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LEAST-SQUARES AND MINIMUM– VARIANCE ESTIMATES FOR LINEAR TIME-INVARIANT SYSTEMS 4.1 GENERAL LEAST-SQUARES ESTIMATION RESULTS In Section 2.4 we developed 2.4-3, relating the 1 1 measureme

LEAST-SQUARES AND MINIMUM–

VARIANCE ESTIMATES FOR LINEAR TIME-INVARIANT SYSTEMS

4.1 GENERAL LEAST-SQUARES ESTIMATION RESULTS

In Section 2.4 we developed (2.4-3), relating the 1
1 measurement matrix

Ynto the 2
1 state vector Xnthrough the 1
2 observation matrix M as givenby

It was also pointed out in Sections 2.4 and 2.10 that this linear time-invariantequation (i.e., M is independent of time or equivalently n) applies to moregeneral cases that we generalize further here Specifically we assume Yn is a

1
ðr þ 1Þ measurement matrix, Xn a 1
m state matrix, and M an

ðr þ 1Þ
m observation matrix [see (2.4-3a)], that is,

Yn ¼

y0

y1

yr

26664

37775n

ð4:1-1aÞ

Xn ¼

x0ðtÞ

x1ðtÞ

xm1ðtÞ

2666

377

155

Copyright # 1998 John Wiley & Sons, Inc ISBNs: 0-471-18407-1 (Hardback); 0-471-22419-7 (Electronic)

and in turn

Nn¼

0

1

r

266

377n

ð4:1-1cÞ

As in Section 2.4, x0ðtnÞ; ; xm1ðtnÞ are the m different states of the targetbeing tracked By way of example, the states could be the x, y, z coordinates andtheir derivatives as given by (2.4-6) Alternately, if we were tracking only a one-dimensional coordinate, then the states could be the coordinate x itself followed

by its m derivatives, that is,

Xn ¼ XðtnÞ ¼

x

D x

Dmx

264

375n

ð4:1-2aÞ

The example of (2.4-1a) is such a case with m¼ 1 Let m0always designate thenumber of states of XðtnÞ or Xn; then, for XðtnÞ of (4.1-2), m0¼ m þ 1 Anotherexample for m¼ 2 is that of (1.3-1a) to (1.3-1c), which gives the equations ofmotion for a target having a constant acceleration Here (1.3-1a) to (1.33-1c)can be put into the form of (2.4-1) with

Xn¼

xn_xn



xn

24

3

Assume that measurements such as given by (4.1-1a) were also made at the Lpreceding times at n 1; ; n  L Then the totality of L þ 1 measurements

can be written as

Yn -

Yn1 -:

YnL

26664

37775

¼

MXn -

MXn1 -

. -

MXnL

266666

377777þ

Nn -

Nn1 -

. -

NnL

266666

377777

ð4:1-5Þ

Assume that the transition matrix for transitioning from the state vector Xn1

at time n 1 to the state vector Xn at time n is given by  [see (2.4-1) ofSection 2.4, which gives  for a constant-velocity trajectory; see also Section5.4] Then the equation for transitioning from Xnito Xn is given by

Xn¼ iXni¼ iXni ð4:1-6Þwhere iis the transition matrix for transitioning from Xnito Xn It is given by

Yn1 -

YnL

266664

377775

¼

MXn -M1Xn -

. -MLXn

2666664

3777775þ

Nn -

Nn1 -

. -

NnL

2666664

3777775

. -ML

2666664

3777775

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

m0

Xnþ

Nn -

Nn1 -

. -

NnL

2666664

3777775

which we rewrite as

YðnÞ¼ T Xnþ NðnÞ ð4:1-11Þwhere

YðnÞ¼

Yn -

Yn1 -

. -

YnL

2666664

3777775

NðnÞ¼

Nn -

Nn1 -

. -

NnL

2666664

3777775

ð4:1-11aÞ

T¼

M -M1 -

. -ML

266666

377777

[In Part 1 and (4.1-4), T was used to represent the time betweenmeasurements Here it is used to represent the observation matrix given by(4.1-11b) Unfortunately T will be used in Part II of this text to represent thesetwo things Moreover, as was done in Sections 1.4 and 2.4 and as shall be donelater in Part II, it is also used as an exponent to indicate the transpose of amatrix Although this multiple use for T is unfortunate, which meaning T hasshould be clear from the context in which it is used.]

