CN108234083B - Method for rapidly determining multi-antenna space-time block code identification threshold - Google Patents

Method for rapidly determining multi-antenna space-time block code identification threshold Download PDF

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CN108234083B
CN108234083B CN201810004625.3A CN201810004625A CN108234083B CN 108234083 B CN108234083 B CN 108234083B CN 201810004625 A CN201810004625 A CN 201810004625A CN 108234083 B CN108234083 B CN 108234083B
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胡国兵
陈正宇
赵嫔娇
陶永会
姜志鹏
杨莉
周睿
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Jinling Institute of Technology
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Abstract

A method for quickly determining the recognition threshold of multi-antenna space-time block codes is provided, which introduces an extreme value statistic theory into the calculation problem of decision threshold in the recognition of MIMO space-time block code patterns based on sequence statistics, and provides a method for quickly determining the threshold capable of being analyzed and expressed, the basic thought is as follows: 1) and (3) auxiliary constant calculation: calculating two normalization constants according to an extreme value theory; 2) establishing an approximate equation: replacing a power function of an incomplete Gamma function in threshold solution by a Gumbel function to obtain an approximate equation of the threshold solution; 3) approximate solution of threshold: and solving an approximate equation according to the set false alarm probability and the auxiliary constant obtained by calculation to obtain an analytic solution of the threshold. Simulation results prove that when the parameters are proper, the method can quickly and accurately calculate the threshold in the MIMO code pattern recognition.

