IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 45, NO. 4, APRIL 2007
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Exploiting Spin Echo Decay in the Detection of Nuclear Quadrupole Resonance Signals Samuel D. Somasundaram, Student Member, IEEE, Andreas Jakobsson, Senior Member, IEEE, John A. S. Smith, and Kaspar Althoefer, Member, IEEE
Abstract—Nuclear quadrupole resonance (NQR) is a radiofrequency technique that can be used to detect the presence of quadrupolar nuclei, such as the 14 N nucleus prevalent in many explosives and narcotics. In a typical application, one observes trains of decaying NQR echoes, in which the decay is governed by the spin echo decay time(s) of the resonant line(s). In most detection algorithms, these echoes are simply summed to produce a single echo with a higher signal-to-noise ratio, ignoring the decaying echo structure of the signal. In this paper, after reviewing current NQR signal models, we propose a novel NQR data model of the full echo train and detail why and how these echo trains are produced. Furthermore, we refine two recently proposed approximative maximum-likelihood detectors that enable the algorithms to optimally exploit the proposed echo train model. Extensive numerical evaluations based on both simulated and measured NQR data indicate that the proposed detectors offer a significant improvement as compared to current state-of-the-art detectors. Index Terms—Explosives detection, multidimensional signal processing, quadrupole resonance, signal detection.
I. I NTRODUCTION
N
UCLEAR quadrupole resonance (NQR) is a radiofrequency (RF) spectroscopic technique that can be used to detect the presence of quadrupolar nuclei, a requirement fulfilled by roughly 50% of the elements in the periodic table. The technique is related to both nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI). Unlike NMR and MRI, NQR does not require a large static magnetic field to split the energy levels of the nucleus, making it attractive as a noninvasive technique for detecting explosives in landmines and unexploded ordnance [1]–[5] or screening baggage for narcotics and explosives at airports [6], as well as a relatively inexpensive analytical technique for the structural characterization of various compounds [7], [8]. NQR signals are acquired by applying pulsed RF radiation to the sample, which drives transitions between the quadrupolar energy levels, and then measuring the responses. Historically, one has primarily measured the free induction decay (FID), which is the response after a single excitation pulse. However, since the advent of Manuscript received September 19, 2005; revised June 15, 2006. This work was supported in part by the Defence Science and Technology Laboratory (DSTL), Fort Halstead, U.K., the Swedish Research Foundation, and the Carl Trygger’s Foundation, Sweden. S. D. Somasundaram, J. A. S. Smith, and K. Althoefer are with the Division of Engineering, King’s College London, Strand, WC2R 2LS London, U.K. (e-mail:
[email protected];
[email protected]; k.althoefer@ kcl.ac.uk). A. Jakobsson is with the Department of Electrical Engineering, Karlstad University, SE-651 88 Karlstad, Sweden (e-mail:
[email protected]). Digital Object Identifier 10.1109/TGRS.2006.890413
multiple-pulse sequences, the trend has instead been to measure echoes, which are the signals measured between a string of pulses [2], [3]. The main advantage of measuring echoes is that signals with higher signal-to-noise ratios (SNRs) can be obtained in a shorter time as compared to measuring FIDs, thereby significantly reducing measurement times. The few publicly available papers discussing the analysis and detection of materials using NQR focus on the use of relatively simple data processing approaches, not fully exploiting the data model of the NQR echo train [9]–[13]. In fact, to the best of our knowledge, no previous work exploits or presents a model of the entire echo train, and only recently have models of individual echoes been exploited [14], [15]. For the majority of cases,1 the echoes will decay over the echo train with a rate described by the spin echo decay time T2e , which is in part due to the loss in longitudinal magnetization (governed by the spin lattice relaxation time T1 ) and in part due to the spinspin relaxation time T2 [3]; consequently, echoes later in the train will have a lower SNR then those earlier. Currently, data processing is usually performed on a single summed echo, being produced by summing up the echoes in an echo train [3], [9], [10], [12]–[15]. In this paper, we propose a new more realistic data model allowing for the full echo train and show how it can beneficially be used in the difficult task of trinitrotoluene (TNT) detection. We stress that the new model can also be exploited in other analytical applications, for instance, to obtain more accurate estimates of the physical NQR parameters (e.g., relaxation times). Herein, we concentrate on the detection of materials using NQR. Early power-based detection algorithms, formed by integrating the power of a predetermined spectral band, required high SNRs to be effective [3], [12]. In [12] and [13], the second-order statistics of the NQR training data were measured and then used in a nonadaptive Bayesian detector. Such an approach should be treated with care as the statistics of NQR signals, between experiments, are nonstationary, e.g., due to variations in environmental parameters, such as temperature, or variations in sample purity. A better approach is to exploit deterministic models of the noiseless NQR signal and then use these together with statistical models of the noise [14]. Early algorithms exploited NQR signal models consisting of sums of either pure sinusoids [9], [10] or damped sinusoids [11]; however, such NQR signal models are only realistic when it is FIDs (and not echoes) that are measured. Recently, we proposed an approximative maximum-likelihood (AML) detector 1 In a few cases, for which T is so short that the nucleus has fully relaxed 1 before application of another pulse, decaying echo trains are not observed.