By way of example of T, assume L¼ 1 in (4.1-11a) and (4.1-11b); then

where I is the identify matrix and  is given by (2.4-1b)

As done in Section 2.4, assume a radar sensor with only the targetrange being observed, with xn representing the target range Then M is given by(2.4-3a) and Yn and Nn are given by respectively (2.4-3c) and (2.4-3b).Substituting (4.1-15) and (2.4-3a) into (4.1-13) yields

1 nT

2666

377

It is instructive to write out (4.1-11) for this example In this case (4.1-11)becomes

YðnÞ¼

yn

yn1

yn2

y0

2666

377

7¼

1 T

1 2T

1 nT

2664

3775

xn_xn

 þ

n

n1

n2

0

2666

377

yni¼ xn iT _xnþ ni ð4:1-22ÞThe above physically makes sense For a constant-velocity target it relatesthe measurement yni at time n i to the true target position and velocity xnand _xnat time n and the measurement error ni The above example thus gives

us a physical feel for the observation matrix T For the above example, the

ði þ 1Þst row of T physically in effect first transforms Xn back in time to time

n i through the inverse of the transition matrix  to the ith power, that is,through iby premultiplying Xn to yield Xni, that is,

Xni¼ iXn ð4:1-23ÞNext Xniis effectively transformed to the noise-free Ynimeasurement at time

n i by means of premultiplying by the observation matrix M to yield thenoise-free Yni, designated as Yni0 and given by

Yni0 ¼ MiXn ð4:1-24ÞThus T is really more than an observation matrix It also incorporates the targetdynamics through  We shall thus refer to it as the transition–observationmatrix

By way of a second example, assume that the target motion is modeled by aconstant-accelerating trajectory Then m¼ 2 in (4.1-2), m0¼ 3, and Xnis given

by (4.1-3) with  given by (4.1-4) From (1.3-1) it follows that

xn1¼ xn _xnTþ xnð1

2T2Þ ð4:1-25aÞ_xn1¼ _xn xnT ð4:1-25bÞ



We can now rewrite (4.1-25a) to (4.1-25c) as (4.1-8) with Xn given by (4.1-3)and

3777

37775

xn_xn



xn

24

0

26664

37775

ð4:1-29Þ

Again, we see from the above equation that the transition–observation matrixmakes physical sense Its (iþ 1)st row transforms the state vector at time Xnback in time to Xni at time n i for the case of the constant-acceleratingtarget Next it transforms the resulting Xnito the noise-free measurement Yni0

n;nfor Xn, which is a linear function

of the measurement given by YðnÞ, that is,

where W is a row matrix of weights, that is, W¼ ½ w1; w2; ; ws , where s isthe dimension of YðnÞ; see (4.1-10) and (4.1-11a) For the least-squares estimate

(LSE) we are looking for, we require that the sum of squares of errors beminimized, that is,

e n;nÞ ¼ en ¼ ½ YðnÞ n;n T½ YðnÞ n;n ð4:1-31Þ

is minimized As we shall show shortly, it is a straightforward matter toprove using matrix algebra that W of (4.1-30) that minimizes (4.1-31) isgiven by

^

W ðTTTÞ1TT ð4:1-32Þ

It can be shown that this estimate is unbiased [5, p 182]

Let us get a physical feel for the minimization of (4.1-31) To do this, let usstart by using the constant-velocity trajectory example given above with T given

by (4.1-18) and YðNÞ given by the left-hand side of (4.1-19), that is,

YðnÞ¼

yn

yn1

yn2

y0

26664

37775

.0;n

26666

3777

7¼

1 T

1 2T

1 nT

26664

37775

n;n

n;n

n;n ð4:1-36Þ

Substituting (4.1-33) and (4.1-36) into (4.1-31) yields

Except for a slight change in notation, (4.1-38) is identical to (1.2-33) of

n j;nand eTby en, but the estimationproblem is identical What we are trying to do in effect is find a least-squaresfitting line to the data points as discussed in Section 1.2.6 relative to Figure

n;n, andn;n In constrast in Section 1.2.6 we represented the linefitting the data by its ordinate and slope at time n 0 and

