A complex pseudo-spectrum based velocity filtering method for track-before-detect

Highlights

• The phase information of complex-valued returns is considered, and a novel complex pseudo-spectrum (CPS) method is proposed for VF-TBD to improve the performance in detecting and tracking weak targets. The procedures for CPS construction and energy integration are presented in detail.

• The detection threshold and receiver operating characteristic (ROC) curve of the proposed CPS based method in complex Gaussian noise are derived theoretically.

• The detailed discussion about the difference between the proposed CPS-VF-TBD method and the conventional methods is provided to account for the improved performance of the proposed method, and simulations are performed in both single-target and multi-target scenarios to demonstrate the superiority of the proposed method.

Abstract

Velocity filtering based track-before-detect (VF-TBD) has advantages in energy integration and multi-target processing. However, in existing VF-TBD methods, real-valued measurements are used to be accumulated rather than complex-valued measurements. This results in significant waste of information and may degrade algorithm performance, since the phase of measurement is discarded. In this paper, a complex pseudo-spectrum (CPS) method is proposed for VF-TBD to improve the performance of weak target detection, based on the utilization of phase information during energy integration. Firstly, a CPS is constructed around the predicted position of each quantized cell with its complex-valued measurement. Samples of the CPSs from the same frame are added onto corresponding cells to obtain intraframe integrated spectrum, where the target echo values sampled on different cells in a frame have the same phase while noise measurement carries random phase, and the phase diversity is used to improve the performance of energy integration. This procedure can be regarded as coherent integration performed intraframe, contributing to improved integration efficiency. Then, amplitude values of the intraframe integrated results from all frames are added to the integration frame to achieve interframe energy integration. The theoretical detection threshold and receiver operating characteristic curve of the proposed CPS based method in complex Gaussian noise are derived in detail. Theoretical analyses and simulation results are provided to demonstrate the superiority of the proposed method.

Introduction

In traditional target tracking algorithms [1], [2], the processors are fed with thresholded data at each scan. However, these methods may incur adverse performance degradation in case of low signal-to-noise ratio (SNR), since the threshold processing may discard detections of weak targets. Although a low threshold can provide a high probability of target detection, the rate of false detections is also increased, which may cause the tracker to form false trajectories.

Track-before-detect (TBD) is an effective strategy for low SNR target detection and tracking [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]. TBD methods can obtain superior detection performance and tracking accuracy by jointly processing multiple consecutive frames of raw sensor data [13], [14], [15], [16], [17] or data pre-processed with a low threshold [18], [19], [20], [21]. In contrast to conventional tracking methods, TBD allows target detection and tracking to be performed simultaneously.

Typical TBD strategies include velocity filtering based TBD (VF-TBD) method [17], [22], [23], [24], [25], [26], dynamic programming based TBD (DP-TBD) method [16], [27], [28], [29], [30], [31], [32], and particle filtering based TBD (PF-TBD) method [4], [33], [34], [35], [36], [37], [38], [39].

DP-TBD methods integrate scoring functions, constructed directly from raw sensor data, along physically feasible trajectories, and declare targets when the integrated value exceeds a given detection threshold. DP-TBD was first proposed to address weak target detection in infrared and optical images [27], [30], and has been widely studied in radar and sonar fields [16], [19], [32]. A disadvantage of DP-TBD is that the target output envelope is extended to a large number of cells, which may lead to degraded performance in detection and estimation of adjacent targets [16], [29]. PF-TBD is implemented by propagating a series of weighted particles recursively, and declare target detections when the existence probability, calculated in terms of particle weights, is above a given threshold. PF-TBD has the advantage of providing a recursive approach for handling targets with complicated motions [36], [39]. PF based methods may be computationally expensive since a great number of particles are required to keep the algorithm robust. Another challenge in DP-TBD and PF-TBD is the curse of dimensionality arising from the large dimension of the state space in multi-target scenario, which is a popular topic and has attracted a number of researches. [16], [19], [29], [38]. In existing DP-TBD and PF-TBD methods, real-valued measurements [16], [31], [32], [36] or complex-valued measurements [7], [8], [32] are used for multi-frame processing. Due to the use of additional phase information, performance improvement can be observed with respect to the case using real-valued measurements.