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[14] and a frequency-selective version termed the frequencyselective AML (FSAML) detector [15]. Both of these detectors exploit the data structure within an echo, allowing the echo signal to be well modeled as a sum of sinusoids that expand then decay and whose frequencies depend, in a known way, on the temperature of the examined sample. As shown in [14] and [15], these detectors will offer a significant detection gain as compared to previous techniques not exploiting the rich data structure. We stress that the aforementioned detectors operate on single echoes and do not exploit the spin echo decay time(s) of the resonant line(s). In this paper, we proceed to extend the AML and FSAML detecto rs to function on a full echo train, thereby producing improved detectors that incorporate the proposed more realistic NQR signal model. We denote the soobtained detectors the echo train AML (ETAML) detector and the frequency-selective ETAML (FETAML) detector. Furthermore, we note that in many applications, it is not possible to shield the sample, causing RF interference (RFI) to be a major concern; for example, in the detection of landmines containing TNT, the relatively weak NQR signal is significantly affected by radio transmissions in the AM radio band [9], [10]. As a result, RFI mitigation techniques would need to be considered in addition to the techniques presented here. There are two main approaches to RFI mitigation, namely: 1) passive and 2) active. Passive methods use specially designed antennas to cancel the far-field RFI [4], [16]–[20], the disadvantage often being some loss in NQR signal strength as compared to using a simple coil. The active methods employ adaptive noise cancellation techniques that require reference antennas to measure the nonstationary background RFI [10], [13], [21]. Such methods have been successfully implemented in [16] and [22], where the system sensitivity is limited only by the instrument’s internal thermal noise. Typically, some RFI will still remain, even after suitable mitigation has been employed [10], and will cause performance degradation in most detection algorithms. For this reason, we also examine here the robustness of the discussed algorithms to such interference. Spurious signals, such as piezoelectric responses from sand and magnetoacoustic responses from magnetic materials, also pose problems for NQR detection. Methods for mitigating these responses (e.g., phase cycling) can be found in [3] and [23]–[26]. This paper is organized as follows. In Section II, we outline data models for NQR signals. Sections III and IV contain derivations for the ETAML and FETAML detectors. In Section V, the performances of the proposed ETAML and FETAML detectors are compared to AML, FSAML, and common demodulation approaches (DMA) using both measured and simulated NQR data. Finally, Section VI contains our conclusions. II. D ATA M ODELS A. Free Induction Decay (FID) A quadrupolar nucleus is a nucleus with spin quantum number I > 1/2. The time-averaged charge distribution of such a nucleus is nonspherical and therefore possesses an electric quadrupole moment. When this type of nucleus is placed in a nonzero electric field gradient (EFG), different quadrupole energy levels arise as some nuclear orientations are energetically
more favorable than others. The EFG seen by the nucleus is a result of the neighboring charges, both electrons and nuclei, and is directly related to the chemical structure of the compound. The compound-specific nature of NQR is due to this relation between the EFG and the chemical structure of the compound, and its uniqueness means that there is little or no interference from other compounds that may be present. The symmetry of the EFG, the magnitude of the nuclear electric quadrupole moment, and the spin quantum number of the nucleus all determine how the energy levels will be split.2 Experimentally, the signal is obtained by applying pulsed RF radiation, at the proper frequency, to drive transitions between the quadrupole energy levels. In a single-pulse experiment, where a single resonant line is excited, the magnetic component of the applied radiation couples with the nuclear magnetic moment and rotates the nucleus away from its equilibrium position in the EFG, exciting the system. The system returns to equilibrium via various relaxation processes inducing an FID in the receiving antenna. As is well known, when d resonant lines (with Lorentzian line shapes) are excited, the measured signal can be well modeled as a sum of d exponentially damped sinusoids [11], i.e., yfid (t) = ρ
d
κfk e−βk t+iωk (T )t + w(t)
(1)
k=1
where t is the FID sampling time, w(t) is an additive colored noise,3 and ρ denotes the common scaling due to the SNR. Commonly, the normalized (complex) amplitudes of the FID, here denoted κfk for the kth line, can be assumed to be known for a given experimental setup. Between experimental setups, these may change, e.g., due to: 1) variations in excitation bandwidth, caused by varying the excitation pulse width; 2) changes in receiver bandwidth, caused by altering the Q-factor of the probe or the impulse response of the antialiasing filter; and/or 3) differing experimental repeat times, which may mean some of the nuclei have not fully relaxed before another excitation pulse is applied. The kth line is ∗ observed to decay with time constant T2,k , known as the FID or spin-phase memory decay time. To simplify our notation, we introduce the sinusoidal damping constant, denoted βk for the kth line, which is related to the spin-phase memory decay ∗ time by βk = Dw /T2,k , where Dw is the spectrometer dwell time. As a result, βk can be assumed approximately known for a given sample. The dependencies of the NQR frequencies ωk (T ) on the (unknown) temperature of the examined sample4 T can often be well modeled as a linear function of temperature [28] ωk (T ) = ak − bk T
(2)
2 The 14 N nucleus, prevalent in most high explosives and narcotics, has spinquantum number I = 1, and the three allowed transitions (for the case of a nonaxially symmetric EFG), commonly labeled v+ (or vx ), v− (or vy ), and v0 (or vz ), are obtained by solving the quadrupolar Hamiltonian for a spin-1 nucleus [27]. 3 The underlying thermal (Johnson) noise at the RF antenna may be well modeled as a white Gaussian noise (WGN) process; however, this is typically colored by the receiver, in which case it may be well modeled as the output of an AR filter driven by WGN [14]. 4 From a notational perspective, we will hereafter indicate a known temperature dependency of a parameter by noting it as a function of T , e.g., ωk (T ).
