spectral estimation notes

8/10/2019 Spectral Estimation Notes

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Spectral Estimation Notes ECE 246/446 Fall 2014

Deterministic Power Spectra and Autocorrelations

If [ ]x n is a signal of finite length, or even is a signal of infinite length, that is absolutely

summable then the Fourier transform ( )X of the signal can be computed in the

ordinary way and, in that case, (by Parseval's formula) the squared magnitude ( ) 2

X of

the transform may be interpreted as the energy distribution of the signal with respect to

frequency !his function is called theDeterministic Power Spectrumof the signal and isdenoted

( ) ( ) 2d

xx X

!he xx subscript indicates that the power spectrum is of the signal [ ]x n , and also that

the spectral values are quantities that are second order in the sense that they are derivedfrom products of pairs of [ ]x n values !he d superscript implies that the powerspectrum is for an isolated signal rather than an estimate of the power spectrum of a

random process (see below)

"otice that ( ) ( ) ( )dxx X X = is the discrete#time Fourier transform of

[ ] [ ]x n x n which is called theDeterministic Autocorrelation of the signal and isdenoted

[ ] [ ] [ ]d

xx n x n x n =

!hus, the deterministic power spectrum and the deterministic autocorrelation are a

discrete#time Fourier transform pair (ie [ ] ( )d dxx xxn )

$hen the original signal is real the deterministic autocorrelation function is

[ ] [ ] [ ] [ ] [ ]dxxm

n x n x n x m x m n

=

= = +

%&pressed in words, [ ]d

xx n is the sum of all products of pairs of signal values that are n#steps apart

A Heuristic Approac to Estimatin! Power Spectra

"ow suppose we want to form an estimate of the power spectrum (ie of the way that

signal energy is distributed with respect to frequency) in a very long signal that

essentially goes on indefinitely !aing the Fourier transform of the entire signal, or of a


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very long section of the signal, is not a good alternative because this transform will be

complicated by variations that represent every detail of the original signal better

approach is to*

+ivide a long section of the signal [ ]x n into blocs of length N *

[ ] [ ] [ ] [ ]{ }

[ ] [ ] [ ] [ ]{ }

[ ] ( ) ( ) [ ]{ }

,

(

(

, , ,

, , , 2

, , , K

x n x x x N

x n x N x N x N

x n x K N x K N x KN

=

= +

= +

K

K

M

K

2 -ompute normali.ed deterministic power spectra*

( ) ( ), , ( (

, ,K K

d d

x x x xN N

K

of the blocs [ ] [ ], (, , Kx n x nK

/ verage the normali.ed power spectra

( )(

,

k k

Kd

x x

k

NK

=

0 1se a large value of Kso that the average closely appro&imates

( ) ( )(

,

lim

k k

Kd

xx K x x

k

NK

=

%

which is the mathematical e&pression for the spectral estimate

veraging together the deterministic power spectra of smaller sections eliminates spectralcharacteristics that vary from section to section, while enhancing characteristics that are

the same in each section

!he normali.ation in step 2 is introduced to scale the total energy to be independent ofN

"andom Si!nals

deeper meaning can be attached to the power spectrum estimate developed above byrelating the estimate to the statistics of a random process !he concept of a random

process is an e&tension of the concept of a random variable which is briefly recalled

2


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random #aria$lex is a variable that assumes specific numeric values depending on

the outcome of a random e&periment For e&ample, rolling a die is an e&periment with

possible outcomes, and a random variable for this e&periment is usually assigned values,2,/,0,3, 4et x denote this random variable ll probabilities and statistics associated

with this random variable are determined by a probability density function ( )xp that, in

this case, specifies that the probability of each assigned value is

For e&ample, the mean x of x is

( )

2 / 0 3 3 /3

x xx p d = = = + + + + + =

and the variance is

( ) ( ) ( )2 22

2 2 2 2 2 2 5323 3 3 3 3 23

x x x x

x p d = =

= + + + + + =

random process [ ]x n is a sequence (there are also continuous random processes) thatassumes specific numeric values at each inde& n depending on the outcome of a randome&periment In this discussion we only consider real#valued random processes

good e&ample of a random process is time#varying thermal noise %ach time the noiseis sampled a different sequence of values is produced, but all the instantiations have

common features

6ne way to thin of the random process is as sequence of random variables (

[ ] [ ] [ ], , , ,x x xK K ) that are characteri.ed by a corresponding sequence of

probability density functions[ ]

( )[ ]

( )[ ]

( )( )( , (

, , , ,x x x

p p p

K K !hese P+Fs produce in

the sequences of means and variances*

[ ] [ ] [ ] [ ]( ) [ ]

[ ] [ ] [ ] [ ]( ) [ ]

[ ] [ ] [ ] [ ]( ) [ ]

22

( ( (

22

, , ,

22

( ( (

,

,

,

x x x

x x x

x x x

x x

x x

x x

= = =

= =

= =

M

M

/


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further assumption is also needed so that the statistics of a random process can be

estimated when only a single instance of the process is available !his assumption, calledergodicity, is that ensembleaverages (denoted by the angle bracets) can be evaluatedusing time averages from a single instance of the process !his means, in particular, that

[ ] [ ]

[ ]( ) [ ]( )

[ ] [ ] [ ] [ ] [ ]

2

(

,

(22

,

(

,

lim

lim

lim

K

x K

k

K

x x K x

k

K

xx K

k

x n x n KNK

x n x n KNK

q x n x n q x n q KN x n KN K

=

=

=

= = +

= = +

= + = + + +

!he term wide sense ergodicrefers to the more limited assumption that time#averaginggives the same results as ensemble averaging in the above three cases

A Second &oo' at te Spectral Estimation Formula

!he spectral estimate that was derived heuristically can now be reconsidered in the

conte&t of a random process


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autocorrelation sequence [ ]xx n m , so that

( ) [ ] ( )( (

, ,

N N j n mxx xx

m n

n m eN

= =

= %

"ow, since the summand of the double sum only depends of the difference n m , we cane&press the double sum as the single weighted sum (where q n m )

( ) ( )( )

[ ](

(

N jqxx xx

q N

N q q eN

=

= %

Finally, if the values of the autocorrelation sequence are only appreciable when q is

much smaller than N then N q N and

( )( )

[ ] ( )

(

(

N

jqxx xx xx

q Nq e

=

= = %

In other words, the estimated power spectrum is equal to the random process powerspectrum

(mportant note* In practice there are many ways to improve on the estimate describedabove, !hese include ways to improve statistical convergence (eg by using blocs of

data that overlap), compensating for the factor ( )N q N , and using parametetricmodels for the Power 8pectrum

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