Multi-bit uniformly quantized ISAC system target detection probability calculation method

Abstract

The invention belongs to the technical field of communication perception integration, and particularly relates to an approximate calculation method for a target detection probability in a multi-bit quantized ISAC system. The method comprises the following steps: firstly, UE (User Equipment) receives a signal sent by a base station to obtain a sensing information matrix; secondly, the sensing information matrix is quantized and then sent to a sensing information processing center; then calculating a characteristic function and a probability density function of an output result of the uniform quantizer at the time of delta-0; determining a probability density function of the quantized perception information; determining a detection model; obtaining an optimal detector and a probability density function of detection statistics by using an NP criterion; and finally, obtaining the target detection probability under the quantitative approximation condition. According to the method, in consideration of the problem of overlarge communication overhead in a sensing information reporting process in a communication and sensing scene, a target detection probability closed-form solution is obtained by utilizing quantitative approximation, and bandwidth required by transmission can be reduced to the maximum extent in a sensing precision range required by a sensing case.

Description

Technical Field

[0001] The present invention belongs to the technical field of communication perception integration, and particularly relates to an approximate calculation method for target detection probability in a multi-bit quantized ISAC system. Background Art

[0002] In next-generation wireless networks, sensing technology plays a key role in intelligent transportation, smart manufacturing, smart cities, and public safety. Integrated Sensing and Communications (ISAC) is considered a core technology for achieving ubiquitous sensing. Its goal is to simultaneously implement sensing and communication functions on shared wireless resources and hardware. This is because sensing and communication share similarities in hardware architecture, channel characteristics, and signal processing processes, resulting in significant advantages such as reduced hardware costs, improved spectrum utilization, and energy efficiency. The sensing modes supported in mobile communication networks are similar to those of radar. Sensing modes can be categorized as bistatic or monostatic, depending on whether the transmit and receive antennas are co-located. While monostatic sensing is subject to strong self-interference caused by transmission leakage, bistatic sensing does not suffer from this problem and does not require corresponding hardware modifications. However, bistatic sensing faces multiple challenges, including clutter suppression, high-precision synchronization, and efficient mechanisms for node selection and switching when the sensing target moves, ensuring the continuity and accuracy of the sensing task. To overcome the limitations of traditional single-base and dual-base sensing, existing research utilizes data from multiple receivers for joint processing to achieve more accurate range and Doppler estimation. This research has largely focused on improving sensing performance, while overlooking the data transmission overhead required to transmit local sensing data to a fusion processing center. Therefore, to reduce the communication burden, receiving devices typically quantize sensing information before transmitting it. Furthermore, in future 6G communications, sensing signal receiving devices, such as mobile phones and vehicles, will likely have limited computing resources and power constraints to achieve the Internet of Everything. Therefore, the receiving end typically quantizes and compresses the received sensing data before transmitting it via a communication loop to a unified sensing data processing center for sensing parameter extraction. Therefore, it is necessary to consider the impact of quantization on sensing performance in ISAC systems.

[0003] Currently, relatively little research has been conducted on quantization in ISAC systems. Existing literature primarily focuses on the impact of single-bit quantization on ISAC system performance. While this research provides preliminary insights into the fundamental impact of quantization on system performance, it is limited to single-bit quantization and fails to fully explore the potential advantages and complexities of multi-bit quantization. Furthermore, in mobile communication networks, the signals used for communication and perception integration are typically OFDM signals. However, current studies of the impact of quantization on perception performance in integrated systems have been based on radar waveforms (e.g., linear frequency modulation signals), which are incompatible with mobile communication networks. Therefore, it is necessary to explore the impact of multi-bit uniform quantization on perception performance in ISAC systems. Summary of the Invention

[0004] The purpose of this invention is to propose an approximate calculation method for target detection probability under multi-bit uniform quantization in an ISAC system. By constructing an approximate model of quantization noise, the target detection probability after multi-bit uniform quantization is derived, and a closed-form detection probability is obtained to facilitate subsequent theoretical analysis and guide system design. The method specifically includes the following steps:

[0005] S1: UE receives the OFDM signal sent by the base station.

[0006] S2: Calculate the channel estimation matrix, i.e., the perception information matrix.

[0007] S3: Quantize the perception information matrix to obtain a quantization result, and the UE sends the quantized data to the perception information processing center;

[0008] S4: For a uniform quantizer with a quantization interval of Δ, when the input is a signal with a Gaussian signal distribution, calculate the characteristic function and the probability density function of the output result of the uniform quantizer when Δ→0.

