Intelligent target spectrum estimation based on OFDM signals for cognitive radar applications

Abstract

Target frequency response estimation in the case of stochastic extended targets is an essential part of a cognitive radar system. A new efficient and practical target spectrum estimation method is proposed based on orthogonal frequency division multiplexing (OFDM) signals. The proposed estimation procedure can entirely suppress any unwanted signal and intelligently estimate the target spectrum in predefined subcarrier frequencies. We show that the proposed estimator is capable to exploit the important target frequency characteristic with high precision in heavy clutter and noise. Computer simulations express a promising precision by comparing the relevant Cramer-Rao bound and the mean square error (MSE) of numerical experiments.

Keywords

Cognitive radar intelligent target spectrum estimation cramer-Rao bound orthogonal frequency division multiplexing stochastic extended target

1 Introduction

Cognitive radar is an innovative paradigm which is the combination of bio-inspired cognitive concepts and advanced system design. All radar parameters are varied on a pulse-by-pulse basis in the current technology. This flexibility offers the opportunity to adopt approaches on both the transmitter and the receiver part [1 –3]. In the practical situation, the radar environment is non-stationary and the requirement to update estimation of the environmental state is necessitated. Cognitive radar can learn from experience on how to deal with different targets, large and small, and at widely varying ranges, all in an effective and robust manner.

In order to achieve the highest level of accuracy in target detection and discrimination, the cognitive radar needs a proper cognition of its operating environment. Hence, a cognitive radar system needs to adapt its transmitter based on the obtained knowledge of the environment especially target parameters. As mentioned in the literature [4 –9], the cognitive radar can adapt its transmitting and receiving parameters such as the transmission power and the operating frequency. To achieve the optimal spectrum utilization, we should have a sufficient control over the radar channel to match the transmitted signal with its surrounded environment. The channel includes different scatterers such as target, clutter, noise and so on. The extended target in the cognitive radar systems is often distributed in more than one resolution cell with unknown power spectral density. Therefore, estimating target scattering coefficients (TSC) in the frequency domain play an important role in cognitive radar applications.

Orthogonal frequency division multiplexing communication techniques have recently received a significant research attention especially for their ability to maintain effective transmission [10 –12]. OFDM signaling scheme [13 –17], which is one way to use several subcarriers simultaneously is commonly used to resolve and exploit the multipath components. It has been shown in some researches that an OFDM signal mitigates possible fading, resolves multipath reflections, and provides additional frequency diversity as different scattering centers of a target resonate at different frequencies.

It was shown in [18] that OFDM provided improved elevation coverage of surface-based radars in multipath conditions, but at the expense of increased transmit power. Furthermore, a Constant-Envelope OFDM was shown to be a viable radar modulation that possesses the added benefits of spectral containment and phase diversity. In [19] a novel method to select proper codes for the synthesis of OFDM signals was derived. The proposed signal can reduce side lobes and achieve to the desired ambiguity function. Sen et al. [20] proposed an algorithm design the spectral parameters of the OFDM waveform for the next coherent processing interval.

Various channel estimation methods for MIMO systems and MIMO OFDM systems have been proposed in the literature [21 –24]. The channel estimation in OFDM systems is usually based on the use of pilot subcarriers in given positions of the frequency-time grid [25, 26]. Channel estimation for OFDM systems using Fourier transform model-based methods has been proposed in [27, 28]. In [29] a neural network channel estimator to estimate channel parameters in MIMO-FDM systems was considered.

Shi et al. [30] presented an estimation approach based on probabilistic neural network (PNN) decision for tracking a target with a specific maneuver. The proposed algorithm offers less computational load compared with traditional multi-model (MM) methods. A model-based 3D pose estimation approach was derived in [31]. In this work, an iterative closest point (ICP) algorithm was proposed with a genetic technique to improve the estimation performance. Meng et al. [32, 33] investigated the high resolution channel estimation. They concluded that the high-resolution channel estimation can be realized by just increasing the transmitter DAC speed while keeping the receiver ADC unchanged.

