CN105182317A - Resource management method based on search pattern of centralized MIMO radar - Google Patents

Abstract

The invention belongs to the technical field of communication radar, in particular to a resource management method based on a new system MIMO radar search mode. The present invention proposes a resource management method based on a centralized MIMO radar search mode by analyzing various parameters that affect MIMO radar search performance, that is, selecting the number of sub-array divisions, search frame period, signal duty cycle and beam Dwell time is a controllable parameter to minimize the consumption of radar resources on the premise of meeting the radar search performance requirements. Among them, the radar search performance not only considers the cumulative detection probability of the target radial direction, but also considers the cumulative detection probability of the target tangential direction for the first time. Finally, the genetic algorithm is used to solve the above optimization problem, and the search parameter configuration results in specific scenarios are obtained.

Description

A Resource Management Method Based on Centralized MIMO Radar Search Mode

technical field

The invention belongs to the technical field of communication radar, in particular to a resource management method based on a new system MIMO radar search mode.

Background technique

In the all-round, multi-level, and three-dimensional modern warfare, in order to cope with the increasingly complex and ever-changing electromagnetic environment, the new system MIMO radar has become the current research focus (He Zishu, Han Chunlin, Liu Bo. Analysis of the concept and technical characteristics of MIMO radar [J ]. Electronic Journal, 2005, 33(12A): 2441-2445.). MIMO radar is a radar system in which multiple transmitting antennas transmit different waveforms independently, and multiple antennas are used to receive and detect at the receiving end. Compared with phased array radar, the sub-array division of MIMO radar is more flexible, the angle resolution and parameter estimation accuracy are higher, and at the same time, it can effectively suppress multipath clutter and improve the detection ability of stealth targets (Lv Hui. Research on signal processing method of MIMO radar [D]. Xi'an: Xidian University, 2011).

According to the different spacing of the antenna array, MIMO radar is divided into centralized (JianLiandP.Stoica.MIMOradarwithcolocatedantennas, IEEESignalProcessingMagazine, vol.24, no.5, pp.106-114, Sep2007.) and distributed (A.M.Haimovich, R.S.Blum, andL . Cimini. MIMO radar with widely separated tantennas, IEEE Signal Processing Magazine, invited publication, vol. 25, no. 1, pp. 116-129, Jan 2008.). Compared with the phased array radar, the working parameters of the centralized MIMO radar increase the number of sub-array divisions, which makes the waveform selection more free and the freedom of resource management is greater. Therefore, it is necessary to implement effective resource management, so that various resources can be fully and efficiently utilized, and the performance advantages of MIMO radar can be better utilized.

In terms of resource management in radar search mode, Billiam (BillamER. Parameter Optimization in Phased Array Radar [A]. IEE Conference Radar-92, 1992, 34-37.) proposed a parameter design criterion for phased array radar, that is, when a given tracking is satisfied, Under the initial performance, the radar consumes the least resources, and on this basis, the optimal search frame period without high priority load is obtained. On the basis of Billiam, Lu Jianbin studied the optimization of phased array radar search parameters under limited search resources with the tracking starting distance as the objective function, and gave the best criteria and methods for calculating the search frame period and dwell time (Lu Jianbin , Hu Weidong, Yu Wenxian. Research on optimal search performance when phased array radar resources are limited [J]. Systems Engineering and Electronic Technology, 2004, 26(10):1388-1390.). Zhang Litao et al. comprehensively introduced various issues involved in the practical application of search performance optimization from a conceptual and systematic point of view, and proposed the selection principles and determination methods of search parameters (Zhang Litao, Li Dun, Wang Guoyu. Research on Search Parameters of Controlled Array Radar [J]. Modern Radar, 2008,30(10):20:25.). The above studies are all carried out for phased array radar, but not for new system MIMO radar.

In terms of MIMO radar search resource management, HanaGodrich et al. took the lead in discussing the issue of power allocation in MIMO radar target detection (AittomakiT, GodrichH, PoorHV, et al. :873-877.). By optimizing the transmission power, antenna position and other factors, the detection probability of the target is maximized, but this research is only for distributed MIMO radar. Literature (Yang Shaowei, Cheng Ting, He Zishu. RF Stealth Algorithm in MIMO Radar Search Mode [J]. Journal of Electronics and Information Technology, 2014,36(5):1017-1022.) established the RF stealth performance in MIMO radar search mode model, but its goal is to optimize the radar RF stealth performance, and the tangential cumulative detection probability is not considered.

