
!k 2 ;
]
(prevents model ambiguities)
f'k g = independent rv's, uniformly distributed 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)Slide L53
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Covariance Function and PSD
Note that:
E E
n
i'p e;i'j o = 1, for p = j e i'p e;i'j o = E nei'p o E ne;i'j o eZ
n
1 i' d' 2 = 0, for p 6= j =2 ; eE xp(t)xj (t ; k) = 2 ei!pk p j pn o
Hence,
r(k) = E fy(t)y (t ; k)g = n=1 2ei!pk + 2 k 0 p pP
and
(! ) = 2
X
n
p=1
2 (! ; !p) + 2 pSlide L54
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Parameter Estimation Estimate either:
f!k
k 'k gn=1 k
2
(Signal Model)
f!k 2gn=1 2 k kOnce
(PSD Model)
^ Major Estimation Problem: f!k g
f!kg are determined: ^X
f ^2g can be obtained by a least squares method from k n 2ei!pk + residuals ^ ^ r(k) = pp=1OR: Both k and 'k can be derived by a least squares method from
f^ g
f^ gX
y(t) =with
n
k=
k=1 k ei'k .
^ ei!k t + residuals k
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L55
Nonlinear Least Squares (NLS) Method
min' g f!Let:
X
N
k k k t=1 
y(t) ;
X
n
k=1 {z F (! ')
k ei(!k t+'k )
2}
= kei'k = 1 : : : n]T Y = y(1) : : : y(N )]T ei!1 ei!n . . . . B = eiN!1 eiN!nk2 6 4
3 7 5
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L56
Nonlinear Least Squares (NLS) Method, con't
Then:
F = (Y ; B ) (Y ; B ) = kY ; B k2 = ; (B B);1B Y ] B B] ; (B B);1B Y ] +Y Y ; Y B(B B);1B YThis gives:
^ = (B B);1B Y !=^ !and
! = arg max Y B (B B );1B Y ^ !
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L57
NLS Properties
Excellent Accuracy:
6 2 (for N var (^k ) = 3 2 ! N k Example: N = 300 SNRk = 2= 2 = 30 dB kq
1)
Then
var
(^k ) 10;5. !
Difcult Implementation: The NLS cost function F is multimodal; it is difcult to avoid convergence to local minima.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L58
Unwindowed Periodogram as an Approximate NLS Method
For a single (complex) sinusoid, the maximum of the unwindowed periodogram is the NLS frequency estimate: Assume: Then:
n=1X
B B=NN t=1
BY=
y(t)e;i!t = Y (!)
(nite DTFT)
1 Y B (B B );1B Y = N jY (!)j2 = ^p(!) = (Unwindowed Periodogram)So, with no approximation,
! = arg max ^p(!) ^ !
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L59
Unwindowed Periodogram as an Approximate NLS Method, con't
Assume: Then:
n>1
f!kgn=1 ' the ^ kprovided that
locations of the peaks of p !
^( )
n
largest
inf j!k ; !pj > 2 =Nwhich is the periodogram resolution limit. If better resolution desired then use a High/Super Resolution method.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L510
HighOrder YuleWalker Method Recall:
y(t) = x(t) + e(t) =
k ei(!k t+'k ) +e(t)  {z } k=1 xk (t)X
n
Degenerate ARMA equation for y (t):n
(1 ; ei!k q;1)xk (t) = k ei(!kt+'k) ; ei!k ei !k(t;1)+'k] = 0o
Let
B (q) = 1 +
4 bk q;k = A(q)A(q) k=1 A(q) = (1 ; ei!1 q;1) (1 ; ei!n q;1) A(q) = arbitraryX
L
Then B
(q)x(t) 0 ) B (q)y(t) = B (q)e(t)Slide L511
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
HighOrder YuleWalker Method, con't
Estimation Procedure: Estimate
f^igL=1 using an ARMA MYW technique b i^( )
Roots of B q give spurious roots.
