%

^k c NBS TECHNICAL NOTE 1021

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards
NATIONAL BUREAU OF STANDARDS

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.

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Fourier Transformation *w 80

c,ec

of the Nonlinear VOR Model °°

T53

to Approximate Linear Form Kf

Dominic F. Vecchia

Statistical Engineering Division

National Engineering Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Prepared for: Department of Defense

Calibration Coordination Group

,<:>..*

U.S. DEPARTMENT OF COMMERCE, Philip M. Klutznick, Secretary

Luther H. Hodges, Jr., Deputy Secretary

Jordan J. Baruch, Assistant Secretary for Productivity, Technology and Innovation

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued June 1980
NATIONAL BUREAU OF STANDARDS TECHNICAL NOTE 1021

Nat. Bur. Stand. (U.S.), Tech. Note 1021, 32 Pages (June 1980)

CODEN: NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1980

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $2.00 (Add 25 percent additional for other than U.S. mailing)
TABLE OF CONTENTS

Page

1 . INTRODUCTION 1

2. NOTATION 2

3. CONTINUOUS TIME NONLINEAR MODEL 3 3.1 Fourier Representation of General Signal 4 3.2 Fourier Representation of VOR Signal 5

4. DISCRETE TIME NONLINEAR MODEL 8

4.1 Fourier Coefficients of VOR Signal 8

4.2 Spectrum of VOR Noise Model 10

4.3 Phase Spectrum Transformation 11

5. LINEAR MODEL FOR PHASE SPECTRUM 13

5.1 VOR Bearing Angle Estimator 14

5.2 Discussion 16

6. ROBUSTNESS OF ASSUMPTIONS 17

6. 1 Signal Offset 17

6. 2 Signal Amplitude 18

7. ACKNOWLEDGMENT 18

REFERENCES 19

APPENDIX 20

LIST OF TABLES

1. Bessel Functions 9

2. Squared Bessel Functions and Weights 15

1X1

FOURIER TRANSFORMATION OF THE NONLINEAR VOR

MODEL TO APPROXIMATE LINEAR FORM

by

Dominic F. Vecchia

This technical note describes a method for transforming a particular nonlinear regression model to a form which is approximately linear in the unknown parameters. The technique involves

computation of the Fourier coefficients for a set of sample data

and uses phase variables to estimate the parameters. The phase

spectrum transformation is employed to obtain bearing angle estimates for a model associated with the Very-High-Frequency OmniDirectional Range (VOR) aircraft navigation system. The transformation provides a model linear in relevant phase parameters.

Thus, estimation of VOR bearing angle utilizes existing

statistical theory. Finally, it is shown that certain generalizations of the VOR model also are reduced to approximate linear form

by the phase spectrum transformation.

Key Words: Fourier coefficients; linear model; nonlinear model;

phase spectrum transformation; spectrum; VOR aircraft navigation

system; white noise.

1. INTRODUCTION

There are many different reasons for making a transformation of variables

in the statistical analysis of data. This technical note discusses an unusual

type of transformation useful in connection with a particular nonlinear

regression model for audiofrequency signals from the Very-High-Frequency

Omni-Directional Range (VOR) air navigation system. The model, which is

considered in a more general form than the VOR requires, is intrinsically

nonlinear in the unknown parameters. By intrinsically nonlinear we mean that

a single observation cannot be transformed into linear form. For example,

consider the two models

Y = exp(B]_ + 62 x + £ )

Y = [e 1 /(S 1 -6

2 > ] [exp(-B2 x) - exp(-BlX )] + e ,

where 3} and $2 are unknown parameters, x is an independent variable, and e is a random error term. Both models are nonlinear in &\ and 32

, but the first

is intrinsically linear because the transformed variable lnY is linear in $1

and 3 2

« However, the second model is intrinsically nonlinear because it is

impossible to convert the model into a form linear in the parameters. For a discussion of these concepts see reference [1].

Usually, it is not useful to transform a model of the second type because

it remains nonlinear whatever transformation is applied. The transformation

introduced in this paper is unusual because it involves computation of the
Fourier coefficients of the sample data and uses phase variables to estimate

parameters. For this reason the procedure to be described is called the phase

spectrum transformation. The method will be demonstrated for the model

specific to the VOR air navigation system.

The VOR is a fundamental component of the present-day air navigation

system. A feature of the VOR system which provides much versatility for

defining controlled airways is that the facility emits an infinite number of

radial courses providing aircraft bearing information. This information is

contained in the phase angles of two 30 Hz audiofrequency signals. The first

has a constant phase at all points around a VOR station and is called the

reference signal. The other, called the variable signal, has a phase equal to

the bearing angle to (or from) the VOR transmitter. In the aircraft, bearing

information is determined by measuring the phase difference between the two

component signals.

The importance of the accuracy of bearing angle estimation devices cannot, of course, be overstated. At present, measurement accuracy for VOR test

instruments depends upon calibration with commercial equipment designed for

that purpose. As system requirements become more severe because of increasing

traffic in the air lanes, it is clear that both the accuracy and precision of

present VOR calibration equipment will require additional scrutiny. Hopefully, this will increase the safety and efficiency of aircraft operations.

For a general discussion of the VOR system, see [2].

This paper presents a statistical technique for estimation of VOR bearing

angle and gives the corresponding precision of the estimated angle. The

general method is based on regression analysis of samples taken by a sampleand-hold amplifier and an analog-to-digital converter. The method provides a linear model in relevant phase parameters. Thus, the bearing angle estimation

utilizes existing statistical theory.

In section 3 of this technical note, the nonlinear regression model is

represented in continuous time. Fourier coefficients are obtained for the

noise-free signal, and results for the special case of the VOR signal are

stated. The results for the VOR application are extended without proof to the

discrete time sample model in section 4. The spectrum for the VOR model with

noise is derived and the phase spectrum transformation is defined. In section

5 the approximate linear model for transformed variables is used to estimate

unknown parameters and the usefulness of the transformed model is discussed.

Section 6 is a limited discussion concerning the properties of estimators if some assumptions specific to the VOR model are invalid.

2. NOTATION

For X, a random variable with probability density f (x) , we denote the mean and variance of X by E[X] and Var[X] , and the covariance between X and a random variable Y is denoted by Cov[X,Y].

