Basic Definitions
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 L1–1
Informal Definition of Spectral Estimation
Given: A finite record of a signal.
Determine: The distribution of signal power over
frequency.
spectral density
signal
t
ω
t=1, 2, ...
ω ω+∆ω
π
! = (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 L1–2
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
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L1–3
Deterministic Signals
fy(t)g1
t=,1 =
If:
Then:
discrete-time deterministic data
sequence
1
X
t=,1
jy(t)j2 < 1
Y (!) =
1
X
t=,1
y(t)e,i!t
exists and is called the Discrete-Time Fourier Transform
(DTFT)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L1–4
Energy Spectral Density
Parseval's Equality:
1
1
2
jy(t)j = 2 , S (!)d!
t=,1
Z
X
where
4 jY (!)j2
S (! ) =
= Energy Spectral Density
We can write
S (!) =
where
(k) =
1
X
k=,1
1
X
t=,1
(k)e,i!k
y(t)y(t , k)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L1–5
Random Signals
Random Signal
probabilistic statements about
future variations
random signal
t
current observation time
1
X
Here:
t=,1
jy(t)j2 = 1
E jy(t)j2 < 1
n
But:
o
E fg = Expectation over the ensemble of realizations
E jy(t)j2
n
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 L1–6
First Definition of PSD
(!) =
where r
1
X
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)j
Note that
1
r(k) = 2 , (!)ei!k d!
Z
(Inverse DTFT)
Interpretation:
1
2
r(0) = E jy(t)j = 2 , (!)d!
n
so
(!)d! =
o
Z
infinitesimal 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 L1–7
Second Definition of PSD
8
>
<
X
N
1
,i!t
(!) = Nlim
E
y
(
t
)
e
!1 N t=1
>
:
Note that
1
2
(!) = Nlim
E
j
Y
(
!
)
j
N
!1 N
where
YN (!) =
is the finite DTFT of
N
X
t=1
2
9
>
=
>
;
y(t)e,i!t
fy(t)g.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L1–8
Properties of the PSD
P1:
(!) = (! + 2) for all !.
Thus, we can restrict attention to
! 2 [,; ] () f 2 [,1=2; 1=2]
P2:
(!) 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 L1–9
Transfer of PSD Through Linear Systems
System Function:
where q ,1
H (q) =
1
X
k=0
hk q,k
= unit delay operator: q,1y(t) = y(t , 1)
e(t)
e (! )
y (t) y (! ) = jH (! )j2e (! )
H (q )
Then
y(t) =
1
X
k=0
H (!) =
hk e(t , k)
1
X
k=0
hk e,i!k
y (!) = jH (!)j2e(!)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L1–10
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 definitions.
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 L1–11
Periodogram
and
Correlogram
Methods
Lecture 2
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–1
Periodogram
Recall 2nd definition of (! ):
8
>
<
X
N
1
,i!t
(!) = Nlim
E
y
(
t
)
e
!1 N t=1
>
:
Given :
Drop “
2
9
>
=
>
;
fy(t)gNt=1
lim
N !1
” and “E
fg” to get
^p(!) = N1
X
N
t=1
y(t)e,i!t
2
Natural estimator
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 L2–2
Correlogram
Recall 1st definition of (! ):
(!) =
1
X
k=,1
r(k)e,i!k
Truncate the “ ” and replace “r
P
^c(!) =
(k)” by “^r(k)”:
NX
,1
k=,(N ,1)
^r(k)e,i!k
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–3
Covariance Estimators
(or Sample Covariances)
Standard unbiased estimate:
N
1
^r(k) = N , k
y(t)y(t , k); k 0
t=k+1
X
Standard biased estimate:
N
1
r^(k) = N
y(t)y(t , k); k 0
t=k+1
X
For both estimators:
^r(k) = ^r(,k); k < 0
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–4
^ (! ) and ^ (! )
Relationship Between p
c
^(k) is used in ^c(!),
If: the biased ACS estimator r
Then:
X
N
,i!t
^p(!) = 1
y
(
t
)
e
N t=1
=
NX
,1
k=,(N ,1)
2
^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
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–5
^ (! ) and ^ (! )
Statistical Performance of p
c
Summary:
Both are asymptotically (for large N ) unbiased:
E ^p(!) ! (!) as N ! 1
n
o
Both have “large” variance, even for large N .
^ (!) and ^c(!) have poor performance.
Thus, p
Intuitive explanation:
r^(k) , r(k) may be large for large jkj
Even if the errors f^r(k) , r(k)gNjkj,=01 are small,
there are “so many” that when summed in
p ! ! , the PSD error is large.
[ ^ ( ) , ( )]
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–6
Bias Analysis of the Periodogram
NX
,1
E ^p(!) = E ^c(!) =
E f^r(k)g e,i!k
k=,(N ,1)
N ,1
=
1 , jNkj r(k)e,i!k
k=,(N ,1)
1
=
wB (k)r(k)e,i!k
k=,1
n
o
n
o
!
X
X
wB (k) =
=
8
<
:
1 , jNkj ; jkj N , 1
0;
jkj N
Bartlett, or triangular, window
Thus,
1
^
E p(!) = 2 , ( )WB (! , ) d
n
Ideally:
o
Z
WB (!) = Dirac impulse (!).
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–7
Bartlett Window WB (! )
2
1
sin(
!N=
2)
WB (!) = N sin(!=2)
"
0
#
WB (!)=WB (0), for N = 25
−10
dB
−20
−30
−40
−50
−60
−3
−2
−1
0
1
ANGULAR FREQUENCY
Main lobe 3dB width
For “small” N , WB
2
3
1=N .
(!) may differ quite a bit from (!).
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–8
Smearing and Leakage
Main Lobe Width: smearing or smoothing
(!) separated in f by less than 1=N are not
Details in resolvable.
