John L. Brown

Uniform linear prediction of bandlimited processes from past samples (Corresp.)

For x(t) either a deterministic or stochastic signal band-limited to the normalized frequency interval \mid\omega\mid \leq \pi , explicit coefficients \{ a_{kn} \} are exhibited that have the property that \begin{equation} lim_{n \rightarrow \infty} \parallel x(t) - \sum_{1}^n a_{kn} x(t - kT) \parallel = 0 \end{equation} in an appropriate norm and for any constant intersample spacing T satisfying 0 ; that is, x(t) may be approximated arbitrarily well by a linear combination of past samples taken at any constant rate that exceeds twice the associated Nyquist rate. Moreover, the approximation of x(t) is uniform in the sense that the coefficients \{ a_{kn} \} do not depend on the detailed structure of x(t) but are absolute constants for any choice of T . The coefficients that are obtained provide a sharpening of a previous result by Wainstein and Zubakov where a rate in excess of three times the Nyquist rate was required.

First-order sampling of bandpass signals--A new approach (Corresp.)

The first-order sampling of a bandpass signal when the upper cutoff frequency is a multiple of the bandwidth is reduced to an application of the low-pass sampling theorem. In the general case, a simple band-imbedding procedure restores the positioning constraint and yields the expansion.

On mean-square aliasing error in the cardinal series expansion of random processes (Corresp.)

An upper bound is derived for the mean-square error involved when a non-band-limited, wide-sense stationary random process x(t) (possessing an integrable power spectral density) is approximated by a cardinal series expansion of the form \sum^{\infty}_{-\infty}x(n/2W) sinc 2W(t-n/2W) , a sampling expansion based on the choice of some nominal bandwidth W > 0 . It is proved that \lim_{N \rightarrow \infty} E \{|x(t) - x_{N}(t)|^{2}\} \leq \frac{2}{\pi}\int_{| \omega | > 2 \pi W}S_{x}( \omega) d \omega, where x_{N}(t) = \sum_{-N}^{N}x(n/2W) sinc 2W(t-n/2W) , and S_{x}(\omega) is the power spectral density for x(t) . Further, the constant 2/ \pi is shown to be the best possible one if a bound of this type (involving the power contained in the frequency region lying outside the arbitrarily chosen band) is to hold uniformly in t . Possible reductions of the multiplicative constant as a function of t are also discussed, and a formula is given for the optimal value of this constant.

On the expansion of the bivariate Gaussian probability density using results of nonlinear theory (Corresp.)

REFERENCES 111 A. Dvoretsky, “On stochastic approximation,” Pmceedings of the Third Berkeley Symposium OR Mathematical Statistics and Probability, vol. 1. Berkeley, C&f.: University of California Press. 1956, pp. 39-55. I21 C. Derman, “Stochastic Approximation,” Ann. Math. Statistics, vol. 27 pp. 879-886, 1956. IN K. S. Fu. 2. J. Nikolio. Y. T. Chien, and W. G. Wee, “On the Stochaatia approximation and related learning techniques,” School of Elec. Engrg., Purdue University, Lafayette, Ind.. Rept. TR-EE66-6. 1966. 81 C. Blaydon, “On a pattern classification result of Aizerman, Braverman and Rmonoer,” IEEE Trans. Information Theory (Correspondence), vol. IT-12. pp. l3283, January 1966. ISI K. S. fi and Y. T. Chien, “On Bayesiau learning in stochastic approximation,” Proc. 4th Annual AU&on Conf. on Cinzuit and System Theory, University of Illinois. Urbana, 1966. IU Yu-Chi Ho and C. Blaydon, “On the abstraction problem in pattern Claasification,” Cruft Lab., Harvard University, Cambridge, Mass., Tech. Rept. 476, October 1965.

Sampling theorem for finite-energy signals (Corresp.)

It is shown that the Shannon sampling expansion for finite-energy bandlimited signals is a special case of the Parseval relation for complex Fourier series.

Robust prediction of band-limited signals from past samples (Corresp.)

For a complex-valued deterministic signal of finite energy band-limited to the normalized frequency band |w| \leq \pi explicit coefficients \{a_{kn}\} are found such that for any T satisfying 0 , \left| f(t)-\sum^{2n}_{k=1}a_{kn}f(t - kT)\right| \leq E_{f}\cdot \beta^{n} where E_{f} is the signal energy and \beta \doteq 0.6863 . Thus the estimate of f(t) in terms of 2n past samples taken at a rate equal to or in excess of twice the Nyquist rate converges uniformly at a geometric rate to f(t) on (- \infty , \infty) . The suboptimal coefficients \{a_{kn}\} have the desirable property of being pure numbers independent of both the particular band-limited signal and of the selected sampling rate 1/T . It is also shown that these same coefficients can be used to estimate the value of x(t) of a wide-sense stationary random process in terms of past samples.

Relation between outputs of a full-wave and half-wave linear rectifier for a class of non-Gaussian inputs (Corresp.)

We have formed the probability density function of the test ending time under the assumption that the test ends at exactly the moment when the likelihood ratio exceeds one of the thresholds. Two types of error are incurred by neglecting the excess over the thresholds. The test, in fact, ends only at the discrete time of observation following the crossing of the threshold, and then only if the value of the likelihood ratio still exceeds the threshold. The first type of error is avoided by the following computational procedure: the computed probability density function is replaced by a distribution which takes values only at the discrete sampling times. The value of this distribution at time N is found by integrating the density function from time N 1 to time N. This is akin to computing N = Cc=r N[F(N) F(N I)], instead of fi = sr Nf(N) dN. There is no simple method of avoiding the second type of error. This remark applies to the analysis of the moments of any sequential test in discrete time, and thus our curves avoid the unwarranted optimism caused by neglecting the excess of the likelihood ratio over the thresholds. Two other tests have been considered for this same detection problem. Swerling and Marcus4 have considered a test using as the test function the likelihood function averaged over the k cells. However, they were not able to obtain analytic results for the average sample number and were forced to use a simulation. Preston6 appears to be the first to consider the maximum likelihood detector for this problem, but he succeeds in actually designing the test only under rather restrictive assumptions.