f!kgn=1, along with L ; n ^ k
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L512
HighOrder and Overdetermined YW Equations
ARMA covariance:
r (k ) +
X
L
i=1
bir(k ; i) = 0 k > L
In matrix form for k2 6 6 6 4 
= L + 1 ::: L + M3 7 7 7 5 2 6 6 6 4 3 7 7 7 5 }
: : : r(1) r(L + 1) : : : r(2) b = ; r(L + 2) . . . . . . r(L + M ; 1) : : : r(M ) r(L + M ) 4 4 = ={z }  {z
r(L) r(L + 1)
This is a highorder (if L > n) and overdetermined (if M > L) system of YW equations.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L513
HighOrder and Overdetermined YW Equations, con't Fact: SVD of rank :
( )=n =U V
U = (M n) with U U = In V = (n L) with V V = In
= (n n), diagonal and nonsingularThus,
(U V )b = ;The MinimumNorm solution is
b = ; y = ;V ;1U Important property: The additional (L ; n) spurious zeros of B (q ) are located strictly inside the unit circle, if the MinimumNorm solution b is used.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L514
HOYW Equations, Practical Solution
Let
^= ^ ^ ^ ^
but made from
f^(k)g instead of fr(k)g. r, V from the
Let U , , V be dened similarly to U , SVD of . Compute
^ = ;V ^ ;1U ^ ^ ^ b ^(q) that
Then !k n=1 are found from the n zeroes of B k are closest to the unit circle.
f^ g
When the SNR is low, this approach may give spurious frequency estimates when L > n; this is the price paid for increased accuracy when L > n.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L515
Parametric Methods for Line Spectra Part 2
Lecture 6
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L61
The Covariance Matrix Equation Let:
a(!) = 1 e;i! : : : e;i(m;1)! ]T A = a(!1) : : : a(!n)] (m n)rank
Note: Dene
(A) = ny(t) y(t ; 1)
(for m3 7 7 7 5
n)
2
46 y(t) = 6 ~6 4
y(t ; m + 1)
. .
= Ax(t) + ~(t) ~ e
where
x(t) = x1(t) : : : xn(t)]T ~ ~(t) = e(t) : : : e(t ; m + 1)]T e4 R = E fy(t)~ (t)g = APA + 2I ~ y
Then
with2
P = E fx(t)~ (t)g = ~ x
6 4
2 1
0
...
02 n
3 7 5
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L62
Eigendecomposition of R and Its Properties
R = APA + 2I (m > n)Let:
1
2 :::
m: eigenvalues of R
fs1 : : : sng: orthonormal eigenvectors associated with f 1 : : : ng fg1 : : : gm;ng: orthonormal eigenvectors associated with f n+1 : : : m g S = s1 : : : sn] (m n) G = g1 : : : gm;n] (m (m ; n))Thus,2
R = S G]
6 4
1
3
"
...
m
7 5
S G
#
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L63
Eigendecomposition of R and Its Properties, con't As rank
(APA ) = n: k > 2 k = 1 ::: n k = 2 k = n + 1 ::: m 0 1; 2 . .. = = nonsingular 0 n; 22 6 4 3 7 5 2 6 4
Note:
RS = APA S + 2S = S
1
0
...
0n
3 7 5
with C (since rank S , Therefore, since S G
j j 6= 0
4 ;1) = AC S = A(PA S
( ) = rank(A) = n). =0 A G=0Slide L64
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
MUSIC Method
a (!1) . . A G= G=0 a (!n) ) fa(!k )gn=1 ? R(G) k2 6 4 3 7 5
Thus,
f!kgn=1 are the unique solutions of k a (!)GG a(!) = 0.Let:
1 N y(t)~ (t) ^ ~ y R = N t=m ^^ S G = S G made from the ^ eigenvectors of RX
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L65
Spectral and Root MUSIC Methods Spectral MUSIC Method:
f!kgn=1 = the locations of the n highest peaks of the ^ kpseudospectrum function:
1 ^G a(!) ! 2 ; a (!)G ^Root MUSIC Method:
]
f!kgn=1 = the angular positions of the n roots of: ^ k^^ aT (z;1)GG a(z) = 0that are closest to the unit circle. Here,
a(z) = 1 z;1 : : : z;(m;1)]TNote: Both variants of MUSIC may produce spurious frequency estimates.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L66
Pisarenko Method
Pisarenko is a special case of MUSIC with m (the minimum possible value). If:
=n+1
m=n+1
Then:
^ g G = ^1, ) f!kgn=1 can be found from the roots of ^ k aT (z;1)^1 = 0 gno problem with spurious frequency estimates computationally simple (much) less accurate than MUSIC with m
n+1
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L67
MinNorm Method
Goals: Reduce computational burden, and reduce risk of false frequency estimates. Uses m n (as in MUSIC), but only one vector in G (as in Pisarenko).
