We occasionally get feedback from our readers. Often, the feedback is positive, letting us know that they find reading these columns interesting, enjoyable, and most importantly, useful. Getting responses like that also makes writing these columns interesting and enjoyable, and makes us feel useful, too. Occasionally, one or more readers finds an error (horrors!) that requires a correction (more horrors!), or presents an alternative approach to, or interpretation of, our discussion. When we wrote our 14-part column on the analysis of noise in spectroscopy, a few readers wrote in, letting us know that they felt that some parts of the derivations were weak, and perhaps less than perfectly rigorous. We took their comments to heart and here present the first of a two-part discussion incorporating the comments. This could be considered a (hide your eyes!) correction, or perhaps, simply another approach to analyzing the situation.

Jerome Workman, Jr.

Some time ago, we published a subseries of columns within this series (1–14) giving rigorous derivations for the expressions relating the effect of instrument noise and other types of noise to their effects on the spectra we observe. During and after the publication of those columns, we received several comments discussing various aspects, including some errors that crept in. Most of those were minor and can be ignored (for example, a graph axis labeled "Transmission" when the axis went from 0–100%, and therefore, the axis should have been labeled "% Transmission").

Howard Mark

One comment, however, was a little more significant, and therefore, we need to take cognizance of it. One of our respondents noted that the analysis performed could be done in a different way, a way that might be superior to the way we did it. Normally, if we agree with someone who takes issue with our work we would simply publish a correction (assuming their comments are persuasive). In this case, however, that seems inappropriate for several reasons. First, the original analysis was published a long time ago and cannot be dispensed with easily. Second, we're not convinced that our original approach is wrong; therefore, it is not clear that a correction is warranted. Third, some of our readers might wish to compare the two approaches for themselves to decide if the original one is actually wrong or simply not as good, or whether, in fact, the new analysis is better. Therefore, we present a new, alternate analysis, along the lines recommended by our respondent.

Alternate Analysis

The point of departure is from our original column (5), which dealt with the effect of random, normally distributed noise whose magnitude (in terms of standard deviation) is independent of the strength of the optical signal. Here we present the revised analysis of this situation of the effect on the expected noise level of the computed transmittance when the signal noise is not small compared to the signal level E_r.

Before we proceed, however, there is a technical point we need to clear up, and that is the numbering of the equations in this column series. Ordinarily, when continuing a subject through several columns, we simply continue the numbering of equations as though there were no break. The column representing our point of departure ended with equation 63. Therefore, it is appropriate to begin this alternate analysis (5) with equation 64, as we would normally. However, equation number 64 (and subsequent numbers) was already used; therefore, we cannot simply repeat using the same equation numbers that we used already. Neither can we simply continue from the last number used in the original analysis (5) because they would also conflict with the equation numbers already used for other purposes.

We resolved this dilemma by adding the suffix "a" to the equation numbers used in this column. Therefore, the first new equation we introduce here will be equation 64a. Fortunately, none of the equations developed in the column with the original analysis, nor the figures, used any suffix, as was done occasionally in other columns (we will copy equation 52b from the previous column, but the "b" suffix does not signify a new equation because it is the equation used previously; also, a "b" suffix is not indicative of a copy of an equation number here, only an "a" suffix). Therefore, we can distinguish the numbering of any equations or other numbered entities in this section by appending the suffix "a" to the number without causing confusion with other corresponding entities.

Now we are ready to proceed.

We reached this point from the discussion just before equation 64 (5). A reader of the original column felt that equation 64 was being used incorrectly. Equation 64, of course, is a fundamental equation of elementary calculus and is itself correct. The problem pointed out was that the use of the derivative terms in equation 64 implicitly indicates that we are using the small-noise model, and especially when changing the differentials to finite differences in equation 65, results in incorrect equations.