Recently, several papers concerning with adaptive waveform design methods for the estimation of target scattering matrix [34, 35]. An estimation method based on Kalman filtering (KF) in the frequency domain was proposed in [35] which exploits thetemporal correlation of the target scattering coefficients in the frequency domain.

Zhu et al. [36] proposed a novel method for suppressing clutter and generating clear microwave images of targets, which combines synthetic aperture radar (SAR) principles with the recursive method and waveform design theory. A closed-form approximate maximum likelihood (AML) estimator was derived based on the maximum likelihood principle [37]. It has been shown that the proposed estimation method is efficient for wireless communications.

In the current research, a novel frequency-based technique is introduced for estimating the spectrum of extended targets based on designing OFDM radar signals. The presented structure for target spectrum estimation can suppress any unwanted signal like noise and heavy clutter, completely exploit the target frequency response and adequately estimates their parameters including amplitude and range. The estimated target frequency response is then utilized in the transmitter via a feedback from the receiver to the transmitter. Based on this information, the cognitive radar can learn from interactions with the environment and updated its transmitted waveform.

The structure of the remainder of this paper is organized as follows. In Section 2, the architecture of the OFDM signal model is addressed. Section 3 introduces the proposed structure based on the presented signal model for extended target estimation and parameter extraction. We further derive the Cramer-Rao lower bound (CRLB) for the estimation problem. In Section 4, numerical examples are given to illustrate the performance of the proposed approach compare with the AML estimator and the CRLB. Finally, Section 5 gives the conclusions.

2 Measurement model

A multicarrier OFDM radar pulse which is consists of K subcarriers at baseband can be described as: $x (t) = \sum_{m = 1}^{M} [\sum_{k = 0}^{K - 1} m_{k} a_{k, m} exp (j 2 π f_{k} t)] s (t - (m - 1) t_{c})$ (1) where $s (t) = {\begin{matrix} 1, & 0 < t < t_{c} \\ 0, & otherwise \end{matrix}$ (2) and $f_{k} = \frac{k}{t_{c}}, k = 0, 1, \dots, K - 1$ (3)

In this model t_c is the chip width and a_k,m denotes the code sequence on the k-th carrier where each carrier is phase modulated using a sequence of M bits.

2.1 Extended target model

A target consisting of a definite number of individual scatterers is called extended or complex target. As indicated in [3, 4], an extended target is modeled by L points as: $h (t) = \sum_{l = 1}^{L} σ_{l} δ (t - τ_{l} - \frac{2 vt}{c})$ (4) where σ_l are the complex reflection coefficients of each target point, τ_l are their corresponding delays and v is the radial velocity of the target.

2.2 Clutter and noise model

The signal-dependent noise (clutter) is supposed to be the output of a linear time-invariant filter with a stochastic impulse response driven at the input by the transmitted waveform. Hence we model such a clutter by impulse response c (t) as: $c (t) = \sum_{v = 1}^{V} ξ_{v} δ (t - τ_{v})$ (5) where ξ_v are the complex reflection coefficients of clutter in each range cell and τ_v are their corresponding delays. The clutter is considered as a zero-mean complex wide sense stationary (WSS) Gaussian random process withknown power spectral density (PSD) P_c (f). Besides, the additive noise n (t) is assumed to be white and circularly symmetric Gaussian. Thus, the received baseband signal reflected from the environment can be written as:

$\begin{matrix} y (t) = x (t) * h (t) + x (t) * c (t) + n (t) \\ = \sum_{k = 0}^{K - 1} \sum_{m = 1}^{M} \sum_{l = 1}^{L} a_{k, m} σ_{l} exp (j 2 π f_{k} (t - τ_{l})) s \\ \times (t - (m - 1) t_{c} - τ_{l} - \frac{2 vt}{c}) \\ + \sum_{k = 0}^{K - 1} \sum_{m = 1}^{M} \sum_{v = 1}^{V} a_{k, m} ξ_{v} exp (j 2 π f_{k} (t - τ_{v})) \\ \times s (t - (m - 1) t_{c} - τ_{v}) + n (t) \end{matrix}$ (6) where * denotes linear convolution.

