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By maintaining an estimate of its own estimation uncertainty and the relativeuncertainty in the various sensor outputs, the Kalman ®lter is able to combine allsensor information ``optima

Kalman Filter Basics

7.1 INTRODUCTION

7.1.1 What is a Kalman Filter?

It is an extremely effective and versatile procedure for combining noisy sensoroutputs to estimate the state of a system with uncertain dynamics

For our purposes in this book:

 The noisy sensors may include GPS receivers and inertial sensors ometers and gyroscopes, typically)but may also include speed sensors (e.g.,wheel speeds of land vehicles, water speed sensors for ships, air speed sensorsfor aircraft, or Doppler radar), and time sensors (clocks)

(acceler- The system state in question may include the position, velocity, acceleration,attitude, and attitude rate of a vehicle on land, at sea, in the air, or in space, butthe system state may include ancillary ``nuisance variables'' for modelingcorrelated noise sources (e.g., GPS Selective Availability timing errors)andtime-varying parameters of the sensors, such as scale factor, output bias, or (forclocks)frequency Selective Availability has been suspended as of May 1,2000

 Uncertain dynamics includes unpredictable disturbances of the host vehicle,whether caused by a human operator or by the medium (e.g., winds, surfacecurrents, turns in the road, or terrain changes), but it may also includeunpredictable changes in the sensor parameters

179

Mohinder S Grewal, Lawrence R Weill, Angus P Andrews

Copyright # 2001 John Wiley & Sons, Inc Print ISBN 0-471-35032-X Electronic ISBN 0-471-20071-9

More abstract treatments of the Kalman ®lter are presented in [18, 19, 40, 46, 67, 69,

71, 72], and a more basic introduction can be found in [31]

7.1.2 How it Works

The Kalman ®lter maintains two types of variables:

1 Estimated State Vector The components of the estimated state vector includethe following:

(a)The variables of interest (i.e., what we want or need to know, such asposition and velocity)

(b)``Nuisance variables'' that are of no intrinsic interest but may be necessary

to the estimation process These nuisance variables may include, forexample, the selective availability errors of the GPS satellites Wegenerally do not wish to know their values but may be obliged to calculatethem to improve the receiver estimate of position

(c)The Kalman ®lter state variables for a speci®c application must include allthose system dynamic variables that are measurable by the sensors used inthat application For example, a Kalman ®lter for a system containingaccelerometers and rate gyroscopes must contain acceleration and rotationrate components to which these instruments respond The acceleration andangular rate components do not have to be those along the sensor inputaxes, however The Kalman ®lter state variables could be the componentsalong locally level earth-®xed coordinates, even though the sensorsmeasure components in vehicle-body-®xed coordinates

In similar fashion, the Kalman ®lter state variables for GPS-only navigationmust contain the position coordinates of the receiver antenna, but these could

be geodetic latitude, longitude, and altitude with respect to a referenceellipsoid or geocentric latitude, longitude, and altitude with respect to areference sphere, or ECEF Cartesian coordinates, or ECI coordinates, or anyequivalent coordinates

2 A Covariance Matrix: a Measure of Estimation Uncertainty The equationsused to propagate the covariance matrix (collectively called the Riccatiequation)model and manage uncertainty, taking into account how sensornoise and dynamic uncertainty contribute to uncertainty about the estimatedsystem state

By maintaining an estimate of its own estimation uncertainty and the relativeuncertainty in the various sensor outputs, the Kalman ®lter is able to combine allsensor information ``optimally,'' in the sense that the resulting estimate minimizesany quadratic loss function of estimation error, including the mean-squared value ofany linear combination of state estimation errors The Kalman gain is the optimal

weighting matrix for combining new sensor data with a prior estimate to obtain anew estimate.

7.2 STATE AND COVARIANCE CORRECTION

The Kalman ®lter is a two-step process, the steps of which we call ``prediction'' and

``correction.'' The ®lter can start with either step, but we will begin by describing thecorrection step ®rst

The correction step makes corrections to an estimate, based on new informationobtained from sensor measurements

The Kalman gain matrix K is the crown jewel of Kalman ®ltering All the effort

of solving the matrix Riccati equation is for the sole purpose of computing the

``optimal'' value of the gain matrix K used for correcting an estimate ^x,

The derivation begins with background on properties of Gaussian probabilitydistributions and Gaussian likelihood functions, then development of models fornoisy sensor outputs and a derivation of the associated maximum-likelihood estimate(MLE)for combining prior estimates with noisy sensor measurements

