現(xiàn)代數(shù)字信號(hào)處理AdvancedDigitalSignalProcessingch4pse

Advanced

Digital

SignalProcessing(Modern

Digital

Signal

Processing)Chapter

4

Power

SpectrumEstimationAmplitude

Spectrum

&

Power

SpectrumAmplitude

spectrum

density

■Signal

amplitudeat

each

frequencyFinite-energy

Fourierdeterministic

signal

transform4.1

IntroductionAmplitudespectrum(including

phase)Power

spectrum

density

(PSD)Real

stationaryrandom

signalFouriertransformPower

spectrum(without

phase)AutocorrelationfunctionSignal

powerat

eachfrequencyWiener-Khintchine

TheoremFourier

transformInverse

Fourier

transf

PSD

of

Ergodic

Stationary

RandomSignalPower

Spectrum

Estimation

(PSE)Estimating

the

PSD

of

real

ergodicstationary

random

signal

with

finiteobservations

(sample)Classical

&

Modern

PSE

MethodsClassical

(linear)

PSEPSE

based

on

Fourier

transform,

non-parametric

model

methodModern

(nonlinear)

PSEPSE

based

on

signal

model

(parametricmodel

method)Desirable

properties

of

PSEUnbiasedConsistentEfficientHigh

frequency

resolutionNarrow

main

lobe,

low

side

lobeSmall

sample

lengthBlackman-Tukey

(BT)

Method

for

PSE4.2

Classical

PSEThe

Fourier

transform

of

the

biasedestimation

of

autocorrelation

function.Periodogram

Method

for

PSEPeriodogramThe

average

energy

spectrum

of

finite

lengthsample

is

an

estimation

of

PSDFFT

The

periodogram

PSE

is

an

asymptoticallyunbiased

but

not

a

consistent

estimationBartlet

windowLimitations

of

Classical

PSELow

frequency

resolutionCaused

by

the

effects

of

data

window:The

degradation

in

resolution

by

mainlobe;The

power

leakage

by

side

lobe

(inter-spectrum

interference).Inconsistent

estimationThe

PSE

of

Harmonic

Process

With

PeriodogramThe

PSE

of

White

Noise

With

PeriodogramModifications

of

Classical

PSEAchieving

low

variance

at

the

expense

ofbias

and

frequency

resolutionAveraging

Periodogram

(Bartlet

method)N:

data

length,

N=L×MM

M

M

MPeriodogramPeriodogramPeriodogramPeriodogramAveragingAveraging

PeriodogramIts

bias

is

larger

than

the

periodogram

while

its

variancthan

the

Periodogram:Modified

PeriodogramThe

window

will

smooth

the

PSD

acquired

by

periodogram.

Ifunction

is

similar

to

a

lowpass

filter.

Averaging

modified

periodogram

(Welchmethod)N:

data

length,

N=L×MM

M

M

MModifiedPeriodogramAveragingAveraging

PeriodogramModifiedPeriodogramModifiedPeriodogramModifiedPeriodogramThe

PSE

of

Harmonic

Process

With

WelchThe

sequence

is

divided

into

eight

sections

with

50%overlap,

each

section

is

windowed

with

a

HammingwindowThe

PSE

of

White

Noise

With

WelchThe

sequence

is

divided

into

eight

sections

with

50%overlap,

each

section

is

windowed

with

a

Hammingwindow4.3

Parameter

Model

Methodsfor

PSEBasic

PrinciplesClassical

PSEAutocorrelationfunctionPSDFouriertransformObservationsxN(n)Estimationof

signalmodel

H(z)Parameter

model

methodsPSDObservationsNx

(n)Linear

systemwith

transferfunction

H(z)White

noisew(n)

The

Time

Series

Model

of

StationaryRandom

SignalStationary

randomsequence

x(n)MA(q)

model

(all-zero

model)Suitable

for

signals

whose

power

spectrahave

vales

but

no

peaks

AR(p)

model

(all-pole

model,

most

widelyused)Suitable

for

signals

whose

power

spectrahave

peaks

but

no

vales,

but

be

widelyused

since

the

linear

relation

between

itsparameters

and

the

signal

autocorrelationfunctionARMA(p,q)

model

(zero-pole

model)Suitable

for

signals

whose

power

spectrahave

vales

and

peaksModel

parameters

to

be

estimatedMA(q):AR(p):ARMA(p,q):

The

Relation

between

the

AutocorrelationFunction

&

the

Model

ParametersARMA(p,q)

modelInverse

z

transformGeneralized

Yule-Walker

equations:

a

nonlinear

eqset,

but

the

equations

are

linear

when

m>q.MA(q)

modelAR

(p)

modelInitial

valutheoremYule-Walkerequation:a

linearequation

setAR

model

power

spectrum

estimation

(AR

PSE)Observations

xN(n)Estimation

ofautocorrelation

functionEstimation

of

ARmodel

parametersAR

model

estimationProperties

of

AR

PSEThe

implied

autocorrelation

function

extensiWith

the

p+1

samples

of

autocorrelation

functio(ACF)

estimationThe

AR

model

parameter

estimation

is

obtainedby

solving

the

Yule-Walker

equation

(m=0,1,…,pFor

m>p,

thecan

be

extrapolated

fromthose

ACF

estimations

of

m≤p

byi.e.,

extrapolating

fromtoMESEKnown

ACF

estimationMaximum

entropyextrapolationUnknown

ACFestimationACFs

withmaximumuncertaintyMESE

of

zero-mean

Gaussian

random

sequencePDF

of

N-dimension

Gaussian

random

sequenceandand

so

on.The

equivalence

between

the

AR

PSE

and

theMESE

of

Gaussian

random

sequenceAR

PSE

for

p=N:MESE

of

Gaussian

random

sequenceFor

Gaussian

random

sequenceMaximum

entropyextrapolationAR

PSE

impliedextrapolation=MESEAR

PSE=There

are

no

poles

of

its

AR

modeloutside

the

unit

circle,

elseThe

stability

of

AR

modelStationary

random

sequence

x(n)The

AR

model

of

stationary

randomsequence

is

stable

(minimum

phase

model)

The

relationship

between

the

AR

PSE

and

thelinear

predictionOne-step

pure

linear

optimal

prediction

filterA

one-step

pure

linear

optimal

prediction

filtethe

solution

of

the

Yule-Walker

equation:AR(p)

modelThe

AR(p)

parameters

could

be

obtained

asthe

coefficients

that

minimized

the

predictionerror

power

of

a

p-th

order

linear

predictor.Methods

of

AR

PSE

(Solutions

of

Y-K

Equation)Levinson-Durbin

recursive

algorithmp

order

AR

model

equationp+1

order

AR

model

equationLetLetExpanded

equationPreparative

equationIfTheni.e.The

predictive

erroris

reduced

graduallyas

p

increases.