A target detection method for communication and perception integrated system
By adopting virtual local echo technology in the integrated multi-antenna communication and perception system, the detection performance problem of single-antenna object detection receiver in a multi-antenna transmission signal environment is solved, and higher target detection accuracy and reliability are achieved.
Patent Information
- Application Number
- CN202411856388.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2024-12-17
- Publication Date
- 2025-03-11
- Estimated Expiration
- 2044-12-17
AI Technical Summary
In the integrated multi-antenna communication and perception system, it is difficult for a single-antenna object detection receiver to effectively handle the degradation of target detection performance caused by multi-antenna transmitted signals, especially due to the randomness of the transmitted signals and the influence of random additive noise.
Virtual local echo (VLE) technology is used to reduce the impact of target direction on echo in a single-antenna target detection receiver, and the detection capability is enhanced through the combined use of multiple VLEs.
Through VLE technology, the impact of the target direction on echo can be effectively removed, and the accuracy and reliability of target detection can be improved, especially under low signal-to-noise ratio conditions, the detection performance is significantly improved.
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Figure CN119322323B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of signal processing and target detection, and in particular to a target detection method for a communication-perception integrated system. Background Art
[0002] The ISAC system uses shared waveforms to simultaneously perform communication and perception tasks, which can improve the utilization of time and frequency resources while reducing the hardware cost of the system. It is one of the current hot spots in the development of 5G and 6G wireless communication technologies.
[0003] The perception function of the ISAC system usually involves four main steps: target detection, parameter estimation, identification and tracking. As the first step of the perception function, target detection is crucial to the ISAC system. Target detection refers to the perception receiver judging whether there is a target in the echo signal based on the received signal. Most of the existing methods for realizing the target detection function are based on classic energy or power detection, and do not care about the waveform characteristics of the transmitted signal itself. However, in the communication-perception integrated system centered on communication performance, an important reason for the decline in target detection performance is that the transmission signal of the communication-perception integrated system is a random communication signal. This random characteristic causes the system's detection capability to be unable to achieve the optimal. Wang Jiale and others from Northwestern Polytechnical University proposed a detection algorithm that uses the statistical characteristics of the echo to improve the detection performance of underwater acoustic communication signals. This method uses the difference in random statistical characteristics between the transmitted signal and the echo signal to achieve target detection, effectively counteracting the reduction in detection performance caused by the randomness of the transmitted signal. However, the method is used for a single-input single-output system. For a multi-antenna ISAC system, it is still difficult to effectively improve the target detection capability.
[0004] In the communication-centric multi-antenna ISAC system, in addition to the above problems that lead to reduced detection performance, there is also a new problem: the multi-antenna ISAC system generally uses an antenna array as the transmitting antenna to support multi-user communication. Correspondingly, this also makes the transmitting antenna of the target detection module a multi-antenna system; but the detection receiver usually uses a single antenna to receive signals. This is because the target detection function is the primary function of the radar system and usually has an early warning function. Therefore, the target detection system is required to be able to operate normally for a long time with low power consumption. The use of a single antenna reception can reduce the power consumption and complexity of the system, thereby achieving long-term early warning capabilities. However, in the multi-antenna ISAC setting, the echo signal received by the single-antenna detection receiver is often closely related to the target direction. Since multiple antennas at the transmitting end radiate signals at the same time, the echo phases of different antennas irradiating the target are related to the target orientation. The existing detection algorithms based on energy or correlation lack consideration of the impact of the target direction on the echo, resulting in a single antenna being unable to distinguish the difference in echoes from different directions. Therefore, it cannot handle the impact of directional information, which greatly reduces the target detection capability of the multi-antenna system. Summary of the invention
[0005] In view of the target detection problem existing in the communication-centric multi-antenna ISAC system, the present invention proposes a target detection method for a communication-awareness integrated system. In this method, a virtual local echo (VLE) is used at a single-antenna target detection receiver to reduce the influence of the target direction on the received echo. In order to further solve the problem of detection performance degradation caused by random communication signals and random additive noise, multiple VLEs are used jointly for target detection to enhance the detection capability.