Vectors and matrices are denoted by underlined letters, for example, _9 and _V. If 9 denotes a vector, then 9' will denote the transpose of _9. An

estimator of 9 will be denoted by 9.
3. CONTINUOUS TIME NONLINEAR MODEL

The nonlinear model considered in this report will be represented, initially, as a continuous function of time. In a later section the mathematical

results obtained for the continuous time model are extended to the case where

the data are equally spaced observations from the continuous signal.

To represent the deterministic component of the model requires two periodic functions described below in (3.0.1). These functions are added to

obtain the expected (ideal) value of the output signal in a nonlinear regression model. The two component functions are v(t;M) = «! cosUTTf^t+fi) + <b {

] and (3.0.1)

s(t;S,J>) = a2 cos[2TTf 2 (t+6) + <j> 2 + 6 sin[2irf ^t+6) + <j> 3

] ] .

The waveform generated by the sum of v(t;S,<J>) and s(t;<5,<f>)> with some parameters assumed known, may be used to represent the ideal audiofrequency signal

for the VOR aircraft navigation system. In this context, v(t;6,_^) is called

the variable phase signal , and s(t;<5,_<J>) is the frequency modulated subcarrier

signal . The frequency modulating sinusoid, contained in the argument of

s(t;6,j0, is called the reference phase signal . Equations for the component

signals have been presented in a more general form than the VOR application

requires. However, the above terminology is used throughout this paper. Following are descriptions of the model parameters:

o^ = amplitude of variable phase signal;

a2 = amplitude of subcarrier signal;

ll = variable (and reference) signal frequency;

f 2 = subcarrier frequency;

3 = modulation index;

<5 = arbitrary fixed time offset;

<J>1 = phase angle of variable phase signal;

<J> 2 - phase angle of subcarrier signal;

<j>3 = phase angle of reference phase signal.

The fixed time offset, 6, is included in (3.0.1) because the output signal

will be observed and sampled from an unknown starting point in the waveform.

We cannot, in general, be assured that observation of the signal begins at a zero crossing on the time axis.

Realistically, measurement of the composite signal involves random

measurement error in some form. In this paper the random error process, e(t),

is assumed to be additive white noise [3], and the resulting process Y(t) can
be represented by

Y(t) = V + v(t;6,_£) + s(t;M) + e(t) , (3.0.2)

where P is a fixed but unknown offset. Unless otherwise specified, e(t) is

not assumed to be Gaussian white noise. However, for the distributional

result obtained in the appendix, we require normality and independence of the

discrete time error series to be as described in subsection 4.1.

In subsection 3.2, a form of the model specific to VOR navigation system

is discussed. For this application some of the parameters in the general form

of the model are assumed to be fixed, known values. On this basis a statistical method for VOR bearing angle estimation will be derived. Because

the relevant angle for the VOR application is C 4*1 — 4>3 ) » it is sufficient to

consider a reparameterized form of the model where we define new parameters _8 by

©1 = 4>l — <t>3

9 2 = 2TTf

L 6 + <f>3

83 = 27rf 2

6 + <j>2 .

For this parameterization the general model becomes

Y(t) = u + v(t;_6) + s(t;_6) + e(t) (3.0.3)

where

v(t;_6) = <xi cos[2Trf 1 t + 9j + 2 ]

and

s(t;_9) = ot2 cos[2irf2t + 83 + 3 sin[2irf 1 t + 82 ] ] .

In the following section, we obtain the Fourier sine and cosine transforms of

E[Y(t)]=y+v(t;_8)+s(t;j3) under the assumption that f 2 =mfi« Hence, E[Y(t)] is

periodic with period (1/fj). Utilizing the general result, the specific

transform for the VOR signal is determined in subsection 3.2.

3.1 Fourier Representation of General Model

Let y(t;_8) denote the expected value of Y(t). The deterministic function

y(t;_8) is given by

y(t;_8) = v + v(t;_8) + s(t;_8).

Suppose that the frequencies f^ and f 2 in the definition of v(t;_8) and s(t;_8) are such that f 2 =mf^ for some positive integer m. Thus, y(t;_6) is periodic

with period (1/f^). Under this assumption, the Fourier coefficients of y(t;_8) can be obtained from the real and imaginary parts of the complex integral
l/2f 1 s k

= 2fj / y(t;_8) exp[i2uf 1 kt] dt, k=0,±l,±2, ... -l/2fj

Let ak and b^. denote the real and imaginary parts of s k , so that s k

= a^+ib^.

That is, a k and b^ are the Fourier cosine and sine transforms, respectively,

of y(t;_8). In the appendix, it is shown that the Fourier coefficients ak and

bk

for k=0,l,..., are

and

r

V , k=0

a k

=

"S al cos [8^+62] + a2 ak

(s) , k=l

^ a2 ak

(s) , k>2

'0

, k=0

-o^ sin[ 8^+62] - a2 b^Cs) , k=l

(3.1.1)

bk =<

v.-a2 t^Cs) , k>2

where, for k>l,

ak(s) = J k_m(B)cos[83 + (k-m)82

] + J_k

_m( B)cos [ 83 - (k+m)82 ]

and

b^s) = J k

_n (B)sin[83 + (k-m)82

] + J-k

_m( B)sin[ 83 - (k+m)82 ]

The notation Jn

(z) denotes a Bessel function of the first kind [4],

Equations (3.1.1) represent the Fourier coefficients for the mean value

function of the general continuous time model where the subcarrier frequency

f2 is an integer multiple of fj, the frequency of the variable phase signal.

For the VOR model, where some parameters in the general model are assumed

known, we will see that the Fourier coefficients can be greatly simplified.

The form of the coefficients for this special case will suggest a method for

estimating the unknown parameters.

3.2 Fourier Representation of VOR Signal

The VOR audio frequency waveform consists of a 30 Hz variable phase

signal linearly added to a frequency modulated 9960 Hz subcarrier signal. The

modulation index for the reference phase signal is assumed to be fixed and

known, as are the amplitudes of each signal. Specifically, parameter values

assumed to be known are:
u =

<*! = 2 1/2

a2 = 2^2

f x = 30

f 2 = 9960 (so ra = f 2 /fi - 332)

3 = 16 .

These specifications define the VOR model

Y(t) = y(t;_6) + e(t)

where

y(t;_9) = 2 1/2 {cos[2iT30t + B

l + 9 2 ] + cos[2Tr9960t + G 3 + 16sin[2Tr30t + 9 2 ]]}.

