LFM-based waveform design for cognitive MIMO radar with constrained bandwidth
- Shuangling Wang^{1},
- Qian He^{1}Email author and
- Zishu He^{1}
https://doi.org/10.1186/1687-6180-2014-89
© Wang et al.; licensee Springer. 2014
Received: 2 February 2014
Accepted: 24 May 2014
Published: 12 June 2014
Abstract
Waveform design is studied for a cognitive multiple-input multiple-output (MIMO) radar system faced with a combination of additive Gaussian noise and signal dependent clutter. The linear frequency modulation (LFM) signals are employed as transmitted waveforms. Based on the sensed statistics of the target and clutter-plus-noise, assuming the LFM waveforms transmitted at different transmitters can have different starting frequencies and bandwidths, these waveform parameters are designed to maximize the signal-to-clutter-plus-noise ratio at the receiver of the cognitive MIMO radar system. The constraints of the allowable range of operating frequency and total transmit energy are considered. We show that in the tested examples, the designed waveforms are nonorthogonal which leads to superior performance compared with that of the frequency spread LFM waveforms commonly used in the traditional MIMO radar systems.
Keywords
1 Introduction
The advantages of multiple-input multiple-output (MIMO) radar have drawn considerable attention in the last decade [1–7]. MIMO radar systems employ multiple antennas on both the transmit and receive sides. The antennas can be either co-located or widely separated. Geometry gains can be obtained for the former since the antennas are located in several different directions with respect to a target, while waveform gains can be produced for the latter by sending different waveforms with different antennas.
Waveform design is a key issue in radar signal processing. The transmit waveforms of MIMO radar are usually optimized for specific goals, such as improving the signal-to-clutter-plus-noise ratio (SCNR) [8], increasing the resolution in the spatial and temporal domains, enhancing the detection performance [5], reducing the estimation error when approximating a desired beam-pattern [4], or maximizing the mutual information (MI) between the random target impulse response and the reflected waveforms [9].
The concept of cognitive radar (CR) was proposed in [10] for optimizing the performance of a radar system faced with interference and the constraint of limited resources. The CR system can intelligently learn the state of the environment and store the information in the database. The stored information can be used as an available prior knowledge for the designs of radar systems and transmit waveforms, which is helpful for improving the performance of target detection and parameter estimation. There have been many researches on waveform design for CR systems [11–14]. In [11], the transmit signals are designed by minimizing the mean-square error of the estimate of the target reflection coefficient.
In [12], the waveform is designed to minimize the average ambiguity of the transmitted signal over certain range Doppler bins. In [13, 14], the waveform is optimized by maximizing the signal-to-interference-plus-noise ratio for CR radar systems.
The cognitive technique has been introduced to MIMO radar systems [15–18] to enhance the robustness and adaptability. During the learning process of a cognitive MIMO radar system, the information of the environment, such as the prior knowledge of clutter and target impulse responses and noise, are collected by multiple receive antennas, which are transferred to multiple transmit antennas through a feedback mechanism for adjusting system parameters. In [15], the authors present an adaptive waveform design method to improve the target recognition performance of the cognitive MIMO radar. In [16], artificial intelligence algorithm is employed to improve the robustness of target detection for the cognitive MIMO radar system. In [17], the authors optimize the waveforms based on the Bayesian Cramer-Rao bound and the Reuven-Messer bound for cognitive MIMO radar systems. In [18], a waveform optimization approach is provided for cognitive MIMO radar based on the MI between the target impulse response and the received echoes and the MI between successive backscatter signals.
As a very common waveform for radar system, the linear frequency modulation (LFM) signal can provide advantages of high-resolution, anti-jamming, far detecting distance, etc. [19–23]. Moreover, the LFM signal can be conveniently generated and it has constant modulus. For MIMO radar, the frequency spread (FS) LFM signals are usually employed as a set of orthogonal signals for transmission [24, 25]. The orthogonality of the FS LFM signals can be obtained by increasing the frequency offset between two waveforms transmitted by the adjective antennas [26]. However, the maximum allowable operating bandwidth for radar system is often limited as the frequency band source becomes more and more crowded with the development of the communication and radar applications. Therefore, it is important to know how one can improve the radar performance through waveform design when the range of operating frequency is constrained.
In this paper, the waveforms are designed for the cognitive MIMO radar system. Since the detection probability is a nondecreasing function of the SCNR under the log-likelihood ratio test [27], we employ the SCNR maximization as the objective of the optimization problem to improve the detection performance. LFM signals are employed as the transmit waveforms. Unlike the FS LFM signals usually adopted in MIMO radars [24, 25], where each of the transmit LFM signals has identical bandwidth and transmit energy and equally spaced starting frequencies, we propose to construct the transmit waveforms as a set of LFM signals whose starting frequencies, bandwidths, and energies can be different and are to be optimized. The prior information about the target and clutter obtained by the cognitive process is used for the waveform optimization. The constraints of the total transmit energy and the allowable range of operating frequency impose restrictions on our optimization problem.
