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516 M. Daszykowski et al.

Elimination of Noise from Analytical Signals

To suppress white (Gaussian) noise in analytical signals, different types of digital filters can be used either in time or in the freąuency domain [17,18], e.g., moving-average, Savitsky-Golay filters, etc. Recently, denoising of chromatographic signals by use of wavelets has attracted much attention, because this approach can efficiently deal with white noise in non-stationary signals [19-21]. Chromatograms are typical examples of such in-strumental signals, because they contain components of very different fre-quencies.

The wavelet transform operates on a single analytical signal, x, and it linearly transforms this signal from its original domain to the wavelet domain. Wavelets are a set of specific functions or basis vectors with orthogo-nal and local properties. For this reason wavelets have much potential to model different signal components and enable a signal to be studied at its different resolution levels. In the waveiet domain, each signal is described by a set of the so-called wavelet coefficients, c:

c = WT_X    (2)

where W is a matTix that contains the basis vectors in columns, and x is an analytical signal.

Processing of instrumental signals in the wavelets domain can be com-putationally very efficient when the so-calied pyramid algorithm is used. However, this algorithm can only deal with signals of length equal to an in-teger power of two [22]. Several approaches can be used to adjust signals to a desired length. The signals are then decomposed in several consecutive steps. The time domain of a signal is recursively cut into halves using a low-pass filter and a high-pass filter until only one point remains in the proc-essed signal [23]. At each decomposition level, two sets of wavelet coefficients are obtained, i.e. n/2 of the low-frequency coefficients (the so-called approximations) and n/2 of the high-frequency coefficients (the so-called details). Details obtained from the first level decomposition of a signal can support estimation of its noise level. For this purpose, two avaiiable thresh-olding policies (soft and hard) and several threshold criteria can be used. On this basis it is possible to determine and eliminate the details represent-ing signal noise [21]. After removal of these details, the signal is trans-formed back in the time domain, eventually resulting in a noise-free signal.