SOMASUNDARAM et al.: EXPLOITING SPIN ECHO DECAY IN THE DETECTION OF NQR SIGNALS
Fig. 1. Temperature dependency of the NQR frequencies in the region of 830–880 kHz for monoclinic TNT at likely landmine temperatures. Of these, the two higher frequency components are very weak when the excitation frequency is close to 841.5 kHz and have therefore been omitted from the estimation.
TABLE I TEMPERATURE SHIFTING FUNCTION CONSTANTS FOR THE d = 4 NQR FREQUENCIES OF MONOCLINIC TNT, FOR AN E XCITATION F REQUENCY OF 841.5 kHz, IN THE R EGION OF 830–860 kHz
where ak and bk are known constants. This dependency is illustrated in Fig. 1 for monoclinic TNT. For completeness, Table I summarizes the values of ak and bk for the fourline region of monoclinic TNT using an excitation frequency of 841.5 kHz. One of the relaxation processes by which the nucleus returns to its equilibrium orientation in the EFG is by losing energy to the thermal motions of the solid. This process is governed by the spin lattice relaxation time T1 . It is worth noting that a wait time of 5T1 is normally required to produce a fully relaxed system, i.e., after the acquisition of an FID, one must wait 5T1 before applying another excitation pulse. This is a limiting factor in the detection of TNT as 5T1 can be up to 30 s. One way to reduce this limitation is to use multiple pulse sequences to generate a train of echoes. B. Echoes and Echo Trains Echoes, produced by using multiple pulse sequences, are beneficial in applications as they enable the NQR signal to be sustained for longer than an FID before the wait time of 5T1 seconds has to be adhered to. There are several relaxation processes that contribute to the decay of the FID. One of these processes is caused by the inhomogeneous nature of the
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sample, which effectively means that for a given resonant line, there is not one but rather a range of resonant frequencies. As a result, after excitation, the signals from the isochromats will become out of phase with each other and lose coherence, resulting in a loss of coherency of the transverse magnetization [3]. However, it is possible to use certain pulse sequences to refocus these signals and bring them back into phase. This is the basis for the formation of echoes. An echo can be produced by applying a second pulse at time tsp after the initial pulse, often with its phase shifted5 by 90◦ (with respect to the first pulse) [2]. This has the effect of refocusing the dephased signals; consequently, at time tsp after the second pulse, the signals from the isochromats are all back in phase, corresponding to the peak of the echo. By applying successive pulses in a sequence known as a pulsed spin locking (PSL) sequence, an echo train is produced. The echo train cannot (usually) be sustained indefinitely due, for example, to saturation effects. The time constant for this process following two pulses is the spin–spin relaxation time, generally denoted T2 . However, in a PSL sequence, for example, an even longer decaying echo train is often found with a time constant T2e (T ) [2]. As an echo is a refocused FID, it can be assumed that, first, an echo consists of a set of sinusoidal components with the same frequencies as those seen in the FID, and second, that from the peak at the center of an echo to the end of the echo, these sinusoids have ∗ the same damping factors (related to T2,k ) as seen in the FID. Therefore, the second half of each echo may be modeled as an FID, as given in (1). Furthermore, due to the way the echo is formed, it can be assumed that the way the signal expands from the beginning of the echo to the center is related to the way it then decays in the second half of the echo. If, for the moment, we ignore the loss in longitudinal magnetization over an echo, then the expansion up to the peak of the echo can be viewed as due to the recovery of the transverse magnetization and the following decay due to its subsequent loss. As a result, if there was no loss in longitudinal magnetization, the echo may be well modeled as being symmetric about the echo center. This is how the NQR signal has been modeled in [14] and [15]. The region of expanding modes will be determined by the pulse sequence parameters, allowing for a precise modeling of the envelope of each echo. To also account for the loss of magnetization over the whole echo train, each echo envelope should further be damped by the spin echo decay time. To stress that the spin echo decay time will be different for each resonant line, we denote it as T2e,k (T ), also noting that its dependency on temperature can often be written as T2e,k (T ) = ck e−dk /T
(3)
where ck and dk are known constants. We note that T2e,k (T ) is also dependent on the echo spacing, the RF excitation pulse power, and the RF pulse width [3]. To simplify our notation, we use ηk (T ) to denote the observed echo train damping parameter for the kth resonant line, noting that it is inversely proportional to T2e,k (T ), i.e., ηk (T ) = Dw /T2e,k (T ). Fig. 2 illustrates that an echo train is not a continuous function of time. Between 5 It is understood that the pulses in NQR spectroscopy consist of RF-modulated pulses, and it is the phase of the RF that is shifted.
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with (·)T denoting the transpose. Using (5), an echo train consisting of M echoes can be written as
yN M =
0 yN
T
...
M −1 T (yN )
T
= ρ vec[AT,β Q] + wN M
(8)
where wN M is defined similar to yN M , and the operation vec[X] stacks the columns of the matrix X on top of each other. Furthermore
Q = [Ψ0
...
ΨM −1 ].
(9)
As is well known, the maximum-likelihood estimate of θ can be found as (see, e.g., [29]) ˆ = arg min yN M − ρ vec[AT,β Q] 2 θ Rw
Fig. 2. Illustration of the real part of a typical echo train.
θ
consecutive echoes, there is a delay, where no data are acquired, to allow for the required refocusing pulse to generate the next echo. The echo spacing, here denoted µ, for a PSL sequence is set to µ = 2tsp . In summary, we propose that the noiseless mth echo of an echo train may be well modeled as y m (t) = ρ
d
κk e−ηk (T )mµ ζkt e−βk |t−tsp |
where P2W = P∗ W−1 P, and θ=[ρ T
(4)
∗ Rw = E{wN M wN M}
T y m (tN −1 )
m = ρAT,β Ψm + wN
˜ + eN M zN M = ρ vec[AT,β Q]
(13)
˜ = [Ψ ˜0 Q
...
