現(xiàn)代數(shù)字信號(hào)處理AdvancedDigitalSignalProcessingch2WienerKalman

Advanced

Digital

SignalProcessing(Modern

Digital

Signal

Processing)Chapter

2

Discrete

Wiener

Filterand

Discrete

Kalman

Filterv(n)s(n):

signalv(n):

noisex(n):

observation

or

measurement:

estimation

of

s(n)h(n):

estimator

or

filter2.1

The

Wiener

Filtering

ProblemState

(Wave)

Estimation

Problemh(n)s(n)x(n)Optimum

Estimation

&

Linear

EstimationLinear

estimationOptimum

estimationThe

estimation

is

optimal

under

certain

criterionWiener

FilterThe

linear

estimator

of

stationary

random

signawith

least

square

error

(LSE,

the

MMSEwithout

prior

PDF)2.2

Normal

Equations

for

Wiener

FilterOrthogonal

PrincipleThe

relationship

between

e(n)

and

x(n)

orfilterin

Wieneri.e.

the

filter

h(n)

is

optimal

iff

the

e(n)

and

x(n)

is

orUnderstanding

of

the

Orthogonal

PrinciplWiener-Hopf

EquationThe

requirement

for

the

coefficients

ho(n)

of

Wiener

filtWiener-Hopf

Equation

in

z-domainIf

s(n)

is

uncorrelated

with

the

v(n),

i.e.then

Wiener-Hopf

Equation

in

Frequency-domainIf

s(n)

is

uncorrelated

with

the

v(n),

i.e.thenIntuitive

Interpretation

of

the

Wiener

F2.3

Solutions

for

Wiener

Filter

Causal

FIR

Wiener

Filter

(Solution

inTime

Domain)Causal

FIR

filterWiener-Hopf

equation

with

causal

FIR

filterorSolution

of

causal

FIR

Wiener

filterIf

Rxx

is

nonsingular,

thenHowever,

such

a

solution

is

often

computatioimpractical

when

the

N

increases.

Estimation

error

(MSE)

of

causal

FIRWiener

filterThe

mean

square

error

(MSE)

for

causal

FIRfilter

is

a

quadratic

function

of

the

filter

covector

h

and

has

a

single

minimum

pointFor

example,

when

N=2,

i.e.

h=[h(0),

h(1)],

the

MSEfunction

is

a

bowl-shaped

surface

with

a

single

minipoint.Non-causal

IIR

Wiener

FilterSolutionIf

s(n)

is

uncorrelated

with

the

v(n)Estimation

error

(MSE)If

s(n)

is

real-valued

and

uncorrelated

with

the

v(n)hence,Causal

IIR

Wiener

FilterSpectral

factorization

theoremRational

power

spectrum

signalA

random

signal

whose

power

spectrum

is

ais

called

arational

function

ofrational

power

spectrum

signal.Spectral

factorization

theoremA

real-valued

stationary

random

sequencewith

rational

power

spectrum

can

berepresented

by

a

time

series

modelWhitening

filterB(z)w(n)

x(n)B-1(z)x(n)w(n)signal

model

(invertible)whitening

filterCausal

IIR

Wiener

filter

with

a

white

noiseas

inputWiener-Hopf

equationSolutionG(z)w(n)u(j)

denotes

unit

step

sequen

Causal

IIR

Wiener

filter

with

a

rationalpower

spectrum

signal

as

inputH

(z)x(n)B-1(z)x(n)B(z)w(n)

Steps

to

compute

the

causal

IIR

WienerfilterSpectral

factorizationCausal

decompositionCompute

the

causal

IIR

Wiener

filterImpulse

responseEstimation

error

(MSE)

MSE

Comparison

of

Different

WienerFilter2.4

Example

for

Solving

Causal

IIRWiener

FilterGiven:

signal

modelmeasurementmodelAssumptions:White

noisesSignal

model

in

z-domain:Spectral

FactorizationCausal

DecompositionCausal

IIR

Wiener

Filter

Impulse

Response

of

the

Causal

IIRWiener

Filter2.5

Wiener

PredictionIntroductionPredictionEstimating

the

s(n+N),

N>0

based

on

theobservations

x(n),

x(n-1),

….Basic

requirements

for

predictionThe

signal

to

be

predictable

is

correlatedwith

the

observations,

i.e.

the

future

of

thesignal

is

relevant

with

its

past

(white

noiseis

completely

unpredictable).Wiener

predictionThe

linear

estimation

of

the

s(n+N)

withminimum

mean

square

error.Basic

Form

of

Wiener

PredictionWiener-Hopf

equation

in

predictionNon-causal

IIR

Wiener

predictorCausal

IIR

Wiener

predictor

Pure

Wiener

Prediction

(withoutObservation

Noise)Non-causalCausal

IIR

pure

Wiener

predictorPrediction

error

(MSE):

One-steplinear

predictor

(one-step

causalFIR

pureWiener

predictor,

solution

in

time-domain)i.e.The

Yule-Walker

equation

set

is

homogeneous

except

forfirst

equation.It

does

not

need

the

cross-correlation

between

signal

sobservation

x(n).It

has

p+1

equations

and

p+1

unknowns.

