Cross-Correlation Noise Studies in Atomic Magnet-Optic Rotation Spectroscopy

Turk J Chem 25 (2001) , 135 – 143. c TÜBİTAK Cross-Correlation Noise Studies in Atomic Magnet-Optic Rotation Spectroscopy∗ Ahmet T. İNCE 26 Ağustos Yeditepe University Campus, Arts and Science Faculty, Department of Physics, Kayışdağı Caddesi, Kayışdağı, 81120, Erenköy, İstanbul-TURKEY Received 05.10.2000 Analytical signals in an Atomic Magneto-Optic Rotation spectrometer are buried in noise at the limit of detection. The noisy analytical signals were analysed by carrying out mathematical correlation of their time domain waveforms. The noise components of signals were removed by auto-correlation to simplify the study. If noise interferes in analytical signals whose source is unclear, a cross-correlation of the output waveform with noise source may identify the source, e.g., mains frequencies and background radio signals. A cross-correlation will reveal whether the two signals are derived from the same source. This can also lead to an improvement in the signal detection limit. Either of the two above situations can occur in studying analytical signals. In this study, both auto-and cross-correlation studies were carried out on analytical signals which had discrete noise sources present in their waveforms. Key Words: Atomic magneto-optic rotation spectrometer, auto-and cross-correlation Introduction The detection limits of analytical spectroscopic measurements are ultimately limited by the presence of broad band system noise. An analytical spectrometer designed to measure Atomic Magneto-Optic Rotation (AMOR) of light through analytes was found to suffer considerably from interfering frequencies on detected spectroscopic signals1−4. The presence of noise in analytical signals can be reduced if their sources are identified. This often is a trial and error procedure, and is time consuming. A more systematic approach is to mathematically correlate the signal waveforms with suspected sources of noise in the system or those present in the surroundings. Auto-correlation and cross-correlation can both be carried out on the signal waveforms. Correlation can identify sources of interfering frequencies related to each other; others have even improved the signal-tonoise ratio of the analytical signal5 . The purpose of this paper is to describe this method of analysis, which is largely independent of the instrumentation5 . ∗ This paper has beed presented at MBCAC III (3rd Mediterranean Basin Conference on Analytical Chemistry) 4-9 June, 2000 Antalya-Turkey 135 Cross-Correlation Noise Studies in Atomic Magnet-Optic..., A. T. İNCE, Theory Similarity and association are good intuitive definitions for the mathematical operation of correlation. The mathematical definition of correlation in the time-domain is given by6 , r(τ ) = lim 1 T ∫ x(t).y(t + τ ).dt T →∞ 2T −T (1) Where r (τ ) is the correlation function formed by summing the lagged products of two waveforms x (t) and y (t), and τ is the time lag between x (t) and y (t). Correlation is a mathematical similarity test between waveforms, it is simplified using a Fast Fourier Transform (FFT). In the frequency domain, it may be represented as7 R (f)=X (f) * Y(f) (2) R (f) is the frequency correlation, and * is used to denote conjugation. R (f) is then inverse transformed back to the time domain to give r (τ ). If the two waveforms are the same, i.e., x (t) = y (t), then an auto-correlation is performed. If the two waveforms are different, i.e., x (t) 6= y (t), a cross-correlation is performed. The FFT of a correlation function is a power spectrum. The FFT of an auto-correlation function is an auto-spectrum, noise power spectrum or power spectral density (PSD). The FFT of a cross-correlation function is a cross-spectrum. Experimental The studies presented here involve investigation in the time domain of the noise sources within a 200 Hz spectral range for an AMOR spectrometer set-up in the Faraday configuration and employing an offset polariser method (θ=45o ). The experimental set-up and D.C. power supply circuit used for the magnet assembly are shown in Figures 1 (a) and (b) respectively. A Rochon prism was used as the analysing polariser since it generates two orthogonally polarised beams with a small angle of separation from the incident beam. These two rays are focused onto the entrance slit of a monochromator, one above the other. On leaving the exit slit of the monochromator, they are reflected by a plane mirror and thus further separated into two rays. Each ray then enters identical side-window PMT tubes. The output of the PMT tubes form the input to a Solartron 1200 model signal processor. Details of the AMOR apparatus are explained in a previous paper2 . Magnesium was the analyte used to carry out the correlation studies. When the rotated plane polarised light traverses the offset polariser, light is split into two rays and the noise sources carried by the rotated plane polarised light are split between these two rays. 136 Cross-Correlation Noise Studies in Atomic Magnet-Optic..., A. T. İNCE, (a) L1 Glan-air prism L2 Magnet solenoids F L3 .... .. .. ...... .. ...... ... ..... .... .. .. ... .. ...... ... . .. ... .. ........ ... .... ..... . .... . .. . . HCL PSU Rochon prism L4 Monochromator Nebuliser Sample POWER PMT PMT OUTPUT (b) POWER OUTPUT Fuse (35A) Transformer variac + a.c. 60A SIL Bridge 60/68 a.c. - d.c. Capacitor Megnet coils 2x200 turns 240V Figure 1. (a) Block diagram showing the AMOR spectrometer arrangement. HCL: hollow cathode lamp; PSU: power supply unit; L1, L2, L3, L4 lenses; F: Flame; PMT: photomultiplier tube (b) D.C. power supply circuit used to drive the electromagnet assembly Noise sources detected in the signals were white noise, a 50 Hz frequency (probably from the hollow cathode lamp’s DC power supply), a flame feature frequency, and a 100 Hz frequency, which is thought to be due to field modulation4. The 50Hz frequency is generated outside the AMORS system by the hollow cathode lamp’s power supply, and the flame feature frequency and field modulation frequency are generated in the AMORS system by the sample. These latter two interference frequencies suffer from noise introduced onto them as a result of the sample introduction system, which is not continuous, but delivers samples only intermittently into the AMOR system. The electrical signal waveforms derived from these two rays form the inputs to a Solartron 1200 signal processor. Auto-correlation and cross-correlation studies of the noise sources were carried out. Autocorrelation of the noise sources was carried out to detect interference frequencies buried in noise. Frequencies which are not of interest may then be removed so that the signal–to-noise ratio can be improved. Frequencies found to be derived from the analytical signal may then be cross-correlated and possibly used for further analytical study. Cross-correlation of the waveforms of both rays reveal frequencies derived from a c (...truncated)