Fig 1 - uploaded by Zhengyuan Xu
Content may be subject to copyright.
Source publication
The Ziv-Zakai bound (ZZB) provides a general mean-square error analytical baseline to evaluate time delay estimation (TDE) techniques for a wide range of time-bandwidth products and signal-to-noise ratios, but generally can only be numerically evaluated. The Weiss-Weinstein bound (WWB) further improves characterization of the attainable system perf...
Context in source publication
Context 1
... of (11) shows that sinc(ωH τ ) changes more rapidly than sinc(ωL τ ) when ωH ωL. This fact permits us to similarly introduce the concept of phase and envelope ambiguities for the UWB case. The phase ambiguity points are caused by ωHsinc(ωH τ )/π with period 2π/ωH , and the envelope ambiguity points are caused by ωLsinc(ωL τ )/π with period 2π/ωL. Fig. 1 demonstrates this transition behavior of Rss( τ ) with ω0 = 20W , ω0 = 5W and ω0 = W respectively. In order to obtain an analytically tractable expression for (6) suitable for UWB signals, we will partition the integration interval [0, D] into sub-intervals separated by s1, s2 and s3, dened below. These points will designate the ...
Similar publications
Error performance of noncoherent detection of on-off frequency shift keying (OOFSK) modulation over fading channels is analyzed when the receiver is equipped with multiple antennas. The analysis is conducted for two cases: 1) the case in which the receiver has the channel distribution knowledge only; and 2) the case in which the receiver perfectly...
Citations
... Typically, the accuracy and reliability of conventional localization techniques degrade in wireless environments due to biases in SVEs caused by multipath propagation and nonline-of-sight (NLOS) conditions. Performance limits on ranging were established in [115]- [127], while tractable models for range information were derived in [51]. To cope with wireless propagation impairments, conventional localization approaches focus on improving the estimation of single values [97]- [102], [128], [129]. ...
Location awareness is vital for emerging Internet-of-Things applications and opens a new era for Localization-of-Things. This paper first reviews the classical localization techniques based on single-value metrics, such as range and angle estimates, and on fixed measurement models, such as Gaussian distributions with mean equal to the true value of the metric. Then, it presents a new localization approach based on soft information (SI) extracted from intra- and inter-node measurements, as well as from contextual data. In particular, efficient techniques for learning and fusing different kinds of SI are described. Case studies are presented for two scenarios in which sensing measurements are based on: 1) noisy features and non-line-of-sight detector outputs and 2) IEEE 802.15.4a standard. The results show that SI-based localization is highly efficient, can significantly outperform classical techniques, and provides robustness to harsh propagation conditions.
... The ZZLBs are very tight; they detect the ambiguity region roughly and the asymptotic region accurately. Some applications of the ZZLBs, discussions and comparison to other bounds can be found in10111229303132333435. In [36, pp. ...
In nonlinear deterministic parameter estimation, the maximum likelihood
estimator (MLE) is unable to attain the Cramer-Rao lower bound at low and
medium signal-to-noise ratios (SNR) due the threshold and ambiguity phenomena.
In order to evaluate the achieved mean-squared-error (MSE) at those SNR levels,
we propose new MSE approximations (MSEA) and an approximate upper bound by
using the method of interval estimation (MIE). The mean and the distribution of
the MLE are approximated as well. The MIE consists in splitting the a priori
domain of the unknown parameter into intervals and computing the statistics of
the estimator in each interval. Also, we derive an approximate lower bound
(ALB) based on the Taylor series expansion of noise and an ALB family by
employing the binary detection principle. The accurateness of the proposed
MSEAs and the tightness of the derived approximate bounds are validated by
considering the example of time-of-arrival estimation.
... N ONLINEAR deterministic parameter estimation is subject to the threshold effect Ziv and Zakai [1969], Chow and Schultheiss [1981], Weiss and Weinstein [1983], Weinstein and Weiss [1984], Zeira andSchultheiss [1993, 1994], Sadler and Kozick [2006], Sadler et al. [2007]. Due to this effect the signal-to-noise ratio (SNR) axis can be split into three regions as illustrated in Fig. 1(a): ...