Noting that y(t;9) is nonlinear in the unknown parameters 9"=[ 8^,

9

2

,

9

3 ]

, we will show in section 4 that the spectrum of Y(t) can be used to transform a set of sample data to new observations satisfying a model approximately

linear in 8. The transformation to linearity will depend on the simplified

form of the Fourier coefficients for y(t;9) when known values of parameters in

the VOR model are substituted in the general equations (3.1.1).

Substituting known values in the expressions for ak

(s) and ty^s), we obtain for k>l,

ak

(s) = Jk_3 32 (16)cos[93+(k-332)e2

] + J_k_332 ( 16)cos [

9

3 -(k+332)

9

2 ]

and

bk

(s) = J k

_332 (16)sin[93+(k-332)92

] + J_k

_ 332 ( 16) cos [

8

3 -(k+332)

9

2

]

.

We need the following results for Bessel functions of the first kind.

Lemma 3.1 ; [4, page 358] For integer n, J n

(z) satisfies

J_n

(z) = (-l) n Jn

(z) .

Lemma 3.2 ; [4, page 365] For fixed z, as n-*-00 through real positive values,

Jn

(z) S (2Tin)- 1/2 (ez/2n) n .
From lemmas 3.1 and 3.2, it follows that

J_k

_ 332 (16) = (-l) k+332Jk+332 (16)

= (-l) k+332 [2ir(k+332)]- 1/2 [16e/2(k+332)] k+332

=

Clearly, the value of Jk+33 2(16) i- s immeasurably small. Note, for example,

333 that if k=l, J k+332 (16) ~ • 02 (

065 ) • Similarly, for small k, we have that

Jk-332 - °-

Using the above results, we get

and

ak

(s) =

, k=l

Jk_332 (16)cos[e3 + (k-332)92 ] , k^2

T>k(s) =

, k=l

Jk_ 332 (16)sin[63 + (k-332)92 ] , k>2

Substituting in the general expressions (3.1.1), the approximate Fourier

coefficients for y(t;9) in the VOR model become

and

r o

ak M

1/2 2 iz cos[e 1 + e 2 ]

k=0

, k=l

^_ 2 i/z Jk

_ 332 (16)cos[63 + (k-332)G2 ] , k>2

r

1/2 bk 2< -2 W/ sintSj + 9 2 ]

, k=0

, k-1

v

-2 1/2 Jk

_332 (16)sin[63 + (k-332)G2 ] , k>2.

Because omitted terms are negligible, in the following sections we consider

the Fourier coefficients to be exact.
DISCRETE TIME NONLINEAR MODEL

The general model and corresponding Fourier transforms introduced in

section 3 will facilitate a later discussion about errors in assumptions for

the VOR model. Because the approximations discussed in the previous section

depend on the particular values of some parameters in the general model, the

phase spectrum transformation will be developed only for the VOR

specifications. In subsection 4.1 we consider the discrete time analogs of

the VOR model and corresponding Fourier coefficients, since a digital phase

estimation technique is desired.

4.1 Fourier Coefficients of VOR Signal

Let Y-;, j=l,...,N, be N equally spaced observations from one period of

the continuous time VOR series. For convenience, N is assumed to be even in

the results to follow. The sample model for VOR applications can be written

Yj = yjd) + ej I

_ > J»k=l,...,N

E[ej]=0; Var[e-j]=a 2

; E[ejek

]=0 if j*k J

where

yj(_9) = 2 1/2 {cos[2tt(j-1)/N + d

1 + 6 2 ]

+ cos[2tt332(j-1)/N + 9 3 + 16sin[2ir( j-l)/N + 62

] ]}.

2 The ei's denote un correlated random error terms with unknown variance a .

The Fourier coefficients (sk

= a^+ib^) for yj(_9) are given by

N

s k

= (2/N) I yi (_6) exp[i2TTk(j-l)/N] . J-l

The ak and b^ follow directly from the continuous time model. Excluding the

coefficients for k=0, which are not useful to estimate _9, equations for ak and

bk are

ak

2 1/2

cos [6! + 8 2 ] , k=l

2 1/2 Jk_3 32 (16)cos[e3 + (k-332)92 ], k=2,3, . . . ,N/2,

and (4.1.1)

f -2 1/2 sinlSj + e 2 ] , k=l

. -2 1/2 Jk

_ 332 (16)sin[e3 + (k-332)62 ], k=2,3, . .

. ,N/2-l . K

Recall that these coefficients can reasonably be considered exact expressions

because omitted terms are negligible. The interesting feature of the

equations for ak and bj^ is the form of the phase of the k

th harmonic. If we let q represent the phase at a chosen harmonic, then the basic equation for

8
q is tan(q)=-b/a. For convenience, we have dropped the subscripts on q, b,

and a. Because Arctan(-b/a) gives the same value for -b and -a as for b and

a, the full solution for q in the interval (-t,tt] is the following

[5, page 12] : r

i* = <

Arctan(-b/a) , a>0

Arctan(-b/a)-ir, a<0, b>0

Arctan(-b/a)+iT, a<0, b<0 (4.1.2)

-tt/2 , a=0, b>0

tt/2 , a=0, b<0

arbitrary , a=0, b=0

The notation Arctan(x) is used to denote the principal value, so that

-ir/2<Ar ctan(x)<iT/2. If arctan(x) denotes any angle whose tangent is x, then

it follows that

arctan(-b/a) = q* + j2ir (4.1.3)

where j is an arbitrary integer. From (4.1.3) and the expressions for a^ and

b^, it then follows that there exist integers Y^ such that qk(_9) =q^*+Yic 2'T is

given by

qk<_8) =

»1 + 9 2 , k=l

(k-332)92 + 9 3 , |k-332|<K , (4.1.4)

where K is a constant chosen to assure that Jk-332^-'-^) ^ s non-negligible. A

zero value for the Bessel function leads to an arbitrary phase because

ak

=

kk=0* Table 1 lists the values of J n

(16) for n=0,...,24. In a later

section it will be shown that a value of K^IO is sufficient for the proposed

estimation of 9.

