Statistical calibration of a vector demodulator - algorithm for statistical demodulator calibration - technical

Statistical Calibration of a Vector Demodulator

VECTOR MODULATION INSTRUMENTS such as the HP 8780A Vector Signal Generator and the HP 8980A Vector Analyzer provide a test system for measurement applications in such fields as digital microwave radio, communications, and radar. Some of the technologically new developments in this series of products are the modulation capabilities (complex, wideband signals), the high-resolution CRT display, and the 350-MHz bandwidth.

The precision measurement capabilities offered in these products raise the issue of the internal calibration of the instruments. All instruments need to be calibrated to account for differences in components and test conditions. The accuracy and precision of these internal self-adjustments are crucial, since they have direct effects on the characterization of the device under test. The hardware measurements made for purposes of internal calibration must be combined in a logical and intelligent fashion to yield the correct calibration factors.

Statistical methods play an important role in the calibration routine. This paper describes a statistical approach to the calibration of the HP 8981A Vector Modulation Analyzer. The next section briefly introduces the use of statistical methods in such situations. The measurement process for a vector demodulator is then described, using the HP 8981A as an illustration. Two sections provide the technical details of the calibration algorithm both with and without an accurate vector signal generator (e.g., the HP 8780A) as a reference. Finally, examples of real data illustrate the computations in the statistical algorithm and formulas for the uncertainties in the calibration factors. It will be shown that the algorithm can be used to calibrate not only the demodulator in the HP 8981A but also external demodulators.

Statistics in Calibration Routines

Calibration consists of comparing a set of measurements from an uncharacterized instrument (e.g., an HP 8981A) with a defined reference standard (e.g., an HP 8780A) according to a measurement algorithm. Thus the calibration model relates the observed measurement readings to the reference standard. A model is never perfect, and the difference between the model and reality can be characterized according to two types of errors that may be present:

* Systematic errors resulting from incomplete specification of the calibration model

* Random errors, or small, unpredictable fluctuations that affect every measurement but are themselves unmeasureable.

Statistical methods can be used to address both types of errors. First, the calibration should be designed to allow identification of possible systematic departures between the model for measurement and the observed data. Second, the influence of random errors can be assessed when estimating the parameters in the model that relate to the measurement process and their uncertainties. In this paper, we shall assume that the random errors are independent and identically distributed according to a symmetric distribution; that is, that the individual errors cannot be predicted in either size or direction, and that the chances of an erroneous measurement being too large or too small are roughly equal. (Diagnostic tools for checking the validity of these assumptions and the consequences of their violation are discussed in connection with the examples later in this paper.)

Rarely is a single estimate of a target quantity sufficient. For example, reporting a sample mean without its standard error provides no information on the reliability of the data that went into that sample mean. The same is true for estimates of the parameters describing the measurement process. These estimates are the calibration factors, and certain limits of fluctuation may be desirable (e.g., gain adjustment accurate within 1%). Therefore, an important part of the statistics in a calibration algorithm is the derivation of associated measures of uncertainty for the calibration factors.

Vector Analyzer Measurement Process

In the HP 8981A, the phase-modulated signal can be expressed mathematically in terms of its frequency components as s(t) = [sigma].sub.[Kappa]A.sub.[Kappa](T)COS([omega].sub.[Kappa].t + [phi].sub.[Kappa].) = [sigma].sub.[Kappa].I.sub.[Kappa].(t)cos([omega].sub.[Kappa].t) + Q.sub.[Kappa].(t)sin([omega].sub.[Kappa].t)

Equation 1 expresses the signal in terms of its demodulated compnents I.sub.[Kappa].(t) (in-phase) and Q.sub.[Kappa].(t) (quadrature). Components that vary over time can be observed by examinating I.sub.[Kappa].(t) and Q.sub.[Kappa].(t) over time at a particular frequency [omega].sub.[Kappa]. Ideally, at a fixed time, a single-frequency signal with constant amplitude is demodulated at I = cos(a) Q = cos(a - (90[deg.] - [phi]) where a is the angle of the vector signal and [phi], the error in the quardrature phase of the demodulation, is zero.

Since the HP 8981A has the ability to display signals in either time or vector form, amplitude and phase information must be recovered accurately. This requires special software functions to estimate and subsequently account for any amplitude or phase offsets. Proper calibration of the HP 8981A thus allows simple measurements of amplitude and phase modulated signals.

