1. An approximate calculation method for cloud MIMO radar target detection probability is characterized by comprising the following steps:

s1, the cloud MIMO radar receives the signal from the transmitter through the receiver;

s2, calculating the detection statistic of each path between the receiver and the transmitter;

s3, carrying out quantization processing on the detection statistics to obtain quantization results, and transmitting the quantization data to a fusion center by each receiver;

s4, determining H₁Hypothesis sum H₀Assuming the case, the total detected statistic at the center is fused, where H₁Assumed to represent the occurrence of a target, H₀Assume that the representation target is not present;

s5, calculating H₁Hypothesis sum H₀Fusing, under hypothetical conditions, the probability distribution function of the total detection statistic at the center;

s6, calculating the target detection probability;

s7, calculating a characteristic function and a probability density function of an output result of the uniform quantizer when the quantization interval is delta → 0 aiming at the uniform quantizer with the quantization interval delta;

s8, determining an approximate probability density function of a quantization result output by a uniform quantizer in the cloud MIMO radar when the delta → 0 is reached;

s9, determining approximate distribution of total detection statistics, obtaining a detection threshold under the condition of quantization approximation, and calculating approximate target detection probability according to the detection threshold under the condition of quantization approximation.

2. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 1, wherein the step S1 specifically includes:

cloud MIMO radar kT_sThe nth receiver receives the signal r from the mth transmitter at time instant_nm[k]Is composed of

Wherein the mth transmitter is at kT_sSampled value of time beingE is transmissionTotal energy, T_sK is the sampling number for the sampling interval, K is 1, …, K is the total number of samples, f_sIn order to be able to sample the rate,to determine a known target reflection coefficient; tau is_nm，f_nmIndicating the time delay and doppler frequency, w, corresponding to the mth transmitter signal path received by the nth receiver_nm[k]Is that the nth receiver receives the noise of the mth transmitter signal path, and()^*the conjugate of the complex number is represented, delta (k-k ') is a dirac function, the value of the function is 1 when k is k ', and the other cases are 0, and k ' represent sampling time; m represents the total number of transmitters; s_mRepresenting the transmitted signal of the mth transmitter,representing the noise w_nm[k]The variance of (c).

3. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 2, wherein the step S2 specifically includes:

detection statistic T of nm transmission path_nmThe calculation is as follows:

where N is 1, …, N, M is 1, …, M, N and M denote the total number of receivers and transmitters,re {. is said to take the real part of a complex number.

4. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 3, wherein the step S3 specifically includes:

for the detection statistic T_nmCarrying out quantization processing to obtain a quantized result q_nmIs composed of

WhereinRepresenting a quantization operation on a real number;

wherein D is 2^bIs the total number of quantizations, b is the number of quantizations bits, gamma₀,…,γ_DA quantizer threshold screen;

arranging the quantized result of the nth receiver into a column vector

Wherein the superscript isRepresenting a transpose;

all the receivers transmit the quantized data to the fusion center, and the fusion center receives the data as

5. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 4, wherein the step S4 specifically includes:

at H₁Hypothesis sum H₀Assuming the case, the total detection statistic at the fusion center is:

wherein alpha is the false alarm threshold P_FAThe determined detection threshold, T ∈ {0,1, …, NM (D-1) }, sum (y), represents the summation of all elements of the vector y.

6. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 5, wherein the step S5 specifically includes:

calculate H₁And H₀Let q be_nmThe conditional probability distribution function of (a) is:

wherein D-1 is 1, …, D-1 is a possible quantification result, is shown in H_iI is 0,1 assumes the following eventProbability of occurrence, Q (-) represents the cumulative distribution function of a standard Gaussian distribution, defined as

Calculate H₁And H₀Assume that the probability distribution function of the total detection statistic T at the fusion center is:

wherein z ∈ {0,1, …, NM (D-1) }, z_nm∈{0,1，…，(D-1)}，n＝1,…,N,m＝1,…,M。

7. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 6, wherein the step S6 specifically includes:

the detection probability is calculated according to the following formula:

wherein alpha is the false alarm threshold P_FAThe determined detection threshold is set to a value that is less than the detection threshold,

8. the approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 7, wherein the step S7 specifically includes:

for a uniform quantizer with a quantization interval Δ, the quantizer input is the mean μ variance σ²With respect to the gaussian signal x of (1),

the approximate probability density function of quantization error ε -q-x when Δ/σ → 0 is