0 0, respectively A line is defned by its ordinate and slope at any time.Hence it does not matter which time we use, time n¼ n or time n ¼ 0 (Thecovariance of the state vector, however, does depend on what time is used.) Thestate vector estimate gives the line’s ordinate and slope at some time Hencethe state vector at any time defines the estimated line trajectory At time n¼ 0the estimated state vector is

At time n it is given by (4.1-34) Both define the same line estimate

To further clarify our flexibility in the choice of the time we choose for thestate vector to be used to define the estimating trajectory, let us go back to(4.1-9) In (4.1-9) we reference all the measurements to the state vector Xn attime n We could have just as well have referenced all the measurementsrelative to the state vector at any other time n i designated as Xni Let uschoose time n i ¼ 0 as done in (4.1-39) Then (4.1-9) becomes

Yn

Yn1

.

Y1 Y

266666664

377777775

¼

MnX0 -Mn1X0 -

. -MX0 -MX

266666664

377777775þ

Nn

Nn1

.

N1 N

266666664

377777775

ð4:1-40Þ

This in turn becomes

Yn

Yn1

.

Y1

Y0

266666664

377777775

¼

Mn -Mn1 -

. -M

M

-266666664

377777775

X0þ

Nn

Nn1

.

N1

N0

266666664

377777775

ð4:1-41Þ

which can be written as

YðnÞ¼ T X0þ NðnÞ ð4:1-42Þwhere YðnÞ and NðnÞ are given by (4.1-11a) with L¼ n and T is now definedby

T¼

Mn -Mn1 -

. -M

M

-266666666

377777777

ð4:1-43Þ

In Section 1.2.10 it was indicated that the least-squares fitting line to the data

of Figure 1.2-10 is given by the recursive g–h growing-memory memory) filter whose weights g and h are given by (1.2-38a and 1.2-38b) Theg–h filter itself is defined by (1.2-8a) and (1.2-8b) In Chapters 5 and 6 anindication is given as to how the recursive least-squares g–h filter is obtainedfrom the least-squares filter results of (4.1-30) and (4.1-32) The results are alsogiven for higher order filters, that is, when a polynominal in time of arbitrarydegree m is used to fit the data Specifically the target trajectory xðtÞ isapproximated by

trajectory is being fitted to the data For this case the transition–observationmatrix is given by (4.1-18) If a constant-accelerating target trajectory is fitted

to the data, then, in (4.1-2) and (4.1-44), m¼ 2, and T is given by (4.1-28) Inthis case, a best-fitting quadratic is being found for the data of Figure 1.2-10.The recursive least-square filter solutions are given in Chapter 6 for

m¼ 0; 1; 2; 3; see Table 6.3-1 The solution for arbitrary m is also given ingeneral form; see (5.3-11) and (5.3-13)

n;ngiven above by (4.1-30) and(4.1-32) requires a matrix inversion in the calculation of the weights In Section5.3 it is shown how the least-squares polynomial fit can be obtained without amatrix inversion This is done by the use of the powerful discrete-timeorthogonal Legendre polynomials What is done is that the polynomial fit ofdegree m of (4.1-44) is expressed in terms of the powerful discrete-timeorthogonal Legendre polynomials (DOLP) having degree m Specifically(4.1-44) is written as

of degree k with kðrÞ orthogonal to jðrÞ for k 6¼ j; see (5.3-2) Using thisorthogonal polynomial form yields the least-squares solution directly as a linearweighted sum of the yn; yn1; ; ynL without any matrix inversion beingrequired; see (5.3-10) and (5.3-11) for the least-squares polynomial fit,designated there as n

voltage-processing method, is presented, which also avoids the need to do amatrix inversion Finally, it is shown in Section 14.4 that when a polynomial fit

