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Home , Autocorrelation, Estimation theory

Power Sp ectrum Estimation from Noisy

Auto correlations Values

L Reb ollo Neira

A G Constantinides

Department of Electrical and Electronic Engineering Imp erial College

Exhibition Road London SW BT England

Abstract

The problem of estimating the p ower sp ectrum from noisy auto correlation values is

considered in this pap er and it is prop osed that in order to reduce errors

of the available should b e employed The oversampling problem is dis

cussed from the Frame Theory p oint of view and it is shown that the frame reconstruction

represents an improvement up on the standard windowing and autoregressive

mo delling approaches

Intro duction

Estimation of the p ower sp ectrum is a fundamental problem in pro cessing and there are

many techniques available for its solution A summary of these with many references can b e

found in There are also b o oks completely dedicated to the sub ject

In this pap er we wish to address the p ower sp ectrum estimation problem with the additional

complication that the auto correlation values are themselves noisyWe maintain that to esti

mate the p ower sp ectrum from a single realization of each measurement a price to b e paid is

oversampling the data at a density larger than the Nyquist one

The oversampling problem is discussed from the Frame Theory p oint of view Within

this framework p ower sp ectrum reconstruction app ears as a tight frame sup erp osition that

lo oks very much like the classical correlogram estimation from oversampling data In fact Fast

CICPBACONICET

Fourier Transform can also b e used to obtain this reconstruction but we wish to stress the

considerable theoretical dierence b etween these two cases the oversampling problem involves

a set of functions which constitute a frame but not a basis Indeed this implicit redundancy

of the frame app ears to b e the main reason for the reduction but is also the reason that

the representation is not unique

The pap er is organized as follows the problem to b e addressed is intro duced in section and

some comments ab out classical p ower sp ectrum estimation are given in section In section

theoversampling problem is discussed as a particular case of the general Frame Theory

Simulations are included in section where the frame reconstruction is compared with the

correlogram and windowing approaches and also with autoregressive AR mo delling

Finally conclusions are drawn in section

The problem to b e addressed

For a zero widesense stationary random pro cess xt the auto R

is dened in terms of ensemble averages as

R E xtxt

where

E xtxt

The normalized p ower sp ectral density function P is the of the auto

correlation function ie

Z



P R expi d



The normalized p ower sp ectral density henceforth referred to as p ower sp ectrum is a p ositive

function which is normalized to unity ie

P

Z



P d



If the pro cess is ergo dic the auto correlation function which is dened as in can b e estimated

as the time average

T

R lim R

T 

where

Z

T 

T

R xtxt dt

T

T

is an unbiased of R as

T

R E R

T

However for T xed the approximation R holds not for every but only for T Al

T T

though lim R R the convergence is not uniform in For near T R

T 

T

do es not tend to P is not a reliable estimator of R and the Fourier Transform of R

T

as a consequence of the large of R

The auto correlation function often in practice can b e directly measured through some exp er

iment for example as in a Michelson interferometer and therefore the measurements are

inuenced by noise or imprecisions The problem we address then maybeformulated as fol

o

lows Given a nite set of observed values of the function R i N

i

o o

whereeach value R is known to within an error our purpose is to estimate robustly

i i

the power spectrum P We deal only with bandlimited auto correlation functions for which

P for j j The mo del for the data at lag will b e taken to b e

C i

Z

C

R P expi d

i i



C

Some comments on classical Power Sp ectrum Esti

mation

A bandlimited auto correlation function of bandwidth can b e uniquely determined in terms

C

n

as of its sample values R

C

n



X sin

C

n

C

R R

n

C C

n

C

C

is the Nyquist Equation enables us to calculate exactly the p ower where

sp ectrum from the Fourier Transform of the discrete data Indeed replacing R byits

equivalent expression in the p ower sp ectrum of wehave

n

Z





sin X

C n

C

P R expi d

n



C C

n

C

n

sin 

C

C

which is equal to The integral in is the Fourier Transform of the function

n



C

C

U

n

C

expi where U is a function that takes the value one if j j or zero

C

C

C C

otherwise Thus the p ower sp ectrum can b e recast as



X

n n U

C

R expi P

C C C

n

It is clear that byknowingtheexactvalues of the auto correlation function sampled at the

Nyquist frequency for n the p ower sp ectrum can b e determined exactlyHow

ever in practice wefacetwo basic limitations First we can only know and pro cess a nite

numb er of data and second these data are noisy

Let us consider rst the nite data problem assuming that wehave at our disp osal the ex

T

n n

C

forj j T or jnj N b c the notation b c is chosen to denote act values of R

C C

the largest integer not exceeding In these circumstances the equation b elow provides the

N

approximate correlogram representation P of P ie

N

X

U n n

C

N

P expi R

C C C

nN

N

In the limit N P P andobviously this would also b e true in the very sp ecial

n

case for which R were zero for jnj N This exceptional case can never b e a consequence