φ(ω)
^φ(ω)
smearing
ω
<1/Ν
Thus:
ω
Periodogram resolution limit =
1=N .
Sidelobe Level: leakage
^φ(ω)≅WB(ω)
φ(ω)≅δ(ω)
leakage
ω
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
ω
Slide L2–9
Periodogram Bias Properties
Summary of Periodogram Bias Properties:
For “small” N , severe bias
As N ! 1, WB (!) ! (!),
^(!) is asymptotically unbiased.
so Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–10
Periodogram Variance
As N
!1
E ^p(!1) , (!1) ^p(!2) , (!2)
2(!1); !1 = !2
= 0;
!1 6= !2
nh
ih
io
(
Inconsistent estimate
Erratic behavior
^φ(ω)
^φ(ω)
asymptotic mean = φ(ω)
ω
+
- 1 st. dev = φ(ω) too
Resolvability properties depend on both bias and variance.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–11
Discrete Fourier Transform (DFT)
Finite DTFT: YN
(!) =
N
X
t=1
y(t)e,i!t
2
,
i
2
Let ! = k and W = e N .
N
Then YN
( 2N k) is the Discrete Fourier Transform (DFT):
Y (k) =
N
X
t=1
y(t)W tk ;
Direct computation of
O N 2 flops
( )
k = 0; : : : ; N , 1
,1 from fy(t)gN :
fY (k)gNk=0
t=1
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–12
Radix–2 Fast Fourier Transform (FFT)
N = 2m
N=2
N
Y (k) =
y(t)W tk +
y(t)W tk
t=1
t=N=2+1
N=2
=
[y(t) + y(t + N=2)W Nk2 ]W tk
t=1
Nk
+1; for even k
with W 2 =
,1; for odd k
Assume:
X
X
X
~ = N=2 and W~ = W 2 = e,i2=N~ .
Let N
For k
4 2p:
= 0; 2; 4; : : : ; N , 2 =
Y (2p) =
For k
N~
X
[y(t) + y(t + N~ )]W~ tp
t=1
= 1; 3; 5; : : : ; N , 1 = 2p + 1:
Y (2p + 1) =
N~
X
t=1
f[y(t) , y(t + N~ )]W tgW~ tp
~ = N=2-point DFT computation.
Each is a N
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–13
FFT Computation Count
Let ck
Then
Thus,
= number of flops for N = 2k point FFT.
k
2
ck = 2 + 2ck,1
k
k
2
) ck = 2
ck = 12 N log2 N
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–14
Zero Padding
Append the given data by zeros prior to computing DFT
(or FFT):
fy(1); : : : ; y(N ); 0; : : : 0g
{z
|
}
N
Goals:
Apply a radix-2 FFT (so N = power of 2)
Finer sampling of ^(!):
2 k N ,1 !
N k=0
2 k N ,1
N k=0
^φ(ω)
continuous curve
sampled, N=8
ω
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L2–15
Improved Periodogram-Based
Methods
Lecture 3
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–1
Blackman-Tukey Method
Basic Idea: Weighted correlogram, with small weight
applied to covariances r k with “large” k .
jj
^( )
^BT (!) =
MX
,1
k=,(M ,1)
fw(k)g =
w(k)^r(k)e,i!k
Lag Window
w(k)
1
-M
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
M
k
Slide L3–2
Blackman-Tukey Method, con't
1
^BT (!) = 2 , ^p( )W (! , )d
Z
W (!) =
=
DTFT
fw(k)g
Spectral Window
^ (!) = “locally” smoothed periodogram
Conclusion: BT
Effect:
Variance decreases substantially
Bias increases slightly
By proper choice of M :
MSE
= var + bias2 ! 0 as N ! 1
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–3
Window Design Considerations
Nonnegativeness:
1
^BT (!) = 2 , ^p( ) W (! , )d
0
Z
| {z }
If W
(!) 0 (, w(k) is a psd sequence)
Then:
^BT (!) 0
(which is desirable)
Time-Bandwidth Product
MX
,1
,1)
Ne = k=,(Mw(0)
w(k)
= equiv time width
1 W (!)d!
2 ,
e =
=
equiv bandwidth
W (0)
Z
Ne e = 1
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–4
Window Design, con't
e = 1=Ne = 0(1=M )
is the BT resolution threshold.
As M increases, bias decreases and variance
increases.
)
Choose M as a tradeoff between variance and
bias.
Once M is given, Ne (and hence e) is essentially
fixed.
)
Choose window shape to compromise between
smearing (main lobe width) and leakage (sidelobe
level).
The energy in the main lobe and in the sidelobes
cannot be reduced simultaneously, once M is given.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–5
Window Examples
Triangular Window, M
0
= 25
−10
dB
−20
−30
−40
−50
−60
−3
−2
−1
0
1
ANGULAR FREQUENCY
Rectangular Window, M
0
2
3
= 25
−10
dB
−20
−30
−40
−50
−60
−3
−2
−1
0
1
ANGULAR FREQUENCY
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
2
3
Slide L3–6
Bartlett Method
Basic Idea:
1
...