˜ M −1 ] Ψ
(14)
where (5)
m m is defined similar to yN , N denotes the echo where wN length, and
(12)
where E{·} and (·)∗ denote the expectation and the conjugate transpose, respectively. As Rw is typically unknown, one is normally forced to use an estimate of Rw in (10). Herein, we will form such an estimate using an approximate low-order noise model derived from measured (noise only) data using the method developed in [14], allowing the additive noise to be approximately described using a sixth-order autoregressive (AR) model. As described in [14], one may therefore form a prewhitened data model such that
Using (4), the mth echo of the echo train can be written as ...
(11)
where the N M × 1 vector eN M is a zero-mean complex white additive noise, and
III. ETAML D ETECTOR
m ym N = y (t0 )
β T ]T
with β denoting a vector formed from the d unknown sinusoidal damping components βk . Furthermore, Rw denotes the noise covariance matrix
k=1
where ζk = eiωk (T )−ηk (T ) , and t = t0 , . . . , tN −1 is the echo sampling time measured with respect to the center of the refocusing pulse, not necessarily being consecutive instances, but typically starting at t0 = 0 to allow for the dead time between the pulse and the first measured sample (after the pulse). For simplicity, we will hereafter assume a uniform sampling starting at t0 . We note that the normalized (complex) amplitudes for the echo train, here denoted κk for the kth component, are different to κfk , the former being also dependent on tsp . Note that the frequencies and complex amplitudes of the d sinusoidal components can be assumed to be constant over each echo.
(10)
T
Ψm = κ1 e−η1 (T )mµ . . . κd e−ηd (T )mµ (6) ··· ζdt0 e−βd |t0 −tsp | ζ1t0 e−β1 |t0 −tsp | .. .. .. AT,β = . . . tN −1 −β1 |tN −1 −tsp | tN −1 −βd |tN −1 −tsp | e · · · ζd e ζ1 (7)
˜ m ]k = κ [Ψ ˜ k e−ηk (T )mµ with [·]k denoting the kth index, and κk C(λk ), for the initial tsp − t0 rows κ ˜k = ˜ k ), otherwise κk C(λ
(15)
(16)
˜ k = eiωk (T )−βk −ηk (T ) . where λk = eiωk (T )+βk −ηk (T ) , and λ Furthermore, C(λk ) denotes the AR prewhitening filter, defined as C(z) =
n k=0
ck z −k
(17)
SOMASUNDARAM et al.: EXPLOITING SPIN ECHO DECAY IN THE DETECTION OF NQR SIGNALS
(see [14] for further details on this model and how to evaluate the AR coefficients). It should be stressed that since the echoes are separate acquisitions, they are filtered separately. Using (13), the minimization in (10) can be written as 2 ˜ min zN M − ρ vec[AT,β Q] (18) θ
F
where · F denotes the Frobenius norm. Thus, the least squares estimate of ρ can be found as [29]
ρˆ = q†T,β zN M = (q∗T,β qT,β )−1 q∗T,β zN M
(19)
where (·)† denotes the Moore–Penrose pseudo-inverse, yielding the maximization max z∗N M ΠqT ,β zN M
(20)
β,T
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each such region is given by the minimal and maximal frequency values for that component considering the measured temperature and the size of the expected temperature uncertainty region. Denoting the measured temperature Tˆs and the temperature uncertainty region ∆Ts , the minimal and maximal frequency values for each component can be determined using (2) with T = Tˆs − ∆Ts and T = Tˆs + ∆Ts , respectively. It should be stressed that each echo should be Fourier transformed individually to avoid discontinuities in the data. The Fouriertransformed (prewhitened) data vector for the mth echo and kth frequency bin can be expressed as ˜ m + Em Zkm = ρvk∗ AT,β Ψ k
where Ekm = vk∗ em N represents the kth frequency bin of the prewhitened noise sequence associated with the mth echo em N , and
where ˜ qT,β = vec[AT,β Q]
(22)
Using the (T, β) pair maximizing (20), we proceed to form the test statistic T (zN M ) as the (approximative) generalized likelihood ratio test (GLRT) for a signal with unknown noise variance, i.e., [30] T (zN M ) = (2N M − 1)
z∗N M ΠqT ,β zN M
z∗N M (I − ΠqT ,β )zN M
vk = 1 ω k
(21)
ΠqT ,β = qT,β q†T,β .
.
...
ωkN −1
T
(26)
with ωk = ei2πk/N . Thus, over the (possibly overlapping) frequency regions of interest, (25) can be expressed as ∗ m ˜ Zm L = ρVL AT,β Ψm + EL
(27)
where
(23)
Using (23), the signal component is deemed present if and only if T (zN M ) > γ, and otherwise not, where γ is a predetermined threshold value reflecting the acceptable probability of false alarm (pf ); here, pf = QF1,2N M −1 (γ), where QFr,p denotes the complementary cumulative distribution function for an F-distribution with r numerator degrees of freedom and p denominator degrees of freedom [30].
(25)
T
m Zm L = Zk1
...
ZkmL
VL = [vk1
...
vk L ]
(28) (29)
m and where Em L is defined similar to ZL . Using (27), the data model for the whole echo train can be expressed as
T −1 T ZM L ∗ ˜ + ELM = ρ vec VL AT,β Q
ZLM =
Z0L
T
...
