A method is presented for determining the maximum entropy spectrumS(\omega)of a process under the constraint that its autocorrelationR(m)is known for everymin a setDof nonconsecutive integers. The method involves a simple steepest ascent technique that is based entirely on Levinson's algorithm.

In the deconvolution problem, the presence of noise introduces large errors in the vicinity of the zero-crossings\omega_{i}of the system function. A simple method is presented for reducing

The system function of an unknown system can be determined in terms of the power spectrum of the output if the

If a signal is estimated by a weighted average of the data in the interval(t - c, t + c ), then the variance\sigma^{2}of the estimate decreases, but its bias b increases, with increasing c. It is shown that in high accuracy estimates, the mean-square error e is minimum ifcis such that\sigma = 2b, regardless of the formh (t)of the smoothing weight. Furthermore, the resultinge_{m}is minimum ifh(t)is a truncated parabola.

The average density\lambda (t)of a Poisson process is estimated by a nonstationary shot noise processs(t). The processs(t)is generated by a time-varying system whose impulse responseh (t, \alpha)is so determined as to minimize the mean-square (ms) estimation error. It is shown that, under realistic assumptions,h (t, \alpha )is a truncated parabola in the variable α . The analysis is extended to two-dimensional processes. The resulting smoothing kernel is an elliptic paraboloid.

The class of window pairsw(t) \leftrightarrowW(\\omega)is considered such thatw(t) = 0for\mid t \mid > MandW(\omega) \geq 0. It is shown that in this class, the window \begin{equation} w_o(t) = \frac{1}{\pi} \biggl| \sin \frac{\pi t}{M} \biggr| + \biggl( 1- \frac{\midt \mid}{M} \biggr) \cos \frac{\pi t}{M}, \qquad \mid t \mid \leq M \end{equation} yields spectral estimates with minimum bias. Furthermore, the variance of the resulting estimate is smaller than that obtained with the known windows of the same size.

The approach to Gaussianity of the outputy(t)of a narrow-band systemh(t)is investigated. It is assumed that the inputx(t)is ana-dependent process, in the sense that the random variablesx(t)andx(t + u)are independent foru > a. WithF(y)andG(y)the distribution functions ofy(t)and of a suitable normal process, a realistic boundBon the differenceF(y) -- G(y)is determined, and it is shown thatB \rightarrow 0as the bandwidth\omega_oof the system tends to zero. In the special case of the shot noise process \begin{equation} y(t) = \sum_i h(t - t_i) \end{equation} it is shown that \begin{equation} \mid F(y) - G(y) \mid\lambda_iis the average density of the Poisson pointst_i.

Upper and lower bounds are established for the mean-square variation of a stationary processX(t)whose power spectrum is bounded by\omega_{c}, in terms of its average powerP_{0}and the average powerP_{1}of its derivative. It is shown that\left( \frac{2}{\pi} \right)^{2} P_{1} \tau^{2} \leq E \{|X(t+\tau )-X(t)|^{2}\} \leq P_{1} \tau^{2} \leq \omega_{c}^{2}P_{0}\tau^{2}where the upper bounds are valid for any\tauand the lower bound for\tau < \pi / \omega_{c}. These estimates are applied to the mean-square variation of the envelope of a quasi-monochromatic process.