Description

Method for rapidly determining multi-antenna space-time block code identification threshold
Technical Field
The invention belongs to the technical field of signal processing, and particularly relates to a method for quickly determining a multi-antenna space-time block code identification threshold.
Background
Gamma distribution, as a common distribution in signal processing. It is noted that the distribution function of the Gamma random variable is the incomplete Gamma function Pn(α, x) and therefore, when Gamma random quantity or its derivative is used as a statistic for signal detection or recognition, it usually involves problems related to threshold determination, statistical analysis of performance, and the like. In recent years, with the development of new generation wireless communication technology, the MIMO transmission system has become 4G anda core technology of wireless transmission in a 5G mobile communication system. Accordingly, the problem of signal identification in the MIMO scenario has become a hotspot problem in the fields of cognitive radio, communication reconnaissance, spectrum monitoring and the like. In the process of identifying MIMO-OFMD-STBC signals, autocorrelation functions of different time delay correlations among different antennas can be used as identification bases, the maximum value of the autocorrelation functions is used as identification characteristics, and the distribution of the characteristics is the product of independent and same-distribution Gamma distribution. In this case, if the threshold for identification is required, the shape of P is necessarily involvednAnd (alpha, x/beta) q, and q is more than or equal to 0 and less than or equal to 1. When n is large and the number of α is greater than 2, the above equation is difficult to solve analytically, thereby increasing the complexity of the correlation process.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a method for quickly determining the identification threshold of a multi-antenna space-time block code, which utilizes an extreme value distribution theory and uses a Gumbel function to approximate the shape of PnQ is more than or equal to 0 and less than or equal to 1, and the left incomplete Gamma function of the equation is raised to the high power, thereby providing a simple and convenient identification threshold analysis approximate solution.
In order to achieve the purpose, the invention adopts the following technical scheme:
a method for rapidly determining a multi-antenna space-time block code identification threshold is characterized by comprising the following steps:
1) and (3) testing the condition adaptability: for the identification of MIMO-OFDM signal space-time block code based on sequence statistics, the number N of receiving antennas is usedrDefining a parameter Nc＝Nr(Nr-1)/2; defining a parameter N according to the length v of a sample length N, OFDM sequence cyclic prefixvN + v, and a false alarm probability value pfaAnd the ranges of the three parameters are checked;
2) and (3) auxiliary constant calculation: calculating the auxiliary constant a according to the extreme value theoryn，bn；
3) Establishing an approximate equation: replacing a power function of an incomplete Gamma function in threshold solution by a Gumbel function to obtain an approximate equation of the threshold solution;
4) approximate solution of threshold: and solving an approximate equation according to the set false alarm probability and the auxiliary constant obtained by calculation to obtain an analytic solution of the threshold.
In order to optimize the technical scheme, the specific measures adopted further comprise:
the step 1) specifically comprises the following steps:
defining a characteristic quantity based on the maximum value of the autocorrelation function between any two receiving antennas:
γ＝maxFc(τ)
wherein the content of the first and second substances,
Figure BDA0001537257050000021
Fij(tau) is a time-delayed autocorrelation function between any two different receiving antennas, subject to a degree of freedom Nc＝Nr(Nr-1)/2, tau is the delay, NrThe number of receiving antennas;
setting false alarm probability P according to constant false alarm criterionfaAnd calculating to obtain a corresponding decision threshold according to the following formula:
(1-pfa)＝PN+v(Nc，λ/2) (1)
wherein λ is a decision threshold, v is the number of OFDM cyclic prefixes, N is the number of samples of the received signal, P isn(α, x) is the parameter α raised to the power n of the incomplete Gamma function;
the corresponding recognition rule is: if gamma is larger than or equal to lambda, the code type of the signal is judged to be STBC, and if not, the code type of the signal is SM code;
in the identification, the threshold is obtained by solving the equation (1), and the false alarm probability P needs to be solved in advancefaParameter NvN + v and the parameter Nc＝Nr(NrThe range of-1)/2 was examined.
The step 2) specifically comprises the following steps:
equation (1) is generalized as:
Pn(α，x/β)＝q，0≤q≤1，n＞＞1，α＞2 (2)
from the parameters α, n, β, a constant a for normalization is calculatedn，bnThe concrete formula is as follows:
bn＝β{ln[n/(α)]+(α-1)ln Bn+[(α-1)2 ln Bn-(α-1)2 ln(α-1)+α-1]/Bn}；
an＝bnβ[bn+β(α-1)]/[(bn)2-β2(α-1)(α-2)]， (3)
wherein, constant Bn＝ln[n/(α)]+ (α -1) ln (α -1), (x) is a Gamma function;
let alpha be Ncβ 2, N + v, substituting formula (3) to obtain a normalization constant
Figure BDA0001537257050000022
The step 3) specifically comprises the following steps:
according to the extreme value theory, n independent and identically distributed Gamma random vectors X with the shape parameter of alpha and the scale parameter of beta1，X2，...，XnThe maximum value Y is max (X)1，X2，...，Xn) The limiting distribution of the distribution function is a Gumbel distribution, i.e. if there is a suitable normalization constant an，bn> 0, there are:
Figure BDA0001537257050000031
wherein a (x) exp [ -exp (-x) ] is a Gumbel function;
equation (2) is converted to the following approximate equation:
Figure BDA0001537257050000032
accordingly, equation (1) is approximated as:
Figure BDA0001537257050000033
the step 4) specifically comprises the following steps:
equation (6) is derived to obtain an approximate solution:
Figure BDA0001537257050000034
the invention has the beneficial effects that: converting the threshold calculation problem when the number of the receiving antennas is more than 2 into a solving problem of an incomplete Gamma function high-order power equation, then associating the solving problem with the maximum value distribution of independent and identically distributed Gamma random variables, and approximating the incomplete Gamma function high-order power by a Gumbel distribution function according to an extreme value theory. As the Gumbel distribution function is an elementary function, a closed approximate solution of an equation can be obtained, the computational complexity is reduced, and the intuitiveness of the analysis of the related problems is improved.
Drawings
FIG. 1 is a flow chart of threshold calculation in MIMO-OFDM space-time block code identification based on sequence statistics.
Fig. 2 is a graph comparing the performance of the invention in the calculation of the MIMO-OFDM space-time grouping identification threshold with the accurate numerical value.
Detailed Description
The present invention will now be described in further detail with reference to the accompanying drawings.