[0006] The technical solution of the present invention is:
[0007] The communication and perception integrated system target detection method comprises the following steps:
[0008] Step 1: Obtain communication data of the communication perception integrated system , and the signal received by the single antenna sensing receiver in the communication sensing integrated system ;
[0009] Step 2: Create a collection and :
[0010]
[0011]
[0012] in and All are set values;
[0013] Step 3: In the collection and Randomly select the For possible target direction parameters ;in is the possible target azimuth, , is the possible target pitch angle, ;
[0014] Step 4: Create a virtual local echo :
[0015]
[0016] in For the direction parameter The launch array guidance vector of the target;
[0017] Step 5: Create intermediate variables :
[0018]
[0019] And Perform a Fast Fourier Transform ;
[0020] Step 6: Repeat steps 3 to 5 for a total of K times to obtain K ,calculate
[0021]
[0022] Pick The maximum value of As a test statistic; according to
[0023]
[0024] Get indicator factor Estimated value of ,in To test the threshold, when the indicator factor estimate is 0, it means that there is no target, and when the indicator factor estimate is 1, it means that there is a target.
[0025] In a further preferred embodiment, the communication sensing integrated system comprises an ISAC transceiver and a communication receiver, wherein the ISAC transceiver comprises a The invention comprises a circular transmitting array with array elements and a single-antenna sensing receiver. The circular transmitting array transmits modulated communication signals for information interaction with users and detection of possible targets. The receiving antenna of the single-antenna sensing receiver is located at the center of the circular transmitting array to receive target reflected echo and background noise. The single-antenna sensing receiver is used to detect whether there is a communication signal echo reflected by the target.
[0026] In a further preferred embodiment, the ISAC signal transmitted by the ISAC transceiver is a standard communication signal, which is encoded in an OFDM modulation format.
[0027] A further preferred solution is: According to the formula
[0028] a d ( i ( k ) , ϕ ( k ) ) = [ e − j 2 π D f r s c s i n i ( k ) c o s ( ϕ ( k ) − f 1 ) e − j 2 π D f r s c s i n i ( k ) c o s ( ϕ ( k ) − f 2 ) ⋮ e − j 2 π D f r s c s i n i ( k ) c o s ( ϕ ( k ) − f L ) ]
[0029] Determine, among which is an imaginary unit, is the frequency interval of OFDM modulation, is the radius of the annular transmitting array, is the propagation speed of electromagnetic waves, It is the first in the ring transmitting array The horizontal angle of the transmitting element, .
[0030] A further preferred solution is: Depend on The received signal on the subcarriers consists of
[0031] y s = [ y s , 0 , y s , 2 , ⋯ , y s , N − 1 ] T = α a R ( r o ) ⊙ S a d ( i o , ϕ o ) + w s
[0032] is the number of subcarriers used in OFDM modulation, The received signal on the subcarrier for
[0033]
[0034] in is the target distance, Representative Matrix No. row vector, For the direction parameter The target's launch array steering vector, is the azimuth of the target, is the pitch angle of the target, is additive noise; is the additive noise vector, is the dot product operator, is the middle vector:
[0035] a R ( r o ) = [ 1 e − j 2 π D f 2 r o c ⋮ e − j 2 π ( N − 1 ) D f 2 r o c ] .
[0036] Beneficial effects:
[0037] The present invention uses a virtual local echo at a single-antenna target detection receiver to reduce the impact of the target direction on the received echo, and smoothes the impact of the direction on the echo, thereby removing the interference of the direction information on the target detection and improving the target detection performance of the communication-aware integrated system. Numerical simulations show that the present invention has obvious advantages over correlation-based or energy-based detection methods in a communication-centric multi-antenna ISAC system.