2.3 Target parameter estimation based on OFDM signals

This section introduces a new approach to estimate the target frequency response in the radar environment consists of heavy clutter and noise.

2.3.1 Generalized the compression of OFDM signals based on DFT

A new approach for compressing the OFDM signals has been introduced in [38] and we generalize this work to the case of multiple pulse (pulse train) processing consists of Doppler processing. A new method based on a discrete Fourier transform (DFT) is proposed to compress the MCPC or OFDM signals. First, the sampling rate of NK/t_c is considered, where N is called the sampling factor and set an integer number. Thus, if P pulse of the OFDM signal, which is defined in the Equation (1) sample with the specified sampling rate, then we have: $\begin{matrix} x [n, p] & = & x ({nt}_{s}) exp (j 2 π \frac{pfd}{PRF} + φ) \\ = & x (n \frac{tc}{NK}) exp (j 2 π \frac{pfd}{PRF} + φ) \\ = & \sum_{k = 0}^{K - 1} \sum_{m = 1}^{M} a_{k, m} exp (j 2 π \frac{nk}{NK}) \\ exp (j 2 π \frac{pfd}{PRF} + φ) s (n \frac{t_{c}}{NK} - (m - 1) t_{c}) \\ n = 0, 1, \dots, NKM - 1, p = 1, 2, \dots, P \end{matrix}$ (7) where f_d is Doppler shift of the moving target and assumed to be known in this paper. The parameter P is the total number of the transmitted pulse. Now by sampling the analog filter impulse response h (t) with rate fs = NK/t_c, we have: $\begin{matrix} h [n, p] = h ({nt}_{s,} p) = x^{*} (({Mt}_{c} - n \frac{t_{c}}{NK}), p) \\ = x^{*} ((({Mt}_{c} - n) \frac{t_{c}}{NK}), p) = x^{*} [(MNK - n), p] \\ = \sum_{k = 0}^{K - 1} \sum_{m = 1}^{M} a_{k, m} exp (- j 2 π \frac{(MNK - n) k}{NK}) \\ \times exp (j 2 π \frac{pfd}{PRF} + φ) s \frac{(MNK - n) t_{c}}{NK} - (m - 1) t_{c}) \\ = \sum_{k = 0}^{K - 1} \sum_{m = 1}^{M} a_{k, m} exp (- j 2 π \frac{(MNK - n - 1) k}{NK}) \\ \times exp (j 2 π \frac{pfd}{PRF} + φ) s \frac{(MNK - n - 1) t_{c}}{NK} \\ - (m - 1) t_{c}) n = 0, 1, \dots, NKM - 1, \\ p = 1, 2, \dots, P \end{matrix}$ (8)

Consequently, the matched filter (MF) output is equal to: $\begin{matrix} y [n, p] = x [n, p] * h [n] = \sum_{z = n + 1 - NKM}^{n - NK (M - 1)} x [z, p] h_{1} [n - z] \\ + \sum_{z = n + 1 - NK (M - 1)}^{n - NK (M - 2)} x [z, p] h_{2} [n - z] \\ + \dots + \sum_{z = n - NK + 1}^{n} x [z, p] h_{M} [n - z] = \sum_{i = 1}^{M} y_{i} [n, p] \end{matrix}$ (9)

A similar procedure has been applied according to [38]. Eventually, we reach to Equation (10). $y [n, p] = \sum_{k = 0}^{K - 1} \sum_{i = 1}^{M} F_{X_{i}} (k, p) a_{k, i}^{*} exp (j 2 π \frac{{pf}_{d}}{PRF} + φ)$ (10) where F_{X
_i} (k, p) is defined as the Fourier transform of vector X_i (p) and equals to: $\begin{matrix} F_{X_{i}} (k, p) = DFT {X_{i} (p)} \\ = \sum_{b = 0}^{NK - 1} x [n + 1 - (M - i + 1) NK + b] \\ exp (- j 2 π \frac{kb}{NK}) k = 0, 1, \dots, NK - 1 \end{matrix}$ (11) where