Table 1. Bessel Functions

n Jn

(16) n Jn

(16)

12

3

4

56

78

9

10

11

12

-.1748990739

.0903971756

.1861987209

-.0438474954

-.2026415317

-.0574732704

.1667207377

.1825138237

-.0070211419

-.1895349656

-.2062056944

-.0682221523

.1124002349

13 .2368225047

14 .2724363352

15 .2399410820

16 .1774531934

17 .1149653049

18 .0668480795

19 .0354428740

20 .0173287462

21 .0078789915

22 .0033536066

23 .0013434266

24 .0005087450
The multiples of 2ir indexed by Y in the expression for q^(_9) are

necessary to adjust the qk

* from the interval ( — tt, tt] to the interval (- 00

, 00 ).

Because the 9's will represent unknown parameters, the Y's are not known in

general. If we assume, however, that 02^0, it is clear from (4.1.4) that the

Y's must satisfy

qj * + Yj2tt < qj * + Y±

2tt

, j<i ; i*l*j ,

so (4.1.5)

(Yj " Yi)2TT < q ± * - qj * , j<i ; i*l#j .

This implies that for | k—332 | <K, the Y^'s can be uniquely determined if, for

example, the following conditions are assumed:

-ir<9

1 +92

<TT

62 >0

-TT<e3

<TT .

With these constraints Yi = Y332= and successive values of Yk near k=332

are determined from (4.1.5). For the extension of the results to the VOR

signal with error we will assume that the Y's are known.

4.2 Spectrum of VOR Noise Model

In the previous subsection Fourier coefficients were obtained for the

deterministic component of the VOR process. For the nonlinear VOR regression

model:

Yj =

yj (_e_) + ej 1

2

> J,k=l,...,N

E[e-j]=0; Var[ej]=a ; E[ejek

]=0 if j*k J

the Fourier coefficients of y i(®) , represented by s^a^+ib^, were shown to

have phase values linear in _6. In this section we prove that random phase

variables derived from Fourier transformation of Y-j, j=l , . . . ,N are appropriately represented by a regression model linear in 9.

Letting the random variables S^=A^+iB}c

, k=l , . . . , N/2-1 , represent the

Fourier coefficients of Y-j, we have

N

Sk

= (2/N) I Yj exp[i2irk(j-l)/N]

j=l

N

= (2/N) I (y i (_9)+e

i )exp[i2TTk(j-l)/NJ

j-l

N

= s k + (2/N) I ej exp[i2irk(j-l)/N]

j=l

10
where sic =aic+ibjc are the Fourier coefficients for y-j(_9) given in (4.1.1). If we let the transform of the random error sequence be denoted by

N

gk + ihk

= (2/N) I eA exp[i2irk(j-l)/N] , j=l

it is well known [6, page 110] that the random variables gk and hk , k=l , . .

.

, N/2-1 are mutually un correlated and

E[gk ] = E[hk] =

Var[gk

] = Var[hk ] = (2/N)</

k=l,... , N/2-1

It therefore follows that the Ak and B k are uncorrelated and satisfy regression models

Ak

= ak + §k

B k

= bk + hk

k=l,... , N/2-1

Linearity of ar ctan(-bk/ak

) in _6 suggests that we consider the phase spectrum

of Yj to estimate _9. In the next section it is shown that the expected values

of phase variables are approximately linear in 8.

4.3 Phase Spectrum Transformation

The definition of phase random variables parallels the description of the

phase qk (_9) for the deterministic component y ^ (_9 ) • Phase variables will be

denoted by Qk and initially are defined using principal values in the interval

( — 7T , tt] . Define

0^.* = Arctan[-Bk/Ak ] , k=l , . . . ,N/2-l . (4.3.1)

In the appendix the distribution of the (\* , k=l , . .

.

, N/2-1 is determined when

the errors are Gaussian. The expected value of 0^.* is not obtained, but the

complexity of the distribution illustrates the usefulness of approximate

moments of Q^* which result from a suitable propagation of errors formula.

To conclude this subsection we obtain approximate formulas for the mean

and variance of Qk

*« These results are the basis for the linear models used

to estimate _9 in Section 5. As defined, Ak and Bk appearing in (4.3.1)

satisfy:

E[Ak

] = ak ; E[Bk] = bk

Var[Ak

] = Var[Bk] = (2/N)a'

Cov[Ak

,Bj] = all j,k

~\

V k,j=l,...,N/2-l

11
For values of k such that a^O^b^., it can be shown [7, page 333] that, to order

N , Qk

* has approximate mean and variance given by:

E[Qk

*] = Arctan[-bk

/ak ]

Var[Qk

*] 3 (2/N)rk

_2

a

2

where

(4.3.2)

2 2 2 rk

= ak + bk

2 , k=l

2J

2k_3 32 (16), |k-332|<K .

The value of K is chosen to assure that Jk-332(16) is nonzero. Note that

E[Qk*]-qk*, where the solution for qk

* in the interval ( — it

, tt] was given in

(4.1.2). Based on the discussion following (4.1.2) we can define Qk

=Qk*+Yk

2T

where the Yk's are (generally) unknown integers such that

E[Qk

] = qk (_9)

Var[Qk

] = (2/N)rk

_2

a

2

where

(4.3.3)

The method of estimating _9 to be outlined in Section 5 is based on the

above results. To the chosen degree of approximation , the important features

are:

1. E[Qk

] is linear in _6 for all permissible k.

2. Var[Qk ] is proportional to a

2 with known constant of proportionality.

3. For all permissible k, the Qk

's are mutually uncorrelated (assuming

white noise errors in the original model).

Because the linear model of Section 5 requires that Qk

be an observable random

variable, it will be necessary to assume that the Yk

's are known integers.

The Yk

's are used to adjust the computed values of the arctan function to

satisfy inequalities implied by (4.3.3). Though it is not obvious that this

correction can be accomplished with the C^, which are subject to error,

computer simulations indicate that the adjustment is possible if the

measurement error variance, a

2

, is small. Values used for a

2 are thought to

be representative of measurement precision for a new system designed to obtain

sample values Y^, j=l,...,N.

12
5. LINEAR MODEL FOR PHASE SPECTRUM

In the previous section approximate formulas for the means of phase

random variables for the nonlinear VOR model were shown to be linear in the

unknown parameters _9. Corresponding variance approximations are unequal at

the harmonics and proportional to a , but do not depend on other unknown

parameters. The additional observation that phase random variables are uncorrelated suggests that 9 and a may be estimated using a linear model in

with known error covariances. For the description to follow the reader is

reminded that expressions for the mean and variance of Q^ are not stated as approximations.

