By William E. Blass

ISBN-10: 0121046508

ISBN-13: 9780121046507

ISBN-10: 032316076X

ISBN-13: 9780323160766

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**Extra resources for Deconvolution of Absorption Spectra**

**Sample text**

Apply convergence/termination tests. If satisfactory, the current Lj is the estimator of L(t) and the procedure is completed. If tests are not satisfactory, set j = j + 1 and return to step B. On exiting from the preceding iterative procedure an estimator of the desired true data set L(t) has been found. Notice that at each step, a j(y)(S ~~ Lj_x θ Τ) is the current correction to the previous iteration. Forming ι " j, Σ 2 (93) [«,O0(s-z,_,er)]2 all points essentially the rms correction to the previous estimator of L{t) yields some idea of convergence of the deconvolution process.

Is given by Eq. (73). Fourier transforming, one obtains /; = s £ « ( l - a O ' , (86) which may be analytically summed and yields /; = ( 4 ) [ l - ( l - « 0 " ] . (87) + 1 This result, which is true at any spectral frequency in transform space, shows (after Frieden [12]) that 1. In the limit as A J - > O O , Ln approaches the inverse filtering estimate (S/t) if |(1 - at)\ < 1. Thus, if at is negative due to noise, then /„ will not converge for that frequency and, more disastrous, neither will its transform £ Λ ( 0 · Α 2.

This algorithm (called the Jansson algorithm in the following chapters) results in Eq. (58) becoming Ln = L„_x + an(y)(S-L„_x®T), (60) where y(t) is the amplitude of Ln_x(t). Jansson's an(y) is shown in Fig. 16. Selection of an χ less than one, for example, uses only a partial correction in Eq. (60). , L„, such that Ln is usually positive or zero in all cases (assuming S was originally positive for all /) and limits the amplitude of Ln{i) to a value j m a . x Since Jansson was dealing with infrared absorption spectra that have no negative amplitudes and are limited to 100% absorption on the other end of the absorption scale, this function an(y) builds in desired constraints.

### Deconvolution of Absorption Spectra by William E. Blass

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