2Μ
Μ
^φ1(ω) ^φ2(ω)
...
average
Ν
^φL(ω)
^φB(ω)
Mathematically:
yj (t) = y((j , 1)M + t) t = 1; : : : ; M
= the j th subsequence
4 [N=M ])
(j = 1; : : : ; L =
^j (!) = 1
M
X
M
t=1
yj (t)e,i!t
2
L
1
^B (!) = L
^j (!)
j =1
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–7
Comparison of Bartlett
and Blackman-Tukey Estimates
L
1
^B (!) = L
j =1
X
=
8
>
<
MX
,1
k=,(M ,1)
MX
,1 8
L
<
1 X
>
:
^rj (k)e,i!k
>
;
9
=
^rj (k) e,i!k
k=,(M ,1) L j =1
M ,1
'
^r(k)e,i!k
k=,(M ,1)
:
9
>
=
;
X
Thus:
^B (!) ' ^BT (!)
with a rectangular
lag window wR k
^( )
Since B ! implicitly uses
method has
()
fwR(k)g, the Bartlett
High resolution (little smearing)
Large leakage and relatively large variance
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–8
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
mainlobe-sidelobe tradeoff capability
1
2
N
. . .
subseq subseq
#1
#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 a temporal
window
^ (!) with a
Welch is approximately equal to BT
non-rectangular lag window.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–9
Daniell Method
By a previous result, for N
1,
f^p(!j )g are (nearly) uncorrelated random variables for
N ,1
2
! = j
j
N j =0
Idea: “Local averaging” of (2J + 1) samples in the
frequency domain should reduce the variance by about
J
.
(2 + 1)
k+J
1
^D (!k ) = 2J + 1
^p (!j )
j =k,J
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–10
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 ,
^D (!) ' ^BT (!) with a rectangular spectral
Hence: Let Z
window.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–11
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
^ (!) by a suitably selected
– Local smoothing/averaging of p
spectral window.
^(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 rectangular
– Implemented by local averaging of periodogram values.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L3–12
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 L4–1
Basic Idea of Parametric Spectral Estimation
Observed
Data
Estimate
parameters
in φ(ω,θ)
Assumed
functional
form of φ(ω,θ)
^θ
Estimate
PSD
^
^φ(ω) =φ(ω,θ)
possibly revise assumption on φ(ω)
Rational Spectra
k e,i!k
j
k
j
m
(!) = P
jkjn 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
sufficiently large.
Note, however:
choice of m and n is not simple
some PSDs are not continuous
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–2
AR, MA, and ARMA Models
(!) can be
By Spectral Factorization theorem, a rational factored as
2
B
(
!
)
(!) = A(!) 2
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
white noise
ARMA:
AR:
MA:
B (q)
A(q)
y(t)y (!) =
B (! ) 2 2
A(!)
ltered white noise
A(q)y(t) = B (q)e(t)
A(q)y(t) = e(t)
y(t) = B (q)e(t)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–3
ARMA Covariance Structure
ARMA signal model:
y(t)+
n
X
i=1
aiy(t , i) =
Multiply by y r(k) +
n
X
i=1
m
X
j =0
bj e(t , j );
(b0 = 1)
(t , k) and take E fg to give:
air(k , i) =
m
X
j =0
bj E fe(t , j )y(t , k)g
m
2
= bj hj ,k
j =0
= 0 for k > m
1
B
(
q
)
where H (q ) = A(q ) =
hk q,k ; (h0 = 1)
k=0
X
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–4
AR Signals: Yule-Walker Equations
AR: m
= 0.
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)
2
1
R = 0
2
32
6
6
6
4
7
7
7
5
"
#
6
6
6
4
"
1
2
a1 = 0
..
..
an
0
3
2
3
7
7
7
5
6
6
6
4
7
7
7
5
#
These are the Yule–Walker (YW) Equations.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–5
AR Spectral Estimation: YW Method
Yule-Walker Method:
(k) by r^(k) and solve for f^aig and ^2:
^r(0) ^r(,1) : : : ^r(,. n) 1
^2
^r(1)
r^(0)
^
a1 = 0
.
..
..
..
... ^
r(,1)
^r(n) : : :
r^(0)
^an
0
Replace r
2
6
6
6
4
32
3
2
3
7
7
7
5
7
7
7
5
6
6
6
4
7
7
7
5
6
6
6
4
Then the PSD estimate is
2
^
^(!) = jA^(!)j2
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–6
AR Spectral Estimation: LS Method
Least Squares Method:
e(t) = y(t) +
n
X
i=1
aiy(t , i) = y(t) + 'T (t)
4 y(t) + y^(t)
=
where '(t) = [y (t , 1); : : : ; y (t , n)]T .
Find = [a1 : : : an]T to minimize
f () =
y=6
4
t=n+1
je(t)j2
^ = ,(Y Y ),1(Y y) where
This gives 2
N
X
y(n + 1)
y(n + 2)
..
.
y(N )
3
7
5
2
;Y
6
=4
y(n)
y(n , 1) y(n + 1) y(n) ..
.
y(1)
y(2)
..
.
y(N , 1) y(N , 2) y(N , n)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
3
7
5
Slide L4–7
Levinson–Durbin Algorithm
Fast, order-recursive solution to YW equations
2
6
6
6
4
|
0 ,1 1 0 . . .
..
... ...
n 1
{z
Rn+1
,n
..
,1
0
32
7
7
7
5
}
6
6
6
4
1
n2
0.
=
.
n
0
3
2
3
7
7
7
5
6
6
6
4
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(n4) flops
Levinson–Durbin Algorithm:
Exploits the Toeplitz form of Rn+1 to obtain the solutions
for k
; : : : ; n in O n2 flops!
=1
( )
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–8
Levinson-Durbin Alg, con't
Relevant Properties of R:
Rx = y $ Rx~ = y~, where x~ = [xn : : : x1]T
Nested structure
2
Rn+2 =
4
Rn+1
n+1
r~n
Thus,
Rn+2 4
where
n = n+1 + ~rn n
3
2
5
4
Rn+1
n+1
0
1
n =
0
2
n+1
~rn
r~n
n+1
~rn
0
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
2
3
5
; ~rn =
32
54
6
4
n
3
..