Consider the n=2+2K equations

r Q

1 + 8 2 + e1

, k=l

Qk =

\I (k-332)92 + 63 + ek , k=332-K, ... ,331,332,333, ... ,332+K

where e^ represents a random error term such that

E[ek]=0 ; Var[ek ]=(2/N)rk

- 2 2

;

Cov[ek ,£j]=0 if k*j ;

and where

(1) the Qk are observable random variables;

(2) the r^ are known constants.

2 (3) 9^, 9 2

, 93, and a are unknown parameters.

The model can be represented as a single matrix equation

Q = X9 + e E[e] = Cov[ e]=o V (5.0.1)

where the vectors and matrices are

Qi

Q332-K

Q

= Q331

Q332

Q333

Q332+K

x =

1 -K

-1

1

K

13
e =

e 332-K

e331

e 332

e 333

E 332+K

V = (1/N)

1

[J_K

(16)]- 2

• • •

[J (16) 1-2

[JK U6)]- 2

For these specifications of the linear model, unbiased estimators of 9 and a are given by [8, page 207]

= -lv\-lv "m-^-i (X'\I~ l X)~ l X'V~ L Q

» 2 a = J_ Q'[v~ 1 -V" 1 X(X'V~ 1 X)~ 1 X'V~ 1 ]q n-3

(5.0.2)

Variances and covariances of the 9's can be estimated by substituting the

estimator of c

2 in

Cov[9] = o

2 (X'V_1X)- 1 (5.0.3)

The quantities listed above are those needed for point estimation and

confidence intervals involving 9 and a

2

.

5.1 VOR Bearing Angle Estimator

According to the reparameterization of the original nonlinear model, the

relevant phase angle for VOR applications is 9^ . The estimator of 9^ can be

obtained by algebraic expansion of the matrix equation in (5.0.2), and its

corresponding variance is the first element of the square matrix in (5.0.3).

The estimator of VOR bearing angle and its estimated variance are

where

K

h = Ql " Ck I j Jj 2 (16)[Q3 32+j - Q332-J

Vad^] = (l/N)(l+CK

)a 2

K

CK

= [2 I j 2Jj 2 (16)]- 1 j=l

(5.1.1)

14
Note that the variance of the bearing angle estimator depends on the selected

number of phase values through C^. Because the Bessel function J-j(16)

approaches zero as j increases, it is clear that C^, and hence Var[ 9jJ , approaches a lower bound as K increases. In practice, computational speed

and/or memory constraints may require that only a few phase observations be

used to estimate 9j . Values of C -j listed in Table 2 suggest that K=10 is

sufficient to minimize C^ .

Table 2. Bessel Functions and Weights

Jj 2 (16) ;2 T 2

J ZJ/(16)

12

34

5

6

78

9 10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

0.0305896861

0.0081716494

0.0346699637

0.0019226029

0.0410635904

0.0033031768

0.0277958044

0.0333112958

0.0000492964

0.0359235032

0.0425207884

0.0046542621

0.0126338128

0.0560848988

0.0742215568

0.0575717228

0.0314896359

0.0132170213

0.0044686657

0.0012561973

0.0003002854

0.0000620785

0.0000112467

0.0000018048

0.0000002588

0.0000000000

0.0081716494

0.1386798547

0.0173034257

0.6570174461

0.0825794204

1.0006489577

1.6322534965

0.0031549718

2.9098037600

4.2520788410

0.5631657107

1.8192690450

9.4783478893

14.5474251288

12.9536376362

8.0613467818

3.8197191653

1.4478476965

0.4534872292

0.1201141759

0.0273766197

0.0054433896

0.0009547344

0.0001490789

61.2080

3.4032

3.0448

.5872

.5360

.2800

.1264

.1264

.0752

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

.0240

However, the apparent gain from using using only a few phase variates is

balanced by a corresponding loss in precision for estimating a^ . According

to the specifications for equipment designed to provide the sample values from

a VOR signal, it is likely that K=10 will provide adequate precision for

estimating 8^ .

The procedure described above can be used to achieve acceptable precision

bounds using only a few lines in the phase spectrum of the observed VOR

signal. An alternative method, which may require fewer observations, is to

obtain an estimate of Q\ from a few phase variables not adjacent to Q332 • To

15
illustrate this approach, values of j J.; 2 (16) that appear in C K are listed in

Table 2. The maximum value of this quantity occurs if j = 14, corresponding to

phase observations at k=318 and k=346. Clearly, to minimize the variance of an estimator of 8^ based on nonadjacent Q^'s, one should add observations in

order of decreasing values on j J -s (16). Thus, phase variate pairs would be

included in the order j=14, 15, 13, 16, etc., corresponding to k=(318,346),

(317, 347), (319, 345) ,(316,348) , etc. It is easy to show that the multiplier

analogous to Cj^ is already near the lower bound of Table 2 after only four

phase pairs are included to estimate 9^ . Hence, if estimation of a

2 is not

severely affected, a significant saving in computational requirements is

achieved using phase variables nonadjacent to Q332 • Assuming that N is

moderately large, the gain is especially desirable if Discrete Fourier

Transforms are used to obtain the Q^.

5.2 Discussion

It can be argued that a transformation of the nonlinear model to linear

form is unnecessary because suitable nonlinear least squares methods can be

applied directly to the sample data. These methods are iterative and require

initial estimates of the unknown phase angles. However, for VOR applications,

software for estimating unknown angles will be implemented on desktop

computers, which will also serve as controllers in VOR calibration systems.

In this case, the phase spectrum transformation and subsequent estimation of

phase angles using the linear models approach is computationally efficient and

is to be preferred if there are no serious deficiencies in the technique.

From a mathematical standpoint, estimation of _8 using the phase spectrum

transformation depends on two related assumptions. First, it was implicitly

assumed that formulas for the mean and variance of phase random variables

approximate the true mean and variance to the extent that departures from the

correct values are negligible. A second assumption, which requires further

study, concerns the adjustment of computed values of the Arctan function.

Recall that Qk^Qk*"^^ 71

' where Q^* is an observable random variable in the

interval ( — tt , tt] . In the derivation of the estimators of _9 and a

2

, it was assumed that the integers (Y^} can be determined from the data. If the Y^'s are not known, then 8 and a

2 in (5.0.2) are not estimators because they are

not observable.

