GEOMETRY-ADAPTIVE DATA COMPRESSION FOR TDOA/FDOA LOCATION Mo Chen and Mark L. Fowler Department of Electrical and Computer Engineering State University of New York at Binghamton {mchen0, mfowler}@Binghamton.edu ABSTRACT The location of an emitting target is estimated by intercepting its emitted signal and sharing them among several sensors to measure the time-difference-of-arrival (TDOA) and the frequency-difference-of-arrival (FDOA). Doing this in a timely and energy efficient fashion, which is especially important for wireless sensor network applications, requires effective data compression. Since the commonly used MSE distortion measure is only weakly related to optimal TDOA/FDOA estimation, in this paper, we derive a new class of non-MSE distortion measures for TDOA/FDOA estimation using the concept of Fisher information. We then use these new distortion measures to compress the data using a wavelet packet transform and show that it improves TDOA/FDOA estimation accuracies relative to using the MSE-based compression. Finally, the scheme of applying our algorithms in a wireless sensor network is proposed, and energy efficiency and accuracy enhancement of the proposed scheme over that of traditional scheme using MSE is shown through the simulations. 1. INTRODUCTION A common way to locate an electromagnetic emitter is to measure the time-difference-of-arrival (TDOA) and the frequency-difference-of-arrival (FDOA) between signals received at pairs of sensors [1],[2]. This requires that the samples of one signal are sent over a data link, where data compression can reduce latency and save energy. Some past results are available on the issue of compression for TDOA/FDOA applications [3]–[7], but only recently have researchers begun to explore other than the standard mean-square-error (MSE) distortion measure [5]–[7]. We introduced a new non-MSE distortion measure [5] that uses a CRLB-based measure for the TDOA-only problem. However, we have found [5] that optimizing this measure was difficult because the form that the CRLB-based distortion metric takes depends on the relationships between the SNRs at the two sensors. More recently we have shown [6] that a Fisher-information-based approach avoids these difficulties and we developed a general method for a single-parameter problem. A key advantage for this approach is that if the noise is uncorrelated between sensors, then the total Fisher information is the sum of the Fisher information from each sensor. Compression only impacts one sensor so we can avoid the complicating cross-sensor couplings. In this paper we attack the two-parameter TDOA/FDOA problem by extending our single-parameter ideas [6] and explore trade-off issues that arise. We assume that the TDOA/FDOA estimation processing and compression processing are not jointly designed – this is motivated by our belief that sensors are likely be called to provide data to other processing systems that are independently designed. With our proposed compression scheme, we develop a new cooperative scheme for the sensor nodes in the wireless sensor network to adapt the compression scheme to the sensor-target geometry. 2. FISHER-INFORMATION-BASED DISTORTION Fisher information matrix (FIM) is a well-known concept in estimation theory [8]. It quantifies how much information a data set provides about the parameters to be estimated. Let wθsx += )( denote a real random vector consisting of a deterministic signal vector )(θs parameterized by 2×1 parameter vector θ, and corrupted by a white noise vector w with variance σ2. The FIM for this θ is the 2×2 matrix )(θJwith elements given by ∂∂∂∂=jTiijJθθσ)()(12θsθs. (1) The FIM specifies an information ellipse – the larger the better – with semi-axes along the FIM’s eigenvectors and whose lengths are the square roots of the eigenvalues. Lossy compression of the data vector x changes the FIM. Namely, it reduces the on-diagonal elements iiJ, which makes the post-compression information ellipse smaller; it is unclear what effect it would have on the cross-information and hence the tilt of the ellipse. In the single-parameter case [6] the way to proceed was clear: compress so as to minimize the reduction in J11 for a given bit budget. But how should we proceed in the two-parameter case? There are several possibilities, but our choice is to minimize the impact on the information ellipse’s semi-axis lengths while neglecting the impact on the ellipse’s tilt: this implies minimizing the reduction of the FIM’s eigenvalues. A simple, effective measure is to minimize the reduction of the sum of the eigenvalues, which is equivalent to minimizing the reduction of the trace of the FIM. However, sometimes TDOA accuracy is more important than FDOA accuracy, or vice versa; so, we use a weighted trace. Thus, our goal is to seek an operational rate-distortion method thatminimizes the reduction in αJ11 + (1-α)J22 with 0 ≤ α ≤ 1 while satisfying a budget on the total number of bits. The signals at sensors S1 and S2 with unknown TDOA of dn and FDOA of dv can be modeled by: 2/,,12/,2/][)]2/([][][)]2/([][2)2/(021)2/(0100NNNnnwennnsnxnwennnsnxnjdnjdddK+−−=+−−=++−=−+νννν (2) where s[n] is a complex baseband signal, v0 and n0 are unknown nuisance parameters that need not be estimated, and w1[n] and w2[n] are uncorrelated complex Gaussian white noise with variances 21σand 22σ, respectively. The signal-to-noise ratios (SNR) for these two received signals are denoted by SNR1 and SNR2, respectively. We assume here that x1[n] is the signal that gets compressed. As shown in [6], the TDOA Fisher information depends on the DFT coefficients (with the DFT frequencies running over both negative and positive frequencies) while the FDOA Fisher information depends on the signal samples themselves. After quantization of the DFT coefficients the data-computable TDOA Fisher information measure is ,)(~~2/2/mJJddddnnNNmnn∑−== (3) with =>+=0 if,00 if,][2)(~212122mmmnnbbqmXmmJddσπ (4) where X1[m] is the normalized DFT of the data at sensor S1 and qm is the quantization noise variance of the mth DFT coefficient when quantized to an allocated bm bits. The tilde (~) in (3) and (4) indicates that the quantity is