VF-TBD is a type of three-dimensional matched filter [22]. The basic idea of VF-TBD is to integrate measurements along the trajectory determined by the velocity assumed in the filter [23], [25]. When the assumed velocity matches the actual one, the integrated target energy reaches the maximum. The target is declared when the integrated value exceeds a given threshold, and its parameters (e.g., position and velocity) are estimated synchronously. Effective energy integration is the key to improving the detectability of weak targets in VF-TBD methods. There are two integration strategies for VF-TBD currently, i.e., cell-to-cell based VF-TBD (CC-VF-TBD) [25], [40], [41], [42] and pseudo-spectrum based VF-TBD (PS-VF-TBD) [43]. The energy integration in CC-VF-TBD is conducted by adding the amplitude of a cell onto the cell closest to its predicted position, while in PS-VF-TBD, samples of each pseudo-spectrum, constructed with amplitude value of a cell, are added to the last frame of the processing batch for multi-frame accumulation. In the case of energy spillover, i.e., the target echo envelope occupies several adjacent cells, PS-VF-TBD can obtain better performance than the former due to the utilization of the spilled energy of target, although it is at the expense of a little higher computational cost. The application of pseudo-spectrum approach is extended to radar systems recently. A pseudo-spectrum approach in mixed coordinates [44] is proposed to deal with raw range-azimuth measurements, and in [45], a pseudo-spectrum based speed square filter is presented for effective weak target detection in range-Doppler plane.

Compared with other TBD strategies, VF-TBD can achieve improved performance in detecting weak targets, along with advantages of improved output SNR and fixed computational complexity [17], [40]. VF-TBD methods can deal with multi-target detection by running a bank of filters, and its well-focused output envelope alleviates the mutual interference of adjacent targets [41], [45]. In addition, the characteristics of target output envelope can be maintained for precise parameter estimation [43]. However, in existing VF-TBD methods, including CC-VF-TBD and PS-VF-TBD, real-valued measurements are used for energy integration, while the valuable phase information is discarded. This leads to a significant waste of information, since complex-valued measurements are available in most radar systems [46].

In this paper, a complex pseudo-spectrum (CPS) method is proposed for VF-TBD (CPS-VF-TBD) to improve the performance in detecting and tracking weak targets. In contrast to the conventional pseudo-spectrum method [43] which handles targets with real-valued measurements, the proposed complex pseudo-spectrum approach takes phase information of complex data into account for algorithm performance improvement. Some DP-TBD and PF-TBD methods [7], [8], [32] use phase information for the likelihood ratio calculation, in this work, the phase is employed in intraframe spectrum integration. Firstly, each quantized cell is predicted to the last frame of the current processing batch according to an assumed velocity and CPS is constructed around the predicted position with the complex measurement value of the cell as the peak intensity. The intraframe integrated spectrum is obtained by adding samples of the CPSs from the same frame onto the corresponding cells. This is actually a coherent integration processing performed intraframe since the sample values of the target echo envelope on cells carry the same phase in a frame while the noise phase is random, and it contributes to improved target energy integration. Then, the interframe energy integration is performed by adding amplitude values of the CPS integration result in each frame to the last frame of the processing batch. The procedure for the energy integration in the proposed CPS-VF-TBD method is presented in detail. The detection threshold is investigated at a constant false alarm rate and the receiver operating characteristic (ROC) curve of the proposed method is derived theoretically in complex Gaussian background noise. Finally, the difference between the proposed CPS-VF-TBD method and the conventional VF-TBD methods, i.e., CC-VF-TBD method and PS-VF-TBD method, is discussed to show that the use of phase information in the proposed CPS-VF-TBD method enables intraframe coherent integration, which is the key to performance improvement. Simulation results illustrate the benefits of the proposed CPS-VF-TBD method in detection performance and estimation accuracy.

The main contributions of this paper are summarized as follows. First, the phase information of complex-valued returns is considered and a novel complex pseudo-spectrum method is proposed for VF-TBD to improve the performance in detecting and tracking weak targets. The procedure for energy integration is presented in detail. Second, the detection threshold is investigated at a given false alarm rate and the theoretical ROC curve is derived in complex Gaussian noise. Third, the detailed discussion about the difference between the proposed CPS-VF-TBD method and the conventional methods is provided to account for the improved performance of the proposed method.

The rest of the paper is arranged as follows. The target dynamic model and measurement model are described in Section 2, and the problem of VF-TBD with complex-valued returns is formulated also. In Section 3, the proposed CPS-VF-TBD method is presented in detail and the theoretical analyses are given. Simulations are performed in Section 4 to verify the proposed CPS-VF-TBD method, followed by conclusions in Section 5.