7.2.1 Gaussian Probability Density Functions

Probability density functions (PDFs)are nonnegative integrable functions whoseintegral equals unity (i.e., 1) The density functions of Gaussian probabilitydistributions all have the form

p…x† ˆ 1

…2p†ndet P

2‰x mŠTP 1‰x mŠ†; …7:3†

where n is the dimension of P (i.e., P is an n  n matrix)and the parameters

ˆdef…

The operator Eh
i is the expectancy operator, also called the expected-valueoperator

The notation x 2 N …m; P† denotes that the variate (i.e., random variable) x isdrawn from the Gaussian distribution with mean m and covariance P Gaussiandistributions are also called normal or Laplace distributions

TABLE 7.1 Analogous Concepts in Three Different Contexts

distribution $ Likelihoodfunction L

a Argmax(f ) returns the argument x of the function f where f …x† achieves its maximum value For example, argmax(sin) ˆ p=2 and argmax(cos)ˆ 0.

7.2.2 Likelihood Functions

Likelihood functions are similar to probability density functions, except that theirintegrals are not constrained to equal unity, or even required to be ®nite They areuseful for comparing relative likelihoods and for ®nding the value of the unknownindependent variable x at which the likelihood function achieves its maximum, asillustrated in Fig 7.1

7.2.2.1 Gaussian Likelihood Functions Gaussian likelihood functionshave the form

7.2.2.2 Scaling of Likelihood Functions Maximum-likelihood estimation

is based on the argmax of the likelihood function, but for any positive scalar c > 0,

Fig 7.1 Maximum-likelihoodestimate.

That is, positive scaling of likelihood functions will have no effect on the likelihood estimate As a consequence, likelihood functions can have arbitrarypositive scaling.

maximum-7.2.2.3 Independent Likelihood Functions The joint probability P…A B†

of independent events A and B is the product P…A&B† ˆ P…A†  P…B† Theanalogous effect on independent likelihood functions LA and LB is the pointwiseproduct That is, at each ``point'' x,

7.2.2.4 Pointwise Products One of the remarkable attributes of Gaussianlikelihood functions is that their pointwise products are also Gaussian likelihoodfunctions, as illustrated in Fig 7.3

Given two Gaussian likelihood functions with parameter sets m A; YA and

fmB; YBg, their pointwise product is a scaled Gaussian likelihood function withparameters fmA B; YA Bg such that, for all x,

&

&

& &

& & &

Fig 7.2 Likelihoodwithout maximum.

One can solve Eq 7.11 for the parameters fmA B; YA Bg as functions of theparameters fmA; YA; mB; YBg: Taking logarithms of both sides of Eq 7.11 willproduce the equation

1

2‰x mA BŠTYA B‰x mA BŠ

ˆ log…c† 1

2‰x mAŠTYA‰x mAŠ1

Next, taking the ®rst and second derivatives with respect to the independent variable

x will produce the equations

of the component likelihood functions (YA and YB in Eq 7.14)

CombinedMaximum-LikelihoodEstimate is a WeightedAverage Equation7.13 evaluated at x ˆ 0 is

which can be solved for

mA B ˆ Yy

ˆ …YA‡ YB†y…YAmA‡ YBmB†; …7:17†where y denotes the Moore±Penrose inverse of a matrix (de®ned in Section B.1.4.7).Equations 7.14 and 7.17 are key results for deriving the formula for Kalman gain.All that remains is to de®ne likelihood function parameters for noisy sensors

7.2.3 Noisy Measurement Likelihoods

The term measurements refers to outputs of sensors that are to be used in estimatingthe argument vector x of a likelihood function Measurement models represent howthese measurements are related to x, including those errors called measurementerrors or sensor noise These models can be expressed in terms of likelihoodfunctions with x as the argument

7.2.3.1 Measurement Vector The collective output values from a multitude `

of sensors are the components of a vector

z ˆdef

z1

z2

z3

z`

2666

377

called the measurement vector, a column vector with ` rows

7.2.3.2 Measurement Sensitivity Matrix We suppose that the measuredvalues zi are linearly1 related to the unknown vector x we wish to estimate,

where H is the measurement sensitivity matrix

7.2.3.3 Measurement Noise Measurement noise is the electronic noise at theoutput of the sensors It is assumed to be additive,

& &

1 The Kalman ®lter is de®ned in terms of the measurement sensitivity matrix H, but the extended Kalman

®lter (described in Section 7.6.4)can be de®ned in terms of a suitably differentiable vector-valued function

h…x†.

where the measurement noise vector v is assumed to be zero-mean Gaussian noisewith known covariance R,

Yzˆ R 1;assuming R is nonsingular

7.2.3.5 Unknown Vector Likelihoods The same parameters de®ningmeasurement likelihoods also de®ne an inferred likelihood function for the truevalue of the unknown vector, with mean

is not unusual for GPS=INS integration

7.2.4 Gaussian MLE

7.2.4.1 Variables Gaussian MLE uses the following variables:

^x, the maximum likelihood estimate of x It will always equal the argmax (meanm)of an associated Gaussian likelihood function, but it can have differentvalues:

^x… †, representing the likelihood function prior to using the measurements.

