Wpływ liczby pobieranych próbek i ich usytuowania w okresie na wyznaczanie wartości skutecznej lub średniej sygnału poliharmonicznego

W artykule przedstawiono zależności określające wartość skuteczną lub średnią bezwzględną sygnału wyliczaną na podstawie próbek pobranych przez przetwornik synchroniczny. Opisują one wpływ częstotliwości harmonicznych zawartych w próbkowanym sygnale na uzyskany wynik. Otrzymane rezultaty stanowią uzasadnienie, sformułowanej przez autorów w pracy [4], tezy o potrzebie i celowości stosowania filtrów dolnoprzepustowych umieszczonych w torze przetwarzania przed przetwornikiem próbkującym.

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A general pricniple in signal sampling is taking the minimum sampling rate more than twice the highest frequency contained in the signal. Does it matter, if on the basis of the collected samples, a root-mean-square (rms) or average signal value is calculated? A synchronous sampling was considered and an effect of sample number on the calculated value was analyzed in this paper. No account was taken - for clarity of the results - of the influence of other factors, such as resolution of converter, aperture time, etc. which also determine accuracy. For the considerationscarried out a stationary input signal and sampling with constant rate equal to multiple of input signal frequency were assumed. In the paper the mathematical equations for the rms value calculated on the basis of sampling were given and the circumstances for the obtainment of a correct rms value were determined. It has been ascertained that for a definite sample number per period of the fundamental frequency, there are a lot of harmonics that do not have to satisfy Shannon's requirement to get a correct result of calculation. But there exist "unlucky" numners of harmonics among them which are determined by a relation between the harmonic number and the number of samples per signal period. Their lowest frequency is equal to half of sampling rate. In this case the error of calculation of rms value can be quite significant. The error value depends on the moment of the first sample taking and on harmonic amplitude, also on itsphase shift in relation to the fundamental. For more than one harmonic contained in the signal, additional errors will come into being, if the sum or the difference of harmonic numbers is a multiple of the sample number in signal period. Calculating of average value on the basis of the taken samples can be done by one of the numerical integration methods. The rectangular method was applied in the paper. From the developed dependencies and computer reckoning it follows that in most cases the calculating error first of all depends on sample number in the signal period. A smaller error is received when the sample number is odd. Just as in calculating the rms, the error value is conditioned by the moment of taking of the first sample as well as by the amplitude and phase of various component harmonics included in the signal. The described examples show that if the frequency of the sampling rate is too low, then the error of computations can be significant under certain circumstances. Also in average value calculating the "unlucky" harmonics occur. Their existence in the signal can lead to big errors which can be even bigger than in the case of removing of a part pf high frequency components from the signal. Then a low-pass filter is desirable in the converter track. Both with the calculation of the rms and the average value, the low-pass filter will not be necessary, provided there is full guarantee that "unlucky" harmonics do not appear in the band of the converted signal. The verification of mathematical dependences was accomplished in MATLAB's environment. The obtained results were presented in the figures. Finally one should answer the questions: "Can the described phenomena be called an aliasing?" and "Can the low-pass filter placed before sampling converter be given the name of antialiasing filter?". Taking into account the opinions presented in the references, e.g. [2], the authors think so.