(30)
IV. FETAML D ETECTOR In this section, the above-presented ETAML detector is extended along the lines in [15] to formulate an FETAML detector. As the temperature of the sample can be assumed to lie in a known temperature range, we may, using (2), determine the range of frequencies each sinusoidal component may be present in. Hence, a frequency-selective detector that only considers these narrow frequency bands can be derived.6 Consider the frequency regions formed by 2πk1 2πk2 2πkL , ,..., (24) N N N with k1 , . . . , kL being L given, not necessarily consecutive, integers selected such that (24) only consists of the possible frequency grid points for each of the d signal components; 6 If
strong RFIs are known to lie in the frequency bands of interest, these could also be omitted to further improve robustness.
where ELM is defined similar to ZLM . Using (30), the minimization in (10) can be approximated as 2 ∗ ˜ min ZLM − ρ vec VL AT,β Q . θ
F
(31)
The GLRT of the FETAML detector is found reminiscent of (23). We note that from a computational point of view, one ∗ ˜ m form a geometshould exploit that the indices of VL AT,β Ψ ∗ ˜ ric series; the lth index of VL AT,β Ψm can be written as7
∗ ˜m VL AT,β Ψ
l
=
d
κk Gk [Ω1 + Ω2 ]
(32)
k=1
7 We here also remark that the frequency domain data vector is typically of a significantly lower dimension than the time-domain representation as L N ; for instance, in our examples, L = 8 and N = 256.
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TABLE II DIFFERENT ETAML AND FETAML DETECTORS
where Gk = e−ηk (T )[mµ+t0 ]+iwk (T )t0 t −t +1 1 − vl,ksp 0 Ω1 = C(λk )eβk (t0 −tsp ) 1 − vl,k ˜ k )e−βk (t0 −tsp ) Ω2 = C(λ
t −t0 +1
ul,ksp
− uN l,k
(33) (34)
1 − ul,k
(35)
vl,k = e−i2πkl /N +iwk (T )−ηk (T )+βk
(36)
ul,k = e−i2πkl /N +iwk (T )−ηk (T )−βk .
(37)
with
We remark that several simplifications can be made to the above-presented detectors. First, for the general case of d unknown sinusoidal damping constants βk , the maximizations in (20) and similarly for FETAML result in (d + 1)-dimensional searches over T and β. However, exploiting the fact that one can often approximate βk ≈ β0 , the maximizations can be reduced to 2-D searches over temperature and the (approximative) common damping constant β0 ; initial estimates for these parameters exist, and only a limited search region is required. Furthermore, similar to the discussion in [14], it is possible to approximate the 2-D searches by two 1-D searches first over T and then over β0 . We refer to these approximative approaches as ETAML-1 and FETAML-1. The full 2-D searches are termed ETAML-2 and FETAML-2. It is worth noting that one may easily include refined searches in ETAML-1 and FETAML-1 by performing a local search around the maximizing parameter values. Such a procedure can be iterated to accurately find the parameters on a very fine grid. Second, we note from Section II that T2e,k (T ) is dependent upon several experimental parameters and that the echo damping shifting functions of temperature, for all possible echo spacings, excitation bandwidths, and frequencies, may not be available. As an alternative, to avoid determining the echo damping shifting functions for a particular experimental setting, one may instead treat the echo damping parameters as unknown constants, each denoted ηk , searching for the values that best fit the measured data. Assuming that the sinusoidal damping parameters have been approximated to be the same, this would lead to a (d + 2)-dimensional search over the common sinusoidal damping constant (β0 ), the d unknown echo damping parameters, and temperature. We denote the approach, where the d + 2-dimensional search is approximated by d + 2 1-D searches, the ETAML-s detector.8 Third, we may approximate all the echo damping constants to be the same. In this case, it may be that an overall echo damping shifting function is available, in which case the search is still a 2-D search over temperature and β0 , and this detector is simply a special case of the ETAML detector. However, if the common echo damping shifting function is unknown, then we treat the echo damping parameter as a single unknown constant, where 8 Numerical simulations carried out in [31] indicate that there is hardly any difference in detection performance when searching over the d echo damping parameters as compared to using known temperature shifting functions.
ηk (T ) ≈ η0 . Here, the search is 3-D over the (approximative) common echo damping constant η0 , the common sinusoidal damping constant β0 , and the temperature T . We denote the approach, where the 3-D search is approximated by three 1-D searches, the ETAML-a detector. We note that similar approaches can be taken for the FETAML detector. Table II summarizes all of the approaches.9 V. N UMERICAL E XAMPLES In this section, we examine the performance of the proposed detectors using both simulated and real NQR data measured at King’s College London. The real data consisted of 1000 data files, 500 with TNT and 500 without, each taking around 1 min to acquire and consisting of four echo trains summed up and phase cycled to reduce baseline offset. The echo trains were generated using a PSL sequence and were made up of 32 echoes, each echo consisting of 256 samples. In general, the first echo of the echo train should be discarded before being input to the algorithms as it can be significantly distorted by contributions from the FID, produced by the preparation pulse in the PSL sequence; however, we have here removed the first four echoes to reduce the SNR enough to allow a meaningful comparison of the algorithms. The sample, which consisted of 180-g creamed monoclinic TNT, was placed inside a shielded solenoidal coil. The Q-factor of the coil and the pulse width were selected to ensure that the excitation bandwidth was sufficient to excite the four-line region of TNT using a single excitation frequency of 841.5 kHz. After preamplification, the analog signal, from the coil, was bandlimited (to avoid aliasing effects during sampling) and demodulated at the excitation frequency prior to sampling at 200 kHz. The temperature of the sample was not artificially controlled but can be assumed to be around 297 K; furthermore, estimates of the individual damping parameters and complex amplitudes were obtained from the summed echo train signal (obtained by averaging all 2000 TNT echo trains) and are shown in Table III. The experimental settings were such that the noise could be assumed white, although we stress that, in general, it may be colored. The detectors were also compared using simulated data with and without residual RFI. The simulated data without RFI, designed to mimic the measured data, was generated using (4) and Tables I and III. The residual RFI was modeled, using the same basic model as described in [9], [10], [15], and [18], as a sum of discrete sinusoids with frequencies uniformly distributed over the interval [−π/4, π/4], thus ensuring that the RFI components were within 25 kHz of the excitation frequency. Typically, 9 It is assumed that β ≈ β for all of the implementations of the ETAML 0 k and FETAML detectors.