7
5
1
1
n
n = 0
0
n
3
2
5
4
2
Slide L4–9
3
5
Levinson-Durbin Alg, con't
2
Rn+2 4
1
n
n = 0 ;
0
n
3
2
5
4
2
3
2
5
Rn+2 4
0
~n = 0
1
n2
3
2
5
4
n
3
2
n
5
Combining these gives:
82
<
4
+2
:
Rn
Thus,
0
1
n + kn ~n
0
1
3
2
5
4
39
=
5
;
2
=
4
n + knn
2
3
0 2 =
n + knn
5
4
2
+1
0
0
kn = ,n=n2 )
~
n
n+1 = 0 + kn 1n
n2+1 = n2 + knn = n2(1 , jknj2)
"
#
"
#
Computation count:
k flops for the step k
2
) n2 flops
This is O
!k+1
to determine fk2; k gn
k=1
(n2) times faster than the direct solution.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–10
3
5
MA Signals
=0
y(t) = B (q)e(t)
= e(t) + b1e(t , 1) + + bme(t , m)
MA: n
Thus,
r(k) = 0 for jkj > m
and
m
2
2
(!) = jB (!)j =
r(k)e,i!k
k=,m
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–11
MA Spectrum Estimation
Two main ways to Estimate (! ):
1. Estimate
fbkg and 2 and insert them in
(!) = jB (!)j22
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
^ (!) with a rectangular lag window of
2 + 1.
^(!) is not guaranteed to be 0
This is BT
length m
Both methods are special cases of ARMA methods
described below, with AR model order n
.
=0
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–12
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
m
,i!k
e
B
(
!
)
k
k
=
,
m
(!) = 2 A(!) = jA(!)j2
P
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
n
X
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 L4–13
ARMA Spectrum Estimation
Two Methods:
2
B
(
!
)
1. Estimate fai; bj ; 2g in (! ) = 2 A(! ) nonlinear estimation problem; can use an approximate
linear two-stage least squares method
^(!) is guaranteed to be 0
2. Estimate
fai; r(k)g in (!) =
m
,i!k
k=,m k e
P
jA(!)j2
linear estimation problem (the Modified Yule-Walker
method).
^(!) is not guaranteed to be 0
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–14
Two-Stage Least-Squares Method
Assumption: The ARMA model is invertible:
e(t) = BA((qq)) y(t)
= y(t) + 1y(t , 1) + 2y(t , 2) + = AR(1) with jkj ! 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 coefficients
modelling techniques.
fkgKk=1 by using AR
1b) Estimate the noise sequence
^e(t) = y(t) + ^1y(t , 1) + + ^K y(t , K )
and its variance
^2 =
N
1
2
j
^
e
(
t
)
j
N , K t=K +1
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–15
Two-Stage Least-Squares Method, con't
fe(t)g by ^e(t) in the ARMA equation,
A(q)y(t) ' B (q)^e(t)
and obtain estimates of fai; bj g by applying least squares
Step 2: Replace
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)]
= [a1 : : : an b1 : : : bm]T
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–16
Modified Yule-Walker Method
ARMA Covariance Equation:
r(k) +
n
X
i=1
In matrix form for k
2
6
4
r(m)
r(m + 1)
..
.
air(k , i) = 0; k > m
= m + 1; : : : ; m + M
: : : r(m , n + 1)
r(m , n + 2)
..
.
...
r(m + M , 1) : : : r(m , n + M )
Replace
=
3
"
7
5
a1
..
.
an
#
=
2
,6
4
fr(k)g by f^r(k)g and solve for faig.
r(m + 1)
r(m + 2)
3
7
..
5
.
r(m + M )
If M
n, fast Levinson-type algorithms exist for
obtaining ai .
f^ g
If M > n overdetermined
squares solution for ai .
f^ g
Y W system of equations; least
Note: For narrowband ARMA signals, the accuracy of
ai is often better for M > n
f^ g
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L4–17
low
low-medium
medium-high
Method
AR: YW or LS
MA: BT
ARMA: MYW
ARMA: 2-Stage LS
medium-high
medium
low-medium
Accuracy
medium
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Computational
Burden
low
Yes
No
No
Guarantee
^(!) 0 ?
Yes
Slide L4–18
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
Summary of Parametric Methods for Rational Spectra
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 L5–1
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(21)
2
, !1
6
(222)
(223)
6
6
!2
!3 -
!
An “Ideal” line spectrum
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–2
Line Spectral Signal Model
Signal Model: Sinusoidal components of frequencies
!k and powers 2k , superimposed in white noise of
power 2 .
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:
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 fe(t)e(s)g = 2t;s E fe(t)e(s)g = 0
(can be achieved by “slow” sampling)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–3
Covariance Function and PSD
Note that:
E
o
,
i'
i'
e p e j = 1, for p = j
n
o
n
o
n
o
,
i'
,
i'
i'
i'
E e pe j = E e p E e j
2
Z
1
i'
= 2 , e d' = 0, for p 6= j
n
Hence,
o
E xp(t)xj (t , k) = 2p ei!pk p;j
n
r(k) = E fy(t)y(t , k)g
= np=1 2p ei!pk + 2k;0
P
and
(!) = 2
n
X
p=1
2p (! , !p) + 2
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–4
Parameter Estimation
Estimate either:
f!k ; k; 'k gnk=1; 2
f!k ; 2k gnk=1; 2
(Signal Model)
(PSD Model)
Major Estimation Problem: f!
^ kg
Once
f!^kg are determined:
f^2k g can be obtained by a least squares method from
n
r^(k) = 2p ei!^pk + residuals
X
p=1
OR:
Both f^kg and f'^kg can be derived by a least
squares method from
y(t) =
n
X
k=1
with k = k ei'k .
k ei!^k t + residuals
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–5
Nonlinear Least Squares (NLS) Method
min
f! ; ;' g
X |
N
k k k t=1
Let:
y(t) ,
n
X
kei(!k t+'k )
k=1{z
F (!;;')
k = k ei'k
= [1 : : : n]T
Y = [y(1) : : : y(N )]T
ei!1 ei!n
..