R( )Let"
1 = the vector in R(G), with rst element equal ^ ^ to one, that has minimum Euclidean norm. g#
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L68
MinNorm Method, con't
Spectral MinNorm
f!gn=1 = ^k
the locations of the n highest peaks in the pseudospectrum
2 1 = a (!) 1 ^ g" #
Root MinNorm
f!gn=1 = ^k
the angular positions of the n roots of the polynomial
aT (z;1) 1 ^ g"
#
that are closest to the unit circle.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L69
Let S
^= S g1 ;1 gm" # " #
MinNorm Method: Determining g ^
Then:
1 2 R(G) ) S 1 = 0 ^ ^ ^ ^ g g ) S ^= ; g" #
MinNorm solution: As:
^ = ;S(S S );1 g
+ S S , (S S );1 exists iff = k k2 6= 1 (This holds, at least, for N 1.) ^^ I=S S=Multiplying the above equation by gives:
(1 ; k k2) = (S S ) ) (S S );1 = =(1 ; k k2) ) ^ = ;S =(1 ; k k2) gLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L610
ESPRIT Method
Let
A1 = Im;1 0]A A2 = 0 Im;1]A
Then A2
Also, let
= A1D, where e;i!1 0 ... D= 0 e;i!n S1 = Im;1 0]S S2 = 0 Im;1]S2 6 4 
3 7 5
Recall S
= AC with jC j 6= 0. Then S2 = A2C = A1DC = S1 C ;1DC{z
}
So has the same eigenvalues as D . determined as
is uniquely
= (S1S1);1S1S2Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L611
ESPRIT Implementation From the eigendecomposition of R, nd S , then S1 and S2 .
^
^
^
^
The frequency estimates are found by:
where
f^kgn=1 are the eigenvalues of k ^ = (S1S1);1S1S2 ^^ ^^
f!kgn=1 = ; arg(^k ) ^ k
ESPRIT Advantages: computationally simple no extraneous frequency estimates (unlike in MUSIC or MinNorm) accurate frequency estimates
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L612
Summary of Frequency Estimation Methods
Method Periodogram Nonlinear LS YuleWalker Pisarenko MUSIC MinNorm ESPRIT
Computational Accuracy / Risk for False Burden Resolution Freq Estimates small mediumhigh medium very high very high very high medium high medium small low none high high medium medium high small medium very high none
Recommendation:
Use Periodogram for mediumresolution applications
Use ESPRIT for highresolution applicationsSlide L613
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Filter Bank Methods
Lecture 7
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L71
Basic Ideas
Two main PSD estimation approaches: 1. Parametric Approach: Parameterize nitedimensional model.
(!) by a
2. Nonparametric Approach: Implicitly smooth ! !=; by assuming that ! is nearly constant over the bands
f ( )g
( )
!;
!+ ] N >1
1
2 is more general than 1, but 2 requires
to ensure that the number of estimated values = = ) is < N . (
=2 2 =1
N > 1 leads to the variability / resolution compromiseassociated with all nonparametric methods.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L72
Filter Bank Approach to Spectral EstimationH 6 (!)1
!c
!Filtered Signal
y (t)
 with varying !Power Calculation
Bandpass Filter c and xed bandwidth

Power in the band

Division by lter bandwidth
^(!c )

6y
y
(! )
;!c
@Q Q !;Z
6 ~ (! ) @ !(b)Z
!c
^FB (!) ' 1 2(a)
;
jH ( )j2 ( )d = ' 21
!+ !;
c ( )d = (') (!)