$\begin{matrix} X_{i} (p) & = & [x [n - (M - i + 1) NP + 1), p] \\ x [n - (M - i + 1) NP + 2), p] \dots \\ x [x [n - (M - i + 1) NP + NP), p]] \end{matrix}$ (12)

The realization of the above equations is shown in Fig. 1. In the first step, the received signal with dimension P (NKM) is separated in distinct pulses. In each pulse, the last received NKM samples are divided into M segments where each one contains NK samples and for each segment, the DFT of length NK is computed. Then, the first K samples of each DFT are demultiplexed and the resulting K different sequences are filtered by K conventional single carrier pulse compression filters that are matched to the corresponding codes modulated on each subcarrier.

Fig.1

Block diagram of the proposed approach to estimate the target frequency response.

3 The proposed intelligent target spectrum estimator

To estimate the extended target spectrum, as the received signal includes clutter and noise, a robust intelligent procedure is required to suppress these undesirable signals and appropriately detect the target parameters. We suppose that clutter and target are separated in the frequency domain. In order to minimize the effect of clutter, a high pass filter (h_MTI (n)) is applied in the pulse dimension to eliminate the clutter for each subcarrier separately. It is also important to remark that when the target Doppler frequency is below the cutoff frequency of the proposed filter, or the clutter energy does not concentrate on the low-frequency domain, the performance of the proposed structure is degraded. Therefore, the output of the moving target indicator filter (h_MTI (n)) is given by: $\tilde{S} (k) = S (k) * h_{MTI} (t) k = 0, 1, \dots, K - 1$ (13) where S (k) is the output of the matched filter in each subcarrier frequency. The moving target indicator (MTI) filter is designed in such a way that the clutter power has been significantly weakened in the output of this filter and it will be negligible compared to noise. At the output of the MTI filter, the pulse integration block is used to reduce the noise effect as Equation (14). ${\hat{S}}_{k} = \frac{\sum_{p = 1}^{P} {\tilde{S}}_{p} (k)}{P}$ (14)

Adding these values for all subcarriers results: $S_{t} = \sum_{k = 0}^{K - 1} {\hat{S}}_{k}$ (15)

Here, a large peak will appear in this vector for each target which is located in its predefined range cell. To maintain a constant false alarm rate or, equivalently, a fixed P_fa, a constant false alarm rate (CFAR) detector is applied to estimate statistics of the interference from radar measurements and to adjust the detector threshold [39]. By considering a fixed value for P_fa, the threshold will be computed to make a decision. Changes in the target background characteristics nullify using the fixed detection threshold and due to this fact, adaptive techniques are required to maintain a constant false alarm rate regardless of the circumstances. To conduct our survey and generalizing the issue, we apply such an adaptive technique.

Next, based on the output of the CFAR detector, the target range cell will be determined. Following that, the target spectrum will be estimated as follows: the information gathered at each subcarrier (the vector S_t feeds the target estimator. Therefore, if the output of each subcarrier divides to the relevant fading coefficient and pre-defined code sequence, we have: $\hat{H} (k) = 0.5 ex {{\hat{S}}_{k} |}_{Rangecell = R^{'}} - 0.1 em / - 0.15 em 0.25 ex a_{k, m} k = 0, 1, \dots, K - 1$ (16) where R’ is the determined target range cell. Therefore, the estimated target frequency response based on the number of subcarriers will be obtained.

The proposed approach has been applied for target spectrum estimation and if we want to estimate the clutter spectrum, the same approach should carry out with the difference that the MTI filter replaces by a low-pass filter in Equation (13).