The rest of the paper is organized as follows. In Section 2, the signal model of the cognitive MIMO radar is introduced. In Section 3, the LFM-based waveform design for limited maximum allowable frequency band and total transmit energy is presented, and the algorithm for solving the optimization problem is given. In Section 4, we show the superior SCNR performance of our designed waveforms over the FS LFM signals through numerical examples. The effects of the number of transmit antennas are also analyzed. Finally conclusions are drawn in Section 5.
Notation: Throughout this paper, we use superscripts (·) ^{ H }, (·) ^{∗}, and (·) ^{T} to denote the complex conjugate transpose, conjugate, and transpose of a matrix, respectively. The ⌊·⌋ denotes the operation of rounding down the value to the nearest integer. The (i mod j) represents the remainder of division of i by j. We use $\mathbb{E}\left\{\xb7\right\}$for expectation with respect to all the random variables within the brackets. The symbol $\u229b$stands for the convolution operator and ⊗ for the Kronecker product operator. We let diag{·} denotes diagonal matrix. Finally, (A)_{ ij }denotes the ij th entry of A , and I_{ N }denotes the identity matrix of size N × N.
2 Signal model
The H_{ t }and H_{ c }in (2) are LN_{ R }× M N matrices, the expressions of which are given in Appendix 1.
As a cognitive MIMO radar system has the ability of learning, the prior knowledge of the environment state can be obtained from previous measurements. Using a specific environment database which contains the statistics of the target and clutter impulse responses and noise, the statistics of the target return x_{ t }and the clutter return x_{ t }can be derived. Next, we discuss the statistics of the target impulse response h_{ t }[n], the clutter impulse response h_{ c }[n], and the noise z.
2.1 Statistics of target return
From (2), the statistic of the target return vector x_{ t }is determined by the statistic of the target impulse response vector h_{ t }, as described in the following lemma.
Lemma 1
where $a=\left(N-{n}_{\tau}+\u230a(i-1)/L\u230b-1\right)\mathit{\text{ML}}+(m-1)L+\stackrel{~}{l}$, $b=\left(N-{n}_{\tau}^{\prime}+\u230a(j-1)/L\u230b-1\right)\mathit{\text{ML}}+({m}^{\prime}-1)L+{\stackrel{~}{l}}^{\prime}$, $\stackrel{~}{l}=\left(\right(i-1\left)\phantom{\rule{2.83795pt}{0ex}}\phantom{\rule{0.2em}{0ex}}\text{mod}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{2.83795pt}{0ex}}L\right)+1$, and ${\stackrel{~}{l}}^{\prime}=\left((j-1)\phantom{\rule{2.83795pt}{0ex}}\phantom{\rule{0.2em}{0ex}}\text{mod}\phantom{\rule{0.2em}{0ex}}\phantom{\rule{2.83795pt}{0ex}}L\right)+1$.
Proof of Lemma
See (S Wang, Q He, Z He, RS Blum, Waveform design for MIMO over-the-horizon radar detection, submitted).
can be computed using Lemma 1.
2.2 Statistics of clutter return
From (2), the statistic of the clutter return vector x_{ c }is determined by the statistic of the clutter impulse response vector h_{ c }, as described in Lemma 1.
can be computed using Lemma 1.
2.3 Statistics of noise
The noise term z = (z_{1} [0],…,z_{ L }[0],…,z_{1} [N_{ R }-1],…,z_{ L }[N_{ R }-1])^{T} is assumed to obey complex Gaussian distribution with mean zero and covariance matrix ${\mathbf{R}}_{z}=\mathbb{E}\left\{\mathbf{z}{\mathbf{z}}^{H}\right\}={\sigma}_{z}^{2}{\mathbf{I}}_{{\mathit{\text{LN}}}_{R}}$, which is assumed to be independent of the target and clutter returns.
3 Waveform design with constrained bandwidth
In this section, waveform design for the SCNR maximization problem is introduced for cognitive MIMO radar systems. The waveforms are constructed as LFM signals where the starting frequencies and bandwidths for each of the LFM signals will be optimized. The optimization problem with the constraints of the allowable range of operating frequency and total transmit energy is presented. The method to solve the optimization problem is given subsequently.
3.1 Waveform design for SCNR maximization
where E_{0} is the total transmit energy. The objective function in (12) is not a convex function, so that the convex optimization approaches do not work for this problem. Instead, we adopt the iterative approach proposed in [28] to solve the optimization problem in (12). We first fix the waveform p to optimize the receiver impulse response h_{ r }, and then fix h_{ r }to optimized p. These two steps are executed iteratively until the stopping criterion is met.