to the data is being made, the alternate voltage-processing method is equivalent

to using the orthogonal discrete Legendre polynomial approach

In Sections 7.1 and 7.2 the above least-squares polynomial fit results areextended to the case where the measurements consist of the semi-infinite set yn,

yn1, instead of Lþ 1 measurements In this case, the discounted squares weighted sum is minimized as was done in (1.2-34) [see (7.1-2)] toyield the fading-memory filter Again the best-fitting polynomial of the form,given by (4.1-45) is found to the data In Section 1.2.6, for the constant-velocitytarget, that is m¼ 1 in (4.1–44), the best-fitting polynomial, which is a straightline in this case, was indicated to be given by the fading memory g–h filter,whose weights g and h are given by (1.2-35a) and (1.2-35b) To find the best-fitting polynomial, in general the estimating polynomial is again approximated

least-by a sum of discrete-time orthogonal polynomials, in this case the orthonormaldiscrete Laguerre polynomials, which allow the discounted weightings for thesemi-infinite set of data The resulting best-fitting discounted least-squares

polynomial fit is given by (7.2-5) in recursive form for the case where thepolynomial is of arbitrary degree m For m¼ 1, this result yields the fading-memory g–h filter of Section 1.2.6 Corresponding convenient explicit resultsfor this recursive fading-memory filter for m¼ 0, , 4 are given in Table 7.2-2.

In reference 5 (4.1-32) is given for the case of a time-varying trajectorymodel In this case M, T, and  all become a function of time (or equivalently n)and are replaced by Mn and Tn and ðtn; tn1Þ, respectively; see pages 172,

173, and 182 of reference 5 and Chapter 15 of this book, in which the varying case is discussed

time-From (4.1-1) we see that the results developed so far in Section 4.1, andthat form the basis for the remaining results here and in Chapters 5 to 15, applyfor the case where the measurements are linear related to the state vectorthrough the observation matrix M In Section 16.2 we extend the results ofthis chapter and Chapters 5 to 15 for the linear case to the case where Yn isnot linearly related to Xn This involves using the Taylor series expansion tolinearize the nonlinear observation scheme The case where the measurementsare made by a three-dimensional radar in spherical coordinates while thestate vector is in rectangular coordinates is a case of a nonlinear observationscheme; see (1.5-2a) to (1.5-2c) Similarly, (4.1-6) implies that the targetdynamics, for which the results are developed here and in Chapters 5 to 15,are described by a linear time differential equation; see Chapter 8, specifically(8.1-10) In Section 16.3, we extend the results to the case where thetarget dynamics are described by a nonlinear differential equation In thiscase, a Taylor series expansion is applied to the nonlinear differentialequation to linearize it so that the linear results developed in Chapter 4 can

be applied

There are a number of straightforward proofs that the least-squares weight is

n;nand set the result equal to zero to obtain

den n;n

¼ TT½YðnÞ n;n ¼ 0 ð4:1-46Þ

n;n yields (4.1-32) as we desired to show

In reference 5 (pp 181, 182) the LSE weight given by (4.1-32) is derived bysimply putting (4.1-31) into another form analogous to ‘‘completing thesquares’’ and noting that e n;nÞ is minimized by making the only termdepending on W zero, with this being achieved by having W be given by(4.1-32) To give physical insight into the LSE, it is useful to derive it using ageometric development We shall give this derivation in the next section Thisderivation is often the one given in the literature [75–77] In Section 4.3 (andChapter 10) it is this geometric interpretation that we use to develop what iscalled the voltage-processing method for obtaining a LSE without the use of thematrix inversion of (4.1-32)

4.2 GEOMETRIC DERIVATION OF LEAST-SQUARES

SOLUTION

We start by interpreting the columns of the matrix T as vectors in ans-dimensional hyperspace, each column having s entries There are m0 suchcolumns We will designate these as t1; ; tm 0 For simplicity and definitenessassume that s¼ 3, m0¼ 2, and n ¼ 3; then

24

3

5 t2 ¼ T0

2T

24

3

and

X3¼ x3_x

 

ð4:2-7Þ

In Figure 4.2-1 we show the vectors t1, t2, and Yð3Þ The two vectors t1and t2define a plane Designate this plane as Tp (In general Tp is an m0-dimensionalspace determined by the m0 column vectors of T ) Typically Yð3Þis not in thisplane due to the measurement noise error NðnÞ; see (4.1-11).

