City Research Online

Abstract

The main objectives of the research programme were :

(1) Developing a discrete mathematical model describing the electro optic process in the spectrophotometer which includes three subprocesses :

(a) The absorption process (Beer - Lambert law)
(b) The slit transfer function
(c) The detector noise

The model provides the theoretical tool in creating signal processing techniques that are applied to spectrophotometry instrumentation, in particular to deficiencies like low resolution, noise and distortion.

(2) Studying and developing signal processing techniques that estimate the concentration of one compound in a mixture of several.

The estimation techniques that are based on the model created in (1), exploit well known concepts in statistical estimation theory (i.e The Maximum Likelihood principle) and give as a result the vector where each component C is the estimate of the compound j.

A discrete formulation has been chosen for the above in order to achieve easy computer implementation both in testing the technique itself and in later practical use of the algorithms developed.

The first step in modelling the electro optic process in the spectrophotometer was to derive the differential equation of a process given its impulse response in a polynomial form or in a gaussian pulse shape. The differential equation found is given in operator form using operators like D the differentiator or 6 the central difference.

The second step was implementation of the differential equation in a discrete form. A standard form of finite series in the discrete Operator (the average operator ) j=1,10 was chosen and derived. The implementation was carried out on both the forward transfer function and the inverse transfer function (with both gaussian and triangular impulse response). The implemented transfer functions were tested for their accuracy as a function of the signal frequency. The concept of the partial deconvolution formula was suggested and implemented. The idea that underlies it is to implement a discrete transfer function that reduces the slit width by only a certain amount and thus allows a trade off between the noise level and the resolution.

In the second part of the work two estimation algorithms are proposed. The two algorithms are based on the Maximum Likelihood Principle which in our case (noncorrelated gaussian noise components) yields the minimum rms error. The first algorithm (called the MLE) is treated as an optimization problem in n dimension. The result, the estimation vector C, gives the minimum rms error. The algorithm is based on an iteration process where a correction vector △C is found at each step until no further correction is possible. The equation for the correction C and for the error (the estimation variance) △C△C' are derived. It is shown that the expression of the error has the same form as in the linear case.

The algorithms were tested with a signal spectrum of a mixture of two gas compounds (two concentrations) and with a perfect resolution (the effect of the spectrophotometer slit width is ignored) over a range of several S/N ratios and concentrations, Also, the calculated rms error was compared to the value given by the theoretical expression derived. It was shown that the two are in good agreement.

The second algorithm (called SE) was developed as a sequential algorithm in order to provide easy and fast hardware implementation. The algorithm based on linearization of the estimation problem and on the Maximum Likelihood principle, processes the signal samples as they enter the signal processing unit, and thus avoids the need for storing the signal.

Since both algorithms are based on the MLE principle it means that the algorithms have a built-in filtering mechanism. Therefore, the spectrophotometer scanning speed is not limited by the time constant of an additional filter but only by the detector time response.

Publication Type:	Thesis (Doctoral)
Subjects:	Q Science > Q Science (General) Q Science > QA Mathematics Q Science > QC Physics
Departments:	School of Science & Technology > School of Science & Technology Doctoral Theses Doctoral Theses

Export

Downloads