[0009] S5: Consider the quantization process as superimposing quantization noise on the input signal and determine the approximate expression of the quantized data.

[0010] S6: After converting the quantized data into the delay-Doppler domain, a detection model and an optimal detector are determined.

[0011] S7: Under the assumptions of H1 and H0, calculate the detection probability based on the approximate distribution of the detection statistic.

[0012] As a further optimization, step S1 specifically includes:

[0013] The signal received by the UE is:

[0014]

[0015] Among them, τ and f Dis the time delay and Doppler frequency of the path from the transmitter to the receiver through the target, is the OFDM frame signal, is the complex attenuation factor of the corresponding perceived target channel, Δf is the subcarrier spacing, T c represents the OFDM symbol period, f D It represents the Doppler shift. represents clutter plus noise, The mean is 0 and the variance is Complex Gaussian white noise;

[0016] As a further optimization, step S2 specifically includes:

[0017] The perception information matrix is ​​calculated as follows:

[0018]

[0019] in Because the modulation system is normalized to unit power,

[0020] As a further optimization, step S3 specifically includes:

[0021] Perception Information Matrix Perform quantization processing to obtain quantized results for:

[0022]

[0023] in Represents uniform quantization of a real number.

[0024] As a further optimization, step S4 specifically includes:

[0025] For input, the mean is μ and the variance is σ 2 For a Gaussian distributed signal x, when Δ / σ is small enough, we can model the quantization error generated by the Gaussian distributed input as a uniformly distributed additive white noise that is unrelated to the input. The PDF of the quantization error is shown below:

[0026]

[0027] The output of the quantizer, y = x + ε, is the sum of two independent Gaussian random variables and a uniform random variable. The characteristic function of y is:

[0028]

[0029] According to the following formula, when Δ / σ is small enough, the probability density function of the uniform quantizer output y is:

[0030]

[0031] As a further optimization, step S5 specifically includes:

[0032] After the perceptual information matrix is ​​quantized, the quantization noise caused by the separate quantization of the real and imaginary parts is introduced. Quantized output Approximately:

[0033]

[0034] in

[0035] As a further optimization, step S6 specifically includes:

[0036] Calculate the quantized output The data converted to the delay-Doppler domain for:

[0037]

[0038] in k is the delay index, k = 1, 2, ..., L, m is the Doppler index, m = 1, 2, ..., S. The target detection task is to determine whether there is a target in the delay-Doppler domain unit (k, m). The detection problem is described as follows:

[0039]

[0040] in, According to the NP criterion, the optimal detector is:

[0041]

[0042] The detection amount α is the detection threshold determined by the false alarm probability P FA =P(T(Y)>α|H0) is determined.

[0043] As a further optimization, step S7 specifically includes:

[0044] The approximate distribution of the test statistic T is calculated as:

[0045]

[0046] in

[0047] The detection threshold under quantization approximation is: α = σ all Q -1 (P FA )

[0048] Then the target detection probability is:

[0049]

[0050] The beneficial effects of the present invention are: for uniform quantization scenarios, a closed-form analytical expression for target detection probability is derived, which provides theoretical guidance for parameter configuration of perceptual data transmission in ISAC target detection scenarios and improves the efficiency of perceptual data transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 Flowchart of a method for calculating the number of quantization bits required in the ISAC system of the present invention;

[0052] Figure 2 It is the probability density curve of the quantizer input and output in the present invention.

[0053] Figure 3 This is the ISAC system model diagram in the present invention;

[0054] Figure 4 This is a curve showing how the detection probability changes with the false alarm probability under different quantization bits in the present invention. DETAILED DESCRIPTION

[0055] The following will clearly and completely describe the technical solutions in the embodiments of the present invention in conjunction with the accompanying drawings. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative efforts are within the scope of protection of the present invention.

[0056] For the convenience of description, we first make the following definitions: * Re{·} represents the real part of a complex number, and Im{·} represents the imaginary part of a complex number.