However, there is no research on resource management of centralized MIMO radars at this stage.

Contents of the invention

In view of the above-mentioned problems or deficiencies, the present invention proposes a resource management method based on the centralized MIMO radar search mode by analyzing various parameters that affect the MIMO radar search performance, that is, selecting the number of sub-array divisions, the search frame period , signal duty cycle, and beam dwell time are used as controllable parameters to minimize radar resource consumption on the premise of meeting radar search performance requirements. Among them, the radar search performance not only considers the cumulative detection probability of the target radial direction, but also considers the cumulative detection probability of the target tangential direction for the first time. Finally, the genetic algorithm is used to solve the above optimization problem, and the search parameter configuration results in specific scenarios are obtained.

A resource management method based on a centralized MIMO radar search mode, comprising the following steps:

Step 1. Establish a MIMO signal processing model: For a MIMO radar where the transceiver arrays are co-located, the array is divided into K sub-arrays, and each sub-array contains L array elements, so the array contains a total number of array elements M=KL. The signal-to-noise ratio of the radar received signal is deduced as

Among them, the peak power of a single radar antenna is P _t , the transmitting antenna gain is G _t , the antenna receiving gain is G _r , the radar cross-sectional area (RCS) of the target is σ, λ is the signal wavelength, N ₀ is the noise power spectral density, R is the radial distance from the target to the radar, η is the duty cycle of the signal, and t _B is the dwell time of the beam.

Step 2. Calculate the tangential cumulative detection probability: Let the search range of MIMO radar be [θ ₁ , θ ₂ ], and define the tracking start angle θ _max according to the concept of tracking start distance,

θ _max = (θ ₂ -θ ₁ )Q,Q∈(0,1)(2)

That is, for a given target, when the tangential cumulative detection probability of the radar reaches the cumulative detection probability threshold, the angle at which the target flies. The time period corresponding to the angular flight θ _max is [t ₀ ,t _max ]. in,

The calculation steps of tangential accumulation detection probability are as follows:

A. Calculate the number of search beams launched by the radar within the time period [t ₀ , t _max ]: the time it takes for the target to fly into the search area tangentially to reach the cumulative detection probability is [t ₀ , t _max ]. During this period, the radar transmits a total of The number of search beams I=I ₁ +I ₂ . in,

At time t ₀ , the beam counter value n _count , n _count ∈ [0, I-1] starts counting, until n _count = I-1, the following calculation is repeated.

B. Calculate the dwell time of each search beam on the target and the single detection probability. When the target flies into the search area t ₀ , the angular position is θ ₁ , and the number of the search beam is n ₀ .

When launching the nth _count radar search beam, the beam pointing range is

[θ _s1 (t ₀ ),θ _s2 (t ₀ )]＝[θ ₁ +(n _search (t ₀ )-1)B _w ,θ ₁ +n _search (t ₀ )B _w ](6)

The corresponding search beam execution time is

Among them, the search frame period counter for

The search wave number n _search is (n _search ≥ 1)

The functional relationship between the angle of the moving target and the time is:

Then, solve for the dwell time of the search beam on the target.

make The solution can be obtained separately and Judging the irradiation time of the n _count beam and Whether there is an intersection, if there is an intersection, it means that the target is irradiated by the beam, and the length t _B ′ of the intersection represents the time length of being irradiated. If t _B ′≠0, P _d (t _B ′)=0. in,

C. The target detection probabilities of the I search beams are non-coherently accumulated to obtain the target tangential cumulative detection probability, as shown in Equation 11.

D. Let t ₀ be uniformly distributed in [0,T _f ], then, with As the tracking start time (t _max >T _f ), the radar’s average tangential cumulative detection probability of the target is:

Step 3. Construct the optimization model in MIMO radar search mode:

The ratio of the time used to search beams in one frame to the search frame period describes the resource distribution of the system in the time domain, the signal duty cycle describes the energy consumption of each search, and the number of sub-array divisions describes the consumption of radar hardware resources , the sum of the three according to their respective weights is used as the objective function to describe the MIMO radar resource consumption. At the same time, in order to maintain the search performance of MIMO radar, the radial cumulative detection probability of the target, the tangential cumulative detection probability of the target is greater than the cumulative detection probability threshold, and the rationality of time resource consumption are the constraints. The optimization model is

Step 4. Solve the above optimization problem by using a genetic algorithm.