Wherein q is the output of the quantizer;

the characteristic function of the uniform quantizer output q when Δ → 0 is obtained according to:

wherein the content of the first and second substances,as a characteristic function of the quantization error e,a characteristic function of the input x to the quantizer;

the probability density function of the uniform quantizer output q when Δ → 0 is obtained according to:

9. the approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 8, wherein the step S8 specifically includes:

the calculation is obtained when a uniform quantizer is adopted in the delta → 0 time cloud MIMO radar_nmApproximate probability density function of (1):

wherein

10. The approximate calculation method for the cloud MIMO radar target detection probability as claimed in claim 1, wherein the step S9 specifically includes:

the approximate distribution of the detection statistic T is calculated as:

wherein

The detection threshold in the case of quantization approximation is:

wherein P is_FAIs the false alarm probability, Q^-1(. h) is the inverse of Q (·);

the approximate detection probability is obtained according to the following formula:

Background

In cloud mimo (Multiple Input Multiple out) radars, the signals returned from the targets are received by each local sensor, which may communicate with the Fusion Center (FC) through a backhaul network. The number of local sensors may be large and the receiving antennas may be distributed over a large geographical area where no wired backhaul network is available, so the local sensors communicate with the FC over a wireless network and the samples collected by the receivers are typically sent to the FC after each receiver is quantized to reduce the communication burden.

Usually, the quantized output is a discrete value, but for some special quantizers, such as uniform quantizers, the output can sometimes be modeled as a continuous value of the input plus quantization noise. In document 1(a. sripad and d. snyder, "interference and knowledge regulation to be from surfaces and white," IEEE Transactions on optics, Speech, and Signal Processing, vol.25, No.5, pp.442-448,1977), the essential condition is given that a scalar uniform quantizer output can be modeled as an input plus uniformly white quantization noise, and that when the quantization interval is sufficiently small, the quantization error of a zero-mean gaussian input Signal can be approximated as uniformly distributed noise. In document 2(S.Khali, O.Simeon, and A.M.Haimovich, "Cloud radio-multistative radio: join optimization of code vector and background quantization," IEEE Signal Processing Letters, vol.22, No.4, pp.494-498,2015) and document 3 (W.J.A.M.Haimovich, "Joint optimization of wave form and quantization in spectral constraints," in 201852nd analyte constraints, Systems, and Computers, pp.1894-1898,2018), the quantized output of the optimal vector quantizer is expressed as an input vector plus Gaussian quantized noise vector. The gaussian quantization error approximation introduces additive gaussian noise to discuss the quantization effect, i.e. the quantized output is modeled as an input plus gaussian noise, so that the output is still a continuous random variable and further theoretical analysis is easier to perform.

Target detection as one of the key functions of Radar, in document 4 (s.khalii, o.simeon, and a.m.haimovich, "Cloud radio-multistative Radar: joint Optimization of code vector and background quantification," IEEE signal Processing Letters, vol.22, No.4, pp.494-498, April 2015) and document 5(s.jeong, o.simeon, a.haimovovich, and j.kang, "Optimization of multipoint radio with multiple-access waveform 2015," in IEEE radio Conference (ary Conference), May, plot, etc.), the detection of a signal using the reception of a single transmitter and multiple receivers is optimized using the emission of a quantized signal. In document 6(w.jiang and a.m. haimovich, "wave form optimization in closed Radar with spectral constraints," in 2019IEEE Radar reference (RadarConf), April 2019, pp.1-6), optimization of Radar Waveform and relay gain in a cloud Radar system to maximize average mutual information was studied based on quantized received signals. In document 7(z.wang, q.he, and r.s.blum, "Sampling rate and bits per sample rate of closed MIMO radar target detection," in 201927th European Signal Processing reference (EUSIPCO),2019, pp.1-5), the detection performance of the cloud MIMO radar is obtained using the quantized received Signal, and a compromise between the Sampling rate and the bit rate of each sample is proposed.

Because the number of antennas in the cloud MIMO radar is large and the distances are generally far away, the local receiver transmits data to the fusion center through a wireless channel. However, the wireless resources are increasingly tight, and the data needs to be quantized at the local receiver and then transmitted to the fusion center. However, it is often difficult to obtain closed-form detection performance based on quantized discrete output. Therefore, it is necessary to discuss the quantized approximate output and obtain the closed detection probability in the cloud MIMO radar. In most existing literature on distributed parameter estimation based on quantized data, there is no detailed discussion about the approximate distribution of quantized output at non-zero mean gaussian input of uniform quantizer.

