Method for realizing MIMO radar true and false target identification based on communication signal

Abstract

The application discloses a method for realizing MIMO radar true and false target identification based on communication signals, and comprises the following steps: at the sending end of a multiple-input multiple-output radar system, a radar communication integration scheme is adopted to embed communication information and spread spectrum information in a radar waveform; at the radar receiving end, a decision problem selected from two possible hypotheses, i.e., a zero hypothesis or a standby hypothesis, is constructed; radar receiving signals under the two decision hypotheses are subjected to correlation processing, bispectrum transformation and diagonal slicing to extract six effective characteristic parameters; data is normalized, support vector machines are used for decision-making, average classification recognition rates under different signal-to-noise ratios are obtained through ten-fold cross-validation, and it is decided whether a target at the same position is a false target or a real target, so that true and false target identification is realized.

Description

TECHNICAL FIELD

[0001] The present application belongs to the field of radar communication, and particularly relates to a method for identifying true and false targets of MIMO (Multi Input Multi Output) radar based on communication signals. BACKGROUND

[0002] With the rapid development of digital radio frequency memory technology, a false target generator (FTG) can generate false targets in random range bins and Doppler cells using intercepted radar signals, so as to achieve the purpose of deceiving a radar system. Since the false target generator is usually located in the main lobe of the radar system, these false targets cannot be easily suppressed in a single domain such as time, frequency or space, and therefore how to identify, suppress and eliminate the deceptive jamming of false targets from the received signals at the radar receiving end becomes a difficult problem in actual systems.

[0003] Most of the prior art is based on traditional radar systems, and generally requires strong anti-interference capability in the radar transmission receiving control and signal processing flow. The literature shows that the theoretical circles first carried out in-depth research on the signal processing flow of the radar. The commonly used algorithms include signal processing algorithms and pattern recognition algorithms, and the main idea is to carry out a series of research work on the different characteristics of the deceptive jamming and target echoes. For example, blind signal separation is introduced, mixed signals are extracted and separated into two kinds of signals, and the deceptive jamming and target echoes are classified and identified according to the different phase quantization characteristics; the different characteristics of the deceptive jamming and target echoes are used to aggregate feature parameters; the correlation between the deceptive jamming and target echoes received by different receivers, the different instantaneous characteristics of the time-frequency distribution images of the deceptive jamming and target echoes are used for identification; a multi-hypothesis tracking algorithm based on multi-function fusion is used to distinguish the deceptive jamming and target echoes by extracting sparse decomposition coefficients and dual-spectrum features.

[0004] Because MIMO radar system has excellent clutter interference suppression capability, can improve target detection performance and angle estimation precision, therefore in radar transmission receiving control aspect, the researchers mostly put the eyes on MIMO radar system, utilize system itself unique characteristics to reach the purpose of identification and suppression deception jamming.For example, in order to utilize frequency diversity to improve spectral efficiency, under the MIMO radar of frequency diversity array (FDA) for transmitting array, detect target signal, rely on the angle steering vector generated by FDA itself to carry out false target identification;According to the time delay in MIMO radar, distinguish true and false target, and pass through adaptive two-dimensional matching filter to suppress false target;According to the time difference difference generated by false target in MIMO radar, distinguish true and false target in joint transmitting receiving space frequency domain;According to the Doppler frequency difference between real target and false target interference, construct frequency resonance matrix, utilize frequency subspace projection method to suppress interference;Utilize the geometry propagation characteristics of deception pulse not satisfying distributed MIMO system to detect and identify;Jointly utilize polarization diversity and frequency diversity method, further improve the performance of suppressing deception jamming.MIMO radar anti-deception interference develops rapidly, and the research achievement is also very remarkable.

[0005] Because the realization function is different, radar and communication system have been independently researched.As the continuous development of radar and communication technology, the boundary between the two systems gradually fades, and more system-level similarities gradually appear.The existing false target suppression and identification technology has two limitations.The first, they mostly focus on radar system itself, and do not consider combining radar and communication to improve the identification ability of false target by utilizing the advantages of communication system;The second, the signals sent by radar transmitter are mostly traditional FDA signals, which are easy to be captured, demodulated and imitated by FTG, causing interference difficult to suppress and eliminate. SUMMARY

[0006] Invention purpose: In view of the defects in the prior art, the present application provides a method for realizing MIMO radar true and false target identification based on communication signals, which decides whether the target is a real target or a false target by radar receiving signals, to realize true and false target identification.

