CN107918116A - A kind of multiple target radar waveform design method based on radar and communications coexistence of systems - Google Patents

A kind of multiple target radar waveform design method based on radar and communications coexistence of systems Download PDF

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CN107918116A
CN107918116A CN201710990099.8A CN201710990099A CN107918116A CN 107918116 A CN107918116 A CN 107918116A CN 201710990099 A CN201710990099 A CN 201710990099A CN 107918116 A CN107918116 A CN 107918116A
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mrow
msub
radar
msup
mtd
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CN107918116B (en
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时晨光
汪飞
李海林
周建江
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Nanjing University of Aeronautics and Astronautics
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Nanjing University of Aeronautics and Astronautics
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    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/418—Theoretical aspects
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01S—RADIO DIRECTION-FINDING; RADIO NAVIGATION; DETERMINING DISTANCE OR VELOCITY BY USE OF RADIO WAVES; LOCATING OR PRESENCE-DETECTING BY USE OF THE REFLECTION OR RERADIATION OF RADIO WAVES; ANALOGOUS ARRANGEMENTS USING OTHER WAVES
    • G01S7/00—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00
    • G01S7/02—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00
    • G01S7/41—Details of systems according to groups G01S13/00, G01S15/00, G01S17/00 of systems according to group G01S13/00 using analysis of echo signal for target characterisation; Target signature; Target cross-section
    • G01S7/414—Discriminating targets with respect to background clutter

Abstract

The present invention provides a kind of multiple target radar waveform design method based on radar and communications coexistence of systems, this method is in the case where radar and communications system works in same frequency range, to maximize the total Signal to Interference plus Noise Ratio of radar system multi-target detection as optimization aim, on the premise of the total emitted energy limitation of radar is met, the transmitted waveform to relatively each target of radar carries out adaptive optimal controls.The advantages of invention is the communication quality that both ensure that communication system, and improves the multi-target detection performance of radar system.The reason for producing the advantage is that present invention employs radar optimum waveform design method, in the case where considering clutter PSD and signal of communication PSD, the multiple target radar optimum waveform based on radar and communications coexistence of systems is established to design a model, by optimizing the transmitting signal ESD of relatively each target of radar, to maximize the total SINR of radar system multi-target detection.

Description

A kind of multiple target radar waveform design method based on radar and communications coexistence of systems
Technical field：
The invention belongs to the technical field of radar waveform design, specifically proposes a kind of based on radar and communications coexistence of systems Multiple target radar waveform design method.
Background technology：
The transmitted waveform of radar has very big associate with the information that it is delivered.Said from practice, the conjunction of transmitted waveform Reason design not only directly affects the performances such as radar system resolution performance, measurement accuracy, interference free performance, target component estimation, and And influence the complexity of signal processing algorithm；In addition, also to take into account the complexity that hardware produces the waveform.Accordingly, it is capable to no conjunction It is particularly important that transmitted waveform of reason design radar is in radar system design.In addition, it is increasingly sophisticated with Battle Field Electromagnetic, Radar waveform design under the conditions of intensive spectrum is become for an important and extremely challenging task.Traditional solution radar Method with wireless communication system radio frequency (Radio Frequency, RF) frequency spectrum congestion is to separate both working frequency range, with Avoid to forming interference each other.However, in face of the increasingly extension sharply increased with bandwidth of operation of wireless RF equipment quantity, tradition Method has been increasingly difficult to meet the actual demand of radar system.In this context, the radar in frequency spectrum share environment With wireless communication system using technical works such as waveform optimization designs in same frequency range, and can be effectively prevented to mutual Working performance impacts.
Environment residing for function that the optimization design of radar emission waveform to be completed with radar, target and target is wanted Ask as foundation, the purpose is to can extract target information in complex environment exactly.In fact, adaption radar Waveform Design Not only constrained by system condition, while need to carry out under Waveform Design criterion.The constraints of system is by modern signal What treatment technology and hardware condition limited, such as limitation of energy limit, bandwidth, time width limitation and permanent mould limitation；And Waveform Design The factors such as criterion and the task of radar, working environment are closely related, for target detection, usually with Signal to Interference plus Noise Ratio (Signal-to-Interference-plus-Noise Ratio, SINR), detection probability, detection time, signal and clutter Correlation etc. is design criteria；It is that the mutual information between tracking error, echo and target is accurate as design mostly for target following Then；For target identification, the mutual information between distance measure, target and echo, target impulse typically between target classification are rung The evaluated error answered is design criteria.
