ROBUST NQR SIGNAL DETECTION S. D. Somasundaram ∗ , A. Jakobsson † and E. Gudmundson ‡ ∗ King’s College London, Division of Engineering, Strand, London, WC2R 2LS, UK † Karlstad University, Department of Electrical Engineering, SE-651 88 Karlstad, Sweden ‡ Uppsala University, Department of Information Technology, Box 337, SE-751 05 Uppsala, Sweden ABSTRACT Nuclear quadrupole resonance (NQR) is a spectroscopic technique that can be used to detect many high explosives and narcotics. Un- fortunately, the measured signals are weak, thereby inhibiting the widespread use of the technique. Current state-of-the-art detectors, which exploit realistic NQR data models, assume that the complex amplitudes of the NQR signal components are known, to within a multiplicative constant. However, these amplitudes are typically prone to some level of uncertainty, thus leading to performance loss in these algorithms. Herein, we develop a frequency selective algo- rithm, robust to uncertainties in the assumed amplitudes, that offers a signiÝcant performance gain over current state-of-the art techniques. Index Terms— Signal detection, robust methods, nuclear quadrupole resonance 1. INTRODUCTION Nuclear Quadrupole Resonance (NQR) is a radio frequency (RF) spectroscopic technique that can be used to detect the presence of quadrupolar nuclei, such as the 14 N nucleus prevalent in many high explosives and narcotics [1–4]. The sample is irradiated with a spe- cially designed sequence of RF pulses and the responses, between pulses, are then measured. The NQR response is highly compound speciÝc, making the technique an important detection tool. Unfor- tunately, the success of NQR has been hindered by the low signal- to-noise ratio (SNR) signals that are typically observed, especially in the low frequency region, for compounds such as trinitrotoluene (TNT) [5]. Current state-of-the-art detectors, which exploit real- istic NQR signal models, assume the complex amplitudes of the NQR signal components are known, to within a multiplicative con- stant [5–9]. These complex amplitudes are typically obtained from laboratory measurements; however, several factors may cause differ- ences between the assumed complex amplitudes and those observed. For example, in a landmine detection scenario, the Ýeld at the sample will vary due to varying distances between the antenna and the mine; consequently, for the same pulse sequence parameters and RF power, the Þip-angle(s) of the excited resonant line(s) will also vary, caus- ing variations in the NQR signal amplitudes. Typically, such varia- tions will reduce the performance of these detectors (see also [10]). Therefore, we here proceed to derive a robust algorithm that Ýnds the best complex amplitude vector within a hypersphere of uncertainty around a speciÝed complex amplitude vector. The approach allows for inclusion of prior information, both via the speciÝed complex amplitude vector and via the selection of the size of the uncertainty This work was supported in part by the Defence Science and Technology Laboratory (DSTL) at Fort Halstead, UK, and the Swedish Research Council. ‡ Corresponding author. E-mail: erik.gudmundson@it.uu.se. region. We also propose a method for selecting the size of the uncer- tainty region, based upon knowledge of the uncertainties of the com- plex amplitudes. Furthermore, to reduce computational complexity and provide robustness to RF interference, the detector is formed using only those spectral bands where the NQR signal components are expected to lie. Extensive numerical analysis using both simu- lated and measured data, obtained from a sample of TNT, indicate that the proposed detector offers a signiÝcantly increased detection performance as compared to current state-of-the-art detectors. 2. DATA MODEL In [5–7], a model for the NQR echo train, as produced by a pulsed spin locking sequence, was presented. The NQR signal is typi- cally embedded in coloured noise, which can be well modelled as an autoregressive (AR) process. Without loss of generality, the pre- whitened data model may be written as z m (t)= ρ d k=1 ¯ Cκ k e -η k (t+mμ) e -β k |t-tsp|+iω k (T )t + e m (t), (1) where t = t0,...,tN-1 is the echo sampling time. Furthermore, m =0,...,M - 1 is the echo number; tsp is the echo peak offset; the echo spacing μ =2tsp; ρ is the common scaling due to the signal power; κ k , β k and η k denote the normalised (complex) amplitude, the sinusoidal damping constant and echo train damping constant of the kth NQR frequency, respectively. Often, information about each κ k is available for a given substance and experimental set-up. The sinusoidal and echo damping constants, β k and η k , are here mod- elled as unknown parameters. Furthermore, ω k (T ) is the frequency shifting function of the kth NQR frequency component which, in general, depends on the unknown temperature, T , of the examined sample. An important point to note is that the number of sinusoidal components, d, as well as the frequency shifting function for each spectral line, ω k (T ), may be assumed to be known. For many sub- stances, such as TNT, the frequency shifting function(s), over the temperatures of interest, can be well modelled as linear functions, i.e., [4] ω k (T )= a k - b k T, (2) where a k and b k , for k =1,...,d, are given constants. The com- plex scaling, due to the prewhitening operation, may be written as ¯ C = C(λ k ) for ⌊t - tsp⌋ < 0, otherwise ¯ C = C( ˜ λ k ), where ⌊x⌋ denotes the integer part of x, λ k = e iω k (T )+β k -η k and ˜ λ k = e iω k (T )-β k -η k . Furthermore, C(λ) denotes the AR prewhitening Ýlter (see [8] for further details). Finally, e m (t) is an additive white noise. 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