Mathematical Problems in Engineering

Volume 2015, Article ID 302083, 10 pages

http://dx.doi.org/10.1155/2015/302083

## System Optimization for Temporal Correlated Cognitive Radar with EBPSK-Based MCPC Signal

School of Information Science and Engineering, Southeast University, Nanjing 210096, China

Received 19 August 2014; Accepted 21 September 2014

Academic Editor: Yudong Zhang

Copyright © 2015 Peng Chen and Lenan Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The system optimization is considered in cognitive radar system (CRS) with extended binary phase shift keying- (EBPSK-) based multicarrier phase-coded (MCPC) signal. A novel radar working scheme is proposed to consider both target detection and estimation. At the detection stage, the generalized likelihood ratio test (GLRT) threshold is deduced, and the GLRT detection probability is given. At the estimation stage, an approach based on Kalman filtering (KF) is proposed to estimate target scattering coefficients (TSC), and the estimation performance is improved significantly by exploiting the TSC temporal correlation. Additionally, the optimal waveform is obtained to minimize the mean square error (MSE) of KF estimation. For the practical consideration, iteration algorithms are proposed to optimize the EBPSK-based MCPC signal in terms of power allocation and coding matrix. Simulation results demonstrate that the KF estimation approach can improve the estimation performance by 25% compared with maximum a posteriori MAP (MAP) method, and the KF estimation performance can be further improved by 90% by optimizing the transmitted waveform spectrum. Moreover, by optimizing the power allocation and coding matrix of the EBPSK-based MCPC signal, the KF estimation performances are, respectively, improved by 7% and 8%.

#### 1. Introduction

The cognitive radar system (CRS) as the future trend of the radar systems, compared with the traditional ones, mainly includes three different aspects [1]: (1) CRS can sense the targets and environment; (2) the transmitted waveform is adaptively optimized to improve the detection and estimation performance; (3) the transmitter, environment, and receiver form a closed loop feedback system. Therefore, optimizing the transmitted waveform based on the working environment becomes a popular research direction in the CRS [2–8].

For the modeling of CRS, a point is utilized to model the target [9]. The echo waveform from the point target is only the delay or Doppler frequency shift of the original one. However, when the size of target is large enough to occupy many resolution cells, the echo signals from different scattering cells are superimposed, and this type of target is modeled as an extended target (ET) [10]. In addition, the target impulse response (TIR), which is often referred to as the high resolution range profile (HRRP) in the automatic target recognition (ATR) problem [11], can be used to describe ET based on the assumption of the linear time-invariant target [12, 13]. However, the view angle between the target and radar changes, and the ET does not satisfy this assumption. Hence, an exponential correlation model is proposed to describe this time dynamic characteristic, where the TIR is stationary with time and uncorrelated among different resolution cells, namely, wide sense stationary-uncorrelated scattering (WSSUS). Therefore, the TIR of this type target can be estimated by Kalman filtering (KF) [5, 14], where only the estimation is taken into consideration and under the present assumption of the target. However, the practical radar systems should detect the target before estimating.

The performance of the target estimation and detection can be significantly improved by optimizing the transmitted waveform in the CRS. During the stage to estimate the target scattering coefficients (TSC), which is essential the Fourier transform of TIR, the performance can be improved by optimizing the transmitted waveform to maximize the mutual information (MI) between the echo signal and TSC [12]. Furthermore, when the precise priori knowledge of the target cannot be obtained, an eigensubspace projection-based method is proposed to enlarge the separation between the echo signals from different targets in [15, 16]. During the detection stage, in order to maximize the probability of target detection, more power is concentrated on frequencies with relatively large TSC in the additive white Gaussian noise (AWGN) systems or relatively high ratio between the target and clutter in the clutter interference systems [17–20]. However, these optimized waveforms have arbitrary nature and are not conveniently generated in the practical radar systems.