Fig. 1 is a block diagram for a simplified demodulator, such as the HP 8981A, which decomposes the modulation on a signal into its in-phase and quadrature components. Accuracy and precision in this decomposition are essential to assess degradation of the signal's amplitude, phase, or frequency, or to assess cross-talk between the two channels. Since I and Q signals are used often in measurement systems, we will discuss the calibration scheme for the Q-vs-I vector representation.

Six sources for adjustments in a vector demodulator can be identified, based on the block diagram of the instrument. These six sources correspond to stages in the transformation of the ideal I/Q signal into the measured, or actual, I/Q signal. The calibration process uses statistics to estimate the sizes and directions of the adjustments attributable to these sources and then corrects the output so that the demodulated I/Q signal agrees with the ideal I/Q signal.

Let (I,Q) denote the pair of points representing the I and Q channels at a particular point in time at a given frequency. The HP 8981A hardware transforms the coordinates (I,Q) into measured coordinates (x,y) through the following series of operations: (I,Q) [right arrow] Rotation [right arrow] Axis Sheer [right arrow] Compression [right arrow] dc Offsets [right arrow] (x,y)

Rotation can be viewed as a phase change in the ideal signal, which can be introduced by any of the components before the demodulation. The axis sheer creates two additional adjustments: quadrature adjustment, resulting from inaccuracy in the 90[deg.] split, and gain imbalance, resulting from different power levels in amplifiers or unequal power losses in other components after the split. The I/Q measurements are rescaled by amplifier compression, and dc offsets may be added to the measurements at any stage.

Notationally, let us represent these adjustments as: [gamma] " gain imbalance (I-channel/Q-channel) [phi] = angular difference between ideal quadrature (90[deg.]) and actual quadrature [theta] = angle of rotation (sometimes called lock error) [rho] = scaling resulting from compression I.sub.0 = dc offset in I-channel Q.sub.0 = dc offset in Q-channel

The four steps in transforming (I,Q) to (x,y) can be expressed mathematically as follows:

* Rotation through angle [theta]:

This transforms (I,Q) into (Icos[theta] - Qsin[theta] + Qcos[theta]).

* Sheer transformation (quadrature adjustment [theta], gain imbalance [gamma]):

This causes two additional adjustments: the ratio in the two channels (I/Q) is now [gamma], not 1, and the tangent of the angle between the two axes (I,Q) is now (90[deg.] - [theta]), not 90[deg.].

*The measurements are uniformly compressed according to a factor [rho].

* The measurements are offset by a fixed amount (I.sub.0 in the I-channel, Q.sub.0 in the Q-channel).

Putting these four effects together, the resultant point (x,y) can be expressed as a function of the ideal input (I,Q) as:

The unobserved variables [epsilon].sub.x., [epsilon].sub.y. are assumed to be uncorrelated random errors in the measurement of the x and y coordinates. That is, x = I.sub.0 + [gamma]([rho]cos[theta])I - [gamma]([rho]sin[theta])Q + [epsilon].sub.x = [alpha].sub.0 + [alpha].sub.1.I + [alpha].sub.2.Q + [epsilon].sub.x y = Q.sub.0 + ([rho]cos[theta]sin[phi] + [rho]sin[theta]cos[phi])I + (-[rho]sin[theta]sin[phi] + [rho]cos[theta]cos[phi])Q + [epsilon].sub.y = [beta].sub.0 + [beta].sub.1.I + [beta].sub.2.Q + [epsilon].sub.y where: [alpha].sub.0 = I.sub.0 [alpha].sub.1 = [gamma][rho]sin[theta] [alpha].sub.2 = -[gamma][rho]sin[theta] [beta].sub.0 = Q.sub.0 [beta].sub.1 = [rho]cos[theta]sin[phi] + [rho]sin[theta]cos[phi] = ([alpha].sub.1.sin[phi] - [alpha].sub.2.cos[phi])/[gamma] [beta].sub.2 = [rho]sin[theta]sin[phi] + [rho]cos[theta]cos[phi] = ([alpha].sub.2.sin[phi] + [alpha.sub.1.cos[phi])/[gamma]