By solving the optimization process of the optimization model, a set of parameters [K _opt , η _opt , t _Bopt , T _fopt ] is obtained, which consumes the least resources on the premise of meeting the search performance requirements.

The working principle of the present invention is as follows: assuming that the MIMO radar transceiver arrays are co-located, the arrays are divided into K sub-arrays, and each sub-array contains L array elements, then the array contains a total number of array elements M=KL. Each sub-array transmits mutually orthogonal waveforms, and the transmitted signal of the kth sub-array is recorded as s _k (t), and the transmitted waveforms of each array element in the sub-array are the same. Then, the composite signal formed by the transmitted signal of the kth sub-array in the space domain can be expressed as:

The phase difference between each sub-array is φ _L = Lφ, then the composite signal formed by K sub-arrays transmitting signals in the space can be expressed as:

After the synthesized signal is scattered by the target, the echo signal is received by the receiving array. If the loss caused by the transmission process and target scattering is ignored, the signal received by the mth array element can be expressed as:

y _m (t)=x(t)e ^-j(m-1)φ +v _m (t)(16)

Among them, v _m (t) represents the noise of the mth receiving channel.

The received signal of each receiving channel is matched with all the transmitted waveforms, and the result of matched filtering between the signal of the mth receiving channel and s _k (t) is:

Among them, v _mk is the matched filtering result of v _m (t) and s _k (t), and E _s is the energy of the transmitted waveform.

Add the results of the matched filtering of the mth array element, that is, perform equivalent transmit beamforming, as follows:

Finally, receive beamforming is performed, as follows:

According to the above signal processing process, the output signal-to-noise ratio can be obtained:

If the total power of radar transmission is P _t , the gain of transmitting antenna is G _t , the radar cross-sectional area (RCS) of the target is σ, G _r is the antenna receiving gain, λ is the signal wavelength, B _r is the equivalent noise bandwidth of the radar receiver, N ₀ is the noise power spectral density. Then the signal-to-noise ratio of the radar received signal satisfies:

For MIMO radar, P _t =Mp _t , where p _t is the peak power of the transmit waveform of a single array element. Processing gain τB _r due to matched filtering and equivalent transmit beamforming, where τ is the signal pulse width. In addition, multi-pulse accumulation is a common method for the system to improve the signal-to-noise ratio. Assuming that the system adopts coherent accumulation technology, after N _p pulse accumulation, the signal-to-noise ratio will be increased by N _p times. therefore,

It is assumed that the controllable parameters of the MIMO radar include the number of sub-array divisions K, the signal duty cycle η, the beam dwell time t _B and the search frame period T _f . Firstly, the relationship between controllable parameters and radar search performance is established, and a search performance optimization model is constructed.

The parameters of the surveillance region search task are denoted as (K, η, t _B , T _f ). The search frame period describes the resource distribution of the system in the time domain, the signal duty cycle describes the energy consumption of each search, and the number of sub-array divisions describes the consumption of radar hardware resources. A system with good search performance should minimize the consumption of time, energy and hardware resources under the condition of ensuring the effectiveness of radar task execution. Therefore, the following formula is used to describe the objective function, that is, the radar resource consumption:

Among them, _NB represents the number of beams that reside, which is related to the number of sub-array divisions. c ₁ , c ₂ , and c ₃ are three set weighting coefficients, and the values reflect the degree of attention to system time, energy, and hardware resource consumption.

