An approach to the modulation recognition of MIMO radar signals

Multiple-input multiple-output (MIMO) radar is a new radar system and draws more and more attentions in recent years. Along with the development of MIMO radar, the MIMO radar countermeasure is brought into being. Since the modulation recognition is of great significance in the electronic reconnaissance, this article presents a modulation recognition method for the signals of the emerging MIMO radar. Signals of interest are classified into three categories based on instantaneous autocorrelation spectrum analysis first. Then non-coding MIMO radar signals are discriminated by spectrum analysis, and coding MIMO radar signals are recognized by source number estimation algorithm. Meanwhile, the sub-carrier numbers of some kinds of MIMO radar signals are estimated. Simulation results verify the effectiveness of the method and the overall correct recognition rate is over 90% when the value of SNR is above 0 dB. Keyword: Modulation recognition, MIMO radar signal, Instantaneous autocorrelation spectrum, Source number estimation, Sub-carrier number


Introduction
The concept of multiple-input multiple-output (MIMO) radar, which comes from communication system, has drawn considerable attention in recent years from both researchers and practitioners [1]. It builds a bridge between the research of radar and communication. MIMO radar is generally divided into two categories, one is statistical MIMO radar with widely separated antennas, and the other is coherent MIMO radar with co-located antennas. In both categories of MIMO radar system, multiple transmit antennas are employed to emit specific waveforms and multiple receive antennas process the reflected signals jointly. MIMO radar offers quite a lot of advantages, such as more degrees of freedom, higher resolution, and sensitivity and better parameter identifiability [2][3][4]. These advantages mostly result from waveform diversity. Due to the waveform diversity, intercepted signals in reconnaissance receiver are multi-carrier signals. Accordingly, signal detection, parameter estimation, and modulation recognition are vastly different from single-carrier (SC) signals adopted by conventional radars. As a result, it poses an emerging and powerful challenge in electronic countermeasures.
In order to occupy an advantageous position in the future electronic warfare, we need to investigate the feature of MIMO radar and study its electronic countermeasures. This article discusses the modulation recognition of MIMO radar signals, which is of great significant in the electronic reconnaissance. Since there are some actual difficulties in engineering practice for statistical MIMO radar, hereinafter we focus on coherent MIMO radar.
The reconnaissance technology of MIMO radar is rarely studied in the published literatures. Liang [5] and Xing et al. [6] discussed about the electronic reconnaissance technology of MIMO radar at system design and conceptual angle. Tang et al. [7] provided a new reconnaissance technology for MIMO radar. The aim of the article is to discriminate whether it is MIMO radar by the number of orthogonal waveforms. However, the location of suspicious radar is as known information, which is usually unknown in actual environment. Besides, Chen et al. [8] and Hassan et al. [9] were about the modulation identification of MIMO system, which is adopted in the wireless communication field. In [8], the combination of second-and fourthorder of cumulants was used as the feature parameters, which were utilized to discriminate the orthogonal frequency division multiplexing signals from the SC modulations. In [9], high-order statistics and neural networks were employed to identify the modulation type of MIMO system with and without channel state information.
Based on the instantaneous autocorrelation spectrum of received signals, an approach to the modulation recognition of MIMO radar signals is proposed in this article. For conventional radar, SC signal is often adopted, such as monopulse (MP) signal, linear frequency modulation (LFM) signal, phase-coded (PC) signal, and frequency-coded (FC) signal. For MIMO radar, four basic modulation types [10][11][12] are involved in this article: MP-MIMO (orthogonal MP signal in MIMO radar), LFM-MIMO (orthogonal LFM signal in MIMO radar), PC-MIMO (orthogonal PC signal in MIMO radar), and FC-MIMO (orthogonal FC signal in MIMO radar). Here, we need to discriminate MIMO radar signal from conventional radar signal and recognize the modulation type of MIMO radar signal.
The remainder of this article is organized as follows. In Section 2, four basic emitting signal models of MIMO radar are given. The recognition method is introduced in Section 3. Instantaneous autocorrelation spectrum analysis, frequency spectrum analysis, and source number estimation algorithm (SNEA) are involved in this section. Simulation results are given in Section 4 and conclusions are drawn in Section 5.