(a) consistent power calculation
(b) Ideal passband lter with bandwidth
(c)
( ) constant on 2 ! ; 2 ! + 2 ]
Note that assumptions (a) and (b), as well as (b) and (c), are conicting.Slide L73
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Filter Bank Interpretation of the Periodogram
1 N y(t)e;i!t 2 ~ ^p(~) = ! 4 N t=1 2 N 1 ~ = N y(t)ei!(N ;t) t=1X X
= Nwhere(
X
1
k=0
hk y(N ; k)
2
! ;i!k = 1 eiN (~;!) ; 1 H (!) = hk e ! N ei(~;!) ; 1 k=0X
1 ei!k k = 0 : : : N ; 1 ~ hk = N otherwise 01
center frequency of H 3dB bandwidth of H
(!) = ! ~
(!) ' 1=NSlide L74
Lecture notes to accompany Introduction to Spectral Analysis by 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. !0 5 10
15 dB
20
25
30
35
40
3
2
1 0 1 ANGULAR FREQUENCY
2
3
Conclusion: The periodogram p ! is a lter bank PSD estimator with bandpass lter as given above, and: narrow lter passband, power calculation from only 1 sample of lter output.
^( )
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L75
Possible Improvements to the Filter Bank Approach 1. Split the available sample, and bandpass lter each subsample. more data points for the power calculation stage. This approach leads to Bartlett and Welch methods. 2. Use several bandpass lters on the whole sample. Each lter covers a small band centered on ! .
~
provides several samples for power calculation. This multiwindow approach is similar to the Daniell method.
Both approaches compromise 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 ltering.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L76
Capon Method
Idea: Datadependent bandpass lter design.
yF (t) =
X
m
k=0
hk y(t ; k)2 6 4 
. = h0 h1 : : : hm] . y(t ; m) h y(t) ~ E jyF (t)j2 = h Rh R = E fy(t)~ (t)g ~ y{z } 7 5 {z } n o
y(t)
3
H (! ) =where a
X
m
(! ) = 1
k=0 e;i! : : : e;im! ]T
hk e;i!k = h a(!)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L77
Capon Method, con't Capon Filter Design Problem:
min(h Rh) hSolution:
subject to h
a(!) = 1
h0 = R;1a=a R;1ao
The power at the lter output is:
E jyF (t)j2 = h0Rh0 = 1=a (!)R;1a(!) which should be the power of y (t) in a passband centered on ! .n
The Bandwidth
1 ' m+1 = (lter 1 length)
Conclusion Estimate PSD as:
+1 ^(!) = m^;1 a (!)R a(!)with
N 1 ^ y(t)~ (t) ~ y R = N ;m t=m+1X
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L78
Capon Properties
m is the user parameter that controls the compromisebetween bias and variance: as m increases, bias decreases and variance increases.
Capon uses one bandpass lter only, but it splits the N data point sample into (N m) subsequences of length m with maximum overlap.
;
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L79
Relation between Capon and BlackmanTukey Methods
Consider
^BT (!) with Bartlett window: m m + 1 ; jkj ^BT (!) = r(k)e;i!k ^ k=;m m + 1 1 m m ^(t ; s)e;i!(t;s) = m+1 r t=0 s=0 ^ a (!)Ra(!) R = ^(i ; j )] ^ r = m+1X X X
Then we have
^ a (!)Ra(!) ^BT (!) = m+1 m+1 ^C (!) = ^ a (!)R;1a(!)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L710
Relation between Capon and AR Methods Let
2 ^k ^k (!) = ^ 2 jAk(!)j be the kth order AR PSD estimate of y (t).AR Then
^C (!) =
1 1 m 1= ^AR(!) m + 1 k=0 kX
Consequences: Due to the average over k, C ! generally has less statistical variability than the AR PSD estimator. Due to the loworder AR terms in the average, C ! generally has worse resolution and bias properties than the AR method.
^( )
^( )
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L711
Spatial Methods Part 1
Lecture 8
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L81
The Spatial Spectral Estimation ProblemSource 1
vSource 2B B B B N B cccc ccc c c c c c
v
@ ,,, @, ,,@ , ,R , , ,@ ,
v Source n
@; @;Sensor 1
@; @; @; @;Sensor 2 Sensor
m
Problem: Detect and locate 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 name spatial spectral analysis)Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L82
Simplifying Assumptions
Fareld sources in the same plane as the array of sensors Nondispersive wave propagation
Hence: The waves are planar and the only location parameter is direction 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 with known transfer characteristics and known locations (That is, the array is calibrated.)