The errors of interest can be recovered via the equations: I.sub.0 = [alpha].sub.0 (I-channel offset) Q.sub.0 = [beta].sub.0 (Q-channel offset) [rho] = ([beta].sup.2.sub.1 + [beta].sub.2.sub.2).sup.1/2 (Compression) [theta] = tan-.sup.1(-[alpha].sub.2./[alapha].sub.1) (Rotation) [gamma] = [([alpha].sub.2.sub.1 + [alpha].sup.2.sub.2)/([beta].sup.2.sub.1 + [beta].sup.2.sub.2)]1/2 (I/Q gain imbalance) [phi] = tan-sup.1 [([alpha].sub.1.[beta].sub.1 + [alpha].sub.2.[beta].sub.2)/([alpha].sub.1.[beta].sub.1)] (Quadrature error)

Notice that, apart from rotation, the ideal I and Q can be recovered via the equations I = (x =I.sub.0.)/[gamma]p Q = [[gamma](y - Q.sub.0.) - sin [phi] (x - I.sub.0)]/[gamma]p cos [phi].

Estimation with Accurate Reference

At this point, we need to distinguish between the case where ideal (I,Q) values are provided and the situation where only magnitude information about the (I,Q) points is known. Ideal (I,Q) measurements can be obtained using a precise signal generator such as the HP 8780A, so that signals having several accurately defined phase modes can be guaranteed. Since this situation lends itself to the easier algorithm, it will be discussed first.

It is evident that an input signal of constant frequency and magnitude and variable phase, namely s(t) = Acost([omega]t + [phi].sub.t.), t = 0,1, . . . results in an I/Q display for which I.sup.2 + Q.sup.2 = (Magnitude).sup.2., which is a circle. It can be verified that the transformed points (x,y) lie on an ellipse. The parameters of this ellipse are related to the adjustments in the instrument as outlined above.

Since we have six adjustments (parameters), we need a minimum of six phase states to obtain an exact fit for the parameters in equation 2.

The 8PSK mode of the HP 8780A allows rapid generation of an I/Q signal at eight evenly distributed phase states, allowing two extra phase states for model confirmation. The coordinates (I,Q) of the points in any one of these states will be considered ideal, transformed (by the six adjustments) into measured coordinates (x,y). For simplicity, we label the eight ideal states sequentially at those points having phases k(45[deg.]), k = 0. 1, ..., 7 (Fig. 2). Thus, where f = 1[square root]2 = 0.7071068. (Numerical association with each state is arbitrary.)

Suppose n/8 points are measured in each state, so n measurements in all are taken. (This requirement is not essential, but the equations are greatly simplified by encouraging orthogonality into the problem.) Denote the coordinates of these measurements by (x.sub.1., y.sub.1.), . . . , (x.sub.n., y.sub.n). Associate the point with its ideal (I,Q) pair as described above: (I,Q.sub.1), . . . , (I.sub.n., Q.sub.n). Then we need to fit [alpha] = ([alpha].sub.0,[alpha].sub.1,[alpha].sub2)' and [beta] = ([beta].sub.0,[beta].sub.1, [beta].sub.2)' in the two relationships: x.sub.i = [alpha].sub.0 + [alpha].sub.1.I.sub.i + [alpha].sub.2.Q.sub.i + [epsilon].sub.xi y.sub.i = [beta].sub.0 + [beta].sub.1.I.sub.i + [beta].sub.2.Q.sub.i + [epsilon].sub.yi

The estimates of ([alpha].sub.0,[alpha].sub.1,[alpha].sub.2), namely ([alpha].sub.0,[alpha].sub.1,[alpha].sub.2), can be obtained by a least squares algorithm. If we let M be the matrix of n rows and 3 columns corresponding to the n observations and three "carriers" (constant term, I,Q), that is, then it is well known.sub.3 that [alpha] = (M'M).sup.-1.M'x [beta] = (M'M).sup.-1.M'y are the least squares estimates of [alpha] and [beta], where x = (x.sub.1., x.sub.2., . . . , x.sub.n.)' and y = (y.sub.1., y.sub.2., . . . , y.sub.n.)' are column vectors and the prime sign (') indicates transpose. Since there are n/8 measurements in each state, and the states are evenly spaced around the circle, the matrix M is orthogonal, and so Furthermore, and thus [alpha] = (M'M).sup.-1.M'x = [ave(x), 2ave (x.I), 2ave(x.Q)]' [beta] = (M'M).sup.-1.M'y = [ave(y), 2ave (y.I), 2ave(y.Q)]'.

