Abstract:
This paper studies the sequential sampling scheme, as a solution to the problem of aliasing, where the sampling interval is restricted to a minimum allowable value AT. In the sequential sampling, the signal is sampled at intervals of AT, AT*Ai, AT*2Ai, AT*3 At, ...; where At < AT and may be selected as desirable. The sequential sampling is, however, analyzed and it is proven that, when the ratio AT/At is an integral number,the associated spectral estimates give a Nyquist frequency. This2Arsampling scheme can, therefore. be employed to yield a required cut-off frequency. The autocorrelation function estimation from the sequentially sampled data is then considered and the approach to this is discussed. Simulation studies are also used for empirical investigations of the sequential sampling. Furthermore, since with this sampling scheme no estimates of the autocorrelation coefficients are available in the initial time interval from 0 to AT, the initial autocorrelation coefficients estimation method is appl ied to solve this problem. This, in addition, provides some empirical studies ot‘ the latter approach. Moreover, the application of the autocorrelation function extrapolation method is considered, as a means of minimizing the sampling time and costs. The contribution of the sequential sampl ing in obtaining a desired cut-off frequency, is also demonstrated by data simulation
Machine summary:
Sequential Sampling, Spectral Analysis, Autocorrelation Function, Aliasing, Fourier Transform INTRODUCTI ON Some data acquisition systems have a minimu m allowable sampling interval and do not provide a desired sampling period less than a minimum allowable value.
obtainable at discrete values of the time delay given as: (View the image of this page) 1 f the ratio yr is an integral number, then higher val ues or j would also provide more contributions to the autocorrelation estimates at the above time delays i,.
An autocovariance function with discrete values at time delays (tq - t» ), can be estimated from the contributions of the products x(t )x(tq) of the sample values: the autocovariance function may be normalized to yield the autocorrelation coefficients [4].
As many contributions as desired for averaging are, however, acquirable only by repeating the sequence and using the time instants zero The estimation of the lag values can take place as follows.
The data were generated (sampled) at the sequential times given by equation (3) and, were then used to estimate the autocorrelation coefficients according to the specifications already explained in this paper.
I t is shown and compared with the true curve in Figure 6, where the sample size reduction has been reflected in the reduced accuracy of the values estimated directl y trom the data and also the initial coefficient estimates.
It is observed that the estimates given to the missing initial coefficients are strongly pronouncing the statistical inaccuracies of the lag values obtained from the data, which would have increased as a result of reducing the sample size.