Let us go back to the case of arbitrary dimension s for the column space of Tand consider the vector

pT ¼

p1

p2

ps

2664

377

From (4.2-8) we see that the vector pT is a linear combination of the columnvectors of T Hence the vector pT is in the space defined by Tp Now the least-squares estimate picks the Xn that minimizes eðXnÞ, defined by (4.1-31) That

is, it picks the Xn that minimizes

eðXnÞ ¼ ðYðnÞ T XnÞTðYðnÞ T XnÞ ð4:2-9ÞApplying (4.2-8) to (4.2-9) gives, for the three-dimensional case beingconsidered,

eðXnÞ ¼X3

i¼1

ðyi piÞ2 ð4:2-10ÞFigure 4.2-1 Projection of data vector Yð3Þ onto column space of 3
2 T matrix.Used to obtain least-squares solution in three-dimensional space (After Strang [76].)

But this is nothing more than the Euclidean distance between the endpoints ofthe vectors pT and Yð3Þ, these endpoints being designated respectively as p0and

Y0 in Figure 4.2-1

The point p0 can be placed anywhere in the plane Tp by varying Xn Fromsimple geometry we know that the distance between the points Y0 and a point p0

in the plane Tpis minimized when the vector joining these two points is made to

be perpendicular to the plane Tp (at the point p0 on the plane Tp) That is, theerror vector

ðTzÞTðYð3Þ 3;3Þ ¼ 0 ð4:2-13Þ

or equivalently, sinceðTzÞT ¼ zTTT

zTðTTYð3Þ TT

3;3Þ ¼ 0 ð4:2-14ÞBecause (4.2-14) must be true for all z, it follows that it is necessary that

TT 3;3¼ TTYð3Þ ð4:2-15ÞThe above in turn yields

3;3¼ ðTTTÞ1TTYð3Þ ð4:2-16Þfrom which it follows that

^

W ¼ ðTTTÞ1TT ð4:2-17Þ

which is the expression for the optimum LSE weight given previously by(4.1-32), as we wanted to show

Although the above was developed for m0¼ 2 and s ¼ 3, it is easy to see that

it applies for arbitrary m0 and s In the literature the quantity ðTTTÞ1TT is

often referred to as a pseudoinverse operator [78] This because it provides thesolution of YðnÞ¼ T Xn (in the least-squares sense) when T is nonsingular, as

it is when s > m0, so that T1 does not exist and Xn¼ T1YðnÞ does notprovide a solution for (4.1-31) The case where s > m0 is called theoverdeterministic case It is the situation where we have more measurements

s than unknowns m in our state vector Also the LSE given by (4.2-16), orequivalently (4.1-30) with W given by (4.1-32), is referred to as the normal-equation solution [75, 76, 79–82] Actually, to be precise, the normal equationare given by a general form of (4.2-15) given by

TT n;n ¼ TTYðnÞ ð4:2-18Þ

which leads to (4.1-30) with W given by (4.1-32)

A special case is where T consists of just one column vector t For thiscase

^

W¼ ðtTtÞ1tT

¼ tT

By way of example consider the case where

Yni¼ MXniþ Nni ð4:2-21Þwith each term of the above being 1
1 matrices given by

Yn1¼ ½yni ð4:2-21aÞ

target modeled as being stationary For this example

t¼

11

.1

266

37

then

n;n ¼1s

Xs i¼1

which is the sample mean of the yi’s, as expected

Before proceeding let us digress for a moment to point out some otherinteresting properties relating to the geometric development of the LSE Westart by calculating the vector pT for the case X3 3;3 Specifically,

3;3 given by (4.2-16) into (4.2-8) yields

pT ¼ TðTTTÞ1TTYð3Þ ð4:2-24ÞPhysically pT given by (4.2-24) is the projection of Yð3Þ onto the plane Tp; see

T is added to indicate the space projected onto.]