The method for quickly determining the identification threshold of the multi-antenna space-time block code shown in fig. 1 specifically includes the following steps:
first, condition adaptability test
Considering the Identification of STBC codes in MIMO-OFDM communication signal Identification, if a block space-time code Identification algorithm based on sequence statistics is adopted, which is proposed by the literature (y.a. eldermerdalsh, o.a. dobre, b.j.liao, "Blind Identification of SM and Alamouti STBC-OFDM Signals", IEEE Transactions on Wireless Communications, vol.14, pp.972-982, 2015.), the process is as follows:
first, a feature quantity is defined based on a maximum value of an autocorrelation function between any two receiving antennas
γ＝maxFc(τ)，
In the formula:
Figure BDA0001537257050000041
wherein Fij(tau) is a time-delay autocorrelation function between any two different receiving antennas, tau is the amount of time delay, NrThe number of the receiving antennas. Is easy to understand, Fc(τ) approximately obeys a degree of freedom of Nc＝Nr(Nr-a central chi-square distribution of 1)/2.
Then, a smaller false alarm probability P is set according to the constant false alarm criterionfaCalculating according to the following formula to obtain the corresponding decision threshold
(1-pfa)＝PN+v(Nc，λ/2) (1)
In the above formula: lambda is a decision threshold, v is the number of OFDM cyclic prefixes, N is the number of samples of a received signal, PnAnd (α, x) is the parameter α raised to the power n of the incomplete Gamma function.
And finally, if gamma is larger than or equal to lambda, judging the code type of the signal to be STBC, and otherwise, judging the code type to be SM code.
The existing problem transformation solves the problem of equation (1), and the range of each relevant parameter needs to be checked in the solution, which specifically comprises the following steps: for false alarm probability pfaGenerally, the value is between 0.0001 and 0.05; for NvN + v parameter, which is required to be above 50; for parameter Nc＝Nr(Nr-1)/2, generally less than 40. If the parameters are not in the range, the approximation algorithm cannot guarantee the performance, but the general MIMO-OFDM signal identification application occasions can meet the requirement.
Second, calculation of auxiliary constant
Generalizing equation (1) to
Pn(α，x/β)＝q，0≤q≤1，n＞＞1，α＞2 (2)
From its parameters α, n, β, a constant a for normalization is calculatedn，bnWhen n is larger, the distribution of independent and identically distributed Gamma random vectors can be approximated by a Gumbel function after proper normalization. In classical EVT theory, the standard normalization constants are:
bn＝F-1(1-1/n)，an＝1/nf(bn)
however, the above formula requires that the inverse of the Gamma distribution be taken, and no analytical solution is available. In order to solve an analytic solution in a complete sense, the invention obtains an analytic solving method of a normalization constant by means of a tail equivalence principle in an EVT theory. Studies have shown that the tails of Gamma random variables are equivalent to those of the generic Weilbull-like distribution, and both distributions belong to the maximum attraction field of the Gumbel function. Let the distribution function of the generic Weilbull class distribution be:
G(x)＝1-Kxλexp(-Cxτ)，x≥x0
in the formula: k, C, lambda is more than 0, tau is more than or equal to 1, x0And the solid constants are the solid constants when the solid constants are more than 0.
When K is ═ betaα-1(α)]-1，λ＝α-1，C＝1/β，τ＝1，x0When 0, there are:
Figure BDA0001537257050000051
the above formula shows that: the tail of the Gamma random variable is equivalent to the tail of the general Weilbull type distribution, namely, the normalization constant of the Weilbull type distribution function can be used for replacing the corresponding normalization constant of the Gamma random variable. In the literature (Gasull A., L6pez-Salcedo J.A., Utzet F.Maxima of Gamma random variables and other Weibull-like distributions and the Lambert W function [ J ]. Test, 2015, 24 (4): 120.), an approximate analytical solution of Weilbull-like distribution extreme value distribution normalization constants is obtained by using an expansion formula of a Lambert W function, and the specific formula is as follows:
bn＝β{ln[n/(α)]+(α-1)ln Bn+[(α-1)2 ln Bn-(α-1)2 ln(α-1)+α-1]/Bn}；
an＝bnβ[bn+β(α-1)]/[(bn)2-β2(α-1)(α-2)]， (3)
in the formula: constant Bn＝ln[n/(α)]+ (. alpha. -1) ln (. alpha. -1) and (x) is a Gamma function. Obviously, the output of the above constants is calculatedThe input quantity is determined by parameters and powers of incomplete Gamma functions in the equation, and constant calculation is given by a closed formula without iterative solution. Let alpha be NcWhere β is 2, N is N + v, and the formula (3) is substituted to obtain the normalization constant in the above formula
Figure BDA0001537257050000052
Three, approximate equation establishment
In the second step, according to the extreme value theory, for n independent Gamma random quantities X with the same distributed shape parameter as alpha and the scale parameter as beta1，X2，...，XnThe maximum value Y is max (X)1，X2，...，Xn) The limiting distribution of the distribution function is a Gumbel distribution, i.e. if there is a suitable normalization constant an，bn> 0, there are:
Figure BDA0001537257050000053
in the above formula: a (x) exp [ -exp (-x) ] is the Gumbel function. When n is large, equation (2) can be converted into the following approximate equation
Figure BDA0001537257050000054
It can be seen that the basic basis for obtaining the above approximation equation is the extreme value distribution theory, i.e. when n is larger, the distribution of independent and identically distributed Gamma random vectors can be approximated by a Gumbel function after proper normalization. According to this principle, equation (1) can be approximated as:
Figure BDA0001537257050000061
approximate solution of four, threshold
For equation (6), a brief derivation yields an approximate solution:
Figure BDA0001537257050000062
in the formula: ln (x) is a natural logarithm.
Fig. 2 shows the performance comparison of the algorithm when used for the MIMO-OFDM space-time packet identification threshold calculation with the precise numerical solution. In the experiment, the number of transmitting antennas is 2, the number of receiving antennas is 3, 4, 5 and 6 respectively, the number of received signal samples is 256, 512 and 1024 respectively, the false alarm probability is 0.0001, the EVT in the figure represents an approximate analytic solution based on the EVT provided by the invention, and Exact represents a numerical accurate solution. As can be seen from the figure, the algorithm has better approximation performance, and the approximation performance is better along with the increase of the number of samples, so that the performance requirement of the application scene can be met.
The above is only a preferred embodiment of the present invention, and the protection scope of the present invention is not limited to the above-mentioned embodiments, and all technical solutions belonging to the idea of the present invention belong to the protection scope of the present invention. It should be noted that modifications and embellishments within the scope of the invention may be made by those skilled in the art without departing from the principle of the invention.