[0057] The present invention is based on a synaesthesia integrated system; the system includes a base station, a UE, a sensing node, and an environmental scatterer. The base station uses an OFDM signal to sense a signal with a delay τ and a Doppler shift f D The user receives the sensing signal reflected by the target. The receiver has N r antennas, and the transmitting end has a single antenna. Each OFDM frame signal at the transmitting end is Each row of the matrix corresponds to the data on a subcarrier, and each column corresponds to the data on an OFDM symbol. We assume that the modulation system is normalized to unit power. The signal we receive is:

[0058]

[0059] in is the complex attenuation factor of the corresponding perceived target channel, Δf is the subcarrier spacing, T c represents the OFDM symbol period, f D It represents the Doppler shift. represents clutter plus noise. Since there are many factors that affect noise, and these factors are independent and randomly distributed, according to the central limit theorem, we can assume that the noise is Gaussian. In addition, we assume that the clutter background is spatially uniform, that is, the scatterers have the same statistical distribution. Since the clutter received by each receiver is the result of a large number of contributions from different clutter scatterers, it is also asymptotically Gaussian. Therefore, we assume The mean is 0 and the variance is Complex Gaussian white noise. We assume that the receiver can perfectly decode and reconstruct the transmitted signal, and its transmitted signal F Tx Negatively impact perception, Tx Remove and get the perception information matrix:

[0060]

[0061] in Since the modulation system is normalized to unit power, E{|(F Tx ) l,s | 2}=1,

[0062] If the sensing information matrix is ​​not quantized, the sensing data processing center receives the channel response After that, it is transformed into the delay-Doppler domain for target detection. Perform a two-dimensional discrete Fourier transform to obtain the perception information matrix in the delay-Doppler domain, which is expressed as follows:

[0063]

[0064] Since the two-dimensional discrete Fourier transform is a linear transformation, Still obeys the complex Gaussian distribution, the mean remains unchanged, and the variance is σ 2 We consider a specific target detection task, where we find the target in the n-1th target detection task and detect its existence in the nth target detection task. The detection task is to determine whether the target exists in the delay-Doppler domain unit (k, m).

[0065] The detection problem is described as follows:

[0066]

[0067] Assume k = LΔfτ, m = Sfd T c ,but According to the above formula, the corresponding probability density functions under the two assumptions can be written as

[0068]

[0069] The log-likelihood ratio is:

[0070]

[0071] Using the NP criterion, the optimal detector is:

[0072]

[0073] The detection amount, α is the detection threshold determined by the false alarm probability P FA =P(T(Y)>α|H0) is determined,

[0074] In order to enable the receiving end to process the received signal in a more efficient and economical way, it is usually necessary to convert the received perception information matrix The data is transmitted to a unified sensor data processing center through a communication loop for signal processing. In order to reduce spectrum overhead, the data needs to be quantized before transmission at the receiving end. Figure 1 As shown, the present invention uses the following steps to calculate the perception data Quantified target detection probability of the ISAC system.

[0075] Step 1: UE receives a signal from the base station;

[0076] The signal received by the UE is:

[0077]

[0078] Among them, τ and f D is the time delay and Doppler frequency of the path from the transmitter to the receiver through the target, is the OFDM frame signal, is the complex attenuation factor of the corresponding perceived target channel, Δf is the subcarrier spacing, T c represents the OFDM symbol period, f D It represents the Doppler shift. represents clutter plus noise, The mean is 0 and the variance is Complex Gaussian white noise;

[0079] Step 2: Calculate the perception information matrix, quantize the perception information matrix, and obtain a quantized result.

[0080] In this step, the perception information matrix is ​​calculated as follows:

[0081]

[0082] in Since the modulation system is normalized to unit power, E{|(F Tx ) l,s | 2}=1,

[0083] Perception Information Matrix Perform quantization processing to obtain quantized results for:

[0084]

[0085] in It means that a real number is uniformly quantized as follows:

[0086]

[0087] Where b = 2q is the number of quantization levels, and q is the number of quantization bits. b is the quantizer threshold.

[0088] Step 3: For a uniform quantizer with a quantization interval of Δ, when the input is a signal with a Gaussian signal distribution, calculate the characteristic function and probability density function of the output result of the uniform quantizer when Δ→0.

[0089] For a uniform quantizer, the input dynamic range of the quantizer is evenly divided into b intervals, each with a length of Δ. The discrete value x falling into one of the intervals is mapped into a quantization level set Γ = {γ1,γ2,…,γ b A quantization level value γ in i , assuming that represents the quantization rule of the quantizer, then the output of the quantizer is:

[0090]

[0091] Assume that the probability density function f of the input signal x is x (x) Figure 2 As shown, b tends to infinity, the corresponding probability density function of the quantizer output f x' (x) is:

[0092]

[0093] We define a rectangular pulse function as:

[0094]

[0095] Then this function is the same as f x The convolution of (x) is:

[0096]

[0097] Next, we multiply the result of this convolution by the spike train c(x), which is defined as follows:

[0098]

[0099] Here are the results:

[0100]