Furthermore, the existing discussion in cloud MIMO radar is based on quantizing the received signal, but when there are many received samples, it may also bring unacceptable communication burden, and a certain compression needs to be performed on the received signal at the local receiver. Therefore, it is necessary to further discuss some algorithms for reducing communication burden in the cloud MIMO radar.

Disclosure of Invention

The technical problem to be solved by the invention is as follows: the approximate calculation method of the cloud MIMO radar target detection probability is provided, the communication burden between a receiver and a fusion center is reduced, and the closed detection probability can be obtained, so that the theoretical analysis and the system design guidance can be carried out subsequently.

The technical scheme adopted by the invention for solving the technical problems is as follows:

an approximate calculation method for cloud MIMO radar target detection probability comprises the following steps:

s1, the cloud MIMO radar receives the signal from the transmitter through the receiver;

s2, calculating the detection statistic of each path between the receiver and the transmitter;

s3, carrying out quantization processing on the detection statistics to obtain quantization results, and transmitting the quantization data to a fusion center by each receiver;

s4, determining H₁Hypothesis sum H₀Assuming the case, the total detected statistic at the center is fused, where H₁Assumed to represent the occurrence of a target, H₀Assume that the representation target is not present;

s5, calculating H₁Hypothesis sum H₀Fusing, under hypothetical conditions, the probability distribution function of the total detection statistic at the center;

s6, calculating the target detection probability;

s7, calculating a characteristic function and a probability density function of an output result of the uniform quantizer when the quantization interval is delta → 0 aiming at the uniform quantizer with the quantization interval delta;

s8, determining an approximate probability density function of a quantization result output by a uniform quantizer in the cloud MIMO radar when the delta → 0 is reached;

s9, determining approximate distribution of total detection statistics, obtaining a detection threshold under the condition of quantization approximation, and calculating approximate target detection probability according to the detection threshold under the condition of quantization approximation.

As a further optimization, step S1 specifically includes:

cloud MIMO radar kT_sThe nth receiver receives the signal r from the mth transmitter at time instant_nm[k]Is composed of

Wherein the mth transmitter is at kT_sSampled value of time beingE is the total energy of emission, T_sK is the sampling number for the sampling interval, K is 1, …, K is the total number of samples, f_sIn order to be able to sample the rate,to determine a known target reflection coefficient; tau is_nm，f_nmIndicating the time delay and doppler frequency, w, corresponding to the mth transmitter signal path received by the nth receiver_nm[k]Is that the nth receiver receives the noise of the mth transmitter signal path, and()^*the conjugate of the complex number is represented, delta (k-k ') is a dirac function, the value of the function is 1 when k is k ', and the other cases are 0, and k ' represent sampling time; m represents the total number of transmitters; s_mRepresenting the transmitted signal of the mth transmitter,representing the noise w_nm[k]The variance of (c).

As a further optimization, step S2 specifically includes:

detection statistic T of nm transmission path_nmThe calculation is as follows:

where N is 1, …, N, M is 1, …, M, N and M denote the total number of receivers and transmitters,re {. is said to take the real part of a complex number.

As a further optimization, step S3 specifically includes:

for the detection statistic T_nmCarrying out quantization processing to obtain a quantized result q_nmIs composed of

WhereinRepresenting a quantization operation on a real number;

wherein D is 2^bIs the total number of quantizations, b is the number of quantizations bits, gamma₀,…,γ_DA quantizer threshold screen;

arranging the quantized result of the nth receiver into a column vector

Wherein the superscript isRepresenting a transpose;

all the receivers transmit the quantized data to the fusion center, and the fusion center receives the data as

As a further optimization, step S4 specifically includes:

at H₁Hypothesis sum H₀Assuming the case, the total detection statistic at the fusion center is:

wherein alpha is the false alarm threshold P_FAThe determined detection threshold, T ∈ {0,1, …, NM (D-1) }, sum (y), represents the summation of all elements of the vector y.