The estimates of the target parameters (I.sub.0., Q.sub.0., [rho], [theta], [gamma], [phi]) can then be unscrambled using equations in (3) above.

Standard Errors

We need some measure of uncertainty in our estimates of the parameters. For example, after calculating an estimate of [gamma], say [gamma], how far off might that estimate be from the true value? One such measure is the standard error (SE) of the estimate. An approximate 95% confidence interval for the estimate is given by estimate [plus-or-minus]2(SE).

To calculate the stnadard errors for the six parameters, we first need a measure of goodness of fit in in our model. This can be calculated most simply as a "root mean square" of the differences between the observed measurement x.sub.i or y.sub.i and its fitted value predicted by the model. Denote these differences, or residuals, by: e.sub.xi = X.sub.i - [alpha].sub.0 - [alpha].sub.1.I.sub.i - [alpha].sub.2.Q.sub.i e.sub.yi = y.sub.i - [beta].sub.0 - [beta].sub.1.I.sub.i - [beta].2.Q.sub.i and calculate

The divisor is (n - 3) instead of an because three parameters have been estimated in each channel. If the departures from the model have a Gaussian distribution, we would expect most of the e.sub.xi to fall within 2s.sub.x or 3s.sub.x., and likewise for e.sub.yi.. This criterion may serve as a simple check for unusual measurements (but see "A Caution," page 24).

Using the root mean squares of the model departures (residuals), we can calculate the standard errors of the parameters in the model. Again assuming these residuals are Gaussian distributed, the standard errors of ([alpha].sub.0., [alpha].sub.1., [alpha].sub.2.) are given by s.sub.x [square root of] diagonal of (M'M).sub.-1 Thus, SE([alpha].sub.0.) = SE(I.sub.0.) = s.sub.x./[square root of] n SE([beta].sub.0.) = SE(Q.sub.0.) = S.sub.y./[square root of] n

The standard errors for the other target parameters are a little more complicated, because they are functions of [alpha].sub.1., [alpha].sub.2., [beta].sub.1., and [beta].sub.2.. We rely on propagation of error formulas.sup.4 for approximations to the standard errors of functions of variables. The approximations are simplified if 2s.sub.y/.sup.2/(n[rho].sup.2.) <<1, a condition that will often be satisifed if the fit is good (s.sub.y is small) and the number of points is moderately large. Then: SE([rho]) [nearly equals] s.sub.y.[square root of]2/n SE([gamma]) [nearly equals] s.sub.x.[square root of]2/n SE(tan[theta]) [nearly equals] s.sub.x.[rho][gamma][square root of]2/n/[alpha].sup.2/.sub.1 SE(tan[phi]) [nearly equals] [(s.sup.2/.sub.x + s.sup.2/.sub.y.)(1 + 1/[gamma].sup.2.)/n].sup.1/2/ ([rho]cos.sup.2.[phi]) where tan[theta] and tan[phi] are the estimates of tan[theta] and tan[phi]. So approximate 95% confidence intervals for the estimates of the errors are: [gamma] [plus-or-minus] 2SE([gamma]) I.sub.0 [plus-or-minus] 2SE(I.sub.0.) Q.sub.0 [plus-or-minus] 2SE(Q.sub.0.) [rho] [plus-or-minus] 2SE([rho]) tan.sup.-1.[tan[theta] [plus-or-minus] 2SE(tan[theta])] tan.sup.-1.[tan[phi] [plus-or-minus] 2SE(tan[phi])].

This calibration algorithm thus proceeds as follows:

* Measure (x.sub.i., y.sub.i.) for i = 1, ..., n

* Associate state (I.sub.i., Q.sub.i.) for i = 1, ..., n

* Calculate: [alpha] = ([alpha].sub.0., [alpha].sub.u., [alpha].sub.2.)' [beta] = ([beta].sub.0., [beta].sub.1., [beta].sub.2.)'

* Calculate target parameters: [gamma], [phi], [rho], [theta], I.sub.0., Q.sub.0

* Calculate: goodness of fit (s.sub.x., s.sub.y.) and residuals (e.sub.xi., e.sub.yi.)

* Identify unusual residuals

* Calculate standard errors.

Inaccurately Known Phase

We now consider the situation where phase cannot be associated precisely with each measurement. Generally, amplitude information is more reliable, that is. I.sup.2 + Q.sup.2 = (I,Q)' (I,Q) = 1.