..
B =
eiN!1 eiN!n
2
}
2
3
6
4
7
5
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–6
Nonlinear Least Squares (NLS) Method, con't
Then:
F = (Y , B )(Y , B ) = kY , B k2
= [ , (BB),1BY ][BB]
[ , (BB),1BY ]
+Y Y , Y B(BB),1BY
This gives:
^ = (B B ),1B Y !=^!
and
B (B B ),1B Y
!^ = arg max
Y
!
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–7
NLS Properties
Excellent Accuracy:
2
6
!k ) = N 32 (for N 1)
var (^
k
Example: N = 300
2
SNRk = 2
=
= 30 dB
k
q
Then
(^!k ) 10,5.
var
Difficult Implementation:
The NLS cost function F is multimodal; it is difficult to
avoid convergence to local minima.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–8
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=1
B B = N
B Y =
N
X
t=1
y(t)e,i!t = Y (!)
(finite DTFT)
Y B (B B ),1B Y = N1 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 L5–9
Unwindowed Periodogram as an
Approximate NLS Method, con't
Assume:
n>1
Then:
f!^kgnk=1 ' the
locations of the
peaks of p !
^( )
n
largest
provided that
inf j!k , !pj > 2=N
which 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 L5–10
High-Order Yule-Walker Method
Recall:
y(t) = x(t) + e(t) =
n
k ei(!k t+'k) +e(t)
k=1 xk (t)
X
|
{z
}
“Degenerate” ARMA equation for y (t):
(1 , ei!k q,1)xk (t)
= k ei(!kt+'k) , ei!k ei[!k(t,1)+'k] = 0
n
o
Let
L
4 A(q)A(q)
bk q,k =
k=1
A(q) = (1 , ei!1 q,1) (1 , ei!n q,1)
A(q) = arbitrary
B (q) = 1 +
Then B
X
(q)x(t) 0 )
B (q)y(t) = B (q)e(t)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–11
High-Order Yule-Walker Method, con't
Estimation Procedure:
Estimate f^bigLi=1 using an ARMA MYW technique
Roots of B^(q) give f!^kgnk=1, along with L , n
“spurious” roots.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–12
High-Order and Overdetermined YW Equations
ARMA covariance:
r(k) +
L
X
i=1
In matrix form for k
2
6
6
6
4
|
r(L)
r(L + 1)
bir(k , i) = 0; k > L
= L + 1; : : : ; L + M
: : : r(1)
r(L + 1)
: : : r(2) b = , r(L + 2)
..
..
..
r(L + M , 1) : : : r(M )
r(L + M )
4
=
=4 {z
3
2
3
7
7
7
5
6
6
6
4
7
7
7
5
}
|
{z
This is a high-order (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 L5–13
}
High-Order and Overdetermined YW Equations,
con't
(
) = n
Fact:
rank
SVD of
: = U V U = (M n) with U U = In
V = (n L) with V V = In
= (n n), diagonal and nonsingular
Thus,
(U V )b = ,
The Minimum-Norm 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 Minimum-Norm solution b is used.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L5–14
HOYW Equations, Practical Solution
Let
^ = but made from f^r(k)g instead of fr(k)g.
^ ^ ^
^
Let U , , V be defined similarly to U ,
SVD of .
Compute
, V from the
^b = ,V^ ^ ,1U^ ^
^(q) that
Then !k n
k=1 are found from the n zeroes of B
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 L5–15
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 L6–1
The Covariance Matrix Equation
Let:
a(!) = [1 e,i! : : : e,i(m,1)! ]T
A = [a(!1) : : : a(!n)] (m n)
Note:
(A) = n
rank
Define
2
6
4
y~(t) = 6
6
4
where
(for m
y(t)
y(t , 1)
..
y(t , m + 1)
n)
3
7
7
7
5
= Ax~(t) + ~e(t)
x~(t) = [x1(t) : : : xn(t)]T
~e(t) = [e(t) : : : e(t , m + 1)]T
Then
4
R=
E fy~(t)~y(t)g = APA + 2I
with
2
P = E fx~(t)~x(t)g =
6
4
21
0
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
...
0
2n
3
7
5
Slide L6–2
Eigendecomposition of R and Its Properties
R = APA + 2I (m > n)
Let:
1 2 : : : m: eigenvalues of R
fs1; : : : sng: orthonormal eigenvectors associated
with f1; : : : ; ng
fg1; : : : ; gm,ng: orthonormal eigenvectors associated
with fn+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
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
7
5
"
S
G
#
Slide L6–3
Eigendecomposition of R and Its Properties,
con't
(APA) = n:
k > 2 k = 1; : : : ; n
k = 2 k = n + 1; : : : ; m
2
,
0
1
=
...
= nonsingular
0
n , 2
As rank
2
3
6
4
7
5
Note:
1
6
2
RS = APA S + S = S 4
2
0
...
0
n
3
7
5
4
,
1
S = A(PA S ) = AC
j j 6= 0
( ) = rank(A) = n).
=0
AG = 0
with C
(since rank S
Therefore, since S G
,
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–4
MUSIC Method
2
AG =
6
4
3
a (!1)
..
7
5
G=0
a(!n)
) fa(!k )gnk=1 ? R(G)
Thus,
f!kgnk=1 are the unique solutions of
a(!)GGa(!) = 0.
Let:
N
1
R^ = N
y~(t)~y(t)
t=m
S^; G^ = S; G made from the
^
eigenvectors of R
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–5
Spectral and Root MUSIC Methods
Spectral MUSIC Method:
f!^kgnk=1 = the locations of the n highest peaks of the
“pseudo-spectrum” function:
1
;
!
2
[
,
;
]
^
^
a (!)GG a(!)
