Computed transmittance noise

We have analyzed the effect that noise has on the computed transmittance, just as we previously analyzed the effect that the sample transmittance has on the computed noise value. We can experimentally measure the variation in noise level due to the sample transmittance. On the other hand, we will not be able to realize the effect of noise on the computed transmittance, for reasons we will discover in our next chapter, which will deal with the noise of the transmittance when the energy is low, or the noise is high, so that again we cannot make the low noise approximation we made previously. 

Figure 44-lla-l Comparison of computed absorbance noise to the theoretical value (according to equation 44-32), as a function of S/N ratio, for constant transmittance (set to unity), (see Color Plate 12)... 

There is, however, something unexpected about Figure 44-1 la-1. That is the decrease in absorbance noise at the very lowest values of S/N, i.e., those lower than approximately Er = 1. This decrease is not a glitch or an artifact or a result of the random effects of divergence of the integral of the data such as we saw when performing a similar computation on the simulated transmission values. The effect is consistent and reproducible. In fact, it appears to be somewhat similar in character to the decrease in computed transmittance we observed at very low values of S/N for the low-noise case, e.g., that shown in Figure 43-6. 

In equation 46-79, Twu represents the mean computed transmittance for Uniformly distributed noise and the parenthesized (1) in both the numerator and the denominator is a surrogate for the actual voltage difference between successive values represented by the A/D steps essentially a normalization factor for the actual physical voltages involved. In any case, if the actual voltage difference were used in equation 46-79, it would be factored out of both the numerator and the denominator integrals, and the two would then cancel. Since the denominator is unity in either case, equation 46-79 now simplifies to... 

In the previous chapter, we have examined the situation in regard to determining the effect of noise on the computed transmittance. Now we wish to examine the behavior of the absorbance for Poisson-distributed noise when the reference signal is small. Our starting point for this is equation 51-24, which we derived previously [3] for the case of constant detector noise, but at the point we take it up the equations have not yet had any approximations, or any special assumptions relating to the noise behavior ... 

Following our usual sequence, our next step here then, as it was for the previous two cases we treated (constant detector noise and Poisson-distributed noise), is to ascertain the effect of this noise on the expected value of computed transmittance. To do this we start with equation 53-5 (reference [2]) ... 

When the noise is small the multiplication factor approaches unity, as we would expect. As we have seen for the previous two types of noise we considered, the nonlinearity in the computation of transmittance causes the expected value of the computed transmittance to increase as the energy approaches zero, and then decrease again. For the type of noise we are currently considering, however, the situation is complicated by the truncation of the distribution, as we have discussed, so that when only the tail of the distribution is available (i.e., when the distribution is cut off at +3 standard deviations), the character changes from that seen when most of the distribution is used. 

The presence of a non-zero dark reading, E0, will, of course, cause an error in the value of r computed. However, this is a systematic error and therefore is of no interest to us here we are interested only in the behavior of random variables. Therefore we set E0s and Eqj. equal to zero and note, if T as described in equation 41-1 represents the true value of the transmittance, then the value we obtain for a given reading, including the instantaneous random effect of noise, is... 

It is even possible for an individual noise pulse to exceed — ET so that a negative reading of Ex will be obtained. This happens in the real world and therefore must be taken into account in the mathematical description. This is a good place to also note that since the transmittance of a physical sample must be between zero and unity, Es must be no greater than If, and therefore when If is small an individual reading of Es can also be negative. Therefore it is entirely possible for an individual computed value of T to be negative. 

Now let us recap where we came from in our discussion, in this mini-series-within-a-book, and where we are going. In Chapter 41, referenced in [2] we demonstrated that, because previous treatments of this topic failed to take into account the effect of the noise of the reference reading, they did not come up with the rigorously correct formula to describe the effect of transmittance on the computed value of the noise. The rigorously exact solution to this situation shows that the noise level of a transmittance... 

Er has dropped to five standard deviations, the optimum transmittance has dropped to 3.2, and then drops off quickly below that value. Surprisingly, the optimum value of transmittance appears to reach a minimum value, and then increase again as Er continues to decrease. It is not entirely clear whether this is simply appearance or actually reflects the correct description of the behavior of the noise in this regime, given the unstable nature of the variance values upon which it is based. In fact, originally these curves were computed only for values of Er equal to or greater than three due to the expectation that no reasonable results could be obtained at lower values of Er. However, when the unexpectedly smooth decrease in the optimum value of %T was observed down to that level, it seemed prudent to extend the calculations to still lower values, whereupon the results in Figure 45-11 were obtained. 

As we did in the analysis of Poisson-distributed noise, we compute the expected value of T as the weighted sum of the transmittance described by equation 49-5 (reference [10]) ... 

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