The resource management in the radar search mode is to reduce its resource consumption under the condition of ensuring the normal search of the radar system. The document (AM Haimovich, RS Blum, and L. Cimini. MIMO radar with widely separated antennas, IEEE Signal Processing Magazine, invited publication, vol.25, no.1, pp.116-129, Jan2008.) defines the tracking start distance (for a given target when the cumulative detection probability of the radar distance when a given value is reached) to describe radar search performance. Here it is assumed that the radar action distance is R _s , the target radial approach speed is v, the search frame period is T _f , and R _t is the tracking start distance, the average radial detection probability of the target by the radar (Yang Shaowei, Cheng Ting, He Zishu. RF Stealth Algorithm in MIMO Radar Search Mode [J]. Journal of Electronics and Information Technology, 2014,36(5):1017-1022.) is:

in, Indicates rounding down to an integer, Δr=vT _f , indicating the radial flight distance of the target in the process of being illuminated twice. P _dc radar can correctly identify the probability of the target, P _di (r-iΔr) represents the detection probability of a single observation of the target at a distance of r-iΔr by the radar, in the case of a given false alarm probability, the detection of a single observation The relational expression of probability and SNR (subject to the target of Swerling-I fluctuation characteristic) is as follows:

It can be seen that the radial cumulative detection probability is related to all configurable parameters after a given tracking start distance requirement.

However, in actual situations, it is also necessary to consider the situation of the target flying tangentially. Therefore, the present invention refers to the concept of the tracking start distance, and defines the tracking start angle as the angle that the target flies when the tangential cumulative detection probability of the radar reaches a given value for a given target. Two parameters are used to jointly describe the search performance.

In the derivation process of the average tangential cumulative detection probability, it is assumed that the search range of the MIMO radar is [θ ₁ ,θ ₂ ], and the target is expected to be detected when it flies from θ ₁ to the angle θ _s = θ ₁ + θ _max . Therefore, the accumulated detection probability in this process must reach the threshold. Define the tracking starting angle as θ _max , θ _max =(θ ₂ -θ ₁ )Q,Q∈(0,1). If the target flies into the region at time t ₀ and flies to θ _s at t _max , now it is necessary to calculate the cumulative detection probability within [t ₀ , t _max ] to determine whether the detection probability is greater than the threshold. in, R _x is the radial distance of the target when flying tangentially, and v _x is the speed of the target tangentially flying.

The schematic diagram of the calculation principle of target tangential cumulative detection probability is shown in Figure 1. The dotted line in Figure 1 represents the relationship between the search wave position angle and time in each frame, and the straight line segment in [(k-1)t _B ,kt _B ] represents the angle range covered by the beam within the t _B dwell time, which is the period equal to T Periodic function of _f . For radar, the dwell time and angular range of the i-th search beam (i≥1) in the j-th frame (j≥1) search are respectively: [T _f (j-1)+(i-1) t _B ,T _f (j-1)+it _B ], [B _w (i-1)+θ ₁ ,B _w (i)+θ ₁ ]. For example, the irradiation angle range of the search beam within [0,t _B ] is [θ ₁ ,θ ₁ +B _w ]. The dotted line in Figure 1 represents the relationship between the time and angle of the target movement, and the functional relationship between the angle of the target and the time is:

In order to calculate the cumulative detection probability within [t ₀ ,t _max ], it is necessary to judge whether the beam hits the target. If the target is irradiated, calculate how long the target stays in the beam to obtain the single target detection probability of coherent accumulation, and then count the number of search beams irradiated to the target within this period of time for non-coherent accumulation Get the target tangential cumulative detection probability.

Finally, t ₀ , which is considered to be a known quantity in the above analysis, is evenly distributed in [0,T _f ]. If t ₀ is different, the number of beams searching for the target is different, and the dwell time is different. So get As the tracking start time (t _max >T _f ), the average tangential detection probability of the radar to the target is

For the above search process, there are the following time constraints:

It means that the sum of the time utilization ratios of each sub-area during the search process does not exceed 1.

In summary, the optimization problem under the MIMO radar search mode that the present invention proposes can specifically be

Expressed as:

Description of drawings:

Figure 1 is a schematic diagram of the principle of calculation of target tangential cumulative detection probability;

Figure 2 is the configuration result of the search frame period under different tracking start distances;

Figure 3 shows the configuration results of the signal duty ratio under different tracking start distances;

Figure 4 is the configuration result of the number of sub-array divisions under different tracking start distances;

Figure 5 shows the configuration results of the beam dwell time under different tracking start distances;

Figure 6 shows the improvement in resource consumption of MIMO radars with different tracking start distances relative to phased array radars;

Figure 7 is the configuration result of the search frame period under different Rx;

Figure 8 is the configuration result of the signal duty cycle under different Rx;

Figure 9 is the configuration result of the number of sub-array divisions under different Rx;

Figure 10 shows the configuration results of the beam dwell time under different Rx.