Root MUSIC Method:
f!^kgnk=1 = the angular positions of the n roots of:
aT (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 accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–6
Pisarenko Method
Pisarenko is a special case of MUSIC with m
(the minimum possible value).
If:
=n+1
m=n+1
G^ = ^g1,
) f!^kgnk=1 can be found from the roots of
aT (z,1)^g1 = 0
Then:
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
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–7
Min-Norm 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 first 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 L6–8
Min-Norm Method, con't
Spectral Min-Norm
f!^gnk=1 =
the locations of the n highest peaks in the
“pseudo-spectrum”
2
1 = a(!) 1^g
"
#
Root Min-Norm
f!^gnk=1 =
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 L6–9
Min-Norm Method: Determining g
^
^ = S gg 1m , 1
Let S
Then:
"
"
#
1 2 R(G^) ) S^ 1 = 0
^g
^g
) S^g = ,
#
"
#
^ = ,S(SS),1
Min-Norm solution: g
I = S^S^ = + SS, (SS),1 exists iff
= kk2 6= 1
(This holds, at least, for N 1.)
As:
Multiplying the above equation by gives:
(1 , kk2) = (SS)
) (SS),1 = =(1 , kk2)
) ^g = ,S=(1 , kk2)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–10
ESPRIT Method
Let
Then A2
Also, let
A1 = [Im,1 0]A
A2 = [0 Im,1]A
= A1D, where
e,i!1
0
...
D=
0
e,i!n
S1 = [Im,1 0]S
S2 = [0 Im,1]S
2
3
6
4
7
5
= AC with jC j 6= 0. Then
S2 = A2C = A1DC = S1 C ,1DC
So has the same eigenvalues as D . is uniquely
Recall S
|
{z
}
determined as
= (S1 S1),1S1 S2
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–11
ESPRIT Implementation
^
^
^
From the eigendecomposition of R, find S , then S1 and
S2.
^
The frequency estimates are found by:
f!^kgnk=1 = , arg(^k )
where
f^kgnk=1 are the eigenvalues of
^ = (S^1 S^1),1S^1 S^2
ESPRIT Advantages:
computationally simple
no extraneous frequency estimates (unlike in MUSIC
or Min–Norm)
accurate frequency estimates
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L6–12
Use ESPRIT for high-resolution applications
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Use Periodogram for medium-resolution applications
Recommendation:
Method
Periodogram
Nonlinear LS
Yule-Walker
Pisarenko
MUSIC
Min-Norm
ESPRIT
Slide L6–13
Computational Accuracy /
Risk for False
Burden
Resolution Freq Estimates
small
medium-high
medium
very high
very high
very high
medium
high
medium
small
low
none
high
high
medium
medium
high
small
medium
very high
none
Summary of Frequency Estimation Methods
Filter Bank Methods
Lecture 7
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–1
Basic Ideas
Two main PSD estimation approaches:
(!) by a
1. Parametric Approach: Parameterize finite-dimensional model.
2. Nonparametric Approach: Implicitly smooth
! !=, by assuming that ! is nearly
constant over the bands
f ( )g
( )
[! , ; ! + ]; 1
2 is more general than 1, but 2 requires
N > 1
to ensure that the number of estimated values
(
= = ) is < N .
=2 2 =1
N > 1 leads to the variability / resolution compromise
associated with all nonparametric methods.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–2
(a)
^FB (!) ' 21
!c !c y
-
6 ~(!)
@
Filtered
Signal
-!
!+
!,
Z
Division by
- lter
bandwidth
^(!c ) -
( )d= ('c) (!)
Power in
the band
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–3
Note that assumptions (a) and (b), as well as (b) and (c), are conflicting.
( ) constant on 2 [! , 2; ! + 2 ]
(b) Ideal passband filter with bandwidth (c)
(b)
Power
Calculation
jH ( )j2( )d= ' 21
,
Z
,
(a) consistent power calculation
,
y
-!
Bandpass Filter
c and
xed bandwidth
!c
- with varying !
6 (!)
@Q
Q -!
y (t)
1
6H (!)
Filter Bank Approach to Spectral Estimation
Filter Bank Interpretation of the Periodogram
2
1
4
,i!~t
^p(~!) =
y
(
t
)
e
N t=1
2
N
y(t)ei!~(N ,t)
= N1
t=1
X
N
X
= N
1
X
k=0
hk y(N , k)
2
where
1 ei!~k ; k = 0; : : : ; N , 1
hk = N
otherwise
0;
1
iN (~!,!) , 1
1
e
,
i!k
H (!) = hk e
= N ei(~!,!) , 1
k=0
(
X
center frequency of H (!) = !~
3dB bandwidth of H (!) ' 1=N
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–4
Filter Bank Interpretation of the Periodogram,
con't
jH (!)j as a function of (~! , !), for N = 50.
0
−5
−10
dB
−15
−20
−25
−30
−35
−40
−3
−2
−1
0
1
ANGULAR FREQUENCY
2
3
^( )
Conclusion: The periodogram p ! is a filter bank PSD
estimator 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
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–5
Possible Improvements to the Filter Bank
Approach
1. Split the available sample, and bandpass filter each
subsample.
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 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
filtering.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–6
Capon Method
Idea: Data-dependent bandpass filter design.
m
yF (t) =
X
k=0
hk y(t , k)
2
y(t)
3
{z
}
..