Although conventional method proposes the thought of radar waveform optimization design, meeting the condition of radar system energy constraint Under, the target detection performance of system is improved, but these methods are directed to single goal scene, and radar and communications system is not considered The situation of frequency spectrum share.In addition, existing radar waveform design method assumes that radar and communications system is mutually divided on frequency spectrum From, be independent of each other, however, in practical applications, with radar and wireless communication system quantity sharply increase and bandwidth of operation Extension, radar and communications system usually coexists on frequency spectrum, and both sides can have an impact mutual performance.
The content of the invention：
Technical problem underlying to be solved by this invention is：Radar system works with communication system in actual environment is considered In the case of same frequency range, on the basis of the limitation of radar system total emitted energy is met, by radar waveform optimization design, The total SINR of multi-target detection is maximized, so as to lift the multi-target detection performance of radar system.
The present invention is from practical application, it is proposed that a kind of multiple target radar waveform based on radar and communications coexistence of systems Design method, on the basis of the total emitted energy limitation of radar system is met, by radar waveform optimization design, maximizes more mesh Mark detects total SINR, so as to lift the multi-target detection performance of radar system.
The present invention uses following technical scheme to solve above-mentioned technical problem：
A kind of multiple target radar waveform design method based on radar and communications coexistence of systems, includes the following steps：
(1) frequency response of each target with respect to radar is obtainedEnergy round trip propagation loss Lr,i, frequency The corresponding environment clutter power spectrum density of rate f points (Power Spectral Density, PSD) Scc,i(f) and signal of communication PSDScom(f)；
(2) using SINR characterization target detection performances.The total emitted energy E of given radar systemx, establish the optimal transmitting of radar Waveform Xi(f) mathematical model of design is as follows：
In formula, BW represents radar emission waveform bandwidth, NQRepresent the target number of radar system detection, αiRepresent i-th of mesh Target priority level, andLcomRepresent that the energy round trip of communication system to radar receiver is lost, Snn(f) frequency is represented The corresponding noise of radar receiver PSD of f points.
(3) Lagrange multiplier ξ is introduced, it is as follows to build Lagrangian object function：
It is right respectively | Xi(f)|2First-order partial derivative is sought with ξ.
(4) order is passed through
Meet at the same timeCaro need-the Kuhn-Tucker condition solved with nonlinear optimization The necessary condition of (Karush-Kuhn-Tucker, KKT), obtains optimal transmitted waveform energy of the radar system relative to each target Spectrum density (Energy Spectral Density, ESD) | Xi(f)|2Expression formula is：
|Xi(f)|2=max [0, Bi(f)(A-Di(f))]
In formula,
Compared with prior art, the invention has the advantages that：
1. the present invention proposes a kind of multiple target radar waveform design method based on radar and communications coexistence of systems, the party The main task that method is completed is in the case where considering that radar and communications system works in same frequency range, to maximize radar system The total SINR of multi-target detection that unites is optimization aim, on the premise of the total emitted energy limitation of radar is met, to relatively each mesh of radar Target transmitted waveform carries out adaptive optimal controls.
The advantages of invention is the communication quality that both ensure that communication system, and improves the multiple target inspection of radar system Survey performance.The reason for producing the advantage be present invention employs radar optimum waveform design method, consider clutter PSD and In the case of signal of communication PSD, establish the multiple target radar optimum waveform based on radar and communications coexistence of systems and design a model, lead to The transmitting signal ESD of optimization relatively each target of radar is crossed, to maximize the total SINR of radar system multi-target detection.
2. compared with prior art, the multiple target radar waveform proposed by the present invention based on radar and communications coexistence of systems is set Meter method, not only allows for the influence of clutter PSD and signal of communication PSD to radar system, and ensure that the logical of communication system Believe quality, improve the multi-target detection performance of radar system.
Brief description of the drawings：
Fig. 1 is radar waveform design flow diagram.