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TABLE III ESTIMATES OF FREQUENCIES, DAMPINGS, AND COMPLEX AMPLITUDES FOR THE d = 4 NQR C OMPONENTS OF M ONOCLINIC TNT FOR AN E XCITATION F REQUENCY OF 841.5 kHz
current techniques only measure the response of a single a priori known resonance frequency (see, e.g., [9]); to ensure the most beneficial performance for this approach, we will herein allow it to have perfect knowledge of the sample temperature so that the most dominating resonance frequency is exactly known. We denote this the demodulation approach with perfect temperature knowledge (DMA-p). In general, it is difficult to estimate the sample temperature with more than 5◦ accuracy; as a comparison, we therefore also include the estimate for a sample with 5◦ offset, terming this the demodulation approach with realistic temperature knowledge (DMA-r). Both of these approaches are based on evaluating the response of a single resonant frequency. Alternatively, one may allow for a search to find the most dominant peak within the relevant frequency region; such an approach will improve the robustness to errors in the temperature measurement. Hereafter, we denote this approach the DMA-s detector. The AML, FSAML, and DMA detectors are generally applied to echoes or echo trains that have been preprocessed to produce a single echo with stronger SNR [9], [14], [15], while the ETAML and FETAML detectors are applied to unprocessed echo trains. Hereafter, we will apply the AML, FSAML, and DMA detectors to the summed echo train, formed by adding all the M consecutive echoes, while ETAML and FETAML are formed on the full echo train signals allowing for the fine structure between the different echoes. We stress that as the former detectors operate on a single echo, they cannot be formed on the full echo train. Since the temperature uncertainty in a landmine detection application is typically ±5◦ , the AML-based detectors and the DMA-s detector use a search region over temperature of [292, 302] K (in 100 steps); however, depending on the application, this region should be selected according to the uncertainty in the temperature of the sample. Furthermore, the AML-based detectors use a search region of β0 = [0.001, 0.01] (in 100 steps). Here, for simplicity, we will assume the echo damping functions to be unavailable and use a search over each of the d echo train damping parameters of ηk = [0.0002, 0.0004] (in 100 steps), i.e., the (F)ETAML-s implementations were used.10 The search regions, for the damping parameters, should be restricted according to the available prior knowledge; for example, lower bounds on the sinusoidal dampings can be determined from the spin–spin relaxation times. As detection is the problem of interest, we proceed to examine the receiver operator characteristic (ROC) curves for the detectors. Figs. 3 and 4 show the ROC curves of the discussed detectors, for simulated data without RFI, 10 Results in [31] confirm that there is hardly any difference in detection performance when assuming all the echo damping factors to be constant when using the ETAML-a and FETAML-a implementations.
Fig. 3. ROC curves comparing the AML-based detectors, for simulated data without RFI components, at −28 dB.
Fig. 4. ROC curves comparing the DMA-based detectors, for simulated data without RFI components, at −28 dB.
using 1500 Monte Carlo simulations at an SNR of −28 dB, −2 2 2 where the SNR is here defined as SNR = σw σs , where σw and σs2 denote the power of the noise and of the noise-free signal, respectively. Figs. 5 and 6 show the ROC curves for the measured data.11 The figures clearly illustrate the improved performance of the proposed detectors over AML, FSAML, and the DMA approaches. As discussed in Section I, in some applications, it is likely that significant amounts of RFI will remain even if RFI mitigation techniques have been used. Here, we evaluate the performance of the detectors in the presence of such components. Figs. 7 and 8 show the ROC curves of the detectors, for simulated data with RFI present, using 1500 Monte Carlo simulations at an SNR of −28 dB with RFI components 18 dB larger than the NQR signal. The figures clearly show that the FETAML detector is superior to the other detectors and should be used when residual RFI components (of this type) are present in the data. 11 We note that, for the measured data set, DMA-p uses an estimate of the average temperature; therefore, for some probabilities of false alarm, DMA-p does slightly worse than DMA-s.
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Fig. 5. ROC curves comparing the ETAML-s, FETAML-s, FSAML, and AML detectors for partially shielded measured data.
Fig. 8. ROC curves comparing the DMA-based detectors, for simulated data with 18-dB RFI components, at an SNR of −28 dB.
decay time(s) of the resonant line(s). Furthermore, we propose two detectors that exploit the new data model by extending two recently proposed AML detectors that also exploit the temperature dependencies of the NQR signals. The presented detectors ensure accurate detection even in the typical case where the temperature of the sample is unknown. Numerical evaluation using both real and simulated data shows a significantly increased probability of detection, for a given probability of false alarm, for the presented detectors over previous AML-based and demodulation approaches. ACKNOWLEDGMENT
Fig. 6. ROC curves comparing the DMA-based detectors for partially shielded measured data.