3.1 Cramer-Rao lower bound

The CRLB is essential for parameter estimation because it provides a benchmark to evaluate the performance of any unbiased estimator [40]. In this section, the CRLB for our estimation problem will be described. In the considered discrete-time low-pass-equivalent complex baseband OFDM system with K subcarriers, the DFT of the time-domain transmitted signal vector can be modeled by: $X_{k} = \sum_{n = 0}^{N - 1} x_{n} e^{\frac{- j 2 π nk}{K}} k = 0, 1, \dots, K - 1$ (17)

The target impulse response vector h = [h₀, h₁, …, h_L-1] ^T is a wide sense stationary process with L scatterer points. The DFT of the target frequency response vector can be modeled by the Equation (18). $H_{k} = W_{kl} h k = 0, 1, \dots, K - 1$ (18)

In the absence of signal-dependent clutter, the DFT of the time-domain received signal vector y (t) in Equation (6), produces a frequency domain vector Y = [Y₀, Y₁, …, Y_K-1] ^T which can be written as: $Y_{K \times 1} = X_{K \times K} H_{K \times 1} + Z_{K \times 1} = {XW}_{kl} h + Z$ (19) where X = diag {X₀, X₁, …, X_K-1} denotes a diagonal transmitted symbol matrix and Z is complex white Gaussian noise with power $σ_{z}^{2}$ . We rewrite Y as shown in the Equation (20). $\begin{matrix} Y_{k} = \sum_{n = 0}^{N - 1} y_{n} e^{- j 2 π nk / K} \\ = \sum_{n = 0}^{N - 1} (\sum_{k^{'} = 0}^{K - 1} X_{k^{'}} H_{k^{'}} e^{j 2 π {nk}^{'} / K} + z_{n}) e^{- j 2 π nk / K} \end{matrix} = H_{k} X_{k} + \sum_{n = 0}^{N - 1} \sum_{\begin{array}{l} k' = 0 \\ k' \neq k \end{array}}^{K - 1} X_{k'} H_{k' e} j 2 π n (k' - k) / k + z_{k}$ (20)

The vector parameter H = [|H₀|, |, H₁|, …, |H_k-1|] ^T is estimated assuming that the estimator $| \hat{H} |$ is unbiased. The log-likelihood function can be found from the Equation (21). $+ \frac{1}{2 α_{z}^{2}} (\sum_{k = 0}^{k - 1} {(\begin{array}{l} Y_{k} - X_{k} H_{K} - \\ \sum_{n = 0}^{N - 1} \sum_{\begin{array}{l} k' = 0 \\ k' \neq k \end{array}}^{K - 1} (H_{k'} X_{k'}) e^{j 2 π n (k' - k) / K} \end{array})}^{2})$ (21)

It was shown in [39] that if we consider α = g (H) = |H|, where g is a K-dimensional function, then the CRLB is computed by: $C_{\hat{α}} \geq \frac{\partial g (H)}{\partial H} I^{- 1} (H) \frac{\partial g (H)^{T}}{\partial H}$ (22) where I (H) is the K×K Fisher information matrix (details of calculation is derived in Appendix). As shown in Equation (34) in Appendix, the estimation variance due to each subcarrier will be presented as:

$\begin{matrix} var (| {\hat{H}}_{i} |) & = & \frac{σ_{z}^{2}}{{(\sum_{n = 0}^{N - 1} x_{n} e^{\frac{- j 2 π ni}{K}})}^{2}}, \\ i = 0, 1, \dots, K - 1 \end{matrix}$ (23)

4 Simulation results

In our simulation, a synthetic moving target included different scatterer points is constructed with equal arbitrary reflectivity coefficient. These scatterer points are located at the ranges R_i from the radar. The target frequency response is estimated based on K = 16 and 32 subcarriers with uniformly random phases and amplitudes. To make the analysis feasible, it is assumed that the sampling rate used in this study is f_s = 4K/t_c.

In this section, the proposed scheme is investigated regarding target parameter estimation. To do this, a moving extended target with L = 100 scatterer points that is located at the range of 2km from the radar is considered. To show the target detection performance of the proposed structure, a heavy clutter and additive white Gaussian noise are considered in the channel. The OFDM radar parameters have been chosen to realize the overall system asfollows:

