Statistical Accuracies

Formulas for computing the expected variation due to the statistical nature of an experiment are vital in optimum planning. Table 6.16 contains formulas useful in ascertain­ing the precision of the various functions involved in noise analysis. The meaning of the fractional-error formulas is that, if many values of a function were determined (at a particular f or t), 68 3% would he within the average ± this fractional error

In all cases the error varies inversely as the square root of the measuring time and inversely as the square root of either the bandwidth, B, or the resolution, Af

В = upper frequency limit of P(f), which is approximately constant from 0 to В and thereafter near zero (6.43)

or

В = if C(r) is approximately C(0)e t/T(- (6 44)

1

Af = — for the ideal and hamming windows of Tm

Fig 6 16 (6 45)

 

(6 48) (6 49)

  image378

1

(N/M)4

  image379

N_

2T

 

1

MAt

 

(6 50)

 

or

 

image228

Af = 7t times the half-power bandwidth of a

sharply tuned circuit (6.46)

The half-power bandwidth is also defined as the resonant frequency divided by the so-called resonance Q

The above pertains primarily to continuous frequency analysis, however, there is also statistical error associated with sinusoidal excitation experiments because random
where N is the number of data points, T/At, and M is the maximum number of lag intervals, rm/At If, when these equations are applied, it is found that some of the parameters so determined are not readily attainable (if, for example, too many digits are required), then obviously suitable compromises must be made between the limita­tions of the analysis and the desired results In continuous analysis one evidently optimizes just Af and T of Eqs 6.48 and 6 50 somewhat independently of the fmax selected.

An illustration of selections made in an ion-chamber noise analysis90 of the Experimental Boiling Water Reactor to measure a resonance at 1.7 Hz is

N= 3331 Fractional error in P(f) = 03

M = 300 fmax = 10 Hz

At = 0 05 sec fmm = Af = 0 067 Hz

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