Fig. 2 is multiple target radar waveform send-receive functional block diagram.
Fig. 3 is frequency response and clutter PSD of the target 1 relative to radar.
Fig. 4 is frequency response and clutter PSD of the target 2 relative to radar.
Fig. 5 is signal of communication PSD.
Fig. 6 is optimum waveform design result of the radar to target 1.
Fig. 7 is optimum waveform design result of the radar to target 2.
Change curves of the Fig. 8 for SINR under distinct methods with the total emitted energy of radar.
Embodiment：
The structure and the course of work of the present invention are described further below in conjunction with the accompanying drawings.
As shown in Figure 1, the present invention includes the following steps：
1st, each target frequency response and signal of communication PSD are determined
The present invention proposes a kind of multiple target radar waveform design method based on radar and communications coexistence of systems.It is considered The influence of environment clutter PSD and communication system transmitting signal PSD to the optimal transmitted waveform of radar, therefore, should determine each mesh first Mark is relative to the frequency response of radar, the propagation loss of energy round trip, clutter PSD and signal of communication PSD.
2nd, the radiation parameter and the total emitted energy E of radar system of radar system are determinedx
Assuming that radar emission waveform bandwidth is BW, minimum step frequency is Δ f, the transmitter antenna gain (dBi) of radar and receives day Line gain is G, and the power spectrum of additive white Gaussian noise is Snn(f), the total emitted energy E of radar systemx, each target priority grade αi。
3rd, build Lagrangian object function K (| Xi(f)|2, ξ), and determine to meet the total emitted energy E of radar systemxMost Big Signal to Interference plus Noise RatioExpression formula
According to requirement of the radar system to multi-target detection performance, and consider that radar and communications system works in same frequency Section, establishes the multiple target radar waveform X based on radar and communications coexistence of systemsi(f) mathematical model of optimization design, following institute Show：
Lagrange multiplier ξ is introduced, shown in structure Lagrange multiplier formula such as formula (2)：
4th, design can solve nonlinear equation K (| Xi(f)|2, ξ) optimize KKT conditions
To determine optimal transmitted waveform ESD of the radar relative to each target | Xi(f)|2, by K in formula (2) (| Xi(f)|2,ξ) It is right respectively | Xi(f)|2Local derviation is sought with ξ, and is made
Meet at the same timeThe KKT necessary conditions solved with nonlinear optimization, it is as follows：
Wherein, target variable represents the optimal solution of each parameter respectively on all bands " * ".
5th, realize nonlinear equation K (| Xi(f)|2, ξ) optimization
By solving formula (4), the optimal transmitted waveform ESD of relatively each target of radar under the conditions of radar and communications coexistence of systems |Xi(f)|2It is represented by：
In formula,AndIt can represent as follows respectively：
It is a constant, its size depends on the total emitted energy of radar：
Iterated to calculate through dichotomy, will meet the A of formula (8)*It is worth in substitution formula (5), trying to achieve examines radar system multiple target Survey one group of transmitted waveform of SINR maximumsAs optimal solution, and finally determine total SINR value of system.
6th, simulation result
Assuming that the parameter in the 2nd step is as shown in table 1.
1 simulation parameter of table is set
Target 1 relative to radar frequency response and clutter PSD as shown in figure 3, target 2 relative to radar frequency response With clutter PSD as shown in figure 4, signal of communication PSD is as shown in Figure 5.Multiple target radar based on radar and communications coexistence of systems is most Excellent Waveform Design result difference is as shown in Figure 6, Figure 7.Multiple target radar optimum waveform based on radar and communications coexistence of systems is set Meter method is to calculate the optimal transmitting of gained relative to the frequency response of radar, clutter PSD and signal of communication PSD according to each target Waveform.The waveform emitted energy of radar system configures the mainly frequency by each target relative to radar it can be seen from Fig. 3 to Fig. 7 Rate response, clutter PSD and signal of communication PSD determine that in the assignment procedure, emitted energy mainly distributes to target frequency response The target high, clutter PSD is small, signal of communication PSD is small.It is more in order to be maximized under conditions of the constraint of radar system gross energy is met Total SINR of target detection, the multiple target radar optimum waveform design method based on radar and communications coexistence of systems are former according to water filling Reason carries out energy distribution, i.e., at the frequency point corresponding to target frequency response maximum, clutter PSD minimums, signal of communication PSD minimums Distribute most energy.