Carrier frequency f_c = 9 GHz

Pulse repetition interval T_PRI = 1 ms

Chip width t_c = 250 ns

Relative Doppler shift f_d = 300 Hz

Subcarrier spacing Δf = 4MHz

Number of scatterer points L = 100

Number of coherent pulses N = 32

Number of subcarriers K = 16 & 32

Sampling frequency f_s = 4K/t_c = 256 & 512 MHz

Available bandwidth B = 60 & 120MHz

Total transmitted power P_t = 10 dBW

Signal to noise Ratio SNR = 10dB

The clutter is assumed to have a Gaussian shape with power spectrum around zero frequency. Figure 2(a) shows the sum of the power spectrum of target, clutter, and noise. An MTI filter with 11 taps is used to suppress clutter and estimate target the Doppler shift. In Fig. 2(b), a sharp peak can be seen in the output of the designed MTI filter which indicates the Doppler frequency of the target. It can be foreseen that using this procedure we will be able to estimate a favorite Doppler shift.

Fig.2

Simulation of a moving extended target with 100 scatterer points located at range 2km in SNR = 10 dB. a) Received signal in the time domain, b) Final output in the time domain, c) Spectrum of target and clutter, d) Actual and estimated extended target frequency response.

Figure 2(c) depicts the final output of the pulse integration block or the input of the target spectrum estimator. This output is fed to the CFAR detector then the range cell of the target is extracted promisingly. Subsequently, the information gathered will be given to the estimator; consequently, there is a result showing an output of the estimator which is equal to the target frequency response with determined radar parameters. Figure 2(d) indicates the proper estimation of the target frequency response according to the defined parameters.

4.1 Investigation of fading effect on the estimation performance

To investigate the fading effect, it is assumed that the fading coefficients in our formulation aresamples of a uniformly distribution in the interval (0, 1). Figure 3 depicts the fading effect of multipath on the target estimation procedure. As indicated in this figure, the fading occurred in two subcarrier frequencies. Figure 3(a) is actual and the estimated target frequency response without considering any fading. While in Fig. 3(b), the fading effect has been applied to two subcarriers. As we expect, the proposed method is not able to correctly estimate the target frequency response in those subcarriers in which fading effect has appeared.

Fig.3

Actual and estimated target response a) no fading, b) with fading in two subcarriers (SNR = 20 dB and K = 32).

Figures 4 and 5 indicate the estimated and actual target frequency response for the proposed and the AML estimator, respectively, which is obtained by averaging over 1000 simulations each at a particular value of the received signal-to-noise ratio (SNR). The proposed target spectrum estimation is implemented with K = 16 subcarriers and various SNR values ranging from 5 dB to 20 dB. It can be found from Fig. 4 that the target frequency response will provide accurate estimation, especially in high SNRs. It is worth mentioning that this approach provides the possibility to estimate the high resolution target frequency response in which there is a limitation of using subcarriers in practice.

Fig.4

Comparison of the proposed estimated target response and actual target response (with K = 16 subcarriers and L = 50 scatterer points) in different SNRs. a) SNR = 10 dB. b) SNR = 20 dB. c) SNR = 30 dB. d) SNR = 40 dB.

Fig.5

Comparison of the AML estimated target response and actual target response (with K = 16 subcarriers and 50 scatterer points) in different SNRs. SNR = 10 dB. b) SNR = 20 dB. c) SNR = 30 dB. d) SNR = 40 dB.

By comparing Figs. 4 and 5, it can be seen that in SNRs lower than 20 dB, the performance of the proposed estimator is superior compared to the AML estimator, while in SNRs more than 20 dB the situation is reversed and the performance of the AML estimator outperforms the proposed method. It is worth mentioning that the AML approach provides the possibility to estimate the target frequency response with high accuracy in high SNRs while the proposed method could not achieve high accuracy because of quantization error due to the discrete estimation of target amplitude.

The obtained results imply that even in scenarios with a remarkable number of scatterer points and low SNRs, the MSE value is very feasible. Thus, the proposed estimation method can be used as a robust option in high-resolution applications such as cognitive radar in which the target consists of so many scatterer points.