[0038] Additional aspects and advantages of the present invention will be given in part in the following description and in part will be obvious from the following description, or will be learned through practice of the present invention. BRIEF DESCRIPTION OF THE DRAWINGS
[0039] The above and / or additional aspects and advantages of the present invention will become apparent and easily understood from the description of the embodiments in conjunction with the following drawings, in which:
[0040] Figure 1 : Communication perception integrated system model;
[0041] Figure 2 : In the absence of noise, formula (19) corresponds to the spectrum, where the target azimuth information: distance 412.5 meters, horizontal angle 85.7 degrees, pitch angle 139.62 degrees;
[0042] Figure 3 : In the absence of noise, formula (19) corresponds to the spectrum, where the target azimuth information: distance 390 meters, horizontal angle 32.96 degrees, pitch angle 66.03 degrees;
[0043] Figure 4 : Number of VLEs Impact on detection performance: Schematic diagram of detection accuracy;
[0044] Figure 5 : Number of VLEs Impact on detection performance: ROC curve;
[0045] Figure 6 :The method of the present invention is similar to the correlation-based method (COR) in Comparison results when: Detection accuracy diagram;
[0046] Figure 7 :The method of the present invention is similar to the correlation-based method (COR) in Comparison results: ROC curve. DETAILED DESCRIPTION
[0047] Embodiments of the present invention are described in detail below. The embodiments are exemplary and intended to be used to explain the present invention, but should not be construed as limiting the present invention.
[0048] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "length", "width", "thickness", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside", "clockwise", "counterclockwise" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the referred device or element must have a specific orientation, be constructed and operated in a specific orientation, and therefore should not be understood as limiting the present invention.
[0049] In addition, the terms "first" and "second" are used for descriptive purposes only and should not be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, a feature defined as "first" or "second" may explicitly or implicitly include one or more of the features. In the description of the present invention, the meaning of "plurality" is two or more, unless otherwise clearly and specifically defined.
[0050] 1. System model
[0051] Figure 1 The communication-centric multi-antenna ISAC system of this embodiment includes an ISAC transceiver and a communication receiver. The ISAC transceiver includes a The ring-shaped transmitting array consists of 16 elements (ISAC transmitting antennas) and a single-antenna sensing receiver. The ring-shaped transmitting array transmits modulated communication signals for information interaction with users and detection of possible targets. The receiving antenna of the single-antenna sensing receiver is located at the center of the ring-shaped transmitting array to receive target reflected echoes and background noise. The single-antenna sensing receiver is used to detect whether there is a communication signal echo reflected by the target. For the convenience of representation, the receiving antenna of the single-antenna sensing receiver is set as the origin of the coordinate system, and the ISAC transceiver accurately knows the transmitted ISAC signal.
[0052] 1. Transmitting signal model
[0053] In this embodiment, the ISAC signal is a standard communication signal with a bandwidth of The specific process is as follows: first, the communication signal to be sent is modulated in the array element-frequency domain. Depend on A group of random independent binary sequences are mapped to the corresponding constellation diagram to generate, represents the number of subcarriers used in OFDM modulation. Perform an inverse Fourier transform (IFFT) to obtain the time domain transmission signal on each array element, as shown in the following formula
[0054] (1)
[0055] Among them, the Fourier transform matrix Expressed as
[0056] [ 1 1 ⋯ 1 1 e j 2 π D f T s ⋯ e j 2 π D f ( M − 1 ) T s ⋮ ⋮ ⋮ ⋮ 1 e j 2 π ( N − 1 ) D f T s ⋯ e j 2 π ( N − 1 ) D f ( M − 1 ) T s ] (2)
[0057] In the above formula represents the frequency interval of OFDM modulation, is the number of time domain sampling points, is the time domain sampling interval. Represents the baseband time domain transmission signal on all antennas. After the baseband signal is obtained, a cyclic prefix (CP) is added to the baseband signal, digital-to-analog conversion and up-conversion are performed, and the communication signal is transmitted through the circular transmission array.
[0058] 2. Receive signal model
[0059] For the target detection function, the signal received by the single-antenna sensing receiver can be expressed as
[0060] (3)
[0061] in is an indicator factor, with a value of 0 or 1. Represents additive Gaussian noise in the time domain.
[0062] For indicator factors , its two values represent two situations:
[0063] (1) Target does not exist: When When , it means that there is no target and the received signal contains only background noise. , there is no echo signal generated by the target.