Claims (2)

1. A method for rapidly determining a multi-antenna space-time block code identification threshold is characterized by comprising the following steps:
1) and (3) testing the condition adaptability: for the identification of MIMO-OFDM signal space-time block code based on sequence statistics, the number N of receiving antennas is usedrDefining a parameter Nc＝Nr(Nr-1)/2; defining a parameter N according to the length v of the cyclic prefix of the sequence of the sample length N, OFDMvN + v, and a false alarm probability value pfaAnd the ranges of the three parameters are checked; the method specifically comprises the following steps:
defining a characteristic quantity based on the maximum value of the autocorrelation function between any two receiving antennas:
Υ＝maxFc(τ)
wherein the content of the first and second substances,
Figure FDA0002686487210000011
Fij(tau) is a time-delayed autocorrelation function between any two different receiving antennas, subject to a degree of freedom Nc＝Nr(Nr-1)/2, tau is the delay, NrThe number of receiving antennas;
setting false alarm probability P according to constant false alarm criterionfaAnd calculating to obtain a corresponding decision threshold according to the following formula:
(1-pfa)＝PN+v(Nc,λ/2) (1)
wherein λ is a decision threshold, v is the number of OFDM cyclic prefixes, N is the number of samples of the received signal, P isn(α, x) is the parameter α raised to the power n of the incomplete Gamma function;
the corresponding recognition rule is: if upsilon is more than or equal to lambda, the code type of the signal is judged to be STBC, otherwise, the signal is SM code;
in the identification, the threshold is obtained by solving the equation (1), and the false alarm probability P needs to be solved in advancefaParameter NvN + v and the parameter Nc＝Nr(NrThe range of-1)/2 is tested;
2) and (3) auxiliary constant calculation: calculating the auxiliary constant a according to the extreme value theoryn,bn(ii) a The method specifically comprises the following steps:
equation (1) is generalized as:
Pn(α,x/β)＝q,0≤q≤1,n＞＞1，α＞2 (2)
from the parameters α, n, β, a constant a is calculated for normalizationn,bnThe concrete formula is as follows:
bn＝β{ln[n/(α)]+(α-1)lnBn+[(α-1)2lnBn-(α-1)2ln(α-1)+α-1]/Bn}
an＝bnβ[bn+β(α-1)]/[(bn)2-β2(α-1)(α-2)] (3)
wherein, constant Bn＝ln[n/(α)]+ (α -1) ln (α -1), (x) is a Gamma function;
let alpha be Ncβ 2, N + v, substituting formula (3) to obtain a normalization constant
Figure FDA0002686487210000025
3) Establishing an approximate equation: replacing a power function of an incomplete Gamma function in the threshold solution equation by a Gumbel function to obtain an approximate equation of the threshold solution; the method specifically comprises the following steps:
according to the extreme value theory, n independent and identically distributed Gamma random vectors X with the shape parameter of alpha and the scale parameter of beta1,X2,...,XnThe maximum value Y is max (X)1,X2,...,Xn) The limiting distribution of the distribution function is a Gumbel distribution, i.e. if there is a suitable normalization constant an,bn> 0, there are:
Figure FDA0002686487210000021
wherein Λ (x) ═ exp [ -exp (-x) ] is a Gumbel function;
equation (2) is converted to the following approximate equation:
Figure FDA0002686487210000022
accordingly, equation (1) is approximated as:
Figure FDA0002686487210000023
4) approximate solution of threshold: and solving an approximate equation according to the set false alarm probability and the auxiliary constant obtained by calculation to obtain an analytic solution of the threshold.
2. The method for rapidly determining the identification threshold of the multi-antenna space-time block code as claimed in claim 1, wherein: the step 4) specifically comprises the following steps:
equation (6) is derived to obtain an approximate solution:
Figure FDA0002686487210000024
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