C

of a situation suchas R for j j T since the auto correlation function is bandlimited

and is therefore an analytical function in the entire complex plane Such functions are either

identically zero or they have isolated zeros but they cannot vanish in a continuous interval

N

Thus in general for nite data the quantity P can only b e an approximation of P In

this pap er we restrict our consideration to those cases for which the auto correlation function

has fast decay and therefore the truncation error committed by assuming that R for

T can b e disregarded However the purp ose of our contribution is to deal with noisy

N

data for which P above is not a go o d estimator of P In the early work in this area

to reduce the variance of the estimate it has b een prop osed to deemphasize the contribution of

R for near T through the use of a window function W This prop osal accepts a

P calculated as smo othed version P of

W

N

X

U n n n

C

P R W expi

W

C C C C

nN

where W decreases gradually to zero as j jT The window can b e selected from several

functions dened for such a purp ose However within the windowing approach reso

lution is inevitably sacriced In order to reduce the variance of the estimate with signicantly

improved resolution we prop ose to oversample the auto correlation values at a density larger

than the Nyquist In the next section we discuss the oversampling problem within the context

of Frame Theory

The oversampling problem Power Sp ectrum Esti

mation as a tight frame sup erp osition

Frames were intro duced by Dun and Shaer and are also reviewed in Young

More recent results on Frame Theory are given in Here we only give the frame

denition and some of the prop erties related with our problem

Denition A family of functions j in a Hilbert space H is cal ledaframe for

j

H if there exist A B so that for al l f H



X

jhf ij B kf k Akf k

j

j 

where the angle brackets denote inner pro duct and kf k the square norm of f The numb ers

A and B are called frame b ounds

From the denition it is clear that a frame is a complete set of functions since the relations

hf i j imply that f The removal of an element from a frame leaves

j

either a frame or an incomplete set A frame that ceases to b e complete if an arbitrary

function is removed is called exact note that the last prop erty implies that only exact

j

frames are bases in the general case the functions are typically not linearly indep endent

j

If the two frame b ounds are equal A B then the frame is called a tight frame In a tight

frame for all f H



X

jhf ij Akf k

j

j 

which implies that knowing the inner pro ducts hf i f can easily b e reconstructed as

j



X

f hf i

j j

A

j 

Formula is reminiscent of the expansion of f into an orthogonal basis but it is imp ortant

to realize that tight frames are not necessarily orthogonal bases Only in the cases for which

the frame b ound A and k k j w il l theset constitute an orthonormal

j j

basis In most other cases the may not b e linearly indep endent Wenowshow that the

j

oversampling problem involves a set of functions whichformatight frame but not a basis In

order to simplify notation we denote

n

expia

a

C

p

U

n

C

C

a

where a Note that k k

n

Proposition For any function P belonging to a Hilbert spaceof band limited functions

C

a

ie P for j j the set n constitutes a tight frame with frame

C

n



bound A a

This follows from

n

Z

 



X X

expia

a

C

p

jhP ij j P U d j

n

C



C

n n

n

expia

C C C

q

g is an othonormal basis in we can express P U In fact since fU

C

C

a a

a

C

a

as

n



X

expia

C

q

P U U g

C

n

C

a

C

n

a

with

n n

Z Z

C



expia expia

a

C C

q q

P U d P U d g

n

C C

C

C C

 

a

a a

Hence from and the orthonormality condition wehave

Z Z

C



X

C

a

jg j jP j d jP U j d

n

C

C

 

C

n

a

n

Z





X

expia

C

q

j P U d j

C

C



n

a

and replacing in we obtain

Z



X

C

 a

ij d jP j d a kP k jhP

n

a



C

n

Using equation and the assumed mo del we are in a p osition to reconstruct the p ower

sp ectrum as

 

X X

a n

a a a

p

i hP P a Ra

n n n

C

C

n n



If a were to increase to a ka where k is an integer numb er then the resulting situation

a

may b e considered as equivalent to removing some functions from the original set Equation

n

 a

therefore still holds with a new frame b ound a This that the set a no

n

longer constitutes an exact frame and hence the functions are linearly dep endentandthus they

do not form a basis As a consequence equation gives only one way of reconstructing the

power sp ectrum In other words there may exist other sets of co ecients c n

n

for which



X

a

P c

n

n

n

However it is a prop erty of frames that the sup erp osition formula derived from is

a

the most economical in the sense that for any other co ecients for which c ahP i we

n

n

have

 