The matrix I P, where I is the identity matrix (diagonal matrix whoseentries equal one), is also a projection matrix It projects Yð3Þ onto the spaceperpendicular to Tp In the case of Figure 4.2-1 it would project the vector

Yð3Þ onto the line perpendicular to the plane Tp forming the vector

the general form given by (4.2-24) it projects YðnÞ onto the column space of T[76].

A special case of interest is that where the column vectors ti of T areorthogonal and have unit magnitude; such a matrix is called orthonormal Toindicate that the tihave unit magnitude, that is, are unitary, we here rewrite tias

t1; ; tm0 Finally when T is orthonormal so that (4.2-29) applies, (4.2-16)becomes, for arbitrary m0,

n;n ¼ TTYðnÞ ð4:2-34Þ

A better feel for the projection matrix P and its projection pTis obtained by firstconsidering the case m0¼ 1 above for which (4.2-31) and (4.2-32) apply

Equation (4.2-32) can be written as

1YðnÞ as given by(4.2-35)

Physically the amplitude of the projection of YðnÞonto the unitary vector ^t1isgiven by the vector dot product of YðnÞ with ^t1 This is given by

^t1 YðnÞ¼ k^t1k  kYðnÞk cos 

¼ k YðnÞk cos  ð4:2-36Þwhere use was made in the above of the fact that ^t1 is unitary so thatk^t1k ¼ 1; k Ak implies the magnitude of vector A, and  is the angle betweenthe vectors ^t1and YðnÞ If ^t1is given by the three-dimensional t1 of (4.2-2) and

YðnÞ by (4.2-4), then the dot product (4.2-36) becomes, from basic vectoranalysis,

^t1 Yn¼ t11Y1þ t21y2þ t31y3 ð4:2-37Þ

For this case ti1 of (4.2-2) is the ith coordinate of the unit vector ^t1 in somethree-dimensional orthogonal space; let us say x, y, z In this space thecoordinates x, y, z themselves have directions defined by respectively the unitvectors i, j, k given by

i¼

100

24

3

010

24

3

001

24

3

Figure 4.2-2 illustrates this dot product for the two-dimensional situation Inthis figure i and j are the unit vectors along respectively the x and y axes.Let us now assume that t1 is not unitary In this case we can obtain theprojection of YðnÞ onto the direction of t1 by making t1 unitary To make t1unitary we divide by its magnitude:

^t1 ¼ t1

But the magnitude of t1, also called its Euclidean norm, is given by

pt¼ t

T 1ffiffiffiffiffiffiffiffi

tT

1t1p

4.3 ORTHONORMAL TRANSFORMATION AND

VOLTAGE-PROCESSING (SQUARE-ROOT) METHOD FOR LSE

We will now further develop our geometric interpretation of the LSE We shallshow how the projection of YðnÞonto the Tp space can be achieved without theneed for the matrix inversion in (4.2-24) This involves expressing the columnvectors of T in a new orthonormal space, not the original x, y, z space We will

n;n can in turneasily be obtained without the need for a matrix inversion This approach iscalled the voltage-processing (square-root) method for obtaining the least-

Figure 4.2-2 Projection of vector YðnÞonto unit vector t1

squares solution Such approaches are less sensitive to computer round-offerrors Hence these methods should be used where computer round-off errorsare a problem With the rapid development of microcomputer chips that aremore accurate (e.g., 32- and 64-bit floating-point computation chips), thisproblem is being diminished Two voltage-processing algorithms, the Givensand Gram–Schmidt offer the significant advantage of enabling a high-throughput parallel architecture to be used The voltage-processing methodswill be discussed in much greater detail in Chapters 10 to 14 Here we introducethe method Specifically, we introduce the Gram–Schmidt method This method

is elaborated on more in Chapter 13 Chapters 11 and 12 respectively cover theGivens and Householder voltage-processing methods