^x…‡†, representing the likelihood function after using the measurements

P, the covariance matrix of estimation uncertainty It will always equal the inverse

of the information matrix Y of the associated likelihood function It also canhave two values:

P… †, representing the likelihood function prior to using the measurements

P…‡†, representing the likelihood function after using the measurements

z, the vector of measurements

H, the measurement sensitivity matrix

R, the covariance matrix of sensor noise

7.2.4.2 Update Equations The MLE formula for updating the variables ^x and

P to re¯ect the effect of measurements can be derived from Eqs 7.14 and 7.17 withinitial likelihood parameters

the information matrix of the measurements, and

where z is the measurement vector

7.2.4.3 Covariance Update The Gaussian likelihood function with eters fmA B; YA Bg of Eqs 7.14 and 7.17 then represents the state of knowledgeabout the unknown vector x combining both sources (i.e., the prior likelihood andthe effect of the measurements) That is, the covariance of MLE uncertainty afterusing the measurements will be

param-P…‡† ˆ Y 1

and the MLE of x after using the measurements will then be

Equation 7.14 can be simpli®ed by applying the following general matrix formula2:

A 1 ˆ YA, the prior information matrix for ^x

A ˆ P… †, the prior covariance matrix for ^x

B ˆ HT, the transpose of the measurement sensitivity matrix

C ˆ R

D ˆ H, the measurement sensitivity matrix,

so that Eq 7.31 becomes

a form better conditioned for computation

7.2.4.4 Estimate Update Equation 7.17 with substitutions from Eqs 7.27±7.30 will have the form shown in Eq 7.37

Hyz

|{z}

Eq: 7:30

24

7.2.5 Kalman Gain Matrix

Equation 7.45 has the form of Eq 7.1 with Kalman gain matrix

7.3 STATE AND COVARIANCE PREDICTION

The rest of the Kalman ®lter is the prediction step, in which the estimate ^x and itsassociated covariance matrix of estimation uncertainty P are propagated from onetime epoch to another This is the part where the dynamics of the underlying physicalprocesses come into play The ``state'' of a dynamic process is a vector of variablesthat completely specify enough of the initial boundary value conditions forpropagating the trajectory of the dynamic process forward in time, and the procedurefor propagating that solution forward in time is called ``state prediction.'' The modelfor propagating the covariance matrix of estimation uncertainty is derived from themodel used for propagating the state vector

7.3.1 State Prediction Models

7.3.1.1 Differential Equation Models Ever since the differential calculuswas introduced (more or less simultaneously)by Sir Isaac Newton (1643±1727)andGottfried Wilhelm Leibnitz (1646±1716), we have been using ordinary differentialequations as models for the dynamical behavior of systems of all sorts

Example 7.1 Differential Equation Model for Harmonic Resonator Dynamicalbehavior of the one-dimensional damped mass±spring shown schematically in Fig.7.4 is modeled by the equations

d2x

dt2 ‡Cdampingm dxdt‡Cspringm x ˆw…t†m ; …7:48†

where m ˆ mass attached to spring and damper

x ˆ upward displacement of mass from its rest position

Cspringˆ spring constant of spring

Cdampingˆ damping coef®cient of dashpot

w…t† ˆ disturbing force acting on mass

Systems of First-Order Linear Differential Equations The so-called statespace models for dynamic systems replace higher order differential equations withsystems of ®rst-order differential equations This can be done by de®ning the ®rst

n 1 derivatives of an nth-order differential equation as state variables

Example 7.2 State Space Model for Harmonic Oscillator Equation 7.48 is alinear second-order …n ˆ 2† differential equation It can be transformed into a system

of two linear ®rst-order differential equations with state variables

x1defˆx …mass displacement†; x2defˆdxdt …mass velocity†;

Fig 7.4 Schematic model for dynamic system of Example 7.1.