As a further optimization, step S5 specifically includes:

calculate H₁And H₀Let q be_nmThe conditional probability distribution function of (a) is:

wherein D-1 is 1, …, D-1 is a possible quantification result, is shown in H_iI is 0,1 assumes the following eventProbability of occurrence, Q (-) represents the cumulative distribution function of a standard Gaussian distribution, defined as

Calculate H₁And H₀Assume that the probability distribution function of the total detection statistic T at the fusion center is:

wherein z ∈ {0,1, …, NM (D-1) }, z_nm∈{0,1，…，(D-1)}，n＝1,…,N,m＝1,…,M。

As a further optimization, step S6 specifically includes:

the detection probability is calculated according to the following formula:

wherein alpha is the false alarm threshold P_FAThe determined detection threshold is set to a value that is less than the detection threshold,

as a further optimization, step S7 specifically includes:

for a uniform quantizer with a quantization interval Δ, the quantizer input is the mean μ variance σ²With respect to the gaussian signal x of (1),

the approximate probability density function of quantization error ε -q-x when Δ/σ → 0 is

Wherein q is the output of the quantizer;

the characteristic function of the uniform quantizer output q when Δ → 0 is obtained according to:

wherein the content of the first and second substances,as a characteristic function of the quantization error e,a characteristic function of the input x to the quantizer;

the probability density function of the uniform quantizer output q when Δ → 0 is obtained according to:

as a further optimization, step S8 specifically includes:

the calculation is obtained when a uniform quantizer is adopted in the delta → 0 time cloud MIMO radar_nmApproximate probability density function of (1):

wherein

As a further optimization, step S9 specifically includes:

the approximate distribution of the detection statistic T is calculated as:

wherein

The detection threshold in the case of quantization approximation is:

wherein P is_FAIs the false alarm probability, Q^-1(. h) is the inverse of Q (·);

the approximate detection probability is obtained according to the following formula:

the invention has the beneficial effects that:

because the detection statistic is firstly calculated at the local sensor so as to reduce samples needing to be transmitted, the communication burden can be greatly reduced, and the engineering realization is more facilitated. And the closed detection probability can be obtained, the subsequent theoretical analysis and system design can be more conveniently carried out, and the system design can be guided in practical application.

Drawings

FIG. 1 is a flow chart of an approximate calculation method for cloud MIMO radar target detection probability according to the present invention;

fig. 2 is a diagram of system parameter settings for an embodiment in which the number of receivers and transmitters is N-4 and M-8, respectively;

FIG. 3 is a diagram illustrating the detection probability P under different quantization bits in the embodiment_DProbability of false alarm P_FASchematic representation of the variations.

Detailed Description

For convenience of description, the following definitions are first made:

is a transposition of^HIs a conjugate transpose, ()^*Which represents the conjugate of the complex number,representing a mathematical expectation, Re{. denotes taking the real part of a complex number.

In the cloud MIMO radar, M transmitters and N receivers are spatially separated, and in a cartesian coordinate system, an M (M-1, …, M) th transmitting antenna and an N (N-1, …, N) th receiving antenna are respectively locatedAndm transmitter at kT_sSampled value of time beingE is the total energy of emission, T_sFor the sampling interval, K (K is 1, …, K) is the sampling number, K is the total number of samples, f_sFor the sampling rate, it is assumed that the transmitted signals of the different transmitters are orthogonal. Assuming that the target exists, the target is located at (x, y) and the moving speed is (v)_x,v_y) So at kT_sThe nth receiver receives the mth transmitter's signal at time instant,

wherein the noise w_nm[k]Assumed to be complex white Gaussian noise with zero mean and for different sampling points k and kTo simplify the analysis, the emission coefficients are assumedIs to determine the known parameters. Tau is_nmRepresenting the delay, f, associated with the nm path_nmIndicating the doppler shift of the received signal associated with the nm path.

Arranging all time sample signals from transmitter m received by receiver n as a vector r_nm

Wherein the content of the first and second substances,

arranging the sample signals received by all receivers into a vector r

Wherein the content of the first and second substances,

at H₁Assumptions (target present in test cell) and H₀Assuming (target does not exist), the detection problem of centralized processing is:

thus, a probability density function of the received signal r under two hypotheses can be obtained as

Further, a log-likelihood ratio of

The optimal detector under centralized processing according to NP criterion is

Wherein alpha is_OIs composed of false alarm doorLimit of P_FA,OThe determined detection threshold is set to a value that is less than the detection threshold,

mean is μ and variance is σ²A gaussian distribution of (a).