As done in Figure 4.2-1, we shall for simplicity initially assume t1, t2, Xn, T,and YðnÞare given by (4.2-1) through (4.2-4) If the column space of T given by

t1 and t2 were orthogonal and had unit magnitudes, that is, if they wereorthonormal, then we could easily project Yð3Þonto the Tpplane, it being given

by (4.2-33), with m0¼ 2 and n ¼ 3 In general t1and t2will not be orthogonal.However, we can still obtain the desired projection by finding from t1 and t2anorthonormal pair of vectors in the Tp plane Designate these unit vectors in the

Tpplane as q1and q2 We now show one way we can obtain q1and q2 Pick q1along t1 Hence

plane; see Figure 4.3-1 Then the projection of Yð3Þonto the Tp plane is given

by the sum of the projection of Yð3Þ onto q1 and q2 as in (4.2-33), that is,

Yð3Þ¼ qT

1Yð3Þq1þ qT

2Yð3Þq2þ qT

3Yð3Þq3 ð4:3-3ÞThe above can be written as a column matrix in this new q1, q2, q3orthonormalcoordinate system given by

37

3

where q is the amplitude of the component of q along the x coordinate, q

that along the y coordinate, and q3jthat along the z coordinate In the new q1,

q2, q3 coordinate system

q1 ¼

100

24

3

5 q2 ¼

010

24

3

5 q3¼

001

24

3

where the coordinates of q1, q2, q3 are expressed in the x, y, z coordinatesystem Note that because the columns of (4.3-8) represent orthogonal unitvectors

3

5 ¼ q

T 1

qT 2

qT3

24

35Yð3Þ¼ QTYð3Þ¼ QT

y1

y2

y3

24

3

5 ð4:3-12Þ

For convenience and clarity let Yð3Þexpressed as a column matrix in the new q ,

q2, q3 coordinate system be written as

is the sought-after orthonormal transformation matrix that transforms Yð3Þfromits representation in the x, y, z orthonormal system to Yð3Þ0 given in the q1, q2, q3orthonormal coordinate system The rows of F are the columns of Q and hencesatisfy the properties given by (4.3-9) to (4.3-11) for an orthonormaltransformation matrix, that is,

n;n Thisn;n that minimizes (4.1-31), or for the special case whererange-only measurements yi are being made by (4.1-37) Let

k FE k2¼ ðFEÞTFE¼ ETFTFE ð4:3-20ÞApplying (4.3-16) to (4.3-20) yields

k FE k2¼ ETE¼k E k2 ð4:3-21Þ

Hence applying an orthonormal transformation to E does not change its

n;n that minimizesk FE k is the same as findingn;n that minimizesk E k From (4.3-18)

FE n;n FYðnÞ

n;n FYðnÞ ð4:3-22ÞFor simplicity let us again assume s¼ 3 and m0¼ 2 with (4.1-1) to (4.1-2a),(2.4-1a) and (2.4-1b) applying For this case E is a 3
1 matrix Let

3

5 ¼ tt1121 tt1222

t31 t32

24

3

We now apply F to (4.3-23) First we apply F to the vectors t1and t2of T Inpreparation for doing this note that, due to our choice of q1, q2, and q3, t1 isonly composed of a component along q1 Thus t1 can be written as

t1 ¼ u11q1þ 0  q2þ 0  q3 ð4:3-24Þ

In turn t2 consists of components along q1 and q2 so that it can be written as

t2 ¼ u12q1þ u22q2þ 0  q3 ð4:3-25ÞFinally Yð3Þconsists of the components y01, y02, y03 along respectively q1, q2, q3

so that it can be written as

Yð3Þ¼ y01q1þ y02q2þ y03q3 ð4:3-26ÞThe values of uijare the amplitudes of the unit vectors q1, q2, q3of which tj

is composed These amplitudes can be easily obtained by applying an pressing such as (4.2-36) This is done in Chapter 13 Now from (4.3-14) thetransformation F applied to the column matrices t1 and t2 transforms thesecolumn vectors from being expressed in the x, y, z coordinate system to the q1,

ex-q2, q3orthogonal coordinate space On examining (4.3-24) and (4.3-25) we thussee that the column matrices for t01 and t2 in the q1, q2, q3 space are given by

Ft1¼

u1100

243