4.2 Evaluating the overall performance of the estimator

The MSE criterion has been utilized to evaluate the performance of the proposed target estimation approach and a Cramer-Rao Lower bound has been derived to estimate the target response to set a lower bound on the variance of the proposed estimator. Simulation results show that the overall performance of the proposed approach is dependent on the number of scatterer points. The performance of the proposed method is compared with the most common estimation method, maximum likelihood estimator, in the same condition. It was shown that having a block diagonal structure, an AML estimator for scattering matrix X was yielded which is definedas [37]: ${\hat{h}}_{AML} = vecb (\hat{H}) = (Γ^{H} Γ) vec (T^{- \frac{1}{2}} Y_{OFDM} Π_{Φ})$ (24) where $\begin{matrix} Γ = [Φ_{1}^{T} \otimes s_{1} Φ_{2}^{T} \otimes s_{2} \dots Φ_{L}^{T} \otimes s_{L}] \\ T = Y_{OFDM} Π_{Φ} Y_{OFDM}^{H} \\ Π_{Φ}^{s} = I_{N} - Π_{Φ} \\ Π_{Φ} = Φ (ν)^{H} {(Φ (ν) Φ (ν)^{H})}^{- 1} Φ (ν) \\ S = [s_{1} s_{2} \dots s_{L}] \\ A = diag (a) \\ Φ (ν) = [Φ_{1}^{T} Φ_{2}^{T} \dots Φ_{L}^{T}] \end{matrix}$ (25)

4.3 MSE performance analysis

Since the estimation of each target spectrum component relative to each subcarrier frequency is affected by the error E_n, we have: ${\hat{H}}_{n} = H_{n} + E_{n}$ (26)

The MSE between predicted and measured values of the target response can be computed as shown in Equation (27). $MSE = E [{∥ H_{n} - {\hat{H}}_{n} ∥}^{2}]$ (27)

Figure 6 shows the MSE performance of the proposed estimator versus the number of scatterer points. The target frequency spectrum is estimated based on K = 32 subcarriers with SNR = 0 dB and the signal-to-clutter ratio (SCR) is equal to 6 dB. As indicated in this figure, as the number of scatterer points grows, the MSE performance decreases.

Fig.6

MSE performance Vs. the number of scatterer points SNR = 0 dB and SCR = 6 dB).

Figure 7 depicts the MSE performance of the proposed method and the AML estimator versus SNR. Figure 7(a) shows the MSE for K = 16 subcarriers. As seen from this figure, the performance of the proposed estimator is better than the AML estimator for SNRs less than 15 dB and when the SNR gets higher than 15 dB, the AML estimator outperforms our method. Figure 7(b) shows the MSE for K = 32 subcarriers. It can be seen that the performance of the proposed estimator is superior as compared to the AML estimator even in the case of high SNRs (between 15 to 25 dB). It is worth mentioning that the slope of the AML estimator MSE curve is more than the proposed method. This is because the proposed scheme has a discrete form and it will not be an unbiased estimator. Therefore, the slope of the MSE curve for the proposed estimator is lower than the AMLestimator.

Fig.7

MSE Vs. SNR for a) K = 16, b) K = 32 subcarriers.

The performance of the proposed approach depends on the number of subcarriers and scatterer points. The proposed structure for target spectrum estimation is simple and can be easily applied in practical implementation. In practice, choosing the number of subcarriers is often hard, so we have to make some compromises in the case of stochastic extended targets to enhance the performance of the cognitive radar system. The estimated target response then could be exploited by the radar transmitter to enhance the overall throughput of the cognitive radar system.

Figure 8 demonstrates the variance of the proposed and AML estimator with obtained CRLB. The implementation is done via K = 16 subcarriers and 1000 simulations were run for each SNR value. It can be found that the proposed estimator outperforms the AML estimator and achieves errors close to CRLB at high SNRs.

Fig.8

Variance of the proposed and the AML estimator Vs. CRLB (L = 100).

5 Conclusion

In this paper, a novel intelligent frequency-based target spectrum estimation method for cognitive OFDM radar has been proposed. The compression of OFDM signals has been generalized based on DFT for signal model description and a new architecture has been proposed to estimate the spectral features of extended targets. The overall performance has been evaluated by comparing the variance of the proposed estimator with the AML estimator and the obtained CRLB. Though the use of OFDM allows one to estimate the target spectrum of the extended target even in low SNRS, however, it results in worse detection performance compared to the AML estimator in high SNRs. Furthermore, the proposed approach has an outstanding novelty for estimating the spectrum of extended targets in heavy noise and clutter environments. In the future, we will investigate the problem of resolving fading effect, focus on the optimal waveform design based on the presented target estimation method and apply the proposed approach in radar imaging to improve the quality of image formation process [41 –45].

