Attainable Mean-square Error Research Articles

A New Approach to Mean Estimation Using Two Auxiliary Variates in Two-Phase Sampling

This paper focuses on estimation of mean using two auxiliary variates in two-phase sampling. The technique of Taylor series expansion is used for obtaining the mean square error (MSE) of the proposed estimator. The expression for minimum attainable MSE is also derived. Some of the estimators of the sampling literature are shown to be the members of the estimator proposed in this paper. An attempt has been made to find optimum sample sizes under a known fixed cost function. The results are extended to the case of multi-auxiliary variates. Efficiency comparisons are made, and an empirical study is carried out in support of theoretical findings.

An Efficient Class of Estimators for the Mean of a Finite Population in Two-Phase Sampling Using Multi-Auxiliary Variates

This paper presents an efficient class of estimators for estimating the population mean of the variate under study in two-phase sampling using information on several auxiliary variates. The expressions for bias and mean square error (MSE) of the proposed class have been obtained using Taylor series method. In addition, the minimum attainable MSE of the proposed class is obtained to the first order of approximation. The proposed class encompasses a wide range of estimators of the sampling literature. Efficiency comparison has been made for demonstrating the performance of the proposed class. An attempt has been made to find optimum sample sizes under a known fixed cost function. Numerical illustrations are given in support of theoretical findings.

Realizable lower bounds for time delay estimation. 2. Threshold phenomena

In time delay estimation of narrowband signals the mean square error (MSE) plotted as a function of the signal to noise ratio (SNR) exhibits threshold phenomena. The thresholds divide the domain of SNR values into several disjoint segments. For very low SNR values the observations are dominated by noise and are essentially useless for delay estimation. Here the MSE is determined largely by the available a priori information. For intermediate SNR values, time delay estimates are possible, but are subject to ambiguities resulting from the oscillatory nature of the signal sample correlation. For very high SNR values, these ambiguities are resolvable and the Cramer-Rao bound (CRB) yields a realistic bound for the attainable performance. A previous paper by Zeira and Schultheiss (see IEEE Trans. Signal Processing, vol.41, p.3102-3113, Nov. 1993) used the Barankin (1949) bound to obtain a realizable lower bound for the intermediate SNR region but left open the question where the transitions from low to medium and medium to high SNR operation occur. Since the attainable MSE in the three regions can be very different, the location of the SNR thresholds is of considerable practical interest. The present paper addresses this issue. It obtains threshold values for the single and two echo problems, with emphasis on the effect of echo separation in the two echo case. It concludes that the location of the thresholds is only weakly dependent on echo separation, even though the attainable MSE in both the intermediate and high SNR regions varies drastically as the echo separation changes. >

Fundamental limitations in passive time delay estimation--Part I: Narrow-band systems

Time delay estimation of a noise-like random signal observed at two or more spatially separated receivers is a problem of considerable practical interest in passive radar/sonar applications. A new method is presented to analyze the mean-square error performance of delay estimation schemes based on a modified (improved) version of the Ziv-Zakai lower bound (ZZLB). This technique is shown to yield the tightest results on the attainable system performance for a wide range of signal-to-noise ratio (SNR) conditions. For delay estimation using narrow-band (ambiguity-prone) signals, the fundamental result of this study is illustrated in Fig. 3. The entire domain of SNR is divided into several disjoint segments indicating several distinct modes of operation. If the available SNR does not exceed SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , signal observations from the receiver outputs are completely dominated by noise thus essentially useless for the delay estimation. As a result, the attainable mean-square error <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\bar{\epsilon}^{2}</tex> is bounded only by the a priori parameter domain. If SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> < SNR < SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> , the modified ZZLB coincides with the Barankin bound. In this regime differential delay observations are subject to ambiguities. If SNR > SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> the modified ZZLB coincides with the Cramer-Rao lower bound indicating that the ambiguity in the differential delay estimation can essentially be resolved. The transition from the ambiguity-dominated mode of operation to the ambiguity-free mode of operation starts at SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> and ends at SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</inf> . This is the threshold phenomenon in time delay estimation. The various deflection points SNR <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> and the various segments of the bound (Fig. 3) are given as functions of such important system parameters as time-bandwidth product (WT), signal bandwidth to center frequency ratio (W/ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> ) and the number of half wavelengths of the signal center frequency contained in the spacing between receivers. With this information the composite bound illustrated in Fig. 3 provides the most complete characterization of the attainable system performance under any prespecified SNR conditions.

Composite bound on the attainable mean square error in passive time-delay estimation from ambiguity prone signals (Corresp.)

The location of a radiating source can be determined by measuring the relative time-delay of its signal wavefront to several spatially separated receivers. A new technique is presented to investigate the performance of time-delay measurement schemes based on a modified version of the Ziv-Zakai lower bound. This technique is shown to yield a tight bound on the attainable mean-square measurement error for any prespecified signal-to-noise ratio conditions.

Optimum Mean-Square Decision Feedback Equalization

In this work we report new results relating to decision feedback equalization. The equalizer and the transmitting filter are optimized in a PAM data communication system operating over a linear noisy channel. We use a mean-square error criterion and impose an average power constraint at the transmitter. Assuming correct past decisions, an explicit formula for the minimum attainable mean-square error is given. The possible advantages of signaling faster than the Nyquist rate while decreasing the number of levels to maintain the same information rate are investigated. It is shown that, in all cases of practical interest, signaling faster than the Nyquist rate, while keeping fixed the information rate, increases the mean-square error. Finally, to illustrate the use of the results, application is made to a cable channel where the loss in dB varies as the square root of frequency. Various asymptotic formulas and curves are provided to exhibit the relationships between the quantities of interest.

Unification of linear information feedback schemes for additive white Gaussian noise channels (Corresp.)

For the transmission of a Gaussian information source over an additive white Gaussian noise channel, several noiseless-feedback linear estimation schemes are shown to be the same in the sense that they not only achieve the rate-distortion bound on the minimum attainable mean-square error, but also possess identical system parameters. Moreover, Butman's noiseless-linear-feedback scheme for the transmission of a digital information source with <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</tex> possible messages over additive white Gaussian noise channels can, in fact, easily be obtained from any of these equivalent schemes by simply imposing a decision structure on it and by using the error probability criterion.

Optimum sampled-data control

A method of rendering feedback control systems amenable to treatment by the Wiener theory is applied to the case in which the controller operates on a sampled measurement. An explicit expression is obtained for the minimum attainable mean-square error for certain classes of system transfer functions and disturbance power spectra, and the form of the optimum controller is derived. The results show the inherent limitations in controllability imposed by the structure of the controlled system and by the sampling process.