Signal models
Assuming that the transmitting arrays of MIMO radar are uniform linear arrays (ULA), in the reconnaissance receiver, the four basic received MIMO signal models can be expressed as follows [10][11][12].
where exp() denotes the exponential function, M is the number of sub-carriers of MIMO radar signal, f 0 is the carrier frequency, f p = 1/T and △ϕ denote the frequency interval and phase difference between adjacent sub-carriers, respectively, u denotes the chirp rate, I is the code length of coding signal, T s = 1/f s is the sampling interval, f s is the sampling rate, . . . ; n I À 1 ð Þ : 1 T 1 g denotes the frequency of sub-pulse i of the mth component, L is the distinct phase number in PC-MIMO, T and T 1 represent the pulse and subpulse width, respectively, g(t), 0 ≤ t ≤ T 1 is the envelope function. Particularly, to ensure the orthogonality of components, ∅ m (i) and f m (i) are usually obtained by intelligent algorithm, such as genetic algorithm and simulated annealing algorithm.

Instantaneous autocorrelation spectrum analyses
The signal pulse parameters can be extracted from instantaneous autocorrelation features. Since the frequency characteristics of different modulation types are diverse, instantaneous autocorrelation function can be utilized to the modulation recognition of MIMO radar signals. The instantaneous autocorrelation function is defined as 3.1.1. S 1 /(S 0 ,S 2 ) selection (a) S 1 set signals analysis The instantaneous autocorrelation of LFM-MIMO signal is where k = m − l, |k| ≤ M − 1. If k = 0, it denotes the signal term. Otherwise, it denotes the cross terms.
For convenience, let Δn = 1, then we can get Accordingly, the signal term is get as The cross term is obtained as From the above, the moduli of signal term and cross term are respectively. If |k| = 1, it is the biggest cross term. Figure 1b shows the instantaneous autocorrelation spectrum of LFM-MIMO signal. The simulated MIMO radar has a ULA comprising four transmitting antennas with half-wavelength spacing between adjacent antennas. The simulated SNR is 5 dB. By the way, for the other parts of Figure 1, the simulation conditions are the same. For Figure 1b, the maximum peak results from signal term while other peaks are from cross terms. As we can see, the biggest cross term appears as the second peak in the autocorrelation spectrum. To extract signal features, we define the feature parameter: the ratio of second peak and maximum peak, which can be expressed as follows: If u = 0, Equation (7) is the instantaneous autocorrelation function of MP-MIMO. Since u is not related to (10), MP-MIMO and LFM-MIMO have the equivalent values of R Amplitute . Comparing Figure 1a with b, the fact is verified since the values of every peak are the same. Given the value ranges of parameters in Equation (10), the minimum value of R Amplitute can be obtained.
In addition, by searching the peaks of instantaneous autocorrelation spectrum of non-coding MIMO radar signal, the number of sub-carriers can be obtained. For Figure 1, the simulated MIMO radar has four transmitting antennas. That is, the sub-carrier number is four. Seven peaks can be seen in Figure 1a,b. Six of them, resulted from cross terms in (9), are symmetrical. Together with the single frequency at center, the distinct frequency number is four. That is, the sub-carrier number is four for the received MIMO radar signal, which is equal to the actual sub-carrier number of non-coding MIMO radar signal. For the instantaneous autocorrelation spectrum of non-coding MIMO radar signal, supposing that the number of peaks is a, the sub-carrier number is (a + 1)/2.
Meanwhile, some signal modulation parameters of non-coding MIMO radar signal can be got from the instantaneous autocorrelation spectrum. For example, the chirp rate u and the frequency interval between adjacent sub-carriers f p . They can be estimated from the location of peaks.