[0064] (2) Target existence: When When , it means there is a target. Therefore, the echo component of the received signal It can be expressed as
[0065] (4)
[0066] The above formula represents the distance , direction is The reflected echo generated by a single static target, where and Indicates the azimuth and elevation angle of the target. is the attenuation coefficient caused by wave propagation and target reflection. Usually, the attenuation factor It is a constant within a certain period of time and only affects the signal-to-noise ratio of the received signal. Therefore, for simplicity, the attenuation factor will not be considered in the following text. In order to reduce the impact of the received signal, only the signal-to-noise ratio of the received signal is studied. Represents the propagation speed of electromagnetic waves. represents the direction of the wave propagating toward the target, expressed as
[0067] a ( i o , ϕ o ) = [ − s i n i o c o s ϕ o , − s i n i o s i n ϕ o , − c o s i o ] T (5)
[0068] Column vector
[0069] p l = r s [ − c o s f l , − s i n f l , 0 ] T (6)
[0070] Represents the first The position of the transmitting array element, is the radius of the circular transmitting array, It is The horizontal angle of the transmitting element. Represents a transpose operation. No. The continuous representation of a row is denoted by , Indicates that from The baseband transmission signal of the array element
[0071] (7)
[0072] in represents the duration of the cyclic prefix, represents the full OFDM symbol duration with cyclic prefix, is a duration of 0 to Rectangular function of ; represent No. Line Elements in a column.
[0073] 3. Problem Formation
[0074] The present invention mainly focuses on the target detection problem in the communication-sensing integrated system, that is, judging whether a target exists based on the signal received by a single antenna sensing receiver. Based on the received signal model, the target detection problem is summarized as the parameter The binary estimation problem is
[0075] (8)
[0076] When the estimated value With actual value If they are the same, it indicates that the target detection system detects correctly. Next, in this embodiment, an effective algorithm will be constructed , used to accurately estimate the parameters .
[0077] 2. Target Detection Method
[0078] Assuming that the length of CP is greater than the echo signal delay caused by the target, the CP of the received signal is removed. Integrate equations (5), (6) and (7) into (4). Then apply fast Fourier transform to equations (3) and (4) to obtain The received signal on the subcarrier for
[0079] (9)
[0080] in Representative Matrix No. row vector, is additive noise, For the direction parameter The target launch array steering vector is:
[0081] a d ( i o , ϕ o ) = [ e − j 2 π D f r s c s i n i o c o s ( ϕ o − f 1 ) e − j 2 π D f r s c s i n i o c o s ( ϕ o − f 2 ) ⋮ e − j 2 π D f r s c s i n i o c o s ( ϕ o − f L ) ] (10)
[0082] Will The received signal on the subcarriers forms a vector
[0083] y s = [ y s , 0 , y s , 2 , ⋯ , y s , N − 1 ] T = α a R ( r o ) ⊙ S a d ( i o , ϕ o ) + w s (11)
[0084] in is the middle vector, is the additive noise vector, is the dot product operator;
[0085] a R ( r o ) = [ 1 e − j 2 π D f 2 r o c ⋮ e − j 2 π ( N − 1 ) D f 2 r o c ] (12)
[0086] Before implementing the target detection function, we first study the definition in formula (12) This can be seen as a frequency value of If this term in equation (11) is removed and FFT is applied to the remaining term, a spectrum with a sharp peak will be obtained. This peak is generated by the sinusoidal characteristics of equation (12).
[0087] However, since the target direction is unknown, it is difficult to effectively remove the direction interference. is impossible. One method is to exhaustively search all possible combinations of target positions. When the target exists, if the exhaustive value is the same as the true value, we can get Item, so as to generate peaks using FFT. When the exhaustive value is different from the true value, it cannot be obtained. Therefore, it is difficult to obtain a sufficiently sharp peak after performing FFT. When the target does not exist, all exhaustive values cannot be generated. Therefore, we can identify whether the target exists by judging the peak of the spectrum generated by all exhaustive results. Although the exhaustive method can effectively solve this target detection problem, the exhaustive method often faces the problems of high computational complexity and grid search dimension, and is not the best choice in actual use. In order to reduce the computational dimension of the exhaustive method and obtain high-performance target detection capabilities, the present invention uses VLE to process the echo signal, analyzes the characteristics of the processed echo signal, and realizes target detection based on the processed echo characteristics, which can greatly reduce the amount of calculation.
[0088] First, create two collections to generate the VLE:
[0089] (13)
[0090] (14)
[0091] gather and The number of elements in are and , and is the setting value; select For possible target direction parameters for
[0092] (15)
[0093] (16)
[0094] Based on For possible target direction parameters, according to the model in formula (11), a new vector is established as
[0095] (17)
[0096] we will It is called a virtual local echo because it represents the echo from By utilizing these virtual local echoes, we can accurately identify the peaks of the spectrum.