[0007] Technical scheme: A method for realizing MIMO radar true and false target identification based on communication signals, comprising the following steps:

[0008] (1) In the radar sending end, a radar communication integrated scheme is adopted, and communication information and spread spectrum information are embedded in the radar waveform;

[0009] (2) In the radar receiving end, a decision problem selected from two possible hypotheses is constructed: zero hypothesis Or alternative hypothesis The zero hypothesis The backup hypothesis for the false target assumption caused by the false target generator sending a fake signal to the radar receiving end through the listener The real target hypothesis for the real target assumption that the transmitted signal is reflected by the real target and received by the radar receiving end

[0010] (3) Correlation processing, bispectrum transformation and diagonal slicing are performed on the radar receiving signals under the two decision hypotheses in step (2), and six effective feature parameters are extracted

[0011] (4) Normalize the data and use support vector machine for decision-making, and obtain the average classification recognition rate under different signal-to-noise ratios through ten-fold cross-validation.

[0012] Further, the step (1) comprises the following steps:

[0013] 1) Based on the traditional FDA waveform, the transmission signal s m (t) of the mth radar transmitting antenna is defined as 0≤t≤T, wherein f m = f c +mΔf is the carrier frequency, m=0, 1, …, M-1, M is the number of radar transmitting antennas, f c is the reference carrier frequency, Δf is the frequency increment, and is negligible compared to f c ; T is the radar pulse period; t is the time index within the radar pulse;

[0014] 2) Phase modulation is performed on the above traditional FDA waveform to obtain the first radar communication integrated waveform Where L represents the number of inserted communication symbols, l represents the lth inserted communication symbol, T p represents the communication symbol period; M p -PSK communication symbol form is inserted, g(t-lT p ) is a rectangular pulse signal with amplitude A and period T p , and g(t-lT p ) has a value of A when lT p ≤t≤(l+1)T p , and has a value of 0 otherwise;

[0015] 3) The spread spectrum signal is embedded into the communication signal to obtain the second radar communication integrated waveform Where K represents the length of the spread spectrum code, bk represents the kth bit of the spread spectrum code sequence, and b k ∈{0,1}; T c represents the spread spectrum code period, represents the phase difference of the lth communication symbol caused by spread spectrum.

[0016] Furthermore, the step (2) includes the following steps:

[0017] The radar transmitter sends the traditional FDA waveform, radar communication integrated waveform and radar communication integrated waveform with spread spectrum respectively, which are reflected by the real target and received by the radar receiver. snm Indicates that the first received signal at this time includes x snm and noise, represented by y1; the monitor forges a false target signal that does not carry communication information through the false target generator, which is received by the radar receiver and represented by x jnm Indicates that the second received signal at this time includes x jnm and noise, represented by y0; the decision problem can be expressed as the decision of y1 and y0.

[0018] Furthermore, the step (3) includes the following steps:

[0019] The processing of radar received signals can first extract its bispectral information; diagonally slice the bispectrum, and the signal after diagonal slicing can be represented by y(n); and extract six characteristic parameters of the bispectral diagonal slice signal, including mean, variance, root mean square, peak-to-peak value, form factor and margin factor, as classification sample data.

[0020] Furthermore, the step (4) includes the following steps:

[0021] First, the sample data was normalized. A simulation experiment was then conducted using ten-fold cross-validation. The classified sample data was then fed into a support vector machine for classification and recognition, achieving the optimal true and false object classification rate. In the simulation, a radial basis function was used as the kernel function, and a grid search method was employed to find the optimal parameters C and gamma. C represents the tolerance to error, and gamma is related to the number of support vectors.

