A Survey of Time Delay Estimation Performance Bounds

Sadler,; Kozick,

doi:10.1109/sam.2006.1677204

Cited by 19 publications

(13 citation statements)

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“…The ZZB is among the best MSE bounds for predicting optimal estimation performance over a wide range of SNRs; e.g., see Van Trees and Bell and references therein [9]. A survey of TDE bounds is given in [10]. ZZBs on TDE have been developed for narrowband frequency hopping channels [11], parallel narrowband channels [12], [13], and for ultra-wideband signals in additive white Gaussian noise (AWGN) channels [14].…”

Section: Introductionmentioning

confidence: 99%

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Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Sadler

Liu

2010

IEEE Trans. Signal Process.

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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 Cramér-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 Cramér-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.

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Section: Introductionmentioning

confidence: 99%

“…ZZBs on TDE have been developed for narrowband frequency hopping channels [11], parallel narrowband channels [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].…”

Section: Introductionmentioning

confidence: 99%

Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Sadler

Liu

2010

IEEE Trans. Signal Process.

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“…; N) hop and kth symbol period is denoted as p i;k (t) = p(t 0 (i 0 1)M T s 0 kT s ), where each hop has M symbols with hop dwell time MT s . The complex envelope of the ith hop waveform is represented as si(t) = K k=0K a i;k p i;k (t) = K k=0K a i;k p(t 0 (i 0 1)M Ts 0 kTs); t 2 [(i 0 1)M Ts; iMTs] (1) in which M = K 1 + K 2 + 1. The symbols a i;k can be taken from a constellation such as PSK or QAM, and we assume they are known to the receiver.…”

Section: Signal and Channel Modelsmentioning

confidence: 99%

“…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].…”

Section: Introductionmentioning

confidence: 99%

Ziv–Zakai Time-Delay Estimation Bounds for Frequency-Hopping Waveforms Under Frequency-Selective Fading

Liu

Sadler

2010

IEEE Trans. Signal Process.

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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.

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“…Performance bounds on TD estimation when all the sensor signals are available at the processing unit have been extensively studied in the past [4]- [8]. In distributed sensor networks however, not all sensor signals may be available at the processing unit, and wireless signal transmission is necessary.…”

Section: Introductionmentioning

confidence: 99%

Cramér-Rao bounds for rate-constrained distributed time-delay estimation

Srinivasan

2010

2010 4th International Symposium on Communications, Control and Signal Processing (ISCCSP)

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The discussion in this paper is limited to the case of two microphones. The following signal model holds for the TD estimation problem:(2) (1) XI(t) = s(t) + nl(t) X2(t) = s(t -T) + n2(t),R(>') = 4~i: max ( 0, log2 ipx~(w)) dMJ, 1r -00The distortion here is the mean-squared error (MSE) between X2 (t) and its reconstruction. Equation (2) provides the relation where Xl (t) and X2 (t) are the received microphone signals, s(t) is the source signal, T is the time delay, and nl (t) and n2(t) are the noise processes at the two sensors. The signals s(t), nl(t), and n2(t) are assumed to be zero-mean stationary Gaussian random processes with power spectral densities~s(w),~nl (w), and e., (w), respectively.It is assumed that the first node is the fusion center and that the second node transmits its signal X2 (t) at a rate R to the first node, where the TD estimate is obtained from the two signals. When the signal X2 (t) is quantized at a rate R, the relation between R and the resulting quantization error can be expressed by the following parametric rate-distortion relation [11]:the communication bit-rate and the resulting accuracy of TD estimation by analyzing the Cramer-Rao lower bound (CRLB) on the variance of the estimation error as a function of the bitrate.For a given observation duration, the CRLB is tight, i.e., the bound is attainable, only if the signal-to-noise ratio (SNR) exceeds a certain threshold [4], [5]. In the rate-constrained case, it is intuitive that the CRLB is attained only beyond certain bit-rates. For a given observation time and SNR, an expression that establishes a condition on the minimum bit-rate required to ensure attainability of the CRLB is also derived in this paper.The remainder of the paper is organized as follows. Section II introduces the signal model employed, and the relevant ratedistortion relations used in the analysis. The CRLB in the rate-constrained case is obtained in Section III. The minimum bit-rate that renders the CRLB tight is derived in Section IV. Conclusions are summarized in Section V.Abstract-This paper investigates time-delay estimation for acoustic source localization in a distributed microphone array. The microphones are assumed to be part of a wireless sensor network, with a constraint on the number of bits that may be exchanged between the sensors. Consequently, at the fusion center, time-delay estimation needs to be performed using quantized signals. In this paper, the relation between the communication bitrate and the Cramer-Rae lower bound (CRLB) on the variance of the time-delay estimation error is explored. The minimum bit-rate required to ensure that the CRLB is attained is also derived.

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A Survey of Time Delay Estimation Performance Bounds

Cited by 19 publications

References 0 publications

Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Ziv–Zakai Time-Delay Estimation Bounds for Frequency-Hopping Waveforms Under Frequency-Selective Fading

Cramér-Rao bounds for rate-constrained distributed time-delay estimation

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A Survey of Time Delay Estimation Performance Bounds

Cited by 19 publications

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Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Ziv–Zakai Bounds on Time Delay Estimation in Unknown Convolutive Random Channels

Ziv–Zakai Time-Delay Estimation Bounds for Frequency-Hopping Waveforms Under Frequency-Selective Fading

Cram&#x00E9;r-Rao bounds for rate-constrained distributed time-delay estimation

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Cramér-Rao bounds for rate-constrained distributed time-delay estimation