Signal Processing for Remote Sensing - Chapter 7

7 Kalman Filtering for Weak Signal Detection in Remote Sensing Stacy L. Tantum, Yingyi Tan, and Leslie M. Collins CONTENTS 7.1 Signal Models.................................................................................................................... 129 7.1.1 Harmonic Signal Model....................................................................................... 129 7.1.2 Interference Signal Model ................................................................................... 129 7.2 Interference Mitigation .................................................................................................... 130 7.3 Postmitigation Signal Models......................................................................................... 131 7.3.1 Harmonic Signal Model....................................................................................... 132 7.3.2 Interference Signal Model ................................................................................... 132 7.4 Kalman Filters for Weak Signal Estimation................................................................. 133 7.4.1 Direct Signal Estimation...................................................................................... 134 7.4.1.1 Conventional Kalman Filter................................................................. 134 7.4.1.2 Kalman Filter with an AR Model for Colored Noise...................... 135 7.4.1.3 Kalman Filter for Colored Noise......................................................... 137 7.4.2 Indirect Signal Estimation................................................................................... 138 7.5 Application to Landmine Detection via Quadrupole Resonance............................. 140 7.5.1 Quadrupole Resonance........................................................................................ 141 7.5.2 Radio-Frequency Interference............................................................................. 141 7.5.3 Postmitigation Signals.......................................................................................... 142 7.5.3.1 Postmitigation Quadrupole Resonance Signal.................................. 143 7.5.3.2 Postmitigation Background Noise...................................................... 143 7.5.4 Kalman Filters for Quadrupole Resonance Detection.................................... 143 7.5.4.1 Conventional Kalman Filter................................................................. 143 7.5.4.2 Kalman Filter with an Autoregressive Model for Colored Noise .................................................................................. 144 7.5.4.3 Kalman Filter for Colored Noise......................................................... 144 7.5.4.4 Indirect Signal Estimation.................................................................... 144 7.6 Performance Evaluation .................................................................................................. 145 7.6.1 Detection Algorithms........................................................................................... 145 7.6.2 Synthetic Quadrupole Resonance Data............................................................. 145 7.6.3 Measured Quadrupole Resonance Data........................................................... 146 7.7 Summary............................................................................................................................ 148 References ................................................................................................................................... 149 ß 2007 by Taylor & Francis Group, LLC. Remote sensing often involves probing a region of interest with a transmitted electro-magnetic signal, and then analyzing the returned signal to infer characteristics of the investigated region. It is not uncommon for the measured signal to be relatively weak or for ambient noise to interfere with the sensor’s ability to isolate and measure only the desired return signal. Although there are potential hardware solutions to these obstacles, such as increasing the power in the transmit signal to strengthen the return signal, or altering the transmit frequency, or shielding the system to eliminate the interfering ambient noise, these solutions are not always viable. For example, regulatory constraints on the amount of power that may be radiated by the sensor or the trade-off between the transmit power and the battery life for a portable sensor may limit the power in the transmitted signal, and effectively shielding a system in the field from ambient electromagnetic signals is very difficult. Thus, signal processing is often utilized to improve signal detectability in situations such as these where a hardware solution is not sufficient. Adaptive filtering is an approach that is frequently employed to mitigate interference. This approach, however, relies on the ability to measure the interference on auxiliary reference sensors. The