[0022] Beneficial Effects: This invention is applicable to MIMO radar systems in the field of radar communications, enabling interference identification and suppression against false target interference from listeners. By adopting an integrated radar and communication solution, communication information is embedded in the radar waveform and utilized to improve identification and suppression performance. The proposed algorithm can effectively identify false target interference; it addresses the limitations and demodulation challenges of using a single radar signal in previous technologies; it transforms radar interference prevention into a detection problem, providing new research avenues; and the integrated radar and communication solution of this invention achieves a compromise between communication and radar performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 is a schematic diagram of a MIMO radar system of the present invention;

[0024] Figure 210-fold cross validation schematic diagram of the present invention;

[0025] Figure 3 This is a comparison chart of the recognition rates of true and false target signals under the single target condition of the present invention;

[0026] Figure 4 This is a comparison chart of the recognition rates of true and false target signals under the dual-target condition of the present invention. DETAILED DESCRIPTION

[0027] The present invention is further illustrated below with reference to the accompanying drawings and specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0028] This invention proposes a method for distinguishing true and false targets based on communication information. When a MIMO radar uses orthogonal transmit waveforms, it proposes embedding a phase-modulated signal (PSK) into the transmit signal from each antenna to achieve information transmission. This method does not destroy the orthogonality of the transmit waveforms and therefore does not affect radar performance. The proposed solution is as follows:

[0029] 1: System model.

[0030] We consider a single-base MIMO radar system that uses echo signals to detect and identify targets. The schematic diagram of the MIMO radar system is shown in Figure 2. Figure 1 As shown, this is a MIMO radar system that uses a unified linear array consisting of M transmitting antennas and N receiving antennas. Its radar transmitter is shared with the radar receiver, and the transmitter periodically transmits pulses carrying communication information for data transmission.

[0031] Based on the traditional FDA waveform, the transmission signal s of the mth radar transmitting antenna m (t) is defined as 0≤t≤T, where f m =f c +mΔf is the carrier frequency, m=0, 1,…,M-1, M is the number of radar transmitting antennas, f c is the reference carrier frequency, Δf is the frequency increment, and is related to f c It can be ignored in comparison; T is the radar pulse period; t is the time index within the radar pulse.

[0032] Phase modulate the above traditional FDA waveform to obtain the first radar communication integrated waveform Where L represents the number of inserted communication symbols, l represents the first inserted communication symbol, T p represents the communication symbol period; Indicates the inserted Mp - PSK communication symbol form, g(t - IT p ) is a rectangular pulse signal with amplitude A and period T p , g(t - IT p ) has a value of A when IT p ≤ t ≤ (l + 1)T p , and has a value of 0 otherwise.

[0033] Since the spread spectrum signal has good autocorrelation and cross-correlation characteristics, it can be used for radar detection, so the spread spectrum signal is embedded into the communication signal to obtain a second radar-communication integrated waveform where K represents the length of the spread spectrum code, bk represents the kth bit of the spread spectrum code sequence, and b k ∈ {0, 1}; T c represents the spread spectrum code period, represents the phase difference of the lth communication symbol due to spread spectrum. In particular, when the second radar-communication integrated waveform is BPSK modulated, M p = 2 and

[0034] 2: Decision problem.

[0035] (1) Real target echo signal

[0036] Consider a real target located at an azimuth angle of θ s and a distance of r s . The time delay of the signal transmitted by the mth antenna, reflected by the real target, and received by the nth antenna is τ snm = (2r s - d T sin(θ s ) - d R n sin(θ s )) / c, where c represents the speed of light, n = 0, 1, …, N-1, d T and d R represent the adjacent antenna spacing distance of the transmitting antenna and the receiving antenna, respectively.

[0037] Consider the case where the propagation delay is less than the radar pulse period. The radar transmitting end respectively transmits the conventional FDA waveform, the radar-communication integrated waveform, and the radar-communication integrated waveform with spread spectrum, and the signal received by the nth receiving antenna after reflection by the real target is where α srepresents the constant related to the attenuation and other factors in the transmission process, and b(t) represents the embedded communication information. The above signal is down-converted, and M correlators are used for correlation processing on each receiving antenna, and is represented by x snm . The first receiving signal at this time includes x snm and noise, and is represented by y1.