The

optimal

one-step

linear

predictor

and

its

prediction

error

(MSE)

canobtained

by

solving

these

equations.It

is

often

used

to

solve

AR

model

parameters.Yule-Walker

Equation:

homogeneous

equation

set

except

thefirst

equationAdditive

Noise

Reduction2.6

Applications

of

Wiener

Filterv(n)h(n)s(n)x(n)An

illustration

of

the

variation

of

Wiener

frequencyresponse

with

signal

spectrum

for

additive

white

noise.Wiener

filter

response

broadly

follows

the

signal

spectrWiener

Channel

EqualiserNoise

v(n)equaliserD-1(z)s(n)x(n)

ChannelChanneldistortionD(z)IntroductionKalman

FilteringA

recursive

numerical

method

to

estimatethe

state

variablesHistorical

perspectiveGenerally

proposed

by

Rudolf

Emil

Kalman

in1960It

is

developed

primarily

for

the

Apollo

spaceprogram

to

solve

the

spacecraft

navigationproblemIt

is

now

widely

used

in

many

military

or

civilareas

such

as

target

tracking,

navigation,optimal

control

etc.2.7

Discrete

Kalman

Filter

An

Example

of

Kalman

Filter

ProblemTo

estimate

the

temperature

of

a

room

withyour

prediction

and

a

thermometer:k:

time

index.A:

system

matrix;

B:

input

matrix;

C:

output

matrix:

state

vector

of

the

system;u:

known

input

vector

to

the

system,

often

let

to

bz:

measured

output

vector.w:

system

noise

vector;v:

observation

(measurement)

noise

vector.System

ModelState

equation

and

measurement

equationNoise

modelwk

and

vk

are

independent

white

noiseswith

Gaussian

distributions.Qk:

Covariance

matrix

(non-negative

definite)

ofsystem

noise

at

instant

k.Rk:

Covariance

matrix

(positive

definite)

ofmeasurement

noise

at

instant

k.Basic

Idea

of

Kalman

FilterRecursive

architecture

of

discrete

KalmanStatepredictionfilterState

estimation

oflast

instant

(known)CurrentmeasurementPrediction

ofcurrent

state(prior

estimation)StateupdatingEstimation

of

currentstate

(posteriorestimation)To

the

nextfilter

period

The

central

problem

of

discrete

Kalmanfilter

is

to

determine

the

Kalman

gain

matrixKkDeduction

of

the

Kalman

Gain

MatrixCovariance

matrix

of

state

estimation

errorCovariance

matrix

of

a

prior

estimation

errorCovariance

matrix

of

posterior

estimation

errorUpdating

of

the

prior

estimationwhere is

the

measurement

innovation(or

residual)

at

instant

k.

It

reflectdiscrepancy

between

the

predicted

and

actualmeasurements.Kalman

gain

matrixThe

Kk

that

minimizing

the

Pk|kIntuitive

interpretation

of

the

Kalman

gainKk

indicates

the

degree

to

update

the

prediction

wthe

measurement

innovationAs

the

measurement

error

covariance

Rk

approaches

zero,

the

Kk

weights

the

residual

moreheavily.As

the

prior

estimation

error

covariance

Pk|k-1approaches

zero,

the

Kk

weights

the

residual

less.

Recursive

Process

of

State

EstimationError

Covariance

MatrixPrediction

of

the

prior

estimation

errorcovariance

()Posterior

estimation

error

covariance(

)Discrete

Kalman

Filtering

AlgorithmOriginal

state &

covariance

matrixStatepredictionCovariance

matrixpredictionKalman

gainmatrixStateupdatingCovariance

matrixupdatingCharacteristics

of

Kalman

FilterState

space

descriptions

of

systemMulti-dimensional

stateRecursive

algorithmNumerical

solution

of

filter

Suitable

for

non-stationary

(time

variant)

st(signal)MMSE

estimation

Optimal

linear

estimation

for

the

state

oflinear

system

with

Gaussian

white

systemand

observation

noiseSuitable

for

the

implementation

by

computerSimulation

Examples

of

Kalman

Filter

Example

1:

estimation

the

states

(positionand

velocity)

of

a

moving

object

with

uniformvelocity

(without

system

noise)System

equation

and

measurement

equationSimulation

settingsMeasurement

noise

is

a

Gaussian

white

noise;Actual

state:Original

estimationExamining

the

simulation

results

with

differenoise

model■Position

estimation

resultPositionestimation

Actual

positionMeasured

positionPositionTimePosition

estimation

errorPosition

estimation

errorPosition

measurementerrorPosition

errorTimeVelocity

estimation

resultVelocityestimationActual

velocityVelocityTime■Position

estimation

result

(more

belief

onmeasurement)Positionestimation

Actual

positionMeasured

positionPositionTimePosition

estimation

errorPosition

estimation

errorPosition

measurementerrorPosition

errorTimeVelocity

estimation

resultVelocityestimationActual

velocityVelocityTime■Position

estimation

result

(more

belief

onprediction)Positionestimation

Actual

positionMeasured

positionPositionTimePosition

estimation

errorPosition

estimation

errorPosition

measurement

errorPosition

errorTimeVelocity

estimation

resultVelocity

estimationActual

velocityVelocityTime■Position

estimation

resultPosition

estimationActual

positionMeasured

positionPositionTimePosition

estimation

errorPosition

estimation

errorPosition

measurement

errorPosition

errorTimeVelocity

estimation

resultVelocity

estimationActual

velocityVelocityTimeExample

2:

tracking

maneuvered

targetSystem

equation

and

measurement

equationSimulation

settingsMeasurement

noise

and

system

noise

areindependent

Gaussian

white

noises;Actual

state:Original

estimationExamining

the

tracking

result

with

accuratenoise

modelTracking

resultPosition

estimationActual

positionMeasured

positionPositionTimeComments

on

Kalman

Filter

The

stable

Kalman

filter

of

stationaryprocess

with

Gaussian

white

noise

issame

as

the

Wiener

filterKalman

filter

with

non-Gaussian

noiseApplicableSuboptimalKalman

filter

with

colored

noiseExtended

Kalman

filter

(EKF)Nonlinear

systemLinearized

with

Taylor

seriesSub-optimal

(after

linearization,

noises

arenon-Gaussian)Data

association

in

target

trackingSingle

target

tracking

with

multi-measurement