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.

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.