Statistical evaluation of digital-analog systems for finite operating time

A frequency-domain mathematical model for comparing the performance of a linear sampled-data channel to a linear continuous channel is developed. The model assumes a suddenly applied stationary random input (input identically zero before the local time origin). The model allows an explicit algebraic definition of ensemble mean-squared error, after a finite operating time, by application of residue theory. The ideal or comparison channel need not be realizable. The quasi-stationary characteristics of the excitation are accounted for by including a starting switch as a variable parameter within the appropriate system weighting functions. The technique developed in this paper is utilized to evaluate the ensemble mean-squared error due to processing a vehicle velocity estimate (as might be supplied by an integrating accelerometer) in a digital computer. The velocity estimate is used to calculate a numerical value of present vehicle position. This problem is of fundamental importance in present-day pure inertial, pure Doppler, or inertial-Doppler navigation systems. The effect of quantization, inherent in the process of analog-to-digital conversion, may be included in this evaluation. It is shown that for cases where system inputs can be approximated by narrow-band, first-order, Markoff-type power spectra, the practical engineering use of the simplest digital integration program (rectangular) or the simplest hold (box-car) is well justified. The examples illustrate the mathematical techniques necessary to compute the ensemble mean-squared error, and illustrate how several simplifying assumptions may be used.