signals measured on the reference sensors are utilized to estimate the interference, and then this estimate is subtracted from the signal measured by the primary sensor, which consists of the signal of interest and the interference. When the interference measured by the reference sensors is completely correlated with the interference measured by the primary sensor, the adaptive filtering can completely remove the interference from the primary signal. When there are limitations in the ability to measure the interference, that is, the signals from the reference sensors are not completely correlated with the interference measured by the primary sensor, this ap-proach is not completely effective. Since some residual interference remains after the adaptive interference cancellation, signal detection performance is adversely affected. This is particularly true when the signal of interest is weak. Thus, methods to improve signal detection when there is residual interference would be useful. The Kalman filter (KF) is an important development in linear estimation theory. It is the statistically optimal estimator when the noise is Gaussian-distributed. In addition, the Kalman filter is still the optimal linear estimator in the minimum mean square error (MMSE) sense even when the Gaussian assumption is dropped [1]. Here, Kalman filters are applied to improve detection of weak harmonic signals. The emphasis in this chapter is not on developing new Kalman filters but, rather, on applying them in novel ways for improved weak harmonic signal detection. Both direct estimation and indirect estimation of the harmonic signal of interest are considered. Direct estimation is achieved by applying Kalman filters in the conventional manner; the state of the system is equal to the signal to be estimated. Indirect estimation of the harmonic signal of interest is achieved by reversing the usual application of the Kalman filter so the background noise is the system state to be estimated, and the signal of interest is the observation noise in the Kalman filter problem statement. This approach to weak signal estimation is evaluated through application to quadru-pole resonance (QR) signal estimation for landmine detection. Mine detection technolo-gies and systems that are in use or have been proposed include electromagnetic induction (EMI) [2], ground penetrating radar (GPR) [3], and QR [4,5]. Regardless of the technology utilized, the goal is to achieve a high probability of detection, PD, while maintaining a low probability of false alarm, PFA. This is of particular importance for landmine detection since the nearly perfect PD required to comply with safety requirements often comes at the expense of a high PFA, and the time and cost required to remediate contaminated areas is directly proportional to PFA. In areas such as a former battlefield, the average ratio of real mines to suspect objects can be as low as 1:100, thus the process of clearing the area often proceeds very slowly. ß 2007 by Taylor & Francis Group, LLC. QR technology for explosive detection is of crucial importance in an increasing number of applications. Most explosives, such as RDX, TNT, PETN, etc., contain nitrogen (N). Some of its isotopes, such as 14N, possess electric quadrupole moments. When compounds with such moments are probed with radio-frequency (RF) signals, they emit unique signals defined by the specific nucleus and its chemical environment. The QR frequencies for explosives are quite specific and are not shared by other nitrogenous materials. Since the detection process is specific to the chemistry of the explosive and therefore is less susceptible to the types of false alarms experienced by sensors typically used for landmine detection, such as EMI or GPR sensors, the pure QR of 14N nuclei supports a promising method for detecting explosives in the quan-tities encountered in landmines. Unfortunately, QR signals are weak, and thus vulner-able to both the thermal noise inherent in the sensor coil and external radio-frequency interference (RFI). The performance of the Kalman filter approach is evaluated on both simulated data and measured field data collected by Quantum Magnetics, Inc. (QM). The results show that the proposed algorithm improves the performance of landmine detection. 7.1 Signal Models In this chapter, it is assumed that the sensor operates by repeatedly transmitting excita-tion pulses to investigate the potential target and acquires the sensor response after each pulse. The data acquired after each excitation pulse are termed a segment, and a group of segments constitutes a measurement. In general, for each potential target there are multiple measurements with each measurement containing many segments. 