Two possible signals that satisfy this requirement are:

* A signal with continuously varying phase (e.g., two tones, offset in frequency)

* A signal with eight discrete phases (e.g., an 8PSK modulated signal and its coherent carrier). Inverting equation 2, and thus (I,Q)'(I,Q) = [rho].sup.-2(x-I.sub.0.-[epsilon].sub.x., y-Q.sub.0.-[epsilon.sub.y.)(S.sup.-1.)(X-I.sub.0.- [epsilon].sub.x., y-Q.sub.0.-[epsilon].sub.y.), because (R'R) = (R.sup.-1.)'(R.sup.-1.) is the identity matrix (R is orthogonal). This can be simplified by noting that

Let us assume that the errors in measuring x and y have zero means, are uncorrelated with one another, and have common standard deviation [sigma]. Then, on average. ave([epsilon].sub.x.) = ave([epsilon].sub.y.) = 0 ave([epsilon].sup.2/.sub.x.) = ave([epsilon.sup.2/.sub.y.) = [sigma].sup.2 and so equation 5 becomes: (x-I.sub.0.).sup.2 - 2[gamma](sin[phi])(x - I.sub.0.)(y - Q.sub.0.) + [gamma].sup.2.(y - Q.sub.0). sup.2 + ([gamma].sup.2 + 1)[sigma].sup.2 - [gama].sup.2 [rho].sup.2.cos.sup.2.[phi] = 0.

If measurement error is negligible compared to the magnitude of the adjustments, the fourth term in equation 6 can be ginored.

Equation 6 is nonlinear in the parameters, of which there are now only five (clearly the rotation, or lock error, cannot be estimated without relative phase information). Algorithms for nonlinear least squares often rely on gradient methods or on linearization of the problem via a Taylor series expansion. In addition, several methods for approximate confidence intervals for the parameters have been proposed. These two issues are described in the Appendix on page 24.

Examples

This first example a serves as a "control." Six (x,y) measurements were taken in each of eight states (Fig. 3). All calibration factors are essentially at their nominal values. ([alpha].sub.0., [alpha].sub.1., [alpha].sub.2.) = (0.000054, 0.16564, - 0.068486) ([beta].sub.0., [beta].sub.1., [beta].sub.2.) = (-0.002694, 0.068058, 0.163904) I.sub.0 = 0.00005 volts (I-channel offset) Q.sub.0 = -0.00269 volts (Q-channel offset) [rho] = 0.17747 volts (Compression) [theta] = tan.sup.-1 (0.39206) = 21.41[deg.] (Quadrature error), Root mean squares: s.sub.x = 0.0012162;s.sub.y = 0.0008686 volts.

Approximate 95% confidence intervals (estimate [plus-or-minus] 2SE): I.sub.0.:(-0.00044, 0.00055) volts Q.sub.0.:(-0.00295, - 0.00243) volts [gamma]:(1.0053, 1.0147) [rho]:(0.17711, 0.17783)volts tan[phi]:(-0.00144, 0.00448) [right arrow] [phi]:(-0.08[deg.], 0.26[deg.]) tan[theta]:(0.39153, 0.39253)[right arrow] [theta]:(21.38[deg.], 21.43[deg.]).

Fig. 4 shows the histograms of the x and y residuals. Clearly there is one aberrant measurement. This measurement occurred in state 4 (I = -1, Q = 0). A comparison of these estimates with those obtained with a more robust procedure (downweighting this point) verified that the point is not so extreme that it affected the quality of the overall fit.

Using the algorithm for unknown phase (see above), I.sub.0.:0.00019 (-0.00044, 0.00055) volts Q.sub.0.:-0.00243 (-0.00295, -0.00243) volts [gamma]:1.0100 (1.0053, 1.0147) [rho]:0.17748 (0.17711, 0.17783) volts tan([phi]):0.00230 (-0.00318, 0.00622) [right arrow] [phi]:0.13[deg.] (-0.18[deg.], 0.36[deg.]).

Notice that these solutions fall well within the confidence intervals of the linear (known-phase) method.