A computer simulation of the VOR signal with independent Gaussian errors was used to determine if the assumptions described above severely limit

applicability of the phase spectrum transformation. Results of this

investigation indicate that linearity of the mean and the ability to adjust

the Arctan function depend, primarily, on variability in measurement errors. The technique was applied consistently for values of less than .001. For

somewhat greater values of a

2

, straightforward determination of the Y's is

usually successful, and indications are that the method can be refined,

perhaps by developing a search technique for the Y's. Analyses were conducted

with N=1024 time samples.

It should be emphasized that values of o^ used in computer simulations

are believed to be representative of expected variability of measurement

errors for a VOR audiofrequency generator currently being constructed.

Computations based on simulated data were used to affirm the mathematical

results of previous sections and are not reported here.

16
To conclude this section we remark that hardware specifications for VOR

generators and the method for sampling the continuous time signal will togeth- er determine the accuracy and reliability of specified values for frequency,

amplitude, offset, and modulation index. Because properties of measurement

errors, such as stability and independence, can be affected by hardware and

software specifications, examination of estimated residuals for the time

samples can be useful to validate assumptions about sampling errors. Estimated residuals for the VOR model are given by

s

j = Yj " y j

( i}

» j =i »---»N •

Plots of residuals and/or tests for serial correlation can be expected to

reveal inconsistent or unusual properties of a particular VOR measurement

system. Detection of a problem may require redesign of the system or a modification of the estimation method developed in this report.

6. ROBUSTNESS OF ASSUMPTIONS

In this section, two generalizations of the VOR model are examined to

understand the consequences if values of some parameters assumed to be known

are in error. To facilitate the discussion, we state a modification of the

VOR model which is sufficient for the generalizations considered in this

section:

Y(t) = v(t;6) + s(t;9) + e (t)

where

and

v(t;_9) = 04 cos[2Tr30t + Q]_ + 6 2 ]

s(t;_6) = ct2 cos[2T:9960t + 16sin[2ir30t + 62 ] ] 1/9 Recall that for VOR applications we assumed that aj_ = a2 =2 1

' .

6. 1 Signal Offset

Let Y(t) denote a signal with a^=a2 =2 1/2

. Suppose that instead of Y(t)

we observe Y(t)+M where M*0. The Fourier transform of the observed process

is:

1/60

60 / JY(t) + u]exp[i2Tr30kt] dt

-1/60

, k=0

, k*0

where S^. is the transform of Y(t). Therefore, signal offset does not affect

the previous results because S was not used tQ estimate _9.

17
6.2 Signal Amplitude

In subsection 3.1 we obtained the Fourier coefficients of a signal more general than the VOR requires. The derivation of essential results that

followed in no way depended on the particular values of the amplitude

parameters a^ and o^. If y(t ;_9)=v(t ;_9)+s(t ;_9) denotes a VOR signal with o^

and oi2 unspecified, and s^a^+ib^ denotes the corresponding transform of

y(t;9), then it is easily shown that

ak

and

bk =

• aj Cos[e 1 + e 2 ] , k=i

- a2 Jk-332( 16 ) cos [ Q 3+( k

"332 ) e2]» k>2

-o^ sin[6 1 + 8 2 ] , k=l

-a2 Jk_332(16)sin[63+(k-332)e2], k>2

Clearly, phase computations using tan(q^)=-bjc /a|c

, which are fundamental to the

estimation method, are invariant with respect to particular values of oij and

c*2, even if otj^o^ .

7. ACKNOWLEDGMENT

This study was supported by the Department of Defense Calibration

Coordination Group, Project No. 79-87. I am indebted to Neil T. Larsen of the

NBS Electromagnetic Technology Division for the opportunity to investigate the

VOR problem and for numerous thoughtful discussions. I also thank

Peter V. Tryon of the NBS Statistical Engineering Division for his helpful

contributions to my understanding of the problem. The initial concept that

phase variables may be useful to estimate VOR bearing angle arose from early

evidence obtained by William L. Gans and Arthur R. Ondrejka of the NBS

Electromagnetic Technology Division. The use of a sampling technique with

subsequent Fourier transform was suggested by Mr. Robert Huenemann of United

Airlines, Maintenance Operations, San Francisco, California. Details of his

method are unknown.

18
REFERENCES

[1] Draper, N. R. , and Smith, H. (1966): Applied Regression Analysis . New

York: Wiley and Sons.

[2] Arthur, M. G. , Halford, D. , and Manny, C. H. Jr. (1973): VOR Aircraft

Navigation System Measurements. NBSIR 73-340, National Bureau of

Standards, Boulder, Colorado.

[3] Jenkins, G. M. , and Watts, D. G. (1969): Spectral Analysis and Its

Applications . San Francisco: Holden Day.

[4] Abramowitz, M. , and Stegun, I. A., eds. (1964): Handbook of Mathematical

Functions , AMS-55, National Bureau of Standards, Washington, DC.

[5] Bloomfield, P. (1976): Fourier Analysis of Time Series: An

Introduction. New York: Wiley and Sons.

[6] Anderson, T. W. (1958): The Statistical Analysis of Time Series.

New York: Wiley and Sons.

[7] Ku, H. H. (1969): "Notes on the Use of Propagation of Error Formulas",

Precision Measurement and Calibration , NBS SP 300, Vol.1,

pp. 331-341.

[8] Graybill, F. A. (1976): Theory and Application of the Linear Model . Belmont, CA: Wadsworth.

19
APPENDIX

The appendix presents the derivation of Fourier coefficients in equation

(3.1.1), and obtains the distribution of phase variables when errors in the

VOR model are Gaussian.

A. 1 Derivation of Fourier Coefficients

In this section we determine the Fourier coefficients for the function

y(t;_6) = P + v(t;_6) + s(t;_9) (A.l.l)

where

v(t;_8) = ax cos^^t + 9

X + 2 ]

and

s(t;_9) = a2 cos^irmfx + 9 3 + 3 sin[2Trf 1 + 9 2 ] ] The following results and trigonometric identities are needed in the

derivation:

l/2f f l/2f , j=k*0

/ cos(2TTfjt)cos(2Trfkt)dt =< (A.1.2)

l/2f lO , j^k

l/2f

/ cos(2Trf jt)sin(2irfkt)dt = 0, all j,k (A.1.3)

•l/2f

cos((*f3) = cos a cos 3 - sin a sin 3 (A. 1.4)

cos a cos 3 = (1/2) [cos(a-3) + cos(oH-3)] (A. 1.5)

IT/cos(nt - z sin t)dt = TfJ n

(z) . (A.1.6)

The Fourier coefficients of y(t;_9) are obtained from the complex integral

l/2f x a k + i\ = 2fj_ / y(t;_e)exp[i2Trf 1 kt]dt

-l/2f!