Footnotes

Appendix (Calculation details of the CRLB)

The K×K Fisher information matrix is given as: (28) $I (θ) = {[\begin{matrix} - E [\frac{\partial^{2} ln p (Y; H)}{\partial H_{0}^{2}}] & \dots & - E [\frac{\partial^{2} ln p (Y; H)}{\partial H_{0} \partial H_{K - 1}}] \\ ⋮ & ⋱ & ⋮ \\ - E [\frac{\partial^{2} ln p (Y; H)}{\partial H_{K - 1} \partial H_{0}}] & \dots & - E [\frac{\partial^{2} ln p (Y; H)}{\partial H_{K - 1}^{2}}] \end{matrix}]}_{K \times K}$

The first derivation with respect to H₀ is:

(29) $\begin{matrix} \frac{\partial ln (P (Y; H))}{\partial H_{0}} \\ = \frac{1}{α_{z}^{2}} (\begin{array}{l} Y_{0} - X_{0} H_{0} - \\ \sum_{n = 0}^{N - 1} \sum_{\begin{array}{l} k' = 0 \\ k' \neq 0 \end{array}}^{K - 1} (H_{k'} X_{k'}) e j 2 π n (k' - k) k \end{array}) (x_{0}) \end{matrix}$

And the second differentiation results in: (30) $\frac{\partial^{2} ln (P (Y; H))}{\partial H_{0}^{2}} = - \frac{X_{0}^{2}}{σ_{z}^{2}} = \frac{- 1}{σ_{z}^{2}} {(\sum_{n = 0}^{N - 1} x_{n})}^{2}$

Taking the expected value yields:

(31) $\begin{matrix} E (\frac{\partial^{2} ln (P (Y; H))}{\partial H_{0}^{2}}) & = & \frac{- 1}{σ_{z}^{2}} E {(\sum_{n = 0}^{N - 1} x_{n})}^{2} \\ = & \frac{- 1}{σ_{z}^{2}} {(\sum_{n = 0}^{N - 1} x_{n})}^{2} \end{matrix}$

Using the same method, it can be easily proved that the elements of the Fisher information matrix for i ≠ j will be equal to zero. Therefore, the Fisher information matrix can be found from (32). (32) $I (H) = {[\begin{matrix} \frac{- 1}{σ_{z}^{2}} {(\sum_{n = 0}^{N - 1} x_{n})}^{2} & 0 & \dots & 0 \\ 0 & \frac{- 1}{σ_{z}^{2}} {(\sum_{n = 0}^{N - 1} x_{n} e^{\frac{- j 2 π n}{K}})}^{2} & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & \frac{- 1}{σ_{z}^{2}} {(\sum_{n = 0}^{N - 1} x_{n} e^{\frac{- j 2 π n (K - 1)}{K}})}^{2} \end{matrix}]}_{K \times K}$

Recall that $\frac{\partial g}{\partial H}$ and I^-1 (H) are both diagonal matrices, thus it follows from (22) that the CRLB computed as shown in Equation (33). (33) $var (| {\hat{H}}_{i} |) = \frac{{(\frac{\partial g_{i}}{\partial H_{i}})}^{2}}{- E [\frac{\partial^{2} ln p (Y; H)}{\partial H_{i}^{2}}]} i = 0, 1, \dots, K - 1$

It can be easily proved that the first derivative of function g (H) = |H| with respect to complex vector H is 1 and ultimately we have: (34) $var (| {\hat{H}}_{i} |) = \frac{σ_{z}^{2}}{{(\sum_{n = 0}^{N - 1} x_{n} e^{\frac{- j 2 π ni}{K}})}^{2}} i = 0, 1, \dots, K - 1$

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