The second example involved a large quadrature adjustment. Eight (x,y) measurements were taken in each of eight states (Fig. 5). ([alpha].sub.0., [alpha].sub.1., [alpha].sub.2.) = (0.089621, 0.143384, -0.292603) ([beta].sub.0., [beta].sub.1., [beta].sub.2.) = (0.011144, 0.227279, 0.244292) I.sub.0 = 0.089621 Volts (I-channel offset) Q.sub.0 = 0.011144 Volts (Q-channel offset) [rho] = 0.33367 Volts (Compression) [theta] = tan.sup.-1 (1.11516) = 48.12[deg.] (Lock error) [gamma] = 0.97656 (I/Q gain ratio) [phi] = tan.sup.-1 (-0.36852) = -20.09[deg.] (Quadrature error),

Root mean squares: s.sub.x = 0.0019728; s.sub.y = 0.0015534 volts.

Approximate 95% confidence intervals (estimate [plus-or-minus] 2SE): I.sub.0.:(0.08913, 0.09011) volts Q.sub.0.:(0.01076, 0.01153) volts [gamma]:(0.97586, 0.97726) [rho]:(0.33312, 0.33422) volts tan[phi]:(-0.36263, - 0.36891)[right arrow]:(-20.25[deg.h, - 19.94[deg.]) tan[theta]:(1.10963, 1.12069)[right arrow]:(47.97[deg.], 48.26[deg.])

Fig. 6 shows the histograms of the x and y residuals.

Using the algorithm for unknown phase (see above), I.sub.0.:0.08871 volts Q.sub.0.:0.017787 volts [gamma]:0.97723 [rho]:0.33978 volts tan([phi]):-0.36460[right arrow] : - 20.03[deg.](-0.18[deg.], 0.36[deg.]).

Notice that a large quadrature adjustment (-19.25[deg.]) and a slight adjustment in the gain imbalance are indicated. The other errors are considered negligible.

A Caution

If the model departures (residuals) are not Gaussian distributed for any reason (e.g., outlying points as in the first example, or misidentified states), or if there are unequal numbers of measurements in each state in the known-phase algorithm, the estimates of the target parameters and their standard errors may be seriously biased. Least squares fitting procedures have a tendency to make all residuals about the same magnitude, even if, in fact, the residuals in all measurements were tiny, except for one. One indication of this case is if the median of the residuals is not zero, even though their average is. But this criterion would be ineffective for detecting two residuals in opposite directions. To reduce the effect of such aberrant measurements, the algorithm is made more robust by assigning a weight to each measurement based on the magnitude of its departure from the model. This feature is essential to avoid serious bias in the calibration factors.sup.6,7

Summary

A vector demodulator can be calibrated using a statistically-based algorithm to determine the size and direction of the adjustments necessary to guarantee accuracy of the output. The uncertainties in these adjustments can be determined based on some assumptions about the measurement process:

* The ideal (I,Q) measurements of the signal in the two channels have been transformed to (X,Y) measurements according to the following series of operations (listed in order of occurence): rotation (lock error), sheer (quadrature, gain imbalance), compression, dc offsets.

* Errors in the measurements in the two channels (x,Y) are uncorrelated.

* When the relative phases among the points are known, data is collected in phase-stepping mode for eight states, and the state (0, ..., 7) can be associated with each measurement. It is convenient if the number of measurements is the same in each of the eight states.

* When phase information is known only imprecisely, measurements are made that ideally lie on a circle in the I-Q plane. Input signals must have constant amplitude and varying phase (e.g., continuously, as in a two-tone setup, or discretely, as for 8PSK).

Based on these assumptions, a straightforward linear least squares algorithm can be used to determine the sizes and directions of the adjustments when relative phases of the incoming signal (such as 8PSK from the HP 8780A) are known. If this is not the case, then a linearization of the nonlinear least squares problem can be solved, with some loss in precision of the adjustments and increased computational time. Since this approach can be used more generally to calibrate external demodulators as well, it is the algorithm currently programmed in the HP 8981A. In both cases, approximate 95% confidence intervals for the adjustments can be derived.

Examples with data collected on the HP 8981A using the HP 8780A as a reference signal illustrate the implementation of the algorithms. The examples illustrate the importance of incorporating an efficient and robust calibration algorithm using statistical principles to control the hardware. Proper modeling of the system combined with the appropriate methodology results in accurate calibration factors as well as an assessment of the uncertainty in these factors.

Acknowledgments

The author would like to thank Kaaren arquez and Eric McHenry for implementing this algorithm in the bread-board phase, for providing data used in the example, and for their comments on earlier drafts of this paper.

COPYRIGHT 1988 Hewlett Packard Company
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