In a cloud MIMO radar system, local sensors send local data to a fusion center over a communication loop. Then, the fusion center processes the received data to complete target detection. In order to reduce the communication burden, the local data needs to be quantized before being sent to the fusion center. Therefore, the optimal detector for the centralized processing in equation (7) is impossible to realize, and from equation (7), we see that the optimal detector calculates the detection statistic T for each transmit-to-receive path_nmFor M-1, …, M calculates and quantizes T at the local receiver n_nmIs reasonable. The quantized detection statistics are sent to the fusion center where all the detection statistics are added and compared to a threshold as shown in equation (7) to make the final decision that the original T in equation (7) is replaced by the quantized detection statistics at the fusion center_nmThis method can greatly reduce the communication load.

In the implementation, as shown in fig. 1, the following steps are adopted in the present invention to calculate the approximate detection performance of the cloud MIMO radar based on the quantized detection statistic:

s1, the cloud MIMO radar receives the signal from the transmitter through the receiver;

in this step, the cloud MIMO radar kT_sThe nth receiver receives the signal r from the mth transmitter at time instant_nm[k]Is composed of

Wherein the m-th transmitter is at kT_sSampled value of time beingE is the total energy of emission, T_sK is the sampling number for the sampling interval, K is 1, …, K is the total number of samples, f_sIn order to be able to sample the rate,to determine a known target reflection coefficient; tau is_nm，f_nmIndicating the time delay and doppler frequency, w, corresponding to the mth transmitter signal path received by the nth receiver_nm[k]Is that the nth receiver receives the noise of the mth transmitter signal path, and()^*the conjugate of the complex number is represented, delta (k-k ') is a dirac function, the value of the function is 1 when k is k ', and the other cases are 0, and k ' represent sampling time; m represents the total number of transmitters; s_mRepresenting the transmitted signal of the mth transmitter,representing the noise w_nm[k]The variance of (c).

S2, calculating the detection statistic of each path between the receiver and the transmitter;

in this step, the detection statistic T of the nm transmission path is calculated_nm，

Where N is 1, …, N, M is 1, …, M, N and M denote the total number of receivers and transmitters,re {. is said to take the real part of a complex number.

S3, carrying out quantization processing on the detection statistics to obtain quantization results, and transmitting the quantization data to a fusion center by each receiver; in this step, the detection statistic T is measured_nmCarrying out quantization processing to obtain a quantized result q_nmIs composed of

WhereinRepresenting a quantization operation on a real number, of

Wherein D is 2^bIs the total number of quantizations, b is the number of quantizations bits, gamma₀,…,γ_DIs a quantizer threshold;

arranging the result after the quantization of the nth receiver into a column vector:

wherein the superscript isIndicating transposition.

All the receivers transmit the quantized data to the fusion center, and the fusion center receives the data as

S4, determining H₁Hypothesis sum H₀Hypothetical situationNext, the total detection statistics at the fusion center:

in this step, H is obtained assuming that the target appears in the test cell₁And assuming that the target does not exist as H₀In the case of (2), the detection problem at the fusion center is

Wherein alpha is the false alarm threshold P_FAThe determined detection threshold, T ∈ {0,1, …, NM (D-1) }, sum (y), represents the summation of all elements of the vector y.

S5, calculating H₁Hypothesis sum H₀Fusing, under hypothetical conditions, the probability distribution function of the total detection statistic at the center;

in this step, H is calculated₁And H₀Let q be_nmIs a conditional probability distribution function of

Wherein D-1 is 1, …, D-1 is a possible quantification result, is shown in H_iI is 0,1 assumes the following eventProbability of occurrence, Q (-) represents the cumulative distribution function of a standard Gaussian distribution, defined as

Calculate H₁And H₀Assuming a probability distribution function of the total detection statistic T at the fusion center as

Where z ∈ {0,1, …, NM (D-1) }, z_nm∈{0,1，…，(D-1)}，n＝1,…,N,m＝1,…,M。

S6, calculating the target detection probability;

in this step, the detection probability is calculated according to the following formula

Where α is the false alarm threshold P_FAThe determined detection threshold is set to a value that is less than the detection threshold,

s7, calculating a characteristic function and a probability density function of an output result of the uniform quantizer when the quantization interval is delta → 0 aiming at the uniform quantizer with the quantization interval delta;

in this step, for a uniform quantizer with a quantization interval Δ, the quantizer input is the mean μ variance σ²With respect to the gaussian signal x of (1),

the approximate probability density function for quantization error ε q-x when Δ/σ → 0 is:

where q is the output of the quantizer.