Systems of linear ®rst-order differential equations represented in ``long-hand''form as

dxn

dt ˆ fn1x1‡ fn2x2‡ fn3x3‡


‡ fnnxn‡ wncan be represented more compactly in matrix form as

f11 f12 f13


f1n

f21 f22 f23


f2n

f31 f32 f33


f3n .

fn1 fn2 fn3


fnn

2666

377

w1

w2

w3

wn

2666

377

7;

respectively

Example 7.3 Harmonic Resonator Model in Matrix Form For the system oflinear differential equations 7.49 and 7.50,

Cdampingm

24

3

m

24

35:

Eigenvalues of Dynamic Coef®cient Matrices The coef®cient matrix F of asystem of linear differential equations _x ˆ Fx ‡ w has effective units of reciprocaltime, or frequency (in units of radians per second) It is perhaps then not surprisingthat the characteristic values (eigenvalues)of F are the characteristic frequencies ofthe dynamic system represented by the differential equations

The eigenvalues of an n  n matrix F are the roots of its characteristicpolynomial

det…lI F† ˆPn

The eigenvalues of F have the same interpretation as the poles of the related systemtransfer function, in that the dynamic system _x ˆ Fx ‡ w is stable if and only if thesolutions l of det…lI F† ˆ 0 lie in the left half-plane

Example 7.4 Damping and Resonant Frequency for Underdamped HarmonicResonator For the dynamic coef®cient matrix

F ˆ C0spring 1

m

Cdampingm

24

35

in Example 7.3, the eigenvalues of F are the roots of its characteristic polynomial

det…lI F† ˆ det Cspringl 1

Cdampingm

24

3

5 ˆ l2‡Cdampingm l ‡Cspringm ;which are

l ˆ Cdamping

12m



C2 damping 4mCspring

q

:

If the discriminant

C2 damping 4mCspring< 0;

then the mass±spring system is called underdamped, and its eigenvalues are acomplex conjugate pair

damping oresonantiwith real part

1

tdampingˆ

Cdamping2mand imaginary part

oresonantˆ2m1 4mCspring C2

damping

q

:The alternative parameter

24

3

So long as the damping coef®cient Cdamping> 0, the eigenvalues of this systemwill lie in the left half-plane In that case, the damped mass±spring system isguaranteed to be stable

Matrix Exponential Function The matrix exponential function is de®ned (inSection B.6.4)as

1 The matrix N ˆ exp…M† is always invertible and N 1ˆ exp… M†

2 If M is antisymmetric (i.e., its matrix transpose MTˆ M), then N ˆ exp…M†

is an orthogonal matrix (i.e., its matrix transpose NT ˆ N 1)

3 The eigenvalues of N ˆ exp…M† are the (scalar)exponential functions of theeigenvalues of M.

4 If M…s† is an integrable function of a scalar s, then the derivative

ddt

where x…t0† is the initial value of the state vector x for t  t0

Time Invariant Systems If the dynamic coef®cient matrix F of Eq 7.51 does notdepend on t (time), then the problem is called time invariant In that case,

If we let ; xk 1; xk; xk‡1; be the corresponding state vector values of alinear dynamic system at those discrete times, then each of these values can bedetermined from the previous value by using Eq 7.56 in the form

Equation 7.59 is the discrete-time dynamic system model corresponding to thecontinuous-time dynamic system model of Eq 7.51.

The matrix Fk 1 (de®ned in Eq 7.60)in the discrete-time model (Eq 7.59)iscalled a state transition matrix for the dynamic system de®ned by F Note that Fdepends only on F, and not on the dynamic disturbance function w…t†

The noise vectors wk are the discrete-time analog of the dynamic disturbancefunction w…t† They depend upon F and w

Example 7.5 State Transition Matrix for Harmonic Resonator Model Theunderdamped harmonic resonator model of Example 7.4 has no time-dependentterms in its coef®cient matrix (Eq 7.53), making it a time-invariant model with statetransition matrix

375; …7:62†

where o ˆ oresonant; the resonant frequency

t ˆ tdamping; the damping time constant

7.3.2 Covariance Prediction Models

The word stochastic derives from the Greek expression for aiming at a target,indicating some degree of uncertainty in the dynamics of the projectile betweenlaunch and impact That idea has been formalized mathematically as stochasticsystems theory, used here for modeling dynamic processes that are not fullypredictable

A stochastic process is a model for the evolution over time of a probabilitydistribution For Kalman ®ltering, this can be viewed as the probability distribution

of the true state of a dynamic process When the underlying probability distribution

is Gaussian, the distribution is completely speci®ed by its mean and its covariance.The mean will be the estimated value of the state vector, and the covariance matrixrepresents the mean-squared uncertainty in the estimate The time update equations

of the Kalman ®lter propagate the estimated value and its associated mean-squareduncertainty forward in time.