(b)S 2 set signals analysis
For PC-MIMO and FC-MIMO, substituting Equations (3) and (4) into Equation (5), respectively, the instantaneous autocorrelation functions are as follows: where the superscript of ∅ and f denote the location of the code. For Equation (11), since φ nþ1 m À φ n l ¼ const in a subpulse, r PC-MIMO [n,1] is constant in a sub-pulse. If n is the last sampling points of a sub-pulse, then n + 1 is in the next sub-pulse. So, the value of φ nþ1 m À φ n l À Á suddenly changes and mutations appear in sub-pulses junctions. The constant values result in zero frequency. The mutations will bring about some low frequencies, the amplitudes of which are very small compared with the zero frequency. That is, only one obvious peak appears at the zero frequency in the frequency spectrum of PC-MIMO.
For Equation (12) The other sub-pulses as well. That is, the energies of the cross terms almost cannot be superimposed. Only one obvious peak appears at the zero frequency in the frequency spectrum of FC-MIMO.
The instantaneous autocorrelation spectrums of PC-MIMO and FC-MIMO signals are shown in Figure 1c,d, respectively. As can be seen, most of the energies gather at the zero frequency while some appear at low frequencies for both PC-MIMO and FC-MIMO, which verify the above analysis. As a result, values of R Amplitute for coding MIMO radar signals are very small, which are less than the value of non-coding MIMO signals. The same to the SC signals, which can be seen in Figure 1e,f.
Consequently, taking the ratio of second peak and maximum peak as the feature parameter, S 1 signals are discriminated from S 0 and S 2 set, which is expressed as follows: & where γ is the value of the threshold.
Assuming that 4 ≤ M ≤ 50, 0.005 ≤ f p /fs ≤ 0.02, which satisfy most of the signal environment, the minimum value of Equation (10) is 0.7502. That is, for non-coding MIMO radar signal, the value of R is always greater than 0.7502. Obviously, for S 0 or S 2 set signals, the value of R is smaller than 0.7502. As a consequence, 0.75, the value of the threshold, permits classifying between S 1 set and other modulation types.
For SC signals, r SC [n, 0] = 1. As a result, the frequency of r SC [n, 0] is zero.
For coding MIMO radar signals, according to (3) and (4), we can get the r[n,0] of PC-MIMO and FC-MIMO signals, which can be expressed as follows: From Equations (14) and (15), we can see that there are some mutations in the correlation function. As a result, they will bring about some low-frequency components. Since the frequency of r SC [n, 0] is zero, this allows us to discriminate SC signals from coding MIMO radar signals. Employing Fourier transform on the r[n,0], which is obtained from S 0 and S 2 set signals, we can obtain its positive frequency spectrum.
where N DFT denotes the points of discrete Fourier transform (DFT). Then we divide the positive frequency into two segments. One is the low-frequency segment; the other is the rest part.
After that, we define the feature parameter R mean as where Mean() is the mean function.  (14) and (15), there are a few mutations in the r[n,0]. Hence, they will lead to some low-frequency components, the values of ratio R mean of coding MIMO radar signals are always greater than 1. This permits us to discriminate SC signals from coding MIMO radar signals. Consequently, setting proper thresholds ς, S 2 set signals are discriminated from S 0 set signals, which is expressed as follows:

MP-MIMO/LFM-MIMO selection
According to the first part, S set signals are divided into three categories. To recognize the modulation type of MIMO radar signal, we need to keep working on S 1 set and S 2 set signals. In this part, our attention is focused on the S 1 set. The goal is to discriminate MP-MIMO signal from LFM-MIMO signal. Selecting two different lengths of time window for the signal and employing DFT on them, we can get S a k ½ ¼ DFT s a n ½ ð Þ j j ; s a n ½ ¼ s n As can be seen in (20), s a is contained in s b and the sampling length of s b is greater than s a .
If the maximum value of S a is noted by Max(S a ), and the corresponding frequency number is k max , then the value of S b [k max ] is obtained. We define the following feature parameter The frequency of MP-MIMO signal is not affected by the time while LFM-MIMO signal is modulated along time. That is, for MP-MIMO signal, the energy at a frequency is increasing with the increase of sampling points while the LFM-MIMO is not, for the energy present at the extra frequencies. Setting a proper threshold ξ, MP-MIMO signal and LFM-MIMO signal are separated.