= [h0 h1 : : : hm]
y(t , m)
h
y~(t)
E jyF (t)j2 = hRh; R = E fy~(t)~y(t)g
{z
|
}
6
4
|
n
7
5
o
H (! ) =
m
X
hk e,i!k = ha(!)
k=0
where a(! ) = [1; e,i! : : : e,im! ]T
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–7
Capon Method, con't
Capon Filter Design Problem:
Rh)
min
(
h
h
Solution:
subject to ha
(!) = 1
h0 = R,1a=aR,1a
The power at the filter 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
o
The Bandwidth
1 =
1
' m+1
(filter length)
Conclusion Estimate PSD as:
^(!) = m^+,11
a (!)R a(!)
with
N
1
R^ = N , m
y~(t)~y(t)
t=m+1
X
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–8
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 bandpass filter 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 L7–9
Relation between Capon and Blackman-Tukey
Methods
^ (!) with Bartlett window:
m m + 1 , jkj
,i!k
^BT (!) =
r
^
(
k
)
e
k=,m m + 1
m m
1
= m+1
^r(t , s)e,i!(t,s)
t=0 s=0
(!)Ra
^ (!) ^
a
= m + 1 ; R = [^r(i , j )]
Consider BT
X
X
X
Then we have
(!)Ra
^
a
(!)
^BT (!) =
m+1
^C (!) = m^+,11
a (!)R a(!)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L7–10
Relation between Capon and AR Methods
Let
2k
^
^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 k
X
Consequences:
Due to the average over k, ^C (!) generally has less
statistical variability than the AR PSD estimator.
Due to the low-order 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 L7–11
Spatial Methods — Part 1
Lecture 8
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–1
The Spatial Spectral Estimation Problem
Source 1
v
@ ,,,
@
,,,
,,
,@
R
,,@
,
vSource 2
B
B
B
B
BN
@@,,
Sensor 1
v Source n
ccc
cc
c
c
cc
cc
c
@@,,
@@,,
Sensor
m
Sensor 2
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 L8–2
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 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 L8–3
Array Model — Single Emitter Case
x(t) =
k =
hk (t) =
ek(t) =
the signal waveform as measured at a reference
point (e.g., at the “first” 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.)
t 2 R (continuous-time 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 fk g with hk (t)
known but x(t) unknown.
Note:
This is a time-delay 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 L8–4
Narrowband Assumption
Assume: The emitted signals are narrowband with known
carrier frequency !c .
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 !ck :
x(t , k ) ' (t) cos[!ct + '(t) , !ck ]
hk (t) x(t , k )
' jHk (!c)j(t)cos[!ct + '(t) , !ck + argfHk (!c)g]
Then:
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) , !ck + arg Hk (!c)] + ek(t)
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–5
Complex Signal Representation
The noise-free output has the form:
z(t) = (t) cos [!ct + (t)] =
= (2t) ei[!ct+ (t)] + e,i[!ct+ (t)]
Demodulate z (t) (translate to baseband):
2z(t)e,!ct = (t)f ei (t) + e,i[2!ct+ (t)] g
n
o
| {z }
|
{z
lowpass
Lowpass filter
}
highpass
2z(t)e,i!ct to obtain (t)ei (t)
Hence, by low-pass filtering and sampling the signal
y~k (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)
|
{z
s(t)
} |
{z
k
Hk (!c)
c
}
c k
or
yk (t) = s(t)Hk (!c) e,i!ck + 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 L8–6
Vector Representation for a Narrowband Source
Let
=
m =
a() =
the emitter DOA
the number of sensors
2
6
4
H1(!c) e,i!c1
..
3
7
5
Hm(!c) e,i!cm
y1(t)
e1(t)
..
..
y(t) =
e(t) =
ym(t)
em(t)
2
3
2
3
6
4
7
5
6
4
7
5
Then
y(t) = a()s(t) + e(t)
()
fkg and fHk(!c)g.
fHk(!c)g do not depend
NOTE: enters a via both
For omnidirectional sensors the
on .
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–7
DEMODULATION
Slide L8–8
-
-
A/D
CONVERSION
- 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
Internal
Noise
External
XXNoise
?
XXXXX
z ll
l - Cabling
and
,
Filtering
,
:
,
Incoming Transducer
Signal
x(t , k )
SENSOR
Analog processing for each receiving array element
Analog Processing Block Diagram
Multiple Emitter Case
Given n emitters with
received signals: fsk(t)gnk=1
DOAs: k
)
y(t) = a(1)s1(t) + + a(n)sn(t) + e(t)
Linear sensors
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 find the dependence of
k on .
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–9
Uniform Linear Arrays
hSource
JJ
JJ
JJ
JJJ^J
JJ
JJ
-
J+J
J] d sin J^
J JJ
JJ
v ]? v J v
v
J
4
1 2 JJJ 3
d -
ppp
vSensor
m
ULA Geometry
Sensor #1 = time delay reference
Time Delay for sensor k:
d
sin
k = (k , 1) c
where c = wave propagation speed
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–10
Spatial Frequency
Let:
d sin = 2 d sin 4 ! d sin !s =
=
2
c c
c=f
c
= c=fc = 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 a sinusoidal time series,
!s = spatial frequency
The function !s 7! a( ) is one-to-one for
sin j 1 d =2
j!sj $ dj=
2
As
d = spatial sampling period
d =2 is a spatial Shannon sampling theorem.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L8–11
Spatial Methods — Part 2
Lecture 9
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–1
Spatial Filtering
Spatial filtering useful for
DOA discrimination (similar to frequency
discrimination of time-series filtering)
Nonparametric DOA estimation
There is a strong analogy between temporal filtering and
spatial filtering.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–2
Analogy between Temporal and Spatial Filtering
Temporal FIR Filter:
mX
,1
yF (t) =
hk u(t , k) = hy(t)
k=0
h = [ho : : : hm,1]
y(t) = [u(t) : : : u(t , m + 1)]T
(t) = ei!t then
yF (t) = [ha(!)] u(t)
If u
|
{z
}
filter 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 L9–3
Analogy between Temporal and Spatial Filtering
Spatial Filter:
fyk(t)gmk=1 =
the “spatial samples” obtained with a
sensor array.
