Given an observation vectory = Ax + \nu, estimate the parameter vectorxsubject to the known constraint relationshipKx = 0. The covariance matrixRof the measurement error\nuandAare known block diagonal matrices. The best linear unbiased estimate ofxand its covariance is obtained and compared with a sequential suboptimal estimate.

This paper presents the equations for the mean-square error of the output of a continuous finite memory filter. The filter output error is unbiased for arbitrary input polynomials up to degreen, and has minimum variance. The input is taken as a polynomial of degreenplus random stationary noise. Noise processes are considered, 1) where the noise is exponentially correlated, and 2) in the white noise case. The solution for a desired output which is an arbitrary fixed linear operation on the input polynomial is given. Tables and graphs of the mean-square error for the derivative and prediction operator for the 0th, 1st, and 2nd derivatives are presented, and for input polynomials up to the 6th degree.

This paper derives the weighting sequence of a linear digital filter whose output is an estimate of the predicted values of the derivatives of the input. The input functions considered are arbitrary linear combinations of n + 1 known functions, plus a random stationary signal and a random stationary noise component. The filter differs from previously considered minimum variance optimum filters in that the primary consideration here is the computational ease with which one can obtain the final solution. An optimization in the minimum variance sense is obtained as a secondary consideration in order to provide some control of the mean square output error. The exponential filter has its simplest form for the class of nonrandom input functions ( h n ) which are the complete solutions of a set of homogeneous linear difference equations of order n with constant coefficients. For this class the input and output are related by a time invariant recursion formula. The output contains a bias error which can be made to approach zero exponentially as the mean square error increases monotonically to a limit with increasing time. A modification of the exponential filter is considered such that the bias error is zero. The solution then involves a recursion formula with time varying coefficients.

An asymptotic solution for the mean-square output noise power of an optimum digital filter is obtained. It is assumed that the input consists of a polynomial plus correlated noise. The asymptotic solution is found by fixing the interval between samples and allowing the number of samples to approach infinity. The solution obtained for the minimum variance filter is compared with the solution as obtained for the "least-squares filter," and it is shown that the latter filter is asymptotically efficient as compared to the former. It is shown for each of the above filters that the mean-square output noise power is proportional to the spectral density function of the correlated noise, evaluated at zero frequency, and that the factor of proportionality is the same.

A growing memory digital filter is defined by considering the input(y_{\nu-u})-output(Z_m)relationship in the formZ_m = \sum_{u=0}^m W_{um} y_{m-u}, m = 0, 1, 2, \cdotswhereW_{um}is the weighting sequence of a linear time varying digital filter. Contained herein are a derivation of an optimum growing memory smoothing and prediction filter in the least squares sense for polynomial input functions, (of degree =K) and a theorem on the class of time invariant sequenceW_u, which are solutions of a difference equation of tiite order, and an application of the theorem to the synthesis of sampled correlated noise by digital processes, using recursion formulas. The recursion formulation represents a practical solution to the generation of a correlated noise sequence on line during simulation studies on digital computers.

Given a set of equally spaced measurements, it is possible to curve fit a "least squares" polynomial to theNobserved data points and obtain estimates of the past, present, or future values of the data or its derivatives by appropriate manipulations of the curve fit. This curve fitting can be accomplished by a linear weighting of the observed data over an interval(n-1) T. If the data is measured in real time such that a new data point is observed eachTseconds, then the desired output (for example, the smooth or predicted value of the data) can be obtained by sliding these fixed number of weights such that the same weight always multiplies the data which is at a fixed lag with respect to the most recent data. Since these weights are zero for lags greater thann, they may be described as a fix-finite memory linear digital filter. In calculating the desired output for each new sample one requires a machine which can storencoefficients,ndata points and performs n multiplications andn - 1additions in at leastTseconds. The coefficients do not change but the multiplications and additions must be performed eachTseconds as a new data point is measured. For large values ofn, and smallT, this may put a severe requirement on the real time solutions of the computer. This paper presents an alternate technique using recursion formulas to obtaining the same results as thenpoint weighting equation. The method has the advantage of requiring considerably less storage, multiplications and additions whenn \gg 1and the degree of the curve fitting polynomial(K)is small.

In a recent article Johnson presents an asymptotic formula for the output noise power of an optimum filter designed to make a zero-lag estimate of either the input or its derivatives. It is assumed that the input function consists of a nonrandom polynomial plus stationary uncorrelated noise. It is the purpose of this paper to present an exact formula for the output noise power for the same input model. The formula presented is more general in that the estimation can be for any lag a with respect to the latest data point. Tables and graphs of the root mean square error for the zero-lag estimation of the 0th, 1st, and 2nd derivative are presented as a function of the input polynomial up to degree 5 and memory spans up to 100 sample points. A comparison is made of the relative error in root mean square using the asymptotic formula derived by Johnson.

The prediction theory presented in this paper is an extension of the prediction theory of Zadeh and Ragazzini. It differs from their theory in that the nonrandom component of the input signal in the Zadeh-Ragazzini model is restricted to a polynomial of known degreen. In the theory developed here, the nonrandom component of the input signal may be any arbitrary linear function of a subset of known analytic functions where the subset of functions are known a priori but the linear relationship need not be. As in the previous solution, the .determination of the impulsive admittance of the optimum predictor reduces to the solution of a modified Wiener-Hopf integral equation.

A method is developed for finding the ordinates of a digital filter which will produce a general linear operator of the signalS(t)such that the mean square error of prediction will be a minimum. The input to the filter is sampled at intervals\Delta t. The samples contain stationary noiseN(j\Delta t), a stationary signal component,M(j\Delta t), and a nonrandom signal component, \begin{equation} P(j\Delta t) = \sum_{k=0}^n a_k P_k (j\Delta t) \end{equation} where the subset of nonrandom functionsP_k (t)are known a priori, but the parameter vectora = (a_o, a_l, \cdots, a_n)need not be. The solution is obtained as a matrix equation which relates the ordinates of the digital filter to the autocorrelation properties ofM(t)andN(t)and the nature of the prediction operation.