PC-MIMO/FC-MIMO selection
The SNEA is employed to recognize the signals of S 2 set in this part. For the sake of convenience, only the first code length signal is chosen. That is, the signal section going to be analyzed is s(n), n < T 1 /T s . We first construct the observation matrix and then get its autocorrelation matrix where N − 1 + M r < T 1 /T s . For PC-MIMO signal, according to Equation (3), we have To simplify expressions, T s = 1 is employed in the calculation. Let Obviously, rank(A 0 ) = 1, where rank(*) denotes the rank of matrix *. Observing Equation (23), we have S ob = A 0 S 0 .
According to the nature of rank, it is easy to get that rank(R) = 1 for PC-MIMO signal. L is the distinct phase number of PC/PC-MIMO, T 1PC and T 1FC are the sub-pulse width of PC/PC-MIMO and FC/FC-MIMO, respectively. The simulation results are based on 1000 Monte Carlo trials for each modulation type and each SNR value. Particularly, we consider that SC signals are correctly recognized when the simulation results are S 0 set signal but not the exact modulation type.    For FC-MIMO signal, according to (4), we have Let   Figure 4 The recognition rate versus SNR (on the condition of 2 for Ω 1 ). Here, Akaike Information Criterion [13] is adopted to calculate the rank of the autocorrelation matrix. By comparing the rank of autocorrelation matrix with 1, PC-MIMO and FC-MIMO are discriminated. That is At last, the modulation recognition method can be summarized as following flowchart in Figure 2

Simulation and analysis
Simulation results are shown in this section. Modulation types in S set, which are given by Ω 1 and Ω 2 , are used to test the validity of the proposed approach. LFM-MIMO (7) MP-MIMO (7) MP-MIMO (10) FC-MIMO (7) FC-MIMO (10) Figure 7 The relative error of estimated sub-carrier numbers of Ω 3 . The figure following each MIMO radar signal denotes the number of sub-carriers. Suppose that the received signals are imbedded in complex AWGN and are rectangular pulse shape. Particularly, experiments are operated on two different simulation conditions, which are showed in Table 1.

The classification results
First, we study the performance of the proposed classifier with several values of SNR. Because of lack of space, simulation results under 0 dB is selected as representatives. Confusion matrices for Ω 1 and Ω 2 are shown in Tables 2 and 3, respectively, on two different conditions listed in Table 1. 1000 signals are utilized for each modulation scheme.
From Tables 2 and 3, we can see that the approach based on instantaneous autocorrelation spectrum successfully classifies the S set signals into three categories. The correct classify rates are all over 90%. It shows that this approach is well prepared for the following modulation recognition. Moreover, the performance shown in Tables 2 and 3 is alike on the constant condition, which indicates that the method is unaffected by sub-carrier numbers of MIMO radar signal.

The recognition results
Figures 3, 4, 5, and 6 represent the final recognition results. As can be seen, the total recognition probability is over 90% when the value of SNR is as low as 0 dB. The results demonstrate that the modulation types are well recognized by the suggested method.
This method presents an excellent performance for non-coding MIMO radar signals, which is of high recognition rate even if the value of SNR is −5 dB. Comparing Figure 3 with Figure 5, the recognition performance is almost the same, which demonstrates that the subcarrier number influences the proposed method slightly. This can also be illustrated by comparing Figure 4 with Figure 6. Comparing Figure 4 with Figure 3, the performance of FC-MIMO signal in low SNR slightly decreased, which results from high code rate. The same conclusion can be obtained by comparing Figure 5 with

The estimation results
As is mentioned above, some parameters of MIMO radar signal can be estimated simultaneously by the recognition method. Here, the estimation results of sub-carrier numbers of non-coding MIMO radar signal and FC-MIMO radar signal are presented as representative.
The simulation is conducted on the condition 1. The signals used to test are Ω 3 and Ω 4 , which can be expressed as  8 show the relative error of estimation. As can be seen, the sub-carrier numbers of MIMO radar signal are almost correctly estimated when the value of SNR above 0 dB. If the values of SNR below 0 dB, the relative error of estimation is acceptable. Comparing Figure 7 with Figure 8, the relative error of Figure 8 is smaller than Figure 7. This results from the calculation methods of relative error. By the proposed approach, for the different sub-carriers, the absolute error changes slightly. Then the relative error decreases with the increases of sub-carriers.

Conclusions
This article presents an approach to recognize the modulation type of MIMO radar signals for the first time. Three feature parameters are proposed in the recognition method. First, the intercepted signal is classified based on the instantaneous autocorrelation spectrum. Then, taking advantage of the difference in frequency domain, MP-MIMO signal and LFM-MIMO signal are discriminated. At last, SNEA is employed to recognize PC-MIMO signal from FC-MIMO signal. Besides, subcarrier numbers of non-coding MIMO radar signal and FC-MIMO signal are estimated simultaneously. Simulation results verify that the proposed method can extract the features of each modulation type, and effectively recognize the signals in the given set. This result can be provided as an analysis reference for the research of MIMO radar countermeasures.