Threshold and ambiguity phenomena are studied in Part 1 of this work where
approximations for the mean-squared-error (MSE) of the maximum likelihood
estimator are proposed using the method of interval estimation (MIE), and where
approximate upper and lower bounds are derived. In this part we consider
time-of-arrival estimation and we employ the MIE to derive closed-form
expressions of the begin-ambiguity, end-ambiguity and asymptotic
signal-to-noise ratio (SNR) thresholds with respect to some features of the
transmitted signal. Both baseband and passband pulses are considered. We prove
that the begin-ambiguity threshold depends only on the shape of the envelope of
the ACR, whereas the end-ambiguity and asymptotic thresholds only on the shape
of the ACR. We exploit the results on the begin-ambiguity and asymptotic
thresholds to optimize, with respect to the available SNR, the pulse that
achieves the minimum attainable MSE. The results of this paper are valid for
various estimation problems.
... In general, for a signal with rectangular 9 As the differentiation order n increases, (5.30) can be used to verify that 3n)/3 "1 tends to 1 for large n, indicating that the improvement in the CRB by increasing n is limited. spectrum, the RMSE gap depends on the ratio between the center frequency fc and the signal bandwidth W as discussed in [77,90]. In particular, having small fc/W leads to a lower RMSE for medium SNR. ...
The need for ubiquitous wireless services has prompted the exploration of using increasingly larger transmission bandwidths often in environments with harsh propagation conditions. However, present analyses do not capture the behavior of systems in these channels as the bandwidth changes. This thesis: describes the development of an automated measurement apparatus capable of characterizing wideband channels up to 16 GHz; formulates a framework for evaluating the performance of wireless systems in realistic propagation environments; and applies this framework to sets of channel realizations collected during a comprehensive measurement campaign. In particular, the symbol error probability of realistic wideband subset diversity (SSD) systems, as well as improved lower bounds on time-of-arrival (TOA) estimation are derived and evaluated using experimental data at a variety of bandwidths. These results provide insights into how the performance of wireless systems scales as a function of bandwidth. Experimental data is used to quantify the behavior of channel resolvability as a function of bandwidth. The results show that there are significant differences in the amount of energy captured by a wideband SSD combiner under different propagation conditions. In particular, changes in the number of combined paths affect system performance more significantly in non-line-of-sight conditions than in line-of-sight conditions. Results also indicate that, for a fixed number of combined paths, lower bandwidths may provide better performance because a larger portion of the available energy is captured at those bandwidths. The expressions for lower bounds on TOA estimation, developed based on the Ziv-Zakai bound (ZZB), are able to account for the a priori information about the TOA as well as statistical information regarding the multipath phenomena. The ZZB, evaluated using measured channel realizations, shows the presence of an ambiguity region for moderate signal-to-noise ratios (SNRs). It is shown that in a variety of propagation conditions, this ambiguity region diminishes as bandwidth increases. Results indicate that decreases in the root mean square error for TOA estimation were significant for bandwidths up to approximately 8 GHz for SNRs in this region.
... Various TDE bounds have been developed [1], and the Ziv–Zakai lower bound (ZZB) [2], [3] is among the best mean-square-error (MSE) bounds for predicting optimal estimation performance over a wide range of signal-to-noise ratios (SNRs), e.g., see [4]. The ZZB approach has also been applied to ultra-wideband signals in additive white Gaussian (AWGN) channels [5], as well as independent parallel frequency hopping and flat fading channels [6], [7]. However, in many cases, as the signal bandwidth grows we encounter a random frequency-selective fading channel [8]. ...
... Braccini and Gambardella introduced the concept of " form-invariant " filters [5]. These are systems such that a scaling of the input gives rise to a scaling of the output. ...