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L83
Array Model Single Emitter Case
x(t) =k=
the signal waveform as measured at a reference point (e.g., at the rst sensor) the delay between the reference point and the kth sensor the impulse response (weighting function) of sensor k noise at the kth sensor (e.g., thermal noise in sensor electronics; background noise, etc.)
hk (t) = ek (t) =Note:
t 2 R (continuoustime signals). Then the output of sensor k is yk (t) = hk (t) x(t ; k) + ek (t) ( = convolution operator).Basic Problem: Estimate the time delays known but x t unknown.
()
f kg with hk(t)
This is a timedelay estimation problem in the unknown input case.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L84
Narrowband Assumption
Assume: The emitted signals are narrowband with known carrier frequency !c . Then:
x(t) = (t) cos !ct + '(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 !ct + '(t) ; !c k ] hk (t) x(t ; k ) ' jHk (!c)j (t)cos !ct + '(t) ; !c k + argfHk (!c)g]
where Hk function
(!) = Ffhk(t)g is the kth sensor's transfer
Hence, the kth sensor output is
yk (t) = jHk (!c)j (t) cos !ct + '(t) ; !c k + arg Hk (!c)] + ek(t)Slide L85
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Complex Signal Representation The noisefree output has the form:
z(t) = (t) cos !ct + (t)] = = (t) ei !ct+ (t)] + e;i !ct+ (t)] 2 Demodulate z (t) (translate to baseband): 2z(t)e;!ct = (t)f ei (t) + e;i 2!ct+ (t)] gn o  {z }  {z }
Lowpass lter
2z(t)e;i!ct to obtain (t)ei (t)
lowpass
highpass
Hence, by lowpass ltering and sampling the signal
yk (t)=2 = yk(t)e;i!ct ~ = yk(t) cos(!ct) ; iyk(t) sin(!ct) we get the complex representation: (for t 2 Z ) yk (t) = (t) ei'(t) jHk (!c)j ei arg H (! )] e;i! + ek (t)
s(t)
{z
} 
Hk (!c)
{z
k
c
}
c k
yk (t) = s(t)Hk (!c) e;i!c k + ek (t) where s(t) is the complex envelope of x(t).Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L86
or
Vector Representation for a Narrowband Source
Let
= m =a( ) =2 6 4
the emitter DOA the number of sensors2 6 4
H1(!c) e;i!c. .
1
3 7 5
Hm(!c) e;i!c m y1(t) e1(t) . . y(t) = e(t) = . . ym(t) em(t)3 7 5 2 6 4
3 7 5
Then
y(t) = a( )s(t) + e(t)NOTE: enters a via both For omnidirectional sensors the on .
()
f kg and fHk(!c)g. fHk(!c)g do not depend
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L87
Analog Processing Block Diagram
Analog processing for each receiving array elementSENSOR DEMODULATION
A/D CONVERSION


Internal Noise External Noise XXXXX ? XX ll z l  Cabling and , Filtering : ,, Incoming Transducer Signal x(t ; k )
Re yk (t)] Lowpass Re yk (t)]  !! ~ Filter A/D cos(!ct) 6 y k (t) Oscillator sin(!ct) ? ~  Im yk (t)] Lowpass Im yk (t)]  !! Filter A/D

Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L88
Multiple Emitter Case Given n emitters with received signals: DOAs:
fsk(t)gn=1 k
k
Linear sensors
) y(t) = a( 1)s1(t) +
+ a( n)sn(t) + e(t)
Let
A = a( 1) : : : a( n)] (m n) s(t) = s1(t) : : : sn(t)]T (n 1)Then, the array equation is:
y(t) = As(t) + e(t)Use the planar wave assumption to nd the dependence of k on .Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L89
Uniform Linear Arrays
Source h
JJ J J
J J
JJ J ^ J
J
JJ
JJ+ JJ
JJ2
] J d sin J ^ JJ J JJ 3 J Jv4
1
v ]? vd 
v
ppp
m
vSensor
ULA Geometry Sensor #1 = time delay reference Time Delay for sensor k:
where c = wave propagation speed
d sin k = (k ; 1) c
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L810
Spatial Frequency
Let:
sin 4 !s = !c d sin = 2 dc=f = 2 d sin c c = c=fc = signal wavelength a( ) = 1 e;i!s : : : e;i(m;1)!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!sj $ dj sin j 1 d =2As
=2
d
=2 is a spatial Shannon sampling theorem.Slide L811
d = spatial sampling period
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Spatial Methods Part 2
Lecture 9
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L91
Spatial Filtering
Spatial ltering useful for DOA discrimination (similar to frequency discrimination of timeseries ltering) Nonparametric DOA estimation
There is a strong analogy between temporal ltering and spatial ltering.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L92
Analogy between Temporal and Spatial Filtering
Temporal FIR Filter:
yF (t) = hk u(t ; k) = h y(t) k=0 h = ho : : : hm;1] y(t) = u(t) : : : u(t ; m + 1)]TIf u
m;1 X
(t) = ei!t then yF (t) = h a(!)] u(t) {z }
lter transfer function
a(!) = 1 e;i! : : : e;i(m;1)! ]T We can select h to enhance or attenuate signals with different frequencies ! .