The multicarrier phase-coded (MCPC) signal is first applied in the wideband radar systems by Levanon [21–23]. Furthermore, the MCPC waveform can achieve the optimal target detection performance with higher spectrum efficiency than the linear frequency modulation (LFM) signal [24, 25]. Unlike the P3 or P4 signal, the MCPC signal has a thumbtack-shaped ambiguity function and is easily generated in the practical radar systems. However, the nonconstant amplitude of MCPC waveform cannot take full advantage of the nonlinear amplifier [26]. On the modulation of MCPC waveform, an extended binary phase shift keying (EBPSK) modulation proposed by Wu et al. is adopted in this work [27, 28]. EBPSK is more flexible than BPSK, where a small angle phase and jump time are utilized to distinguish the modulation waveform of code from that of code , which tightens the spectrum of the transmitted waveform [25, 29, 30]. In the wireless communication system, reference [31] has confirmed that EBPSK can achieve the same theoretical bit error rate (BER) performance as BPSK and achieve higher spectral efficiency with the same bit rate by tuning modulation parameters. Therefore, we first utilize the EBPSK-based MCPC signal in the CRS, since the tight spectrum is easy to optimize.

In this work, the problem of system optimization is considered in the CRS with EBPSK-based MCPC signal, and we propose a new radar working scheme, where both the target detection and estimation are taken into account. During the initial stage of the target detection, the generalized likelihood ratio test (GLRT) algorithm based on constant false alarm rate (CFAR) is utilized. In the presence of ET, the approach based on KF is proposed to estimate TSC by exploiting the temporal correlation, and the optimal spectrum is obtained to minimize the trace of the mean square error (MSE) matrix of KF estimation. In addition, the iteration algorithms are proposed to optimize the amplitudes and coding matrix of EBPSK-based MCPC signal, respectively.

The remainder of this work is organized as follows. In Section 2, the system model of the CRS with EBPSK-based MCPC signal and the new radar working scheme are given. In Section 3, we optimize this radar system, including the GLRT target detection based on CFAR, the maximum a posteriori probability (MAP) receive filter, the TSC estimation based on KF, the optimal power allocation, and coding matrix. Section 4 gives the simulation results. Finally, Section 5 concludes this work.

The notations used in this work are defined as follows. Symbols for matrices (upper case) and vectors (lower case) are in boldface. , , , , , , , and denote the conjugate transpose (Hermitian), the diagonal matrix, the identity matrix of size , the Gaussian distribution with zero mean and covariance being , the absolute value, the norm, the determinant of a matrix, the expectation, and the trace of a matrix, respectively.

#### 2. The System Model of Cognitive Radar with EBPSK-Based MCPC Signal

The system model of cognitive radar considered in this work to detect and estimate the ET is shown in Figure 1, where the transmitted waveform is EBPSK-based MCPC signal and the TIR of ET is during the th pulse. Then the echo signal can be described aswhere denotes the convolution operation, and denotes the AWGN. The EBPSK-based MCPC signal can be expressed aswhere , denotes the entry at th row and th column of the coding matrix , denotes the amplitude of the th subcarrier, and denotes the carrier frequency of the th subcarrier, denotes the carrier frequency of the first subcarrier, and denotes the frequency interval between subcarriers. and are the waveform of code and with EBPSK modulation, respectively,where denotes the time interval of the waveform for one code and denotes the jump interval of the EBPSK modulation. Then we can get the spectrum of transmitted waveform by the Fourier transformwhere and is the Fourier transform of ; then the spectrum of echo signal isThe discrete form in the frequency domain can be obtained aswhere the length of is , denotes the discrete Gaussian noise, denotes the variance of noise, denotes the discrete form of , denotes the TSC, that is, the discrete form of , and the diagonal matrix , where is the discrete form of . According to [14], the exponential correlation model of the TSC iswhere denotes the TSC during the th pulse, denotes the pulse repetition interval (PRI) in the radar system, denotes the temporal decay constant, and follows the Gaussian distribution. When ,and . To simplify the analysis in this work, the correlation between the individual scatters is small enough to assume that is a diagonal matrix; then denotes the variance of , where is the th entry of .