[0038] (2) False target jamming signal

[0039] Consider a listener located at an azimuth angle of θ and a distance of r. The signal captured by the listener is where τ m = -d T msin(θ) / c and τ = 2r / c. Since the listener is unknown to the embedded communication information b(t) of the radar sending end, and cannot demodulate the communication information within the reasonable time delay of the received signal at the receiving end, the FTG is resorted to to forge a false target signal that does not carry communication information, and to interfere with the radar receiving end. The FTG can generate a false target with an appropriate time delay τ0in any distance target box. Assuming that the azimuth angle of the false target forged by the FTG is θ j , the distance is r i , and the false target signal forged, sent by the FTG and received by the nth radar receiving antenna is where α j (j = 1, 2) is the transmission attenuation, A is the power amplification coefficient when the FTG forges the false target, τ n = -d R nsin(θ) / c, and τ jnm = τ + τ m + τ n + τ0= τ j + τ jm + τ jn is the time delay of the false target forged by the FTG, s j (t) is the complex envelope of the false target signal forged by the FTG. Similarly, the above signal is down-converted, and M correlators are used for correlation processing on each receiving antenna, and is represented by x jnm . The second receiving signal at this time includes x jnm and noise, and is represented by y0.

[0040] (3) Decision

[0041] In the above model, the problem to be solved is to determine whether the existing target from the same direction and position is a real target or a false target forged by the FTG. Therefore, we first ensure the position (θ j , r j ) of the false target and the position (θ s, r s ) are the same.

[0042] The decision problem can be expressed as follows: Wherein, y0 represents the receiving signal received by the radar receiving end through the FTG false target signal forged by the listener; y1 represents the receiving signal received by the radar receiving end after the radar waveform embedding communication information sent by the radar sending end is reflected by the real target; e represents a noise term. Our purpose is to analyze the received signals y0 and y1, extract features, put the selected typical features into the support vector machine, and make a decision on the hypothesis Or Realize the identification and suppression of false targets.

[0043] 3: Feature extraction.

[0044] The high-order spectrum is defined as the k-1 dimensional DFT transform of the k-order cumulant, which is the energy spectrum of multiple frequencies of the signal. When k=3, it is defined as the third-order cumulant, also known as bispectrum. Bispectrum is usually a complex function, which contains not only the amplitude information of the signal, but also the phase information of the signal; while the traditional power spectrum only contains the amplitude information of the signal. Therefore, compared with the power spectrum, the bispectrum can obtain more characteristic information, so the invention adopts the strategy of bispectrum transformation on the received signals y0 and y1, analyzes the signals, and selects appropriate feature parameters for feature extraction.

[0045] After the signal received by the radar receiver is processed by the matched filter, it contains various noises, which tend to be Gaussian distribution. Therefore, the processing of the radar received signal can first extract its bispectrum information. As long as the bispectrum information of the received signal is large enough, even in the case of very small signal-to-noise ratio (SNR), the SNR of the random signal after bispectrum processing will be greatly improved, which is beneficial to the detection and analysis of the signal. Bispectrum not only reflects the amplitude and phase distribution characteristics of the signal, but also effectively suppresses the interference of Gaussian noise.

[0046] Bispectrum itself has a typical characteristic, that is, it has symmetry. The six symmetry lines are ω1=ω2, 2ω1=-ω2, 2ω2=-ω1, ω1=-ω2, ω1=0 and ω2=0, which divide the definition region of bispectrum into 12 sectors. Therefore, we can completely describe all bispectra as long as we know the bispectrum of the main domain. Using the symmetry of bispectrum, we can perform diagonal slicing on the bispectrum. Bispectrum diagonal slicing can be obtained by first calculating the bispectrum and then calculating the diagonal line of the bispectrum. Compared with bispectrum, bispectrum diagonal slicing greatly reduces the calculation amount without affecting the signal analysis. The signal after diagonal slicing can be represented by y(n).

[0047] After obtaining the bispectral diagonal slice signal, we can extract the multidimensional information of the radar received signals y0 and y1 by extracting the typical characteristic parameters of the bispectral signal. We selected six features commonly used in signal analysis: mean, variance, root mean square, peak-to-peak value, form factor, and margin factor. These six characteristic parameters of the bispectral diagonal slice signal were extracted for feature extraction, as shown in the following equation, where y(i) is the signal obtained after bispectral diagonal slicing, and i∈[1,n].