Spatial FIR Filter output:
yF (t) =
m
X
k=1
hk yk (t) = hy(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!c2 : : : e,i!cm ]T
The corresponding filter output is
yF (t) = [ha()] s(t)
|
{z
}
filter 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 L9–4
Analogy between Temporal and Spatial Filtering
h
u(t) = ei!t
?
-
z
1
0
@
1
@
?
C
B
h
,
@
C
B
1 q
C
B
?
@
,
i!
[h a(! )]u(t)
R
@
C
e
- B
-
-P C
B
C
B
?
C
B
C
B
u(t)
..
C
B
C
B
.
C
B
C
B
C
B
C
B
?
A
@
,
m,1 q
hm, e,i m{z, ! }
|
?
-
a(!)
s
s
qq
qq
0
1
1
1
(
1)
1
(Temporal sampling)
(a) Temporal filter
narrowband source with DOA=
z
Z Z
ZZ
~
aa 1reference point
!!
0 1 1
B
C
C
aa 2- B
B
C
,
i!
2
!
B
C
e
!
BB
C
C
BB
Cs(t)
.
BB .. C
C
C
B@
C
A
,
i!
m
| e {z }
a()
aa m( )
qq
!!
(Spatial sampling)
( )
h
h
P ha
qq
0
-
?
@
@
1
-
?
@
@
R
@
-
[ - ( )]s(t)
hm, 1
?
(b) Spatial filter
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–5
Spatial Filtering, con't
Example: The response magnitude h a of a spatial
filter (or beamformer) for a 10-element ULA. Here,
h a 0 , where 0
j ( )j
= ( )
= 25
0
−30
a
et
Th
30
)
eg
(d
60
−60
90
−90
0
2
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
4
6
Magnitude
8
10
Slide L9–6
Spatial Filtering Uses
Spatial Filters can be used
To pass the signal of interest only, hence filtering out
interferences located outside the filter's beam (but
possibly having the same temporal characteristics as
the signal).
To locate an emitter in the field of view, by sweeping
the filter 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 L9–7
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 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()Rh() give the
DOAs of the sources.
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–8
Beamforming Method
Assume the array output is spatially white:
R = E fy(t)y(t)g = I
E jyF (t)j2 = hh
n
Then:
o
Hence: In direct analogy with the temporally white
assumption for filter bank methods, y t can be
considered as impinging on the array from all DOAs.
()
Filter Design:
h)
min
(
h
h
subject to
ha() = 1
Solution:
h = a()=a()a() = a()=m
E jyF (t)j2 = a()Ra()=m2
n
o
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–9
Implementation of Beamforming
N
1
R^ = N
y(t)y(t)
t=1
X
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 Blackman-Tukey
periodogram.
Resolution Threshold:
inf jk , pj >
=
wavelength
array length
array beamwidth
Inconsistency problem:
Beamforming DOA estimates are consistent if n
inconsistent if n > .
1
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
= 1, but
Slide L9–10
Capon Method
Filter design:
Rh) subject to ha() = 1
min
(
h
h
Solution:
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 L9–11
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 = 2I
The signal covariance matrix
P = E fs(t)s(t)g
is nonsingular.
Then:
R = E fy(t)y(t)g = APA + 2I
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 accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–12
Nonlinear Least Squares Method
N
1
2
k
y
(
t
)
,
As
(
t
)
k
min
fk g; fs(t)g N t=1
X
|
Minimizing f over s gives
{z
f (;s)
}
^s(t) = (AA),1Ay(t); t = 1; : : : ; N
Then
N
1
f (; ^s) = N k [I , A(AA),1A]y(t)k2
t=1
N
1
= N y(t)[I , A(AA),1A]y(t)
t=1
= trf[I , A(AA),1A]R^g
^g
Thus, f^k g = arg max trf[A(AA),1A]R
X
X
fk g
=1
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 L9–13
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 L9–14
Yule-Walker Method
y
(
t
)
A
y(t) = y~(t) = A~ s(t) + ~ee((tt))
"
Assume:
Then:
#
"
#
"
#
E fe(t)~e(t)g = 0
4
A~ (M L)
, = E fy(t)~y(t)g = AP
Also assume:
M > n; L > n ( ) m = M + L > 2n)
rank(A) = rank(A~) = n
(,) = n, and the SVD of , is
1 g n
0
V
n
n
, = [ U1 U2 ] 0
g L,n
0
V
2
n M ,n
~V2 = 0
Properties: A
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 L9–15
YW-MUSIC DOA Estimator
f^kg =
the n largest peaks of
1=~a()V^2V^2~a()
where
~a(), (L 1), is the “array transfer vector” for y~(t)
at DOA V^2 is defined similarly to V2, using
N
1
,^ =
y(t)~y(t)
X
N t=1
Properties:
Computational complexity: medium
Performance: satisfactory if m 2n
Main advantages:
– weak assumption on fe(t)g
need not be calibrated
– the subarray A
Lecture notes to accompany Introduction to Spectral Analysis
by P. Stoica and R. Moses, Prentice Hall, 1997
Slide L9–16
MUSIC and Min-Norm Methods
Both MUSIC and Min-Norm methods for frequency
estimation apply with only minor modifications to the
DOA estimation problem.
Spectral forms of MUSIC and Min-Norm can be used
for arbitrary arrays
Root forms can be used only with ULAs
MUSIC and Min-Norm break down if the source
signals are coherent; that is, if
(P ) = rank(E fs(t)s(t)g) < n
rank
Modifications 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 L9–17
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)
A2 = A1D, where
e,i!c (1)
0
...
D=
0
e,i!c (n)
() = the time delay from subarray 1 to
Then
2
3
6
4
7
5
subarray 2 for a signal with DOA = :
() = d sin()=c
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 L9–18
ESPRIT Method, con't
ESPRIT Scenario
source
θ
known
subarray 1
displac
ement
v
ector
subarray 2
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 first
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 L9–19