exponential decay and in the semilog plot. Figure 3B shows 20 averages for the same decay. Averaging improves the signal-to-noise ratio (SIN) (Fig. 3A) by about the expected square root of the number of averages (1 - 3, 11, 12). Similar results are shown for a quenched sample (4 µM Ru(phen)a 2 +) with a single decay (Fig. 4A) and five averages (Fig. 4B). Figures 3A and 4A illustrate quantization noise, which is noticeable as the staircase at the ends of the semilog plots. Quantization noise arjses from finite ADC resolution and systematically biases computed results. Quantization noise may be reduced by using a higher resolution ADC or, more simply anq cheaply, by signal averaging in thE:l presence of random noise equal to or greater than the incremental steps of the ADC (16). Signal averaging of a noisy signal interpolates between the quantization levels. Quantization noise is virtually eliminated by 20 averages (Fig. 3B) and is greatly reduced by only five averages in the more severe case (Fig. 4B). The reduction of systematic quantization errors is clearly visible.
2.00
*
1.50
...... I
--
I->
1. 0 0
0
I->
0 .50
q
= 0 .201 kq = 9. 4 1 x 10
Slo p e
s
-1
M-1
0 .00 -.t.hcrrrrrrrrmmrnrnrn~rrrrrrmmrnrnrn~ 0 .00 2.00 4.00 6.0 0 8 .00 10.00
Ru(phen) 32+ (µM)
Quality of Data Fitting
The data are well fit by a single exponential model as shown by the residual plot for the data of Figure 3A (Fig. 3C). The residuals are the differences between the observed and calculated data. Also, note the improved residuals in Figure 3D for the averaged data of Figure 3B. Systematic deviations in the residuals plots are an indication of more complex kinetics or instrument bias. For Poisson statistics, standard deviations are proportional to the square root of the intensity (2, 3, 11, 12). The decreased noise level of Figures 3C and 3D is consistent with a Poissoq noise component associated with the photomultiplier. Comparison of Least Squares and RLD Method
Using the optimum fitting regions, there was no discernible difference in the values of the lifetimes as determined by either the RLD method or LLS. With an IBM clone equipped with a math coprocessor, LLS was so fas t (
L = -' P,
(3)
where Pmin is the minimum pa