[0097] Combine equation (17) with the received signal vector equation (11) to establish the intermediate variable
[0098] (18)
[0099] In formula (18), the element-by-element division operation is used and Perform a Fast Fourier Transform
[0100] (19)
[0101] in is the convolution operator, represents the impulse function, is the frequency variable. After the above processing of the received signal, we consider the signal spectrum characteristics in the two cases of target presence or target absence.
[0102] 1) Target existence: In the communication and perception integrated system used, a unique phenomenon is that if the target exists ( ), in formula (19), the ratio It usually exhibits a strong DC component and a weak broadband component. This characteristic causes a clear peak to appear at the zero-frequency component of its spectrum, such as Figure 2 and Figure 3 As shown. The convolution relationship between the spectrum of and the DC component, Equation (19) will also produce a sharp line spectrum.
[0103] 2) Target does not exist: When no target exists ( ), the received signal is obtained after being processed in equation (19): , the result is still a random signal, which is caused by additive noise is determined by the randomness of.
[0104] From the above analysis, it can be seen that the presence or absence of the target will produce a clear difference in the spectral characteristics of the processed signal, and the target's position information does not need to be known. Based on this difference, we can use the target without exhaustive enumeration without knowing the target's position information. In addition, Figure 2 and Figure 3 In the The DC component size at the value is different. This difference makes it not robust enough to use only a single VLE for target detection. Therefore, this embodiment further uses VLE is used to enhance the reliability of target detection. This process can be expressed as follows:
[0105] (20)
[0106] In the formula Represents the absolute value operation. Note that this does not require VLEs have the same properties and can be randomly drawn from sets (15) and (16) without considering traversal or scanning. Finally, we get the test statistic related to target detection:
[0107] (twenty one)
[0108] represent The maximum value of . According to the test statistic, we can get The estimated value is
[0109] (twenty two)
[0110] It represents the test threshold, which is related to the signal-to-noise ratio of the received signal and is a set empirical parameter.
[0111] (III) Simulation Verification
[0112] Next, the performance of the detection method proposed in the present invention is evaluated by numerical simulation. The numerical simulation parameters are shown in Table 1.
[0113] Table 1 Simulation parameters
[0114]
[0115] The ISAC system studied in this paper is centered on the communication function. Specifically, we use the standard multi-antenna OFDM system communication waveform to achieve the target detection function. Under this system configuration, the adopted ISAC system has the same communication performance as the traditional multi-antenna OFDM system. Therefore, the communication performance of the communication perception integrated system studied will no longer be shown. In order to quantitatively describe the detection performance of the proposed system, we define two indicators to evaluate the detection performance. The first indicator is the detection accuracy, defined as
[0116] (twenty three)
[0117] This represents the probability that the object detector successfully detects the object. Indicates Monte Carlo (MC) experiment. and Respectively represent The estimated and true values of the sub-MC simulation. Another indicator is the receiver operating characteristic (ROC) curve, which shows the different detection thresholds. The performance of the binary classifier, that is, the probability of detection at different thresholds and false alarm probability The relationship between.
[0118] exist Figure 4 and Figure 5 The detection performance and the number of VLEs are shown in The relationship between Figure 4 Detection accuracy When the signal-to-noise ratio is -10dB, the highest detection accuracy is is about 0.7, at this time . Increase the number of VLEs to , the maximum detection accuracy is increased to 0.85. Furthermore, When the SNR is increased to 0dB, the detection accuracy is very close to 1 even at low SNR levels. value, the maximum detection accuracy can basically reach 1. In addition, The value expands the threshold range where the detection accuracy remains 1. These results show that the parameter has a significant impact on the detection performance. Especially at lower signal-to-noise ratio levels, higher The target detection accuracy is significantly improved, and at a higher signal-to-noise ratio level, the larger Increased optimal detection threshold In addition, Figure 4 The ROC curve in shows that the higher Values correspond to better detection performance.