Ziv-Zakai Bayesian lower bounds on time-delay estimation (TDE) are developed for frequency hopping waveforms. These wideband waveforms are readily employed for both communications and TDE. We consider random frequency-selective fading channels, described by a Gaussian tapped delay line model with inter-hop correlation. The receiver does not know the channel realization to assist in estimating the time delay but does know the channel statistics. The development incorporates a random uniform prior on the delay. The bounds provide tight mean-square error prediction for the maximum a posteriori (MAP) estimator performance over a large range of signal-to-noise ratios (SNRs). They also accurately predict the TDE frequency diversity gain from multi-frequency transmission in fading channels. The special case of independent fading is discussed, including both flat Rician and Rayleigh fading, and closed-form expressions enable easy study of the effects of SNR, frequency diversity, and channel statistics on TDE.
... Weiss and Weinstein derived the ZZB for time delay estimation of narrowband [5] and wideband [6] signals. Extending this work, the ZZB for time delay estimation of ultra-wideband signals was studied in [7]. Bell et al. extended the ZZB from scalar to vector parameter estimation [8], and used it to develop a lower bound on the MSE in estimating the 2-D bearing of a narrowband plane wave [9]. ...
This paper presents the derivation of the Ziv-Zakai bound (ZZB) for the localization problem in a MIMO radar system. The target is positioned in the near-field of a network of radars of arbitrary geometry. The radars have ideal mutual time and phase synchronization. The target location is estimated by coherent processing exploiting the amplitude and phase information between pairs of radars. An analytical expression is developed for the ZZB relating the estimation mean square error (MSE) to the carrier frequency, signal bandwidth, the number of sensors, and their location. From numerical calculations of the bound, three regions of signal-to-noise ratio (SNR) can be distinguished in the performance of the location estimator: a noise-dominated region, an ambiguity region, and an ambiguity free region. In the noise-dominated region, the signals received by the radars are too weak, and thus the localization error is limited only by the a priori information about the location of the target. In the ambiguity region, the performance of the location estimator is affected by sidelobes. In the ambiguity free region, estimation errors are very small and the ZZB approaches the Cramer-Rao lower bound (CRLB).
... A survey of TDE bounds is given in [10]. ZZBs on TDE have been developed for narrowband frequency hopping channels [11] , parallel narrowband chan- nels [12], [13], and for ultra-wideband signals in additive white Gaussian noise (AWGN) channels [14]. Bayesian bounds have also been developed by Weinstein and Weiss and applied to TDE [9], [10], [15]. ...
Using the Ziv-Zakai bound (ZZB) methodology, we develop a Bayesian mean-square error bound on time delay estimation (TDE) in convolutive random channels, and compare it with time delay estimator performance and a Crame??r-Rao bound. The channel is modeled as a tapped delay line, whose taps are Gaussian random variables that may be nonzero mean and correlated, a model widely adopted in many applications such as wideband fading in a multipath channel. The time delay has a uniform prior distribution. A ZZB is developed that incorporates the prior on the random time delay, as well as the convolutive random Gaussian channel, and does not assume the receiver has knowledge of the channel realization. The ZZB provides good performance prediction for maximum a posteriori (MAP) time delay estimation, tracking the low, medium, and high signal-to-noise ratio (SNR) regimes. The convolutive channel model includes important special cases, such as narrowband Gaussian channels corresponding to Rayleigh/Rician fading, wideband multipath channels with a power decay profile (such as exponential decay), and known channels. We show that the associated Crame??r-Rao bound is tight only at high SNR, whereas the ZZB predicts threshold behavior and TDE breakdown as the SNR decreases. When compared to a ZZB that assumes knowledge of the channel realization, the ZZB developed here provides a more realistic and tighter bound, revealing the performance loss due to lack of channel knowledge. The MAP estimator incorporates knowledge of the channel statistics, and so performs much better than a maximum likelihood estimator that minimizes mean square error but does not use knowledge of the random channel statistics. Several examples illustrate the estimator and bound behaviors.
