The results confirm the alias-free character of the Poisson sampling scheme even for unbounded spectra. From measurements of power spectra, it has been possible in the case of linear systems to obtain useful estimates of the stochastic relationships that generate such time series (see, for example, reference [19)). Blackman-Tukey closed-form expressions are derived for the Poissson sampling scheme.
A detailed discussion on the averaging power of the spectral windows is provided in Chapter 2. In this chapter we continue to examine the first basic property of the scheme, outlined in Section 1.2: the expected value of the . Due to the stationarity of the sampling process {t}, the summation in equation 2.21 does not depend on k.
For absolute convergence of the convolution integral (see e.g. Apostol [25] p. 490), and since S(f) is assumed to be bounded on (-eo,+oo), it is sufficient that. The bandwidth over which the true spectrum is averaged plays a major role in the stability of spectral estimates. It is simple that the bandwidth Sm(Q) of a spectral window. w) is the base of the rectangle whose height is the peak of ~(w).
3 and analytically calculate the bandwidth and sidelobes for each of the windows for comparative analysis. Absolute integrability follows from the finiteness of the integration limits and ordinary integrability demonstrated above. In the practical estimation of spectra from uniformly sampled data, errors are known to occur in the estimates due to the periodic nature of the sampling scheme.
Fitting the spectral window in this way is usually achieved by weighting the mean-lagged products with a smooth, positive definite function. A general expression for the weighting function of the type proposed by Julius Von Hann and R. A consequence of the following analysis is that the question of whether an estimate is aliased or not can be answered directly from the characteristic spectral window ~(w).
How close to the true value is the estimate available from a finite data size, N ; and what is the covariance of the estimates. For analytical convenience, we will make some of the following approximately equivalent forms of equation 1.26. From the results of the preceding sections, a number of useful results related to assumptions about the spectrum S(f) can be drawn.
Estimation of Slow-varying Spectra
Thus, we see that even with large data sizes, reliable estimates of spectral peaks are difficult to obtain, regardless of the sampling scheme used. For both the periodic and Poisson sampling schemes, we saw in Chapter I I that E{~(fr)} is approximately the same when the spectrum is confined to the Nyquist band and m is very large. The corresponding width of estimate can also be displayed to approximate the corresponding window bandwidths.
Since the window bandwidth is for all practical purposes equal, we look at the quantity o(N.
For readers unfamiliar with the work of Blackman and Tukey [1], the research cited in this thesis is by no means complete. Details of "measurement planning" - "pre-whitening", "post-greening", etc. have been deliberately omitted from this account to avoid undue repetition of the main reference above. Regarding the issue of aliasing, in this task we found a more practical way of testing aliasing - obtaining a graph of the spectral window, ~(rn).
In particular, we verified via an estimation algorithm that the Poisson sampling process is alias-free even for non-band-limited spectra. The Gaussian assumption that most of the analysis in Chapter III has been subjected to is not just for analytical convenience, since we are dealing with very large data size, and the central limit theorem validates this assumption even when the process is not Gaussian. These results, although approximate, are as useful as any others derived in this thesis, as they make use of the governing assumption of very large data size.
Using these results, we further found that the Poisson sampling process achieves less variability than the periodic sampling process for spectra that are very smooth. On the other hand, for rapidly changing spectra, the Poisson is just as unreliable as the periodic scheme. However, this does not mean that Blackman and Tukey's algorithm should be discarded, as we will now see. In the last section, we concluded that for the same number of process samples, the Poisson sampling scheme achieves a smaller variance than the periodic scheme.
We already know that increasing the data size produces smaller variance, so it is quite possible that in a given time T we can obtain more samples by sampling periodically than with Poisson sampling. Even when signal duration is not limited, the question is how much longer does one wait for say, N samples by sampling one way. Now, for the case when the waiting time T, is fixed, the probability of obtaining between N-K and N+K samples by Poisson sampling.
In certain application areas, prior statistical information on sample durations is not available, in which case, it may be necessary to record these sample times simultaneously. In reasonable cases this will require modification of existing recording hardware and possibly more computing time. By sampling a time sequence in a random fashion corresponding to random channel availability instead of periodic samples, the need for buffering can be eliminated.
94- References
Feller, William: Probability Theory and its Applications, Vol.I, John Wiley and Sons, Inc., New York, N.Y. 0.: "A Theoretical Model for Seismic Noise", Basic Geophysics Division, Esso Production Research Co., un- Basic Geophysics Division, Esso Production Research Co., niet-gepubliceerd rapport (zomer 1967).