[0101] Therefore, the PDF output by the quantizer is a series of uniformly spaced Dirac pulses. This output PDF can be obtained by Nyquist sampling the input PDF and convolving it with a uniform PDF distributed between ±q / 2. According to the literature (Widrow B, Kollár I. Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications [M]. Cambridge, UK: Cambridge University Press, 2008.), when the quantization step size Δ is small enough so that 2π / Δ, or the "quantization radian frequency," is at least twice the highest "frequency" component contained in the characteristic function of the input PDF, the input PDF can be completely recovered from the output PDF. This is Quantization Theorem I. This also leads to Quantization Theorem II, which states that when the quantization frequency is at least equal to the highest frequency component contained in the quantizer input, the moment of the quantizer input signal can be completely recovered from the moment of the quantizer output signal. When Quantization Theorem I or Quantization Theorem II is satisfied, the moment of the quantizer's output signal is equal to the moment of the sum of the quantizer's input signal and independent noise uniformly distributed between ±Δ / 2, that is, the quantizer's output can be simulated by adding a uniformly distributed additive white noise that is uncorrelated with the input to the input. Gaussian distributed input does not satisfy the above theorem, but according to the literature (Wang Z, He Q, Blum R S. Target detection using quantized cloud MIMO radar measurements [J]. IEEE Transactions on Signal Processing, 2022, 70: 1-16.) and the literature (Sripad A, Snyder DA necessary and sufficient condition for quantization errors to be uniform and white [J]. IEEE Transactions on Acoustics, Speech, and SignalProcessing, 1977, 25 (5): 442-448.), when Δ / σ is small enough, we can approximate the quantization error generated by the Gaussian distributed input into an additive white noise uniformly distributed and uncorrelated with the input. The PDF of the quantization error is shown below:

[0102]

[0103] The output of the quantizer, y = x + ε, is the sum of a Gaussian random variable and a uniform random variable, which are independent of each other. Therefore, the characteristic function of y is the product of the characteristic functions of x and ε. The characteristic function of ε is:

[0104]

[0105] The characteristic function of y is:

[0106]

[0107] Using Taylor series expansion:

[0108]

[0109] And because:

[0110]

[0111] So we can get:

[0112]

[0113] From this we can get the characteristic function of y to be approximately:

[0114]

[0115] So we can get the output y of the quantizer to be Gaussian distributed with mean μ and variance σ 2 +Δ 2 / 12.

[0116] According to the following formula, when Δ / σ is small enough, the probability density function of the uniform quantizer output y is:

[0117]

[0118] Step 4: Determine the quantized output Approximate probability density function.

[0119] After the perceptual information matrix is ​​quantized, it is equivalent to introducing quantization noise caused by the separate quantization of the real and imaginary parts. Quantized output Approximately:

[0120]

[0121] in

[0122] Step 5: Convert the quantized perception information matrix to the delay-Doppler domain and determine the detection model.

[0123] Calculate the quantized output The data converted to the delay-Doppler domain for:

[0124]

[0125] in k is the delay index, k = 1, 2, ..., L, and m is the Doppler index, m = 1, 2, ..., S. We consider a specific target detection task, where we discover the target in the n-1th target detection task and detect its existence in the nth target detection task. The detection task is to determine whether the target exists in the delay-Doppler domain unit (k, m).

[0126]

[0127] in, Assume k = LΔfτ, m = Sf d T c ,but According to the NP criterion, the optimal detector is:

[0128]

[0129] The detection amount α is the detection threshold determined by the false alarm probability P FA =P(T(Y)>α|H0) is determined.

[0130] Step 6: Determine the approximate distribution of the detection statistic, obtain the detection threshold under the quantitative approximation, and calculate the approximate target detection probability based on the detection threshold under the quantitative approximation.

[0131] The approximate distribution of the test statistic T is:

[0132]

[0133] in

[0134] The detection threshold under quantization approximation is: α = σ all Q -1 (P FA )

[0135] Then the approximate target detection probability is:

[0136]

[0137] Define SQR as the signal-to-quantization noise ratio, A max is the dynamic range of the quantizer and defines the perceptual signal-to-interference-noise ratio Then the target detection probability is simplified to:

[0138]

[0139] In the simulation, it is assumed that the transmitter has a single antenna and the receiver has multiple antennas. The specific scenario is as follows: Figure 3 Assume that the carrier frequency is 5 GHz, the subcarrier spacing is 30 kHz, the number of subcarriers is 128, the number of OFDM symbols is 500, the complex attenuation factor of the channel is 1, and the signal transmission power is 1. The sensing delay of the signal from the transmitter to the sensing target to the receiver is 8.3×10 -6 s, the Doppler frequency shift of the perceived target is 1.8×10 3 Hz. The perceived signal-to-interference-and-noise ratio (SINR) is defined as 5 dB.