X X

a

jhP ij jc j a

n

n

n n

Thus the assumed mo del gives the set of co ecients with minimum norm

Wehave seen that anypower sp ectrum with P forj j can b e exactly re

C

constructed from an innite but discrete set of samples of the auto correlation function ei

ther as a sup erp osition of orthonormal functions a or as a sup erp osition in a tight

frame a In the real situation in which only samples for Tare known wehave

T

N

C

M b c b c data and the approximate representation of the p ower

a a

sp ectrum in terms of M frame elementsis

M

X

n a

M a

p

Ra P

n

C C

nM

Since in this pap er we fo cus our attention on the cases for which the truncation error can b e

disregarded in comparison to the error due to the noise of the data the frame decomp osition

b ecomes much more appropriate

The frame prop erty of reducing errors is p ointed out in and it is extensively studied

in for tightWeylHeisenb er frames where it is showed that the noise contribution to the

reconstruction of a function decreases with the oversampling In the next section we

exemplify this fact bynumerical simulations Moreover we indicate the gain in resolution which

a tight frame reconstruction represents in relation to the windowing approach The results are

also compared to those obtained from AR mo dels which repro duce the exact p ower sp ectrum

only for noiseless auto correlation values

Simulations

The particular error mo del to b e chosen is the standard variance of the estimator Sucha

variance is exactly derivable for a normal pro cess but it is known to b e a go o d approximation

for other pro cesses as well and the form is given by

Z

T 

jj

R R R d

T

R

T T

T

T

The estimator R isunbiased and hence the errors are assumed to b e zero mean and will

then b e simulated as

o o

R R n M

n n n

where is a generated by a Gaussian distribution whose variance

n n

n

n isgiven by With T the T interval is discretized as a

T

n M n

R

C

T

C

M b cFor all the examples we shall set and a in the oversampling

C

a

case or a for the Nyquist density

The continuous curve in Fig a b c and d is the exact p ower sp ectrum we wish to obtain

and that can b e estimated with high accuracy from the noiseless data by all the approaches

that we compare for the case of noisy data

Since in the simulations given here the auto correlation values are real the p ower sp ectrum is

an even function and thus wehave plotted it in the domain only The squares triangles

C

and diamonds of Fig a represent the classic correlogram estimations for three dierent sets of

noisy data sampled at the Nyquist density in the time domain T For the sake of clarity

in Fig a b c d and e the corresp onding results are given for only three set of data The

squares triangles and diamonds of Fig b represent the estimations that are obtained from the

same data as in Fig a but multipling them bytheTukeyHanning window The results

of Fig d are obtained by using Pap oulis window The dotlines of Fig d show the frame

reconstructions for six dierentsetsofoversampled data By comparing Fig d with a b

and c it can b e seen clearly that oversampling leads to a signicant reduction of noise eect

in comparison not only with the correlogram estimation but also with windowing approaches

The squares triangles and diamonds of Fig e represent the estimations obtained from the

same data as in Fig a b and c but from an AR mo del of order As it can b e seen in Fig

e the intro duction of noise renders this mo del as exp ected totally inadequate

The gures a b c d and e are as ab ove but for T

Conclusions

The problem of estimation of the p ower sp ectrum from uncertain auto correlation values has

b een addressed The cases that have b een considered are those in which due to the unrelia

bility of the data the classic correlogram cannot b e applied over a single set of data In those

cases oversampling the time domain of the data has b een prop osed as a way of reducing noise

eects The price to b e paid in order to b e able to determine the p ower sp ectrum using a single

realization of each measurement is to pro cess more samples than the minimum that are needed

if the data are noiseless

The oversampling problem has b een discussed from the frame Theory p oint of view Recon

struction of the p ower sp ectrum as a sup erp osition in a tight frame is in fact a trivial case

within the general Frame TheoryItisvery easy and fast to compute through the use of the

Fast Fourier Transform However from Frame Theory it is p ossible to obtain a deep under

standing of the concepts involved when is applied to estimate the p ower

sp ectrum from oversampling of data

Simulations supp ort the view that oversampling of the data can pro duce imp ortant improve

ment in the sp ectral estimation over other standard approaches

Aknowledgments

L Reb ollo Neira wishes to express her gratitude to Dr T J Seller for his help and assistance

during her stay in England and also to Ing H Jelveh for corrections of the rst draft

L Reb ollo Neira is a memb er of the Research Career of CICPBA Comision de Investigaciones

Cientcas de la Provincia de Buenos Aires Argentina and acknowledges supp ort from the

National Research Council of Argentina CONICET

Figure Captions

Figure a Power sp ectrum vs angular frequency The continuous curve represents the

true p ower sp ectrum P The squares triangles and diamonds are correlogram estimations

for three dierent set of data the T interval at the Nyquist density

Figure b Power sp ectrum vs angular frequency The continuous curve represents the

true p ower sp ectrum P The squares triangles and diamonds are the estimations obtained

from the same data as in Fig a but using TukeyHanning window

Figure c Same description as in Fig b but using Pap oulis window

Figure d Power sp ectrum vs angular frequency The continuous curve represents

the true p ower sp ectrum P The dotslines the frame reconstructions for six dierent set of

data oversampling the T interval

Figure e Power sp ectrum vs angular frequency The continuous curve represents

the true p ower sp ectrum P The squares triangles and diamonds are the the estimations

obtained from an AR mo del of order

Figure a Same details as in Figa but for a T data domain

Figure b Same details as in Figb but for a T data domain

Figure c Same details as in Figc but for a T data domain

Figure d Same details as in Figd but for a T data domain

Figure e Same details as in Fige but for a T data domain and with an AR

mo del of order

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