7.3.2.1 Zero-Mean White Gaussian Noise Processes A zero-mean whiteGaussian noise process in discrete time is a sequence of independent samples ; wk 1; wk; wk‡1; from a normal probability distribution N …0; Qk† withzero mean and known ®nite covariances Qk In Kalman ®ltering, it is not necessary(but not unusual)that the covariance of all samples be the same

Sampling is called independent if the expected values of outer products

for all integer indices i and j of the random process

Zero-mean white Gaussian noise processes are the fundamental random processesused in Kalman ®ltering However, it is not necessary that all noise sources in themodeled sensors and dynamic systems be zero-mean white Gaussian noiseprocesses It is only necessary that they can be modeled in terms of such processes(to be addressed in Section 7.5)

7.3.2.2 Gaussian Linear Stochastic Processes in Discrete Time Alinear stochastic processes model in discrete time has the form

Qk represent the unknown random disturbances Together, they model the tion of the necessary statistical properties of the state variable x

propaga-Example 7.6 Harmonic Resonator with White Acceleration DisturbanceNoise If the disturbance acting on the harmonic resonator of Examples 7.1±7.5were zero-mean white acceleration noise with variance s2

disturbance, then its bance noise covariance matrix would have the form

Forces applied to a rigid body, for example, can affect rotational dynamics as well astranslational dynamics This sort of coupling of common disturbance noise sourcesinto different components of the state dynamics can be represented by using a noisedistribution matrix G in the form

d

where the components of w…t† are the common disturbance noise sources and thematrix G represents how these disturbances are distributed among the state vectorcomponents

The covariance of state vector disturbance noise will then have the form GQwGT,where Qw is the covariance matrix for the white-noise process w…t†

The analogous model in discrete time has the form

xk ˆ Fk 1xk 1‡ Gk 1wk 1; …7:67†where fwkg is a zero-mean white-noise process in discrete time

In either case (i.e., continuous or discrete time), it is possible to use the noisedistribution matrix for noise scaling, as well, so that the components of wk can beindependent, uncorrelated unit normal variates and the noise covariance matrix

Qwˆ I, the identity matrix

7.3.2.4 Predictor Equations The linear stochastic process model parameters

F and Q can be used to calculate how the discrete-time process variables m and Pevolve over time

Using Eq 7.64 and taking expected values,

7.4.2 Common Terminology

The symbols used in Table 7.2 for the variables and parameters of the Kalman ®lterare essentially those used in the original paper by Kalman [71], and this notation isfairly common in the literature

The following are some names commonly used for the symbols in Table 7.2:

H is the measurement sensitivity matrix or observation matrix

H^xk… † is the predicted measurement

z H^xk… †, the difference between the measurement vector and the predictedmeasurement, is the innovations vector

K is the Kalman gain

Pk… † is the predicted or a priori value of estimation covariance

Pk…‡† is the corrected or a posteriori value of estimation covariance

Qk is the covariance of dynamic disturbance noise

R is the covariance of sensor noise or measurement uncertainty

^xk… † is the predicted or a priori value of the estimated state vector

^xk…‡† is the corrected or a posteriori value of the estimated state vector

z is the measurement vector or observation vector

7.4.3 Data Flow Diagrams

The matrix-level data ¯ow of the Kalman ®lter implementation for a time-varyingproblem is diagrammed in Fig 7.5, with the inputs shown on the left, the outputs(corrected estimates)on the right, and the symbol z 1 representing the unit delayoperator

The dashed lines in the ®gure enclose two computation loops The top loop is theestimation loop, with the feedback gain (Kalman gain)coming from the bottom

TABLE 7.2 Essential Kalman Filter Equations Predictor (Time Updates)

Predicted state vector:

^xk… † ˆ Fk^xk 1…‡† Eq 7.68 Predicted covariance matrix:

P k … † ˆ F k P k 1 …‡†F T

k ‡ Q k 1 Eq 7.69 Corrector (Measurement Updates)

Kalman gain:

K k ˆ P k … †HTk …H k P k … †HTk ‡ R k † 1 Eq 7.46 Corrected state estimate:

^xk…‡† ˆ ^x k … † ‡ K k …z k H k ^xk… †† Eq 7.1 Corrected covariance matrix:

Pk…‡† ˆ Pk… † KkHkPk… † Eq 7.47

loop The bottom loop implements the Riccati equation solution used to calculate theKalman gain This bottom loop runs ``open loop,'' in that there is no feedbackmechanism to stabilize it in the presence of roundoff errors Numerical instabilityproblems with the Riccati equation propagation loop were discovered soon after theintroduction of the Kalman ®lter.