[0048] Mean:

[0049]

[0050] variance:

[0051]

[0052] RMS:

[0053]

[0054] Peak-to-peak value:

[0055] pk=max(y(n))-min(y(n))

[0056] Form factor: the ratio of the effective value of the signal to the rectified average value, that is,

[0057]

[0058] Margin factor: the ratio of the peak-to-peak value of the signal to the root square amplitude, that is,

[0059]

[0060] Respectively Decision-making and The radar received signals y0 and y1 under the decision-making process are bispectral transformed to obtain the bispectral signal. Then, diagonal slices are taken to obtain the bispectral signals y0(i) and y1(i), where i∈[1,n]. The characteristic parameter values ​​of the six selected characteristic parameters are obtained and used as the classification sample data.

[0061] 4: Support vector machine classification and recognition.

[0062] For the convenience of data processing, we first normalize the sample data and map the data to the range of 0 to 1 for analysis and processing. Then we use ten-fold cross validation to conduct simulation experiments. Figure 2As shown, the sample data set is divided into 10 parts, and 9 parts are used as training data and 1 part is used as test data in turn for experiment. In each experiment, the corresponding true and false target recognition rate is obtained. The average of the recognition rates of the 10 experiments is used as the estimation of the final true and false target classification recognition rate.

[0063] The classification sample data is input into the support vector machine for classification recognition. The radial basis function of the support vector machine is selected as the kernel function for simulation. The radial basis function has two parameters that affect the classification recognition rate, which are C and gamma. C represents the tolerance of error, and if its value is high, the system is not tolerant to error and is prone to overfitting; if its value is low, it is prone to underfitting. Gamma is related to the number of support vectors, and the number of support vectors affects the speed of training and prediction. If its value is large, the corresponding support vector is less, and vice versa, if its value is small, the support vector is more. Therefore, we need to select the appropriate optimal parameters C and gamma to achieve more accurate classification recognition rate.

[0064] In the present application, the grid search method is used to find the optimal parameters C and gamma. The grid search method is to adjust the parameters in steps within the specified parameter range, train the support vector machine using the adjusted parameters, and find the parameters with the highest accuracy on the validation set from all the parameters as the optimal parameters. After finding the optimal parameters C and gamma, the normalized data is input into the support vector machine, and the ten-fold cross-validation method is used to classify and recognize the feature parameter data to obtain the optimal true and false target classification recognition rate.

[0065] 5: Simulation results.

[0066] In the simulation process, SNR is defined as the ratio of the power of the real target signal to the noise power; SJR is defined as the ratio of the power of the real target signal to the power of the false target signal. In the entire simulation process, SJR is set to 0 dB to ensure that the support vector machine is used to classify and recognize the true and false target signals under the condition that the power of the true and false target signals is the same.

[0067] When there is only one single target in space, the classification recognition rate of the true and false target signals generated by the support vector machine varies with SNR as shown in Figure 3 .

[0068] Case (1): When the radar transmitter sends the radar communication integrated waveform (---), it can be seen that the classification recognition rate is higher than 90%, and when SNR≥12dB, the classification recognition rate can be stabilized at 100%, effectively identifying false target interference;

[0069] Case (2): when the radar sending end sends the radar communication integrated waveform with spread spectrum (---+---), it can be seen that when SNR is greater than or equal to 6 dB, the classification recognition rate can be stabilized at 100%, and false target interference can also be identified, and it is superior to case (1);

[0070] Case (3): when the radar sending end sends the conventional FDA waveform (---o---), because it does not contain communication information, it is easy to be captured, demodulated and imitated, the classification recognition rate is stabilized at about 45%, and it cannot distinguish false target signals, and the interference caused thereby is difficult to suppress and eliminate.

[0071] When there are multiple targets in space, taking two targets as an example, the classification recognition rate of true and false target signals generated by the support vector machine varies with SNR as shown in Figure 4

[0072] Case (4): when the radar sending end sends the radar communication integrated waveform (---*---), it can be seen that the classification recognition rate increases with the increase of SNR, and when SNR is greater than or equal to 16 dB, the classification recognition rate can be stabilized at more than 90%, and the identification of false target interference signals is realized;

[0073] Case (5): when the radar sending end sends the radar communication integrated waveform with spread spectrum (---+---), it can be seen that when SNR is greater than or equal to 12 dB, the classification recognition rate is more than 90%, and when SNR is greater than or equal to 20 dB, it can be stabilized at 100%, and the effect is superior to case (4), and the identification of false target interference signals is realized;

[0074] Case (6): when the radar sending end sends the conventional FDA waveform (---o---), similar to case (3), the classification recognition rate fluctuates at about 35%, and it cannot distinguish false target signals, and the interference caused thereby is difficult to suppress and eliminate.

