FOURIER TRANSFORMATION OF THE NONLINEAR VOR MODEL TO APPROXIMATE LINEAR FORM by Dominic F. Vecchia This technical note describes a method for transforming a The trans- Key Words: Fourier coefficients; linear model; nonlinear model; 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 where B1 and B2 are unknown parameters, x is an independent variable, and € is a random error term. Both models are nonlinear in B1 and B2, but the first is intrinsically linear because the transformed variable lnY is linear in 81 and B2. 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 be cause it involves computation of the Fourier coefficients of the sample data and uses phase variables to estimate 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 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]. Ө Vectors and matrices are denoted by underlined letters, for example, e and V. If denotes a vector, then e will denote the transpose of e. An estimator of 9 will be denoted by 0. 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 and v(t; d,p) = a1 cos [2f1(t+8) + 1] (3.0.1) s(t; $, $) = a2 cos(2πf2(t+8) + 2 + В sin[2f1(t+8) + $3]] • The waveform generated by the sum of v(t;d,p) and s(t; d,p), 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; 8, ) is called the variable phase signal, and s(t;,p) is the frequency modulated subcarrier signal. The frequency modulating sinusoid, contained in the argument of s(t;,), 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. lowing are descriptions of the model parameters: Fol fl The fixed time offset, 8, 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 |