... First, we assumed the TOA estimation CRB can be achieved. However, it was shown in [47], [48] that this CRB is in nature a local bound and not tight for scenarios with low signal-to-noise ratio (SNR). Alternatively, one may consider the improved Ziv–Zakai bound (ZZB) as a better metric. ...
Sensor positioning is an important task of location-aware wireless sensor networks. In most sensor positioning systems, sensors and beacons need to emit ranging signals to each other. Sensor ranging energy should be low to prolong system lifetime, but sufficiently high to fulfill prescribed accuracy requirements. This motivates us to investigate ranging energy optimization problems. We address ranging energy optimization for an unsynchronized positioning system, which features robust sensor positioning (RSP) in the sense that a specific accuracy requirement is fulfilled within a prescribed service area. We assume a line-of-sight (LOS) channel exists between the sensor and each beacon. The positioning is implemented by time-of-arrival (TOA) based two-way ranging between a sensor and beacons, followed by a location estimation at a central processing unit. To establish a dependency between positioning accuracy and ranging energy, we assume the adopted TOA and location estimators are unbiased and attain the associated Cramer-Rao bound. The accuracy requirement has the same form as the one defined by the Federal Communication Commission (FCC), and we present two constraints with linear-matrix-inequality form for the RSP. Ranging energy optimization problems, as well as a practical algorithm based on semidefinite programming are proposed. The effectiveness of the algorithm is illustrated by numerical experiments.
... Weiss and Weinstein derived the ZZB for time delay estimation of narrowband [5] and wideband [6] signals. Extending this work, the ZZB for time delay estimation of ultra-wideband signals was studied in [7]. Bell et al. extended the ZZB from scalar to vector parameter estimation [8], and used it to develop a lower bound on the MSE in estimating the 2-D bearing of a narrowband plane wave [9]. ...
The Ziv-Zakai bound (ZZB) is developed for the estimation error of a radiating source located in a plane, and observed by sensors widely distributed over the same plane. The source is non-cooperative in the sense that the transmitted waveform and its timing are unknown to the sensors. The sensors do have however, information on the power spectral density of the source. Moreover, sensors have ideal mutual time and phase synchronization. The source location is estimated by coherent processing exploiting the amplitude and phase information between pairs of sensors. An analytical expression is developed for the ZZB relating the estimation error to the carrier frequency, signal bandwidth, the number of sensors, and their location. Numerical examples demonstrate that the ZZB closely predicts the performance of the maximum likelihood estimate across the full range of signal to noise ratio (SNR) values. At low SNR, the ZZB bound demonstrates performance dominated by noise, at medium SNR, the performance is dictated by the presence of sidelobes in the localization metric, and at high SNR, it is shown that the ZZB converges to the Cramer-Rao bound.
... Among them the Ziv-Zakai bound (ZZB), with its improved versions such as the Bellini- Tartara bound [9] and the Chazan-Zakai-Ziv bound [10], can be applied to a wider range of SNRs. In [11] the approach developed in [12] for the estimation of the relative delay between two sensors emitting noise-like signals is extended to that emitting UWB signals. The work [13] evaluates the ZZB for Gaussian signals assuming perfect channel knowledge at the receiver. ...
Time-of-arrival (TOA) based localization plays an important role due to the possibility to exploit the fine delay resolution property when wideband signals are adopted. This paper investigates lower bounds on TOA estimation error for wideband and ultrawide bandwidth (UWB) ranging systems operating in realistic multipath environments. In particular, analytical expressions for the Ziv-Zakai bound (ZZB) on TOA estimation error for multipath channels are provided and discussed under different operating conditions. Using these bounds it is shown how the a priori knowledge about the multipath characteristics can be accounted for in the TOA estimation performance. These bounds serve as useful performance benchmarks for the design of practical TOA estimators.