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L93
Analogy between Temporal and Spatial Filtering Spatial Filter:
fyk(t)gm=1 = k
the spatial samples obtained with a sensor array.
Spatial FIR Filter output:
yF (t) =
X
m
k=1
hk yk (t) = h y(t)
Narrowband Wavefront: The array's (noisefree) 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 ]TyF (t) = h a( )] s(t) {z }
The corresponding lter output is
lter transfer function
We can select h to enhance or attenuate signals coming from different DOAs.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L94
Analogy between Temporal and Spatial Filteringu(t) = ei!t
z1
s q; s qq
h
?1
?

m;1
q ;1
? 
1  @ 0 B 1 C h @@ C B B e;i! C @ h a(!)]u(t) R @ C  ? B P C B C B B B .. Cu(t) C B . C C B C B C B C B C B A @ ;i m; ! hm;  e {z } ? a(!)1
?
0
(
1)
1
(Temporal sampling)
(a) Temporal lternarrowband source with DOA=Z
z
Z
Z
Z ~ aa 1reference !!
h
0
0 1 B aa 2 B ;i! Be 2 B !! B B B B .. B . B B @ ;i! m  e {z a( ) aa m!
point
( )
( )
1 C C C C C Cs(t) C C C C C A }

? @ ? @ @
h
1
@ R @ 
P
h a( )]s(t) 
hm;?
1
!
(Spatial sampling)
(b) Spatial lterLecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L95
Spatial Filtering, con't
Example: The response magnitude h a of a spatial lter (or beamformer) for a 10element ULA. Here, h a 0 , where 0
j ( )j
= ( )
= 25
0
30
30
Th et a (d eg )
60
60
90 0 2 4 6 Magnitude 8 10
90
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L96
Spatial Filtering Uses
Spatial Filters can be used To pass the signal of interest only, hence ltering out interferences located outside the lter's beam (but possibly having the same temporal characteristics as the signal). To locate an emitter in the eld of view, by sweeping the lter through the DOA range of interest (goniometer).
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L97
Nonparametric Spatial Methods
A Filter Bank Approach to DOA estimation. Basic Ideas Design a lter h
( ) such that for each
It passes undistorted the signal with DOA = It attenuates all DOAs
6=
Sweep the lter through the DOA range of interest, and evaluate the powers of the ltered signals:
E jyF (t)j2 = E jh ( )y(t)j2 = h ( )Rh( ) with R = E fy (t)y (t)g.n o n
o
The (dominant) peaks of h DOAs of the sources.
( )Rh( ) give theSlide L98
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Beamforming Method
Assume the array output is spatially white:
Then:
R = E fy(t)y (t)g = I E jyF (t)j2 = h hn o
Hence: In direct analogy with the temporally white assumption for lter bank methods, y t can be considered as impinging on the array from all DOAs.
()
Filter Design:
min (h h) hSolution:
subject to
h a( ) = 1
h = a( )=a ( )a( ) = a( )=m E jyF (t)j2 = a ( )Ra( )=m2n o
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L99
Implementation of Beamforming
1 N y(t)y (t) ^ R=N t=1X
The beamforming DOA estimates are:
f^kg =
the locations of the n largest peaks of a Ra .
( )^ ( )
This is the direct spatial analog of the BlackmanTukey periodogram. Resolution Threshold:
inf j k ; pj > =
wavelength array length array beamwidth
Inconsistency problem: Beamforming DOA estimates are consistent if n inconsistent if n > .