[0140] Figure 4 A graph plots the relationship between detection probability and false alarm probability for different quantization bit numbers. The graph uses the unquantized detection probability of the optimal detector to measure the detection performance of the quantized ISAC system. The solid line represents the detection probability obtained using quantization approximation, while the dashed line represents the quantized detection probability obtained through 10,000 Monte Carlo simulations. As can be seen from the graph, as the number of quantization bits increases, the quantization interval gradually decreases, and the corresponding curve obtained by Gaussian approximation of the quantized output gradually approaches the curve obtained by direct analysis of the quantized output. When the number of quantization bits is greater than 3, the approximated result is almost identical to the result obtained by direct analysis. Therefore, when the quantization bit number is large enough, the quantized approximate detection probability is almost the same as the true detection probability.

[0141] The above embodiments further illustrate the purpose, technical solutions and advantages of the present invention in detail. It should be understood that the above embodiments are only preferred implementation plans of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made to the present invention within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for approximate calculation of target detection probability in a multi-bit quantized ISAC system, characterized in that: The following steps are involved: S1: UE receives the OFDM signal sent by the base station; S2: Calculate the channel estimation matrix, i.e., the perception information matrix; S3: Quantize the perception information matrix to obtain a quantization result, and the UE sends the quantized data to the perception information processing center; S4: For a uniform quantizer with a quantization interval of Δ, when the input is a signal with a Gaussian signal distribution, calculate the characteristic function and probability density function of the output result of the uniform quantizer when Δ→0; S5: Consider the quantization process as superimposing quantization noise on the input signal and determine the approximate expression of the quantized data; S6: After converting the quantized data into the delay-Doppler domain, determine the detection model and the optimal detector. S7: Under the assumptions of H1 and H0, calculate the detection probability based on the approximate distribution of the detection statistic.

2. The method for approximate calculation of target detection probability in a multi-bit quantized ISAC system according to claim 1, characterized in that The calculation process of obtaining the perception information matrix in step S2 includes: Step 1: The signal received by the UE is: Among them, τ and f D is the time delay and Doppler frequency of the path from the transmitter to the receiver through the target, is the OFDM frame signal, is the complex attenuation factor of the corresponding perceived target channel, Δf is the subcarrier spacing, T c represents the OFDM symbol period, f D It represents the Doppler shift. represents clutter plus noise, The mean is 0 and the variance is Complex Gaussian white noise; Step 2: The perception information matrix is ​​calculated as follows: in Because the modulation system is normalized to unit power, 3. The method for approximate calculation of target detection probability in a multi-bit quantized ISAC system according to claim 1, characterized in that The determination of the probability density function of the quantized perceptual information in step S5 includes the following calculation process: Step 1: Perception Information Matrix Perform quantization processing to obtain quantized results for: in Indicates uniform quantization of a real number; Step 2: For input, the mean is μ and the variance is σ 2 For a Gaussian distributed signal x, when Δ / σ is small enough, we can model the quantization error generated by the Gaussian distributed input as a uniformly distributed additive white noise that is unrelated to the input. The PDF of the quantization error is shown below: The output of the quantizer y = x + ε is the sum of two independent Gaussian random variables and a uniform random variable. The characteristic function of y is: According to the following formula, when Δ / σ is small enough, the probability density function of the uniform quantizer output y is: Step 3: After the perceptual information matrix is ​​quantized, the quantization noise caused by the separate quantization of the real and imaginary parts is introduced. Quantized output Approximately: in 4. The method for approximate calculation of target detection probability in a multi-bit quantized ISAC system according to claim 1, characterized in that Step S7 is to calculate the probability of quantifying the approximate target detection. The calculation process includes: Step 1: Calculate the quantized output The data converted to the delay-Doppler domain for: in k is the delay index, k = 1, 2, ..., L, m is the Doppler index, m = 1, 2, ..., S, then the target detection task is to determine whether there is a target in the delay-Doppler domain unit (k, m). The detection problem is described as follows: in, According to the NP criterion, the optimal detector is: The detection amount α is the detection threshold determined by the false alarm probability P FA =P(T(Y)>α|H0)determine; Step 2: Calculate the approximate distribution of the test statistic T as: in The detection threshold under quantization approximation is: α = σ all Q -1 (P FA )Then the target detection probability is:

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