7.1.1 Harmonic Signal Model The discrete-time harmonic signal of interest, at frequency f0, in a single segment can be represented by s(n) ¼ A0 cos(2pf0nþ f0), n ¼ 0,1,...,N ÿ1 (7:1) The measured signal may be demodulated at the frequency of the desired harmonic signal, f0, to produce a baseband signal, s(n), s(n) ¼ A0ejf0 , n ¼ 0,1,...,N ÿ 1 (7:2) Assuming the frequency of the harmonic signal of interest is precisely known, the signal of interest after demodulation and subsequent low-pass filtering to remove any aliasing introduced by the demodulation is a DC constant. 7.1.2 Interference Signal Model A source of interference for this type of signal detection problem is ambient harmonic signals.Forexample,sensorsoperatingintheRFbandcouldexperienceinterferencedueto other transmitters operating in the same band, such as radio stations. Since there may be many sources transmitting harmonic signals operating simultaneously, the demodulated ß 2007 by Taylor & Francis Group, LLC. interference signal measured in each segment may be modeled as the sum of the contribu-tion from M different sources, each operating at its own frequency fm, I ¼ X ~m(n)ej(2pfmnþfm), n ¼ 0,1,..., N ÿ1 (7:3) m¼1 where the superscript denotes a complex value and we assume the frequencies are distinct, meaning fi ¼ fj for i ¼ j. The amplitudes Am(n) from a discrete time series. Although the amplitudes are not restricted to constant values in general, they are as-sumed to remain essentially constant over the short time intervals during which each data segment is collected. For time intervals on this order, it is reasonable to assume Am(n) is constant for each data segment, but may change from segment to segment. Therefore, the interference signal model may be expressed as M I ¼ Amej(2pfmnþfm), n ¼ 0,..., N ÿ 1 (7:4) m¼1 This model represents all frequencies even though the harmonic signal of interest exists in a very narrow band. In practice, only the frequency corresponding to the harmonic signal of interest needs to be considered. 7.2 Interference Mitigation Adaptive filtering is a widely applied approach for noise cancellation [6]. The basic approach is illustrated in Figure 7.1. The primary signal consists of both the harmonic signal of interest and the interference. In contrast, the signal measured on each auxiliary antenna or sensor consists of only the interference. Adaptive noise cancellation utilizes the measured reference signals to estimate the noise present in the measured primary signal. The noise estimate is then subtracted from the primary signal to find the signal of interest. Adaptive noise cancellation, such as the least mean square (LMS) algorithm, is well suited for those applications in which one or more reference signals are available [6]. Signal of interest Primary antenna + + − Signal estimate Interference signal Reference antenna Adaptive filter FIGURE 7.1 Interference mitigation based on adaptive noise cancellation. ß 2007 by Taylor & Francis Group, LLC. In this application, the adaptive noise cancellation is performed in the frequency domain by applying the normalized LMS algorithm to each frequency component of the fre-quency domain representation of the measured primary signal. The primary measured signal at time n may be denoted by d(n) and the measured reference signal with u(n). The tap input vector may be represented by u(n) ¼ [u(n) u(n ÿ 1) u(n ÿ M þ 1)]T and the tap weight vector may be given by w(n). Both the tap input and tap weight vectors are of length M. Given these definitions, the filter output at time n, e(n), is e(n) ¼ d(n) ÿ wH(n)u(n) (7:5) The quantity wH(n)u(n) represents the interference estimate. The tap weights are updated according to w(nþ 1) ¼ w(n) þm0Pÿ1(n)u(n)e(n) (7:6) where the parameter m0 is an adaptation constant that controls the convergence rate and P(n) is given by P(n) ¼ bP(nÿ 1) þ (1 ÿb)ju(n)j2 (7:7) with 0 < b < 1 [7]. The extension of this approach to utilize multiple reference signals is straightforward. 7.3 Postmitigation Signal Models Under perfect circumstances the interference present in the primary signal is completely correlated with the reference signals and all interference can be removed by the adaptive noise cancellation, leaving only Gaussian noise associated with the sensor system. Since the interference often travels over multiple paths and the sensing systems are not perfect, however, the adaptive interference mitigation rarely removes all the interference. Thus, there is residual interference that remains after the adaptive noise cancellation. In add-ition, the adaptive interference cancellation process alters the characteristics of the signal of interest. The real-valued observed data prior to interference mitigation may be represented by x(n) ¼ s(n) þ I(n) þ w(n), n ¼ 0,1,..., N ÿ 1 (7:8) where s(n) is the signal of interest, I(n) is the interference, and w(n) is Gaussian noise associated with the sensor system. The baseband signal after demodulation becomes x(n) ¼ s(n) þ I(n) þ w(n) (7:9) where I(n) is the interference, which is reduced but not completely eliminated by adaptive filtering. The signal remaining after interference mitigation is y(n) ¼ s(n) þv(n), n ¼ 0,1,..., N ÿ 1 (7:10) ß 2007 by Taylor & Francis Group, LLC. ... - tailieumienphi.vn