3.2 Optimization with fixed waveform
3.3 Optimization with fixed h_{ r }
where the expressions of $\mathbb{E}\left\{{\mathbf{H}}_{t}^{H}{\mathbf{h}}_{r}{\mathbf{h}}_{r}^{H}{\mathbf{H}}_{t}\right\}$ and are also determined by the statistics of h_{ t }in (6) and h_{ c }in (9), as described in the following lemma.
Lemma 2
where $d=\left(N+{n}^{\prime}-\lfloor (j-1)/M\rfloor -1\right)\mathit{\text{ML}}+\left({l}^{\prime}-1\right)M+{\stackrel{~}{m}}^{\prime}$, $e=\left(N+n-\lfloor (i-1)/M\rfloor -1\right)\mathit{\text{ML}}+\left(l-1\right)M+\stackrel{~}{m}$, $\stackrel{~}{m}=\left(\right(i-1\left)\phantom{\rule{2.83795pt}{0ex}}\text{mod}\phantom{\rule{2.83795pt}{0ex}}M\right)+1$, and ${\stackrel{~}{m}}^{\prime}=\left(\right(j-1\left)\phantom{\rule{2.83795pt}{0ex}}\text{mod}\phantom{\rule{2.83795pt}{0ex}}M\right)+1$.
Proof of Lemma
See (S Wang, Q He, Z He, RS Blum, Waveform design for MIMO over-the-horizon radar detection, submitted to IEEE Transactions on Aerospace and Electronic Systems).
As the problem in (18) is highly nonlinear, it is solved numerically using the interior point method. The optimized solution is denoted as p_{⋆}.
3.4 Summary of the iterative method
The above discussed iterative method that solves the problem in (12) can be summarized in the following algorithm:
Then, both of the optimized waveform parameter vector p_{opt} and receiving impulse response vector h_{r,opt} are achieved. The optimized waveform s_{opt} can be obtained by plugging p_{opt} into (11).
4 Simulations
where ${\sigma}_{S}^{2}=40$ and ρ_{ S }= 0.9 denotes one-lag correlation coefficient. The temporal correlation matrix C_{ T } is considered by the conventional time-varying autoregressive (TVAR) [31, 32] modeling (see [33] for details).
4.1 Optimal waveform design
Starting frequencies, bandwidths, and transmit energies in the waveform parameter vector p for optimized LFM signals
s _{ m } [ n] | f_{ m }/ Hz | b_{ m }/ Hz | E _{ m } |
---|---|---|---|
m = 1 | 170.35 | 1,684.6 | 0.6625 |
m = 2 | 209.00 | 1,621.7 | 2.4429 |
m = 3 | 255.81 | 1,537.0 | 0.7964 |
m = 4 | 1,100.6 | 556.60 | 0.1000 |
where Δ f is the frequency offset between the waveforms sent from two adjacent transmit antennas which is fixed to k/T to attain approximate orthogonality [26], where k is a positive integer. When the operating frequency is constrained in $\mathcal{F}=\phantom{\rule{0.3em}{0ex}}[0,B]$, the B_{ u } should satisfy B_{ u }+ (M-1)Δ f ≤ B. In this example, we let Δ f = 2/T, B_{ u } = B - (M-1) Δ f, and the value of the other parameters are the same as those in the previous example. Obviously, the optimized LFM signals in Figure 3 and the FS LFM signals in Figure 4 are very different. Unlike the FS LFM signals, where each waveform has identical bandwidth and the frequency offset between adjacent waveforms are identical, the optimized LFM signals may have different bandwidths and the offsets between different pairs of adjacent waveforms may be different, providing more degrees of freedom, which may lead to superior system performance.
4.2 SCNR performance
5 Conclusions
In this paper, the waveforms are proposed to be constructed by a group of LFM signals with undetermined starting frequencies, bandwidths, and transmit energies. The waveform parameters and the receiver impulse response are jointly designed by maximizing the SCNR performance of a cognitive MIMO radar system under the constraints of allowable range of operating frequency and total transmit energy. The algorithm for solving the waveform optimization problem was presented. We showed through numerical examples that the systems using the proposed waveforms have superior SCNR performance than the systems using the FS LFM signals.
Appendix 1
Convolution matrices
where s is the overall waveform vector as defined in (3) and H_{ ε }denotes the convolution matrix for target or clutter, in which h_{ε,ml}[n] is defined in (1).
Declarations
Acknowledgements
This work was supported by the National Nature Science Foundation of China under Grants 61032010 and 61102142, the International Science and Technology Cooperation and Exchange Research Program of Sichuan Province under Grant 2013HH0006, and by the Fundamental Research Funds for the Central Universities under Grant ZYGX2013J015.
Authors’ Affiliations
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