Detailed ways:

Based on the detailed technical solution of the present invention, through the optimization of the technical solution for MIMO radar, the resource consumption of the radar can be reduced while ensuring the detection performance of the radar.

The radar is selected as a uniform linear array with M=2048 elements, wavelength λ=5.45cm, Boltzmann constant k=1.38×10 ^-23 J/K, standard temperature T=230K, noise factor F=2, effective area of the antenna The duty cycle η _e =0.5, the total peak power of the signal is P _t =10 ⁵ W, and the operating distance of the radar R _s =200km. In the simulation, the Swerling-I type target is taken as an example. The average RCS of the target is 1m ² , the false alarm probability P _fa =10 ^-6 , and the cumulative detection probability threshold P _D =0.95. The angular range of the region is θ∈[-40°,40°]. Radial velocity v _r =1.5Ma, tangential velocity v _t =1.5Ma. In the genetic algorithm, the population size is set to 500, the maximum genetic algebra is 300, the crossover probability is 0.7, the mutation probability is 0.05, and the generation gap is 0.9.

Objective function weights: c1=0.8, c2=0.1, c3=0.1. The optional parameter set of subarray division number K is {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048}, the duty cycle is η∈[0,0.25], and the beam dwell time t _B ∈[0,26]ms, search frame period T _f ∈[0,20]s.

Simulation 1, assuming Rx=30km, Q=0.2 at this time, under different tracking start distances, the parameter configuration results are shown in Figure 2-Figure 5.

When the starting distance of tracking is small, MIMO radar has more accumulation times, which correspondingly reduces the requirement for the signal-to-noise ratio of a single detection. With the increase of the tracking start distance, the allowed number of divided sub-arrays decreases, the signal duty cycle increases, the dwell time increases and the search frame period decreases to meet the improved detection performance requirements. However, the signal duty cycle, the number of sub-array divisions, the beam dwell time, and the search frame period are mutually restricted, and finally the resource optimization selection is reasonable. Reducing the number of sub-array divisions is beneficial to improve the probability of single detection, but at the same time, the objective function wants to minimize resource consumption, so the number of sub-array divisions fluctuates. When the number of sub-array divisions is reduced, a shorter dwell time is allowed to meet the detection performance requirements, and the change trend of the two is almost the same. The probability is maintained, the search frame period changes little in the whole process, and it is affected by the radial and tangential cumulative detection probabilities at the same time.

In addition, under the parameter configuration of simulation 1, the system resource consumption of MIMO radar and phased array radar is compared. The energy consumption improvement ratio is defined as the difference between phased array resource consumption and MIMO radar resource consumption divided by phased array radar resource consumption. The simulation results are shown in Figure 6. It can be seen that the resource consumption of MIMO radar in search mode is less than that of phased array radar. And as the tracking start distance increases, the resource consumption improvement ratio decreases. This is because as the tracking start distance increases, the dwell time increases, the number of sub-array divisions decreases, and the advantages brought by the number of MIMO radar factor array divisions weaken.

In the specific actual radar design parameter requirements, the tracking starting distance and tracking starting angle requirements are generally given, and the only parameter that changes at this time is the radial distance Rx between the target and the radar when it flies into the surveillance airspace along the tangential direction. In simulation three, when R _t =130km and Q=0.3, the parameter optimization results under different R _x are solved. The parameter configuration results are shown in Figure 7 to Figure 10.

It can be seen from Figure 8-10 that the number of sub-array divisions is greater than or equal to 2 within 90km, and greater than 4 thereafter. Therefore, the radar search parameters are determined by segmenting Rx: when Rx is less than or equal to 90km, the number of subarray divisions is 2, the signal duty cycle is 19.5%, and the beam dwell time is 1ms; when Rx is greater than or equal to 90km, the number of subarray divisions is 2 The number is 8, the signal duty cycle is 20.3%, and the beam dwell time is 2ms. The search frame period takes 50km as the breaking point.

To sum up, it can be seen that the requirement of increasing the tracking start angle on the basis of the tracking start distance has an impact on the parameter assignment results, and it is of practical significance to increase the tangential cumulative detection probability to describe the radar search performance together; and through the phased array radar In contrast, under the same search requirement, the resource consumption of MIMO radar is smaller. Therefore, this method can effectively reduce resource waste, so that the radar system can efficiently process other tasks such as search and tracking.