The authors would like to thank M. D. Rowe, J. Barras, and E. Tate (née Balchin) of King’s College London for their invaluable help in the NQR laboratory. R EFERENCES
Fig. 7. ROC curves comparing the AML-based detectors, for simulated data with 18-dB RFI components, at an SNR of −28 dB.
VI. C ONCLUSION In this paper, we provide detailed descriptions of current NQR data models and propose a new model of the entire echo train, which incorporates knowledge of the spin echo
[1] R. Siegel, “Land mine detection,” IEEE Instrum. Meas. Mag., vol. 5, no. 12, pp. 22–28, Dec. 2002. [2] J. A. S. Smith, “Nitrogen-14 quadrupole resonance detection of RDX and HMX based explosives,” in Proc. Eur. Conv. Secur. and Detect., 1995, vol. 408, pp. 288–292. [3] M. D. Rowe and J. A. S. Smith, “Mine detection by nuclear quadrupole resonance,” in Proc. Detect. Abandon. Landmines (IEE) EUREL, 1996, vol. 43, pp. 62–66. [4] A. N. Garroway, M. L. Buess, J. B. Miller, B. H. Suits, A. D. Hibbs, A. G. Barrall, R. Matthews, and L. J. Burnett, “Remote sensing by nuclear quadrupole resonance,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 6, pp. 1108–1118, Jun. 2001. [5] S. D. Somasundaram, J. A. S. Smith, K. Althoefer, and L. D. Seneviratne, “Detection of landmines using nuclear quadrupole resonance (NQR): An overview,” in Proc. HUDEM Conf., Brussels, Belgium, Jun. 16–18, 2004, pp. 833–840. [6] E. Rao, “Application of an explosive detection device based on quadrupole resonance (QR) technology in aviation security,” in Proc. IEEE 35th Int. Carnahan Conf. Secur. Technol., Oct. 16–19, 2001, pp. 282–288. [7] E. Balchin, D. J. Malcolme-Lawes, M. D. Rowe, J. A. S. Smith, M. J. Bearpark, J. W. Steed, W. Wu, A. J. Horsewill, and D. Stephenson, “The unusual solid state structure of heroin hydrochloride monohydrate and its selective detection using NQR spectroscopy,” New J. Chem., vol. 28, no. 11, pp. 1309–1314, 2004. [8] E. Balchin, D. J. Malcolme-Lawes, I. J. F. Poplett, M. D. Rowe, J. A. S. Smith, G. E. S. Pearce, and S. A. C. Wren, “Potential of nuclear quadrupole resonance in pharmaceutical analysis,” Anal. Chem., vol. 77, no. 13, pp. 3925–3930, Jul. 2005.
SOMASUNDARAM et al.: EXPLOITING SPIN ECHO DECAY IN THE DETECTION OF NQR SIGNALS
[9] Y. Tan, S. L. Tantum, and L. M. Collins, “Cramer–Rao lower bound for estimating quadrupole resonance signals in non-Gaussian noise,” IEEE Signal Process. Lett., vol. 11, no. 5, pp. 490–493, May 2004. [10] ——, “Landmine detection with nuclear quadrupole resonance,” in Proc. Geosci. and Remote Sens. Symp., 2002, vol. 3, pp. 1575–1578. [11] D. J. Malcolm-Lawes, S. N. Mallion, M. D. Rowe, and J. A. S. Smith, NQR Signal Processing, Jan. 18, 2000. Great Britain Patent GB00/02582. [12] F. Liu, S. Tantum, L. Collins, and L. Carin, “Statistical signal processing for the detection of buried landmines using quadrupole resonance,” in Proc. SPIE Conf. Detect. and Remediat. Technol., 2000, vol. 4038, pp. 572–577. [13] S. Tantum, L. Collins, and L. Carin, “Signal processing for NQR discrimination of buried landmines,” in Proc. SPIE Conf. Detect. and Remediat. Technol., 1999, vol. 3710, pp. 474–482. [14] A. Jakobsson, M. Mossberg, M. Rowe, and J. Smith, “Exploiting temperature dependency in the detection of NQR signals,” IEEE Trans. Signal Process., vol. 54, no. 5, pp. 1610–1616, May 2006. [15] A. Jakobsson, M. Mossberg, M. Rowe, and J. A. S. Smith, “Frequency selective detection of nuclear quadrupole resonance signals,” IEEE Trans. Geosci. Remote Sens., vol. 43, no. 11, pp. 2659–2665, Nov. 2005. [16] A. D. Hibbs, G. A. Barrall, P. V. Czipott, A. J. Drew, D. K. Lathrop, Y. K. Lee, E. E. Magnusson, R. Matthews, D. C. Skvoretz, and S. A. Vierkötter, “Man portable mine detector using nuclear quadrupole resonance—First year progress and test results,” in Proc. 2nd Int. Conf. Detect. Abandon. Land Mines, 1998, pp. 138–141. [17] R. M. Deas, I. A. Burch, and D. M. Port, “The detection of RDX and TNT mine like targets by nuclear quadruple resonance,” in Proc. SPIE—Detect. and Remediat. Technol. Mines and Minelike Targets, 2002, vol. 4742, pp. 482–489. [18] J. A. S. Smith and M. D. Rowe, “Apparatus for and method of nuclear quadrupole resonance testing a sample,” U.S. Pat. Off., Washington, DC, U.S. Patent 6 486 838, Nov. 26, 2002. [19] B. H. Suits and A. N. Garroway, “Optimising surface coils and the self-shielded gradiometer,” J. Appl. Phys., vol. 94, no. 6, pp. 4170–4178, Sep. 2003. [20] G. E. Poletto, T. M. Osan, and D. J. Pusiol, “Pulsed 14 N NQR device designed to detect substances in the presence of environmental noise,” Hyperfine Interact., vol. 159, no. 1–4, pp. 127–130, Dec. 2004. [21] Y. Jiang, P. Stoica, and J. Li, “Array signal processing in the known waveform and steering vector case,” IEEE Trans. Signal Process., vol. 52, no. 1, pp. 