Fig. 8 gives under different wave design method total SINR with the change curve of the total emitted energy of radar.Can by Fig. 8 Know, under conditions of the constraint of radar system gross energy is met, total SINR value obtained by the optimal transmitted waveform of radar system is substantially high In total SINR value obtained by homogeneous energy distribution transmitted waveform, this is because homogeneous energy distribution transmitted waveform is not any It is in the case of on prioris such as target frequency response, clutter PSD and signal of communication PSD, radar waveform emitted energy is equal Even to distribute in whole frequency range, therefore, it has worse multi-target detection performance.
From above-mentioned simulation result, the multiple target radar optimum waveform design side based on radar and communications coexistence of systems Method, it is contemplated that the influence of clutter PSD and signal of communication PSD to radar system, it is total to maximize the multi-target detection of radar system SINR is target, adaptive optimal controls radar relative to each target transmitted waveform, so as to both ensure that the logical of communication system Believe quality, and effectively improve the multi-target detection performance of radar.
The operation principle and the course of work of the present invention：
The present invention in the case where radar system and communication system work in same frequency range, according to priori, obtains first Each target is taken relative to the frequency response of radar, propagation loss, environment clutter PSD and signal of communication PSD；Then, with maximum The total SINR of multi-target detection for changing radar system is target, and on the premise of the total emitted energy of radar system is met, foundation is based on The multiple target radar waveform mathematical optimization models of radar and communications coexistence of systems, and model is solved by KKT conditions.Through Dichotomy iterates to calculate, and is chosen at and meets to cause the total SINR of radar system multi-target detection under conditions of radar system energy constraint The maximum optimal transmitted waveform of radarAs optimal solution, and the optimal transmitted wave by radar relative to each target ShapeBring into formula (1), you can obtain the radar system maximum SINR for meeting constraints.
The inventive point of the present invention：
1st, in the case of considering that radar and communications system works in same frequency range in practice, according to priori, obtain each Target calculates radar relative to the frequency response of radar, the propagation loss of energy round trip, environment clutter PSD and signal of communication PSD The total SINR of multi-target detection of system；
2nd, using the total SINR of multi-target detection for maximizing radar system as target, the total emitted energy of radar system is being met Under the premise of, the multiple target radar waveform mathematical optimization models coexisted based on radar-communication system are established, and using formula (1) as mesh Scalar functions, solve this problem using KKT conditions, are iterated to calculate through dichotomy, determine the optimal of each radar in system Transmitted waveform

Claims (3)

  1. A kind of 1. multiple target radar waveform design method based on radar and communications coexistence of systems, it is characterised in that radar with In the case that communication system works in same frequency range, the mathematical model of the optimal transmitted waveform optimization design of radar is established.
  2. 2. a kind of multiple target radar waveform design method based on radar and communications coexistence of systems according to claim 1, It is characterized in that, the mathematical model constrains bar to maximize the total Signal to Interference plus Noise Ratio of radar system multi-target detection as optimization aim Part is the total emitted energy limitation of radar.