Claims (1)

1. A resource management method based on a centralized MIMO radar search mode comprises the following steps:

step 1, establishing an MIMO signal processing model:

aiming at the MIMO radar with the co-located transmitting and receiving array, the array is divided into K sub-arrays, each sub-array comprises L array elements, the array comprises the number M of the array elements in total, which is KL, and the signal-to-noise ratio of the radar receiving signal is deduced to be

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In which the peak power of a single antenna of the radar is p_tGain of transmitting antenna of G_tThe antenna receiving gain is G_rThe radar cross-sectional area (RCS) of the target is σ, λ is the signal wavelength, N₀For noise power spectral density, R is the radial distance from the target to the radar, η is the duty cycle of the signal, t_BIs the beam dwell time;

step 2, calculating the tangential cumulative detection probability:

let search range of MIMO radar be [ theta ]₁,θ₂]Defining a tracking start angle theta based on the concept of the tracking start distance_max，

θ_max＝(θ₂-θ₁)Q,Q∈(0,1)(2)

For a given target, when the tangential cumulative detection probability of the radar reaches the cumulative detection probability threshold, the target flies through the angle; angular fly-through theta_maxCorresponding to a time period of [ t ]₀,t_max]Wherein

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the calculation steps of the tangential accumulation detection probability are as follows:

A. calculate [ t ]₀,t_max]In the time period, the number of the radar transmitting search beams is as follows: the time taken for the target to reach the accumulated detection probability along the tangential flying search area is t₀,t_max]During the period, the total number I of the search beams transmitted by the radar is equal to I₁+I₂Wherein

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t₀time beam counter value n_count,n_count∈[0,I-1]Starting to count up to n_countI-1 cycle the following calculations;

B. calculating the residence time of each search beam on the target and the single detection probability, wherein the target flies into the search area t₀At the moment, the angular position is theta₁When the search beam number is n₀，

At the n-th of transmission_countWhen a radar searches for a beam, the beam is directed in a range of

[θ_s1(t₀),θ_s2(t₀)]＝[θ₁+(n_search(t₀)-1)B_w,θ₁+n_search(t₀)B_w](6)

Corresponding search beam execution time is

<math> <mrow> <mtable> <mtr> <mtd> <mrow> <mo>&lsqb;</mo> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mn>1</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>t</mi> <mrow> <mi>s</mi> <mn>2</mn> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>=</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <msub> <mi>T</mi> <mi>f</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>B</mi> </msub> <mo>,</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <msub> <mi>T</mi> <mi>f</mi> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>-</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <msub> <mi>t</mi> <mi>B</mi> </msub> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow> </math>

Wherein the search frame period counterIs composed of

Search wave position number n_searchIs (n)_search≥1)

<math> <mrow> <msub> <mi>n</mi> <mrow> <mi>s</mi> <mi>e</mi> <mi>a</mi> <mi>r</mi> <mi>c</mi> <mi>h</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>mod</mi> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <msub> <mi>N</mi> <mi>B</mi> </msub> </mfrac> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>&NotEqual;</mo> <msub> <mi>kN</mi> <mi>B</mi> </msub> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>...</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>N</mi> <mi>B</mi> </msub> <mo>,</mo> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>kN</mi> <mi>B</mi> </msub> <mo>,</mo> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>..</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mrow> </math>

The functional relation between the angle of the moving target and the time is as follows:

<math> <mrow> <msub> <mi>&theta;</mi> <mi>t</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>v</mi> <mi>x</mi> </msub> <mrow> <mo>(</mo> <mi>t</mi> <mo>-</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <msub> <mi>r</mi> <mi>x</mi> </msub> </mfrac> <mo>+</mo> <msub> <mi>&theta;</mi> <mn>1</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mrow> </math>

then, solving the dwell time of the search beam on the target;

order to <math> <mrow> <msub> <mi>&theta;</mi> <msub> <mi>s</mi> <mn>1</mn> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>t</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>r</mi> </mfrac> <mo>+</mo> <msub> <mi>&theta;</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>&theta;</mi> <msub> <mi>s</mi> <mn>2</mn> </msub> </msub> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <msub> <mover> <mi>t</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> </mrow> <mi>r</mi> </mfrac> <mo>+</mo> <msub> <mi>&theta;</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </math> The solution can be obtained respectivelyAndjudging the nth_countIrradiation time of individual beamAndwhether an intersection exists or not, if so, the intersection represents that the target is irradiated by the beam, and the length t of the intersection is_B' then represents the length of time that is illuminated, if t_B′≠0，P_d(t_B') 0. Wherein,