[0075] The simulation proves that the method for identifying true and false targets based on communication information proposed in the application can effectively identify false target interference. If spread spectrum is added to the radar communication integrated waveform as the sending signal of the radar sending end, the effect of identifying false target interference will be better than that of the radar communication integrated waveform without spread spectrum. At the same time, the simulation results in single target and multiple target scenarios can infer that the identification ability of the algorithm for false target interference has certain correlation with the number of targets existing in space.​

Claims

1. A method for realizing true and false target identification of MIMO radar based on communication signals, comprising the following steps: (1) At the radar transmitter, a radar communication integration solution is adopted to embed communication information and spread spectrum information into the radar waveform; (2) At the radar receiver, a decision problem is constructed to choose between two possible hypotheses: the null hypothesis or alternative hypothesis The null hypothesis The false target hypothesis is caused by the false target generator sending a false signal to the radar receiver. It is assumed that the transmitted signal is reflected by the real target and received by the radar receiver; (3) performing correlation processing, bispectral transformation, and diagonal slicing on the radar received signals under the two decision assumptions in step (2) to extract six effective feature parameters; (4) The data is normalized and decision-making is performed using support vector machines. The average classification recognition rate under different signal-to-noise ratios is obtained through ten-fold cross validation.

2. The method for realizing true and false target identification of MIMO radar based on communication signals according to claim 1, characterized in that: The step (1) includes the following steps: 1) Based on the traditional FDA waveform, the transmission signal s of the mth radar transmitting antenna m (t) is defined as where f m =f c +mΔf is the carrier frequency, m=0,1,…,M-1, M is the number of radar transmitting antennas, f c is the reference carrier frequency, Δf is the frequency increment, and is related to f c Compared to negligible; T is the radar pulse period; t is the time index within the radar pulse; 2) Phase modulate the above traditional FDA waveform to obtain the first radar communication integrated waveform Where L represents the number of inserted communication symbols, l represents the inserted lth communication symbol, T p represents the communication symbol period; Indicates the inserted M p -PSK communication symbol form, g(t-lT p ) is a signal with an amplitude of A and a period of T. p rectangular pulse signal, when lT p ≤t≤(l+1)T p Time g(t-lT p ) is A, otherwise it is 0; 3) Embed the spread spectrum signal into the communication signal to obtain the second radar communication integrated waveform Where K represents the length of the spreading code, b k represents the kth bit of the spreading code sequence, and b k ∈{0,1};T c represents the spreading code period, Represents the phase difference of the lth communication symbol caused by spread spectrum.

3. The method for realizing true and false target identification of MIMO radar based on communication signals according to claim 2, characterized in that: The step (2) includes the following steps: The radar transmitter sends the second radar communication integrated waveform, which is reflected by the real target and received by the radar receiver. snm Indicates that the first received signal at this time includes x snm and noise, represented by y1; the monitor forges a false target signal that does not carry communication information through the false target generator, which is received by the radar receiver, represented by x jnm Indicates that the second received signal at this time includes x jnm and noise, represented by y0; the decision problem is expressed as the decision of y1 and y0.

4. The method for realizing true and false target identification of MIMO radar based on communication signals according to claim 2, characterized in that: The step (3) includes the following steps: The processing of radar received signals first extracts its bispectral information; diagonally slices the bispectrum, and the signal after diagonal slicing is represented by y(n); six characteristic parameters of the bispectral diagonal slice signal, including mean, variance, root mean square, peak-to-peak value, form factor and margin factor, are extracted as classification sample data.

5. The method for realizing true and false target identification of MIMO radar based on communication signals according to claim 2, characterized in that: The step (4) includes the following steps: First, the sample data is normalized; secondly, a ten-fold cross-validation is used for simulation experiments; the classified sample data is input into the support vector machine for classification and recognition to obtain the best classification recognition rate of true and false targets; in the simulation, the radial basis function is selected as the kernel function, and the grid search method is used to find the optimal parameters C and gamma, where C represents the tolerance to error and gamma is related to the number of support vectors.

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