[0119] In addition, in order to demonstrate the superiority of the proposed method, a comparison is made with the traditional algorithm. The correlation-based (COR) method is used as a benchmark, which first calculates the correlation between the transmitted signal and the sum of the echoes, and then uses the maximum value of the correlation sequence as the test statistic to identify the target. Figure 5 and Figure 6 The performance curves of detection accuracy and ROC are shown in Figure 2. The results show that our proposed method can outperform the COR method even under low SNR conditions, such as -10dB.
[0120] The above experimental results prove that the embodiment proves the validity of the invention content of the scheme published by this method. It should be noted that the above embodiment is only a special case of the scheme published by this patent. All methods proposed by the present invention applied in the design of communication perception integrated waveforms are within the scope of protection of this patent, including but not limited to the following aspects:
[0121] 1. Use a modulation method different from that in this example, such as orthogonal time-frequency modulation (OTFS), orthogonal amplitude-phase modulation (QAM), etc.
[0122] 2. Use a different number of array elements and array formation than in this example;
[0123] 3. Use a communication symbol modulation method different from that of this example and use system parameters different from that of this example.
Claims
1. A communication and perception integrated system target detection method, characterized in that: The following steps are involved: Step 1: Obtain the communication data S of the integrated communication and perception system and the signal y received by the single antenna perception receiver in the integrated communication and perception system s ; Step 2: Create a set Φ θ and Φ φ : Where N θ and N φ All are set values, N θ +1 is the set Φ θ The number of elements, N φ +1 is the set Φ φ The number of elements of ; Step 3: In the set Φ θ and Φ φ Randomly select the kth possible target direction parameter (θ (k) ,φ (k) ), where θ (k) is the possible target azimuth, θ (k) ∈Φ θ ,φ (k) is the possible target pitch angle, φ (k) ∈Φ φ ; Step 4: Create a virtual local echo d (k) (θ (k) ,φ (k) ): d (k) (i (k) ,f (k) )=Sa d (i (k) ,f (k) ) where a d (θ (k) ,φ (k) ) is the direction parameter (θ (k) ,φ (k) ) of the target launch array guidance vector; Step 5: Create intermediate variables And Perform a Fast Fourier Transform Step 6: Repeat steps 3 to 5 for K times to get K calculate Pick The maximum value V max As a test statistic; according to Get an estimate of the indicator factor α Where γ is the test threshold. When the indicator factor estimate is 0, it means that there is no target. When the indicator factor estimate is 1, it means that there is a target.
2. According to claim 1, a communication and perception integrated system target detection method is characterized by: The communication and perception integrated system includes an ISAC transceiver and a communication receiver. The ISAC transceiver includes a circular transmitting array with L array elements and a single-antenna sensing receiver. The circular transmitting array transmits a modulated communication signal for interacting with user information and detecting possible targets. The receiving antenna of the single-antenna sensing receiver is located at the center of the circular transmitting array to receive target reflected echo and background noise. The single-antenna sensing receiver is used to detect whether there is a communication signal echo reflected by the target.
3. According to claim 2, a communication and perception integrated system target detection method is characterized by: The ISAC signal transmitted by the ISAC transceiver is a standard communication signal, which is encoded in OFDM modulation format.
4. According to claim 3, a communication and perception integrated system target detection method is characterized by: a d (θ (k) ,φ (k) ) According to the formula Determine, where j is the imaginary unit, Δf is the frequency interval of OFDM modulation, r s is the radius of the annular transmitting array, c is the propagation speed of the electromagnetic wave, It is the horizontal angle of the lth transmitting element in the circular transmitting array, l = 1, 2,…, L.
5. According to claim 3, a communication and perception integrated system target detection method is characterized by: y s It consists of the received signals on N subcarriers and s =[and s,0 ,and s,2 ,…,and s,N-1 ] T =aaa R (r o )⊙Sa d (i o ,f o )+w s N is the number of subcarriers used in OFDM modulation, and the received signal y on the nth subcarrier s,n for where r o is the target distance, s n represents the nth row vector of matrix S, a d (θ o ,φ o ) is the direction parameter (θ o ,φ o ) of the target’s launch array steering vector, φ o is the target pitch angle, θ o is the target azimuth, w s,n is additive noise; w s is the additive noise vector, ⊙ is the dot product operator, a R (r o ) is the intermediate vector:
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