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### Transcript of Spectral Estimation 2

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:

discrete-time 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 Discrete-Time 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 mainlobe-sidelobe 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 non-rectangular 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