23–35, Jan. 2004. [22] B. H. Suits, A. N. Garroway, and J. B. Miller, “Noise-immune coil for unshielded magnetic resonance measurements,” J. Magn. Reson., vol. 131, no. 1, pp. 154–158, Mar. 1998. [23] M. L. Buess, A. N. Garroway, and J. P. Yesinowski, “Removing the effects of acoustic ringing and reducing temperature effects in the detection of explosives by NQR,” U.S. Pat. Off., Washington, DC, U.S. Patent 5 365 171, Nov. 15, 1994. [24] G. A. Barrall, L. J. Burnett, and A. G. Sheldon, “Method and system for cancellation of extraneous signals in nuclear quadrupole resonance spectroscopy,” U.S. Pat. Off., Washington, DC, U.S. Patent 6 392 408, May 21, 2002. [25] J. B. Miller, A. N. Garroway, and B. H. Suits, “Removing the effects of acoustic ringing and reducing temperature effects in detection of explosives by NQR,” U.S. Pat. Off., Washington, DC, U.S. Patent 5 365 171, Nov. 15, 1994. [26] J. B. Miller and A. N. Garroway, “Nuclear quadrupole resonance method and apparatus,” U.S. Pat. Off., Washington, DC, U.S. Patent 6 777 937, Aug. 17, 2004. [27] J. A. S. Smith, “Nuclear quadrupole resonance spectroscopy,” J. Chem. Educ., vol. 48, no. 1, p. 39, Jan. 1971. [28] J. A. S. Smith, M. D. Rowe, R. M. Deas, and M. J. Gaskell, “Nuclear quadrupole resonance detection of landmines,” in Proc. Int. Conf. Requirements Technol. Detection, Removal, and Neutralization Landmines and UXO, 2003, vol. 2, pp. 715–721. [29] P. Stoica and R. Moses, Spectral Analysis of Signals. Upper Saddle River, NJ: Prentice-Hall, 2005. [30] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume II: Detection Theory. Englewood Cliffs, NJ: Prentice-Hall, 1998. [31] S. D. Somasundaram, A. Jakobsson, J. A. S. Smith, and K. Althoefer, “Detection of explosives and narcotics using nuclear quadrupole resonance,” Karlstad Univ., Karlstad, Sweden, Tech. Rep. KAU EE-2005-05, 2005.
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Samuel D. Somasundaram (S’06) received the B.Sc. degree in chemistry from Imperial College London, London, U.K., in 2001 and the M.Sc. degree in information engineering (specializing in control) from City University, London, in 2002. He is currently working toward the Ph.D. degree at King’s College London, London. His research interests include the application of signal processing to nuclear quadrupole resonance to facilitate the characterization and detection of compounds containing quadrupolar nuclei. Mr. Somasundaram is a member of the IET.
Andreas Jakobsson (S’95–M’00–SM’06) received the M.Sc. degree from the Lund Institute of Technology, Lund, Sweden, in 1993 and the Ph.D. degree in signal processing from Uppsala University, Uppsala, Sweden, in 2000. He was previously with Global IP Sound AB; the Swedish Royal Institute of Technology, Stockholm, Sweden; and King’s College London, London, U.K. He was also a Visiting Researcher with Brigham Young University, Provo, UT; Stanford University, Stanford, CA; Katholieke Universiteit Leuven, Leuven, Belgium; and the University of California, San Diego. He is currently a Professor of signal processing with the Department of Electrical Engineering, Karlstad University, Karlstad, Sweden. He also holds an Honorary Research Fellowship with Cardiff University, Wales, U.K. His research interests include statistical and array signal processing, detection and estimation theory, and related applications in remote sensing, telecommunication, and biomedicine.
John A. S. Smith is an Emeritus Professor with King’s College London, London, U.K., where he is also the Director of the nuclear quadrupole resonance (NQR) Group. He is currently the Project Director of a NATO-funded research program devoted to the detection of abandoned land mines in Southeast Europe. He is the coauthor of over 200 papers and reviews in the scientific and technical literature and joint holder of 15 patents. Recent publications include work on the NQR detection of explosives in vehicles and a study of the structure and the NQR spectrum of heroin hydrochloride. Dr. Smith was the Founder of the International Symposia on Quadrupole Interactions, an Advisor for the GEC-Marconi NQR trials at Heathrow Airport, and has acted as a Consultant for the European Commission, the International Atomic Energy Agency, and the DSTL, Fort Halstead.
Kaspar Althoefer (M’02) received the degree from the Aachen University of Technology, Aachen, Germany, in 1991 and the Ph.D. degree from King’s College London (KCL), London, U.K., in 1996. He is currently a Senior Lecturer with the Department of Mechanical Engineering, KCL. He is the coauthor of over 90 refereed research papers. His main interests lie in sensing, data analysis, and interpretation using neural networks and fuzzy logic as well as robot-based applications. Dr. Althoefer is a member of the IET.