1
= 1, butSlide L910
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Capon Method Filter design:
min(h Rh) subject to h a( ) = 1 hSolution:
h = R;1a( )=a ( )R;1a( ) E jyF (t)j2 = 1=a ( )R;1a( )n o
Implementation:
f^kg =
the locations of the n largest peaks of
^ 1=a ( )R;1a( ):
Performance: Slightly superior to Beamforming. Both Beamforming and Capon are nonparametric approaches. They do not make assumptions on the covariance properties of the data (and hence do not depend on them).Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L911
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 in all sensors:
E fe(t)e (t)g = 2IThe signal covariance matrix
P = E fs(t)s (t)gis nonsingular.
Then:
R = E fy(t)y (t)g = APA + 2IThus: The NLS, YW, MUSIC, MINNORM and ESPRIT methods of frequency estimation can be used, almost without modication, for DOA estimation.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L912
Nonlinear Least Squares Method
1 N ky(t) ; As(t)k2 min(t)g N f k g fs t=1X 
Minimizing f over s gives
f ( s)
{z
}
^(t) = (A A);1A y(t) t = 1 : : : N sX
Then
1 N k I ; A(A A);1A ]y(t)k2 f ( ^) = N s t=1 1 N y (t) I ; A(A A);1A ]y(t) = N t=1 ^ = trf I ; A(A A);1A ]Rg ^ Thus, f ^k g = arg max trf A(A A);1A ]RgX
f kg
For N , this is precisely the form of the NLS method of frequency estimation.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L913
=1
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 by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L914
YuleWalker Method
e y(t) = y(t) = A s(t) + ~(t) ~ y(t) ~ e(t) A" # " # "
#
Assume: Then:
E fe(t)~ (t)g = 0 e
4 E fy(t)~ (t)g = AP A ~ ;= y
(M L)
Also assume:
M > n L > n ( ) m = M + L > 2n)rank
~ (A) = rank(A) = n" # " #
(;) = n, and the SVD of ; is ; = U1 U2 ] 0 n n 0 V1 g n ;n 0 V2 g L n M ;n ~ ~ Properties: A V2 = 0 V1 2 R(A)Then: rank{z} {z}
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L915
YWMUSIC DOA Estimator
f^kg =where
the n largest peaks of
1=~ ( )V2V2 ~( ) a ^^a
~( ), (L 1), is the array transfer vector for y(t) a ~at DOA
^ V2 is dened similarly to V2, using 1 N y(t)~ (t) ^ ;=N y t=1X
Properties: Computational complexity: medium Performance: satisfactory if m
2n
Main advantages: weak assumption on e t the subarray A need not be calibrated
f ( )g
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L916
MUSIC and MinNorm Methods
Both MUSIC and MinNorm methods for frequency estimation apply with only minor modications to the DOA estimation problem. Spectral forms of MUSIC and MinNorm can be used for arbitrary arrays Root forms can be used only with ULAs MUSIC and MinNorm break down if the source signals are coherent; that is, if rank
(P ) = rank(E fs(t)s (t)g) < n
Modications that apply in the coherent case exist.
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L917
ESPRIT Method Assumption: The array is made from two identical subarrays separated by a known displacement vector. Let
m = # sensors in each subarray A1 = Im 0]A (transfer matrix of subarray 1) A2 = 0 Im]A (transfer matrix of subarray 2)Then
A2 = A1D, where e;i!c ( 1) 0 ... D= 0 e;i!c ( n) ( ) = the time delay from subarray 1 to2 6 4
3 7 5
subarray 2 for a signal with DOA = : where d is the subarray separation and is measured from the perpendicular to the subarray displacement vector.Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997 Slide L918
( ) = d sin( )=c
ESPRIT Method, con't ESPRIT Scenariosource
known
subarray 1
displac ement v
ector
subarray 2
Properties: Requires special array geometry Computationally efcient
No risk of spurious DOA estimatesDoes not require array calibration Note: For a ULA, the two subarrays are often the rst m and last m array elements, so m m and
;1
;1 A1 = Im;1 0]A
= ;1 A2 = 0 Im;1]A
Lecture notes to accompany Introduction to Spectral Analysis by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L919