20
where aj. and b^ can be represented by

a k

= a k( y ) + a k^ v ) + ak^ s ) 1

\ k=0,±l,±2,'".

bk

= bk

(y) + bk

(v) + bk

(s) J

The components of ak and b^ represent Fourier coefficients of respective terms

in (A. 1.1). Since p is a constant, it is clear that

f 2p , k=0

ak(M) =

I , k*0

and

bk

(y) , all k

Following is a derivation of ak

(v) and ak

(s). The corresponding Fourier sine

transforms, bk

(v) and ty^Cs), are easily deduced from these results.

For non-negative integer values of k, we have

l/2f x a k

(v) = 2f : / ax cos[2Trf 1 t + 9^ 6 2 ]cos(2Trf x kt)dt

-l/2fj

l/2f ! = 2f 1 a

1 cos(9 1 + 2 ) / cos(2irf 1 t)cos(2TTf 1 kt)dt

-l/2fi

l/2fi

- 2f 1 a1 sin(e 1 + 9 2

) / sin(2irf 1 t)cos(2uf x kt)dt

-l/2f 1

r ax cos(6 1 + 9 2 ) , k=l

I , k*l

where we have used (A. 1.2) to (A. 1.4). Similarly,

r -ax sin(9 1 + 9 2 ) , k=l

bk

(v) =

I , k*l

The Fourier coefficients ak

(s) are given by

l/2f x a k

(s) = 2f x / 02 cos[2Trmf 1 t + 9 3 + 3 sin[2irf 1 t + 92

] ]cos(2Trf

1 kt)dt.

-l/2f x

21
Substituting u=27rf

1 t+92, so dt=(2irfi) 1t we get

IT ak

(s) = ff -1 a2 / cos[mu - m92 +63+3 sin u]cos(ku - k82

) du,

-Tf

where the limits -Tf+0 2 <u<Tf+8 2 can be replaced by -Tr<u<TT because the

integrand has period 2tt. Furthermore, using the identity (A. 1.5), we obtain

TT ak

(s) = (2ir) -1 a2 / cos[(m-k)u - (m-k)62 +63 + 6 sin u]du

-Tf

+ / cos[(m+k)u - (m+k)62 +63+3 sin u]du,

-TT

and using (A. 1.4) we get

Tf ak

(s) = (2Tf) -1a2 cos[03 - (m-k)82 J J cos[(m-k)u + 3 sin ujdu

-Tf

Tf - sin[03 - (m-k)82J J sin[(m-k)u + 3 sin ujdu

-Tf

Tf + cos [83 - (m+k)82J / cos[(m+k)u + 3 sin ujdu

-Tf

Tf - sin[83 - (m+k)62J J sin[(m+k)u + 3 sin ujdu.

-Tf

The second and fourth integrals are zero because the integrands are odd

functions with period 2tt. Since the first and third integrands are even functions, it follows from (A. 1.6) that

ak

(s) = a

2 {jm_k (3)cos[83-(m-k)82

] + Jmhk( 3)cos [e3-(m+k)

2 ] } .

Similarly,

bk

(s) = -a2 {jm_k (3)sin[83 -(m-k)82

] - Jm+k ( 3)sin [ 3 -(m+k)

2 ]

} .

22
We have proved that the Fourier coefficients of y(t;_9) are

r

v , k=0

a k

=

"S al cos [ Q

l + e 2^ + a2 ak^ s ^ » k=1

^ a2 ak

(s) , k>2

and

^ , k=0

bk

= ^~al sin [ e

l + e 2^ ~ a2 bk( s ) > k=1

vr«2 bk( s )

where a^Cs) and b^Cs) are defined above.

, k>2

A. 2 Distribution of Phase Random Variables

A justification for using approximate formulas for the mean and variance

of phase random variables is the complexity of the exact distribution of the

Qk* ' s even when errors are assumed to be Gaussian. For completeness, the

distribution of Ar ctanf-B^/A^] is derived in this section. The subscript is

dropped in the derivation.

If errors are Gaussian, then A and B are independent Gaussian random

variables and

E[A]=a; E[B]=b; a

2+b 2 =r2

;

Var[A]=Var[B]=(2/N)a2

.

Particular values of a and b are given by (4.1.1). The joint distribution of

A and B is

f A)B (u,v) = (N/4na 2 )exp{-(N/4a 2 )[(u-a) 2+(v-b) 2 ]}, -°°<u,v<°°.

Let X=A and Y=Ar ctan[-B/A] . Then because u=x and v=-x tan y, the Jacobian of

the transformation is J=|x|sec y, and the joint distribution of X and Y is

given by

f(x,y) = (N/4a 2

) |x|sec2 y exp {-(N/4a 2

) [(x-a) 2+(x tan y+b) 2 ]}, -°°<x<«>,-7r/2<y<TT/2.

Expanding the exponent and completing the square, we obtain

1/9

f(x,y) = (Nsec2 y/4iTa 2

) exp {(N/4a 2

) [ (a-b tan y)

2 cos2y-r 2

] } 1/9 9

• |x| (Nsec2y/4ira 2

) exp {(-N/4a 2 )sec2 y [x-(a-b tan y)cos 2y] }

= K(y)«|x| <f>[x:(a-b tan y )cos 2y, (2/N) a

2 cos2 y]

where <|>[z:5,t ] denotes the probability density function of a Gaussian

23
distribution with mean £ and variance t' . Integration of f(x,y) over x to

obtain the distribution of Y shows that f (y )=K(y)E [ |x| ] , where X is Gaussian

with mean (a-b tan y)cos y and variance (2/N)o cos y.

If Z is a Gaussian random variable with mean £ and variance i , then

°° E[|Z|] = - /ztf)[z:C,T 2 ]dz + Jz<f.[z:?,T 2 ]dz

-00

00 = .-£ + 2/z<f>[z:£,T 2 ]dz.