  3. 3. a kind of multiple target radar waveform design method based on radar and communications coexistence of systems according to claim 2, It is characterized in that, comprise the following steps that：
    Step 1：Obtain frequency response of each target with respect to radarEnergy round trip propagation loss Lr,i, frequency The corresponding environment clutter power spectrum density S of rate f pointscc,i(f) and communication signal power spectrum density Scom(f)；NQRepresent radar system The target number of system detection；
    Step 2：The total emitted energy E of given radar systemx, establish the optimal transmitted waveform X of radari(f) mathematical model of optimization design It is as follows：
    <mfenced open = "" close = "}"> <mtable> <mtr> <mtd> <munder> <mi>max</mi> <msup> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> </munder> </mtd> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>Q</mi> </msub> </munderover> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <msub> <mo>&amp;Integral;</mo> <mrow> <mi>B</mi> <mi>W</mi> </mrow> </msub> <mfrac> <mrow> <msup> <mrow> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>S</mi> <mrow> <mi>c</mi> <mi>c</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>S</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <msub> <mi>L</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>s</mi> <mo>.</mo> <mi>t</mi> <mo>.</mo> <mo>:</mo> </mrow> </mtd> <mtd> <mrow> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>Q</mi> </msub> </munderover> <msup> <mrow> <mo>|</mo> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>f</mi> <mo>&amp;le;</mo> <msub> <mi>E</mi> <mi>x</mi> </msub> <mo>.</mo> </mrow> </mtd> </mtr> </mtable> </mfenced>
    In formula, BW represents radar emission waveform bandwidth, αiRepresent the priority level of i-th of target, andLcomRepresent communication The energy round trip of system to radar receiver is lost, Snn(f) the corresponding noise of radar receiver power spectrum of frequency f points is represented Degree；
    Step 3：Lagrange multiplier ξ is introduced, it is as follows to build Lagrangian object function：
    <mfenced open = "" close = ""> <mtable> <mtr> <mtd> <mrow> <mi>K</mi> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi>&amp;xi;</mi> <mo>)</mo> </mrow> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>Q</mi> </msub> </munderover> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <msub> <mo>&amp;Integral;</mo> <mrow> <mi>B</mi> <mi>W</mi> </mrow> </msub> <mfrac> <mrow> <msup> <mrow> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <msub> <mi>S</mi> <mrow> <mi>c</mi> <mi>c</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>S</mi> <mrow> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>S</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <msub> <mi>L</mi> <mrow> <mi>c</mi> <mi>o</mi> <mi>m</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mi>&amp;xi;</mi> <mo>&amp;CenterDot;</mo> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mi>x</mi> </msub> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <msub> <mi>N</mi> <mi>Q</mi> </msub> </munderover> <munder> <mo>&amp;Integral;</mo> <mrow> <mi>B</mi> <mi>W</mi> </mrow> </munder> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mi>d</mi> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
    Step 4：Lagrangian object function in step 3 is right respectively | Xi(f)|2First-order partial derivative is sought with ξ；And make：
    <mfenced open = "" close = "}"> <mtable> <mtr> <mtd> <mrow> <mo>{</mo> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>K</mi> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi>&amp;xi;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>/</mo> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>{</mo> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>K</mi> <mo>(</mo> <mrow> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>X</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mn>2</mn> </msup> <mo>,</mo> <mi>&amp;xi;</mi> </mrow> <mo>)</mo> <mo>)</mo> </mrow> <mo>/</mo> <mo>&amp;part;</mo> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>)</mo> </mrow> <mo>}</mo> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mfenced>
    Meet at the same timeThe necessary bar of the Caro need-Kuhn-Tucker condition solved with nonlinear optimization Part, obtains optimal transmitted waveform energy spectral density of the radar system relative to each target | Xi(f)|2Expression formula is：
    |Xi(f)|2=max [0, Bi(f)(A-Di(f))]
    In formula,
    <mrow> <mfenced open = "" close = "}"> <mtable> <mtr> <mtd> <mrow> <msub> <mi>B</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> <mrow> <msub> <mi>S</mi> <mrow> <mi>c</mi> <mi>c</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msqrt> <mfrac> <mrow> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <msub> <mi>S</mi> <mrow> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>D</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mo>|</mo> <msub> <mi>H</mi> <mi>i</mi> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> <mo>|</mo> </mrow> </mfrac> <mo>&amp;CenterDot;</mo> <msqrt> <mfrac> <mrow> <msub> <mi>S</mi> <mrow> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>(</mo> <mi>f</mi> <mo>)</mo> </mrow> </mrow> <mrow> <msub> <mi>&amp;alpha;</mi> <mi>i</mi> </msub> <msub> <mi>L</mi> <mrow> <mi>r</mi> <mo>,</mo> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>A</mi> <mo>=</mo> <msqrt> <mfrac> <mn>1</mn> <mi>&amp;xi;</mi> </mfrac> </msqrt> <mo>,</mo> <mrow> <mo>(</mo> <mi>&amp;xi;</mi> <mo>&gt;</mo> <mn>0</mn> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>.</mo> </mrow>
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