+

+ +

+ +

7.5 ACCOMMODATING CORRELATED NOISE

The fundamental noise processes in the basic Kalman ®lter model are zero-meanwhite Gaussian noise processes:

fwkg, called dynamic disturbance, plant noise, or process noise and

fvkg, called sensor noise, measurement noise, or observation noise

However, it is not necessary that the physical noise processes of the real-worldapplicationÐeither the dynamic disturbance or the sensor noiseÐbe uncorrelated inorder to apply Kalman ®ltering

For applications with uncorrelated noise sources, it is only necessary to model thecorrelated noise process jkˆ vk, wk using a linear stochastic system model of thesort

jk ˆ Fk 1jk 1‡ yk 1;where fykg is a zero-mean white Gaussian noise process, and then augment the statevector by appending the new variable j,

xaugmentedˆ xoriginal

j

…7:70†and modify the parameter matrices F; Q, and H accordingly

7.5.1 Correlated Noise Models

7.5.1.1 Autocovariance Functions Correlation of a random sequence fjkg

is characterized by its discrete-time autocovariance function Pf g‰DkŠ, a function ofjkthe delay index Dk de®ned as

Pf g‰DkŠ ˆjk defEkh…jk mx†…jk‡Dk mx†Ti; …7:71†where mx is the mean value of the random sequence fjkg

For white-noise processes,

P‰DkŠ ˆ 0; Dk 6ˆ 0;C; Dk ˆ 0;



…7:72†where C is the covariance of the process

7.5.1.2 Random Walks Random walks, also called Wiener processes, arecumulative sums of white-noise processes fwkg:

a stochastic process model with state transition matrix F ˆ I, an identity matrix.Random walks are notoriously unstable, in the sense that the covariance of thevariate jk grows linearly with k and without bound as k ! 1 In general, if any ofthe eigenvalues of a state transition matrix fall on or outside the unit circle in thecomplex plane (as they all do for identity matrices), the variate of the stochasticprocess can fail to have a ®nite steady-state covariance matrix However, thecovariance of uncertainty in the estimated system state vector can still converge

to a ®nite steady-state value, even if the process itself is unstable Methodsfor determining whether estimation uncertainties will diverge are presented inChapter 8

7.5.1.3 Exponentially Correlated Noise Exponentially correlated randomprocesses have ®nite, constant steady-state covariances A scalar exponentiallyrandom process fxkg has a model of the sort

where Q is the variance of the scalar zero-mean white-noise process fwkg

The autocovariance sequence of an exponentially correlated random process indiscrete time has the general form

which falls off exponentially on either side of its peak value s2 (the processvariance)at Dk ˆ 0 The parameter Nc is called the correlation number of theprocess, where Ncˆ t=Dt for correlation time t and sample interval Dt

7.5.1.4 Harmonic Noise Harmonic noise includes identi®able frequencycomponents, such as those from AC power or from mechanical or electricalresonances A stochastic process model for such sources has already been developed

in the examples of this chapter

7.5.1.5 SA Noise Autocorrelation A clock dithering algorithm is described

in U.S Patent 4,646,032 [134], including a parametric model of the autocorrelationfunction (autocovariance function divided by variance)of the resulting timing errors

SA was suspended on May 1, 2000, but can be turned on again Knowledge of thedithering algorithm does not necessarily give the user any advantage, but there is atleast a suspicion that this may be the algorithm used for SA dithering of theindividual GPS satellite time references Its theoretical autocorrelation function isplotted in Fig 7.6 along with an exponential correlation curve The two are scaled tocoincide at the autocorrelation coef®cient value of 1=e  0:36787944 , theargument at which correlation time is de®ned Unlike exponentially correlatednoise, this source has greater short-term correlation and less long-term correlation.The correlation time of SA errors determined from GPS signal analysis is on theorder of 102±103s It is possible that the actual correlation time is variable, whichmight explain the range of values reported in the literature

Although this is not an exponential autocorrelation function, it could perhaps bemodeled as such

Exponentially correlated

Relative time scale (tau/NT)

Fig 7.6 Autocorrelation function for pseudonoise algorithm.