Integration by parts gives

E[|Z|] = ?-(l-2$[-C/x:?,x 2 ]) + (2T 2

/7T)

1/2

exp {-? 2/2T 2

}.

w where $(w: ?, t

2

)=/<J>[z: £, T

2 ]dz. It follows that the distribution of —00 Y=Arctan[-B/A] is given by f (y)=K(y)E [ |Z | ] with £=(a-b tan y)cos 2y and

t -(2/N)o cos y. Substition of these values above gives the following

distribution for a phase random variable:

1 / 9 f(y) = exp(-Nr 2 /4a2 ){iT~ 1 +(Ncos 2y/4TTa2

) (a-b tan y)

•(l-$[-(Ncos 2 y/2a2

) 1/2

(a-b tan y):?,x 2 ])

•exp[(Ncos 2y/4a2 )(a-b tan y) ]}, -TT/2<y<Tr/2,

where £ and t^ are defined above.

24
NBS-114A (REV. 9-78)

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET

1. PUBLICATION OR REPORT NO.

NBS TN-1021

2.Gov't Accession No. 3. Recipient's Accession No.

4. TITLE AND SUBTITLE

Fourier Transformation of the Nonlinear VOR Model to

Approximate Linear Form

5. Publication Date June 1980

6. Performing Organization Code

7. AUTHOR(S)

Dominic F. Vecchia

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, DC 20234

10. Project/Task/Work Unit No.

11. Contract/Grant No. Project No. 79-87

12. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City, State, ZIP) Department of Defense Calibration Coordination Group

VOR Subgroup, C. V. Wolf, Chairman

U.S. Army, USAMCC Attn: DRSMI-MME

Huntsville, AL 35809

13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or

literature survey, mention it here.)

This technical note describes a method for transforming

regression model to a form which is approximately linea

The technique involves computation of the Fourier coeff

data and uses phase variables to estimate the parameter

formation is employed to obtain bearing angle estimates

the Very-High-Frequency Omni-Directional Range (VOR) ai

The transformation provides a model linear in relevant

estimation of VOR bearing angle utilizes existing stati

is shown that certain generalizations of the VOR model

mate linear form by the phase spectrum transformation.

a particular nonlinear

r in the unknown parameters.

icients for a set of sample

s. The phase spectrum transfor a model associated with

rcraft navigation system,

phase parameters. Thus,

stical theory. Finally, it

also are reduced to approxi17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the fir3t letter of the first key word unless a proper name;

separated by semicolons) Fourier coefficients; linear model; nonlinear model; phase spectrum transformation;

spectrum; VOR aircraft navigation system; white noise.

18. AVAILABILITY [Jt} Unlimited

I I For Official Distribution. Do Not Release to NTIS

[y] Order From Sup. of Doc, U.S. Government Printing Office, Washington, DC

20402 Q Order From National Technical Information Service (NTIS), Springfield, VA. 22161

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(THIS REPORT)

UNCLASSIFIED

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studies of special interest to physicists, engineers, chemists,

biologists, mathematicians, computer programmers, and others engaged in scientific and technical work.

National Standard Reference Data Series—Provides quantitative data on the physical and chemical properties of materials, compiled from the world's literature and critically evaluated. Developed under a worldwide program coordinated by NBS under

the authority of the National Standard Data Act (Public Law

90-396).

NOTE: The principal publication outlet for the foregoing data is the Journal of Physical and Chemical Reference Data (JPCRD)

published quarterly for NBS by the American Chemical Society

(ACS) and the American Institute of Physics (AIP). Subscriptions,

reprints, and supplements available from ACS, 1 155 Sixteenth St., NW, Washington, DC 20056.

Building Science Series—Disseminates technical information developed at the Bureau on building materials, components,

systems, and whole structures. The series presents research results,

test methods, and performance criteria related to the structural and

environmental functions and the durability and safety characteristics of building elements and systems.

Technical Notes—Studies or reports which are complete in themselves but restrictive in their treatment of a subject. Analogous to monographs but not so comprehensive in scope or definitive in treatment of the subject area. Often serve as a vehicle for final reports of work performed at NBS under the sponsorship of other government agencies.

Voluntary Product Standards—Developed under procedures

published by the Department of Commerce in Part 10, Title 15, of

the Code of Federal Regulations. The standards establish nationally recognized requirements for products, and provide all concerned interests with a basis for common understanding of the

characteristics of the products. NBS administers this program as a supplement to the activities of the private sector standardizing

organizations. Consumer Information Series—Practical information, based on NBS research and experience, covering areas of interest to the con- sumer. Easily understandable language and illustrations provide

useful background knowledge for shopping in today's technological marketplace. Order the above NBS publications from: Superintendent of Documents, Government Printing Office, Washington, DC 20402. Order the following NBS publications—FIPS and NBSIR's—from

the National Technical Information Services, Springfield, VA 22161

.

Federal Information Processing Standards Publications (FIPS

PUB)—Publications in this series collectively constitute the Federal Information Processing Standards Register. The Register

serves as the official source of information in the Federal Govern- ment regarding standards issued by NBS pursuant to the Federal Property and Administrative Services Act of 1949 as amended,

Public Law 89-306 (79 Stat. 1127), and as implemented by Executive Order 11717(38 FR 12315, dated May 11, 1973) and Part 6 of Title 15 CFR (Code of Federal Regulations).

NBS Interagency Reports (NBSIR)—A special series of interim or

final reports on work performed by NBS for outside sponsors

(both government and non-government). In general, initial dis- tribution is handled by the sponsor; public distribution is by the National Technical Information Services, Springfield, VA 22161,

in paper copy or microfiche form.

BIBLIOGRAPHIC SUBSCRIPTION SERVICES

The following current-awareness and literature-survey bibliographies are issued periodically by the Bureau:

Cryogenic Data Center Current Awareness Service. A literature sur- vey issued biweekly. Annual subscription: domestic $25; foreign

$30. Liquefied Natural Gas. A literature survey issued quarterly. Annual

subscription: $20.

Superconducting Devices and Materials. A literature survey issued

quarterly. Annual subscription: $30. Please send subscription or- ders and remittances for the preceding bibliographic services to the National Bureau of Standards, Cryogenic Data Center (736) Boulder, CO 80303.
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