The characteristic function of the uniform quantizer output q when Δ → 0 is obtained according to:

whereinAs a characteristic function of the quantization error e,is a characteristic function of the input x to the quantizer.

The probability density function of the uniform quantizer output q when Δ → 0 is obtained according to:

s8, determining an approximate probability density function of a quantization result output by a uniform quantizer in the cloud MIMO radar when the delta → 0 is reached;

in this step, the quantizer output q when the uniform quantizer is used in the cloud MIMO radar of Δ → 0 is obtained using the result of step S7_nmThe approximate probability density function of (a) is:

wherein

S9, determining approximate distribution of total detection statistics, obtaining a detection threshold under the condition of quantization approximation, and calculating approximate target detection probability according to the detection threshold under the condition of quantization approximation.

In this step, the approximate distribution of the detection statistic T is obtained as follows:

wherein

The detection threshold under the quantization approximation condition is obtained as follows:

wherein P is_FAIs the false alarm probability, Q^-1(. h) is the inverse of Q (-),

the approximate detection probability is obtained according to the following formula:

the working principle of the approximate calculation method for the cloud MIMO radar target detection probability is as follows:

in cloud MIMO radar, in H₁Assumptions (target present in test cell) and H₀The detection problem of centralized processing under the assumption (target does not exist) is

Thus, a probability density function of the received signal r under two hypotheses can be obtained as

Further, a log-likelihood ratio of

The optimal detector under centralized processing according to NP criterion is

Wherein alpha is_OIs determined by a false alarm threshold P_FA,OThe determined detection threshold is set to a value that is less than the detection threshold,

in a cloud MIMO radar system, local sensors send local data to a fusion center over a communication loop. Then, the fusion center processes the received data to complete target detection. In order to reduce the communication burden, the local data needs to be quantized before being sent to the fusion center. Thus, based on equation (7), we can compute the detection statistic T for each transmit-to-receive path at the local sensor_nmAnd will T_nmAnd the quantized data is sent to a fusion center. To simplify the analysis, assuming that the local receiver to fusion center communication loop is ideal, all quantized detection statistics can be added and compared to a threshold at the fusion center to make the final decision, as shown in equation (7). Thus, the detection problem at the fusion center becomes

Where α is the false alarm threshold P_FAA determined detection threshold.

From the distribution of the detection statistics, H can be obtained₁And H₀Let q be_nmIs a conditional probability distribution function of

Further, the probability distribution function of the total detection statistic T at the fusion center under two assumptions can be obtained as

Where z ∈ {0,1, …, NM (D-1) }, z_nm∈{0,1，…，(D-1)}，n＝1,…,N,m＝1,…,M。

Thus, the detection probability is

Where α is the false alarm threshold P_FAThe determined detection threshold is set to a value that is less than the detection threshold,

as can be seen from the formula (12), when we convert T_nmThe quantized output is a discrete random variable that makes it difficult to obtain a closed-form expression of the probability of detection. Therefore, we next discuss the approximate probability distribution of the quantizer output and use this to obtain a closed form of detection probability.

Assuming a uniform quantizer with a quantization interval Δ is used, the quantizer input is the mean μ variance σ²According to the literature (a. sripad and d. snyder, "interference and quantization conditions for quantization errors to be uniform and white," IEEE Transactions on Acoustics, Speech, and Signal Processing, vol.25, No.5, pp.442-448, Oct 1977), the output q of the quantizer can be modeled as the sum of the input and quantization errors epsilon, and thus

ε＝q-x (36)

Wherein the content of the first and second substances,

from this, a cumulative distribution function of the quantization error ε can be obtained

From the cumulative distribution function, a probability density function that yields the quantization error ε is

Defining the function g (epsilon) as

And g (epsilon) is extended from the interval epsilon ∈ delta/2, delta/2 ] to epsilon ∈ [ - ∞, infinity ], and then the extended function with the period delta can be expressed as a Fourier series