C. carrying out incoherent accumulation on the target detection probability of the I search beams to obtain the target tangential accumulated detection probability,

<math> <mrow> <munderover> <mo>&Pi;</mo> <mrow> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>I</mi> </munderover> <mo>&lsqb;</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>P</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <msub> <mi>t</mi> <mi>B</mi> </msub> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> <mo>;</mo> </mrow> </math>

D. let t₀At [0, T_f]Are internally uniformly distributed, thus, inAs a tracking start time (t)_max>T_f) The average tangential cumulative detection probability of the radar to the target is as follows:

<math> <mrow> <msub> <mi>P</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>T</mi> <mi>f</mi> </msub> </mfrac> <msubsup> <mo>&Integral;</mo> <mn>0</mn> <msub> <mi>T</mi> <mi>f</mi> </msub> </msubsup> <mrow> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>-</mo> <munderover> <mo>&Pi;</mo> <mrow> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mi>I</mi> </munderover> <mo>&lsqb;</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>P</mi> <mrow> <mi>d</mi> <mi>c</mi> </mrow> </msub> <mo>&CenterDot;</mo> <msub> <mi>P</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <mrow> <msup> <msub> <mi>t</mi> <mi>B</mi> </msub> <mo>&prime;</mo> </msup> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>,</mo> <msub> <mi>n</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>u</mi> <mi>n</mi> <mi>t</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> <mo>}</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow> <mo>;</mo> </mrow> </math>

step 3, constructing an optimization model under the MIMO radar search mode:

the ratio of the time for searching beams in one frame to the period of the search frame describes the resource distribution of the system in the time domain, the signal duty ratio describes the energy consumption of each search, the number of sub-array partitions describes the consumption of radar hardware resources, the three describe the resource consumption of the MIMO radar by adding the respective weights as a target function, meanwhile, the MIMO radar keeps the search performance by taking the target radial cumulative detection probability, the target tangential cumulative detection probability is greater than the cumulative detection probability threshold value, the time resource consumption rationality is the constraint condition, and the optimization model is

<math> <mrow> <munder> <mi>min</mi> <mrow> <mo>(</mo> <mrow> <mi>K</mi> <mo>,</mo> <mi>&eta;</mi> <mo>,</mo> <msub> <mi>t</mi> <mi>B</mi> </msub> <mo>,</mo> <msub> <mi>T</mi> <mi>f</mi> </msub> </mrow> <mo>)</mo> </mrow> </munder> <msub> <mi>c</mi> <mn>1</mn> </msub> <mfrac> <mrow> <msub> <mi>t</mi> <mi>B</mi> </msub> <msub> <mi>N</mi> <mi>B</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> </mfrac> <mo>+</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mi>&eta;</mi> <mo>+</mo> <msub> <mi>c</mi> <mn>3</mn> </msub> <mfrac> <mi>K</mi> <mi>M</mi> </mfrac> </mrow> </math>

<math> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>R</mi> <mi>t</mi> </msub> <mo>)</mo> </mrow> <mo>&GreaterEqual;</mo> <msub> <mi>P</mi> <mi>D</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>P</mi> <mi>d</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>&theta;</mi> <mi>max</mi> </msub> <mo>)</mo> </mrow> <mo>&GreaterEqual;</mo> <msub> <mi>P</mi> <mi>D</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <msub> <mi>N</mi> <mi>B</mi> </msub> <msub> <mi>t</mi> <mi>B</mi> </msub> </mrow> <msub> <mi>T</mi> <mi>f</mi> </msub> </mfrac> <mo>&le;</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow> </math>

And 4, solving the optimization problem by adopting a genetic algorithm:

obtaining a set of parameters [ K ] by solving the optimization process of the optimization model_opt,η_opt,t_Bopt,T_fopt]And on the premise of meeting the requirement of search performance, the consumption of resources is minimum.