WhereinIs Fourier series, and is expressed as

WhereinRepresenting the characteristic function of the input x. Substitution of formula (42) into formula (41) can be obtained

The probability density function of the quantization error ε obtained by substituting expressions (43) and (40) into expression (39) is

Since the input x is the value μ variance σ²The characteristic function of x is

The probability density function of the quantization error ε obtained by substituting equation (45) into equation (44) is

According to the formula (46), the mean and variance of the quantization error ε are

And

as shown in the formula (46), the second term

Similarly, the second term in the formula (48) satisfies

Next we analyze when Δ/σ →Terms at the right end of formulae (49) and (50) at 0The value of (a). Since the exponential sum is non-negative, therefore

Further obtain the

Wherein, when Δ/σ → 0

According to the formulae (51), (52) and (53), when Δ/σ → 0

When Δ/σ → 0, the results of equations (46) and (48) can be simplified to be those of equations (49), (59) and (54)

And

E[ε²]＝Δ²/12 (56)

therefore, the quantization error ε can be approximated to follow a uniform distribution of the probability density function as equation (55) as long as Δ/σ is sufficiently small.

Since the quantizer output q ═ x + ε can be expressed as the sum of the Gaussian input and the uniformly distributed quantization error, the input and quantization errors are mutually independent, as available according to the literature (M. Bertocco, C. Narduzzi, P. Paglierani, and D.Petri, "A noise model for quantized data," IEEE Transactions on Instrumentation and Measurement, vol.49, No.1, pp.83-86, Feb 2000). Thus, the characteristic function of the quantizer output q can be written as

WhereinAs a characteristic function of the quantization error e,is a characteristic function of the input x to the quantizer. When Δ → 0, it can be obtained by Taylor series expansion

Thus, when Δ → 0, we can get

When equation (60) is substituted into equation (57), the characteristic function of q satisfies the condition of Δ → 0

Characteristic function phi of q as delta approaches 0_q(t) gradually approaches phi (t), and characteristic functions are known from literature (B.V.Gndenko and dA.N.Kolmogorov, "Limit distributions for purposes of Sums of independent random variables," IEEE Transactions on Instrumentation and Measurement, server.Addison-Wesley Mathesics series.Cambridge, MA: Addison-Wesley,1954)φ_q(t) the corresponding probability density function converges to the probability density function corresponding to the characteristic function phi (t). Thus, it can be seen that the probability density function of the uniform quantizer output q when Δ → 0 can be approximated as

Using the result of equation (62) may be a quantizer output q when a uniform quantizer is employed in the cloud MIMO radar when Δ → 0_nmIs approximated by a probability density function of

WhereinFurther, an approximate distribution of the detection statistic T can be obtained as

Wherein

Using the approximate distribution of equation (64), a detection threshold of detection problems in equation (32) can be obtained as

Wherein P is_FAIs the false alarm probability, Q^-1(. h) is the inverse of Q (-),

thus, the approximate detection probability is

Example (b):

in the present embodiment, it is assumed that the cloud MIMO radar system has M-8 transmitters and N-4 receivers, the antenna positions of which are shown in fig. 2, and the detection target is located at (150,130) M and moves at a speed of (25,20) M/s. The transmission signal in the simulation is a frequency-extended Gaussian monopulse signalTake T as 0.01s, Δ f as 150Hz, f_s＝600Hz。

Definition ofAnd set the noise variance toSNR＝-5dB。

FIG. 3 plots the detection probability P based on different quantization bit numbers b_DAnd false alarm probability P_FAA graph of the relationship (c). In the figure, the optimal centralized processing result is labeled with unquantized to measure the detection performance of the quantization system, the result based on the discrete quantization output analysis obtained in step S6 is labeled with direct, and the approximate result after the quantization output is represented by gaussian approximation in step S9 is labeled with aprox. As can be seen from the figure, as b increases, the quantization interval gradually decreases, and the corresponding curve obtained by using gaussian approximation for the quantized output gradually approaches the curve obtained by directly analyzing the quantized output, and when b is not less than 2, the result of approximation is almost the same as that of directly analyzing. Therefore, when the quantization bit is large enough, the approximate detection probability obtained by the gaussian approximation is almost the same as the true detection probability. When the number of quantization bits b is greater than 5, the quantization performance is almost identical to that when it is not quantized, indicating that the performance loss of quantization at this time is almost negligible.