Fractional domain waveform parameter selection method based on goodness-of-fit test

Abstract

The invention discloses a fractional domain waveform parameter selection method based on goodness-of-fit test, and belongs to the technical field of wireless communication. The problem that the probability that transmitted information is detected is still high in an existing method is solved. According to the method, Gaussian white noise is used as a background noise model, the distance between a distribution function of a received sample signal and a Gaussian white noise distribution function is calculated, the distance is compared with a threshold so as to judge the existence of a detection signal, and then waveform parameters are selected according to the detection probability corresponding to each group of waveform parameters. According to the method, the diversity advantage of the waveform subjected to extended weighted fractional Fourier transform is fully exerted, transformation parameter optimization with the advantage of low detection probability is completed, the optimized waveform has the statistical characteristic of Gaussian-like distribution, the detection difficulty of an eavesdropper is increased, and the probability that transmitted information is detected is reduced. The method can be applied to the technical field of wireless communication.

Description

Technical Field

[0001] The present invention belongs to the technical field of wireless communication, and particularly relates to a method for selecting fractional-domain waveform parameters based on goodness-of-fit test. Background Art

[0002] The weighted fractional Fourier transform changes the original time-frequency characteristics of a signal and transforms the signal into a mixed carrier form that contains both single-carrier components and multi-carrier components. Under specific parameters, the constellation diagram in the complex plane exhibits characteristics similar to Gaussian white noise, and the whitening of the communication behavior will increase the difficulty of detection and identification by eavesdroppers, ensuring the secure transmission of information. The extended weighted fractional Fourier transform is a transformation method that expands a single parameter into four parameters based on the classical weighted fractional Fourier transform. More parameters make the transformation form of the signal more flexible. However, not all signals after the extended weighted fractional Fourier transform have the characteristic of Gaussian-like statistical properties. Therefore, how to take advantage of waveform diversity in constructing Gaussian-like signals and propose a new parameter optimization scheme to reduce the probability of detected transmitted information is a problem worthy of research. Summary of the Invention

[0003] The purpose of the present invention is to reduce the probability of detected transmitted information, and a method for selecting fractional-domain waveform parameters based on goodness-of-fit test is proposed.

[0004] The technical solution adopted by the present invention to solve the above technical problems is: a method for selecting fractional-domain waveform parameters based on goodness-of-fit test, the method specifically includes the following steps:

[0005] Step 1: Randomly generate M signal sequences each with a length of 2N points, and denote the d-th signal sequence as s d , s d = [s d,1 , s d,2 ,..., s d,2N , s d,1 , s d,2 ,..., s d,2N represents the 1st, 2nd,..., 2N-th elements in the signal sequence s d ;

[0006] Step 2: Perform QPSK mapping on the signal sequence s d to obtain the mapped N-point baseband signal sequence x d , x d = [x d,1 , x d,2 ,..., x d,N , x d,1 , x d,2 , …, x d,NDenote the first, second, …, Nth elements in the baseband signal sequence x d ;

[0007] Step 3: Establish a waveform parameter database for the extended weighted fractional Fourier transform, and select waveform parameters using the M baseband signal sequences obtained in Step 2 to obtain the selected waveform parameters.

[0008] Furthermore, any set of waveform parameters in the waveform parameter database of the extended weighted fractional Fourier transform is denoted as The waveform parameters satisfy: and

[0009] Furthermore, the process of selecting waveform parameters using the M baseband signal sequences obtained in Step 2 to obtain the selected waveform parameters is as follows:

[0010] Step 3-1: Initialize the number of waveform parameter groups a = 1;

[0011] Step 3-2: Initialize the number of baseband signal sequences d = 1;

[0012] Step 3-3: Obtain the weighting coefficient of the extended weighted fractional Fourier transform according to the ath group of waveform parameters Then perform the extended weighted fractional Fourier transform on the dth baseband signal sequence using the weighting coefficient to obtain the transformed signal

[0013] Step 3-4: For each element X1, X2,... X in the transformed signal N take the square of the modulus respectively, then sort the obtained N results from smallest to largest, and obtain the theoretical distribution function according to the sorting result;

[0014] Step 3-5: Construct the test statistic AD(Z) according to the theoretical distribution function, and then determine the threshold value λ of the test statistic;

[0015] Step 3-6: Compare the test statistic AD(Z) with the threshold value λ to obtain the judgment result; specifically:

[0016] If the test statistic is greater than or equal to the threshold value, then

[0017] If the test statistic is less than the threshold value, then

[0018] Step 3-7: Determine whether d = M is satisfied, where M represents the total number of baseband signal sequences;

[0019] If d = M is satisfied, then execute Step 3-8;

[0020] If d≠M, then let d = d + 1, and return to execute step 33;

[0021] Step 38: Calculate the detection probability of the waveform parameters of the a-th group

[0022] Compare the detection probability with the threshold t0. If it is less than t0, then retain the waveform parameters of the a-th group. If it is greater than or equal to t0, then discard the waveform parameters of the a-th group;

[0023] Then determine whether a = K is satisfied, where K represents the total number of groups of waveform parameters in the waveform parameter database;

[0024] If a = K is satisfied, then execute step 39;

[0025] If a = K is not satisfied, then let a = a + 1, and return to execute step 32;

[0026] Step 39: Obtain the set composed of all the retained groups of waveform parameters.

[0027] Further, obtaining the weighting coefficient of the extended weighted fractional Fourier transform according to the waveform parameters of the a-th group is specifically:

[0028]

[0029] where represents the weighting coefficient of the extended weighted fractional Fourier transform, and i represents the imaginary unit.

[0030] Further, the specific process of step 34 is:

[0031] Step 341: Let X1, X2,... X N all be complex Gaussian distribution variables, X n = b n + ic n , n = 1, 2,..., N. The real part b n and the imaginary part c n are independent and identically distributed real Gaussian random variables. The real part b n and the imaginary part c n obey the Gaussian distribution N(0, σ 2 ), and σ 2 is the variance of the Gaussian distribution. Take the square of the modulus of the variable X n :

[0032]

[0033] Step 342: For |X n |2 Perform an ascending sort, and denote each element in the sorting result as X1′, X2′, …, X′ N , and then calculate the variable U according to each element in the sorting result n :

[0034]

[0035] Step 343, variable U n , n = 1, 2, …, N follows a chi-square distribution with 2 degrees of freedom, then the theoretical distribution function of variable U n is:

[0036]

[0037] where F(·) represents the theoretical distribution function, e represents the base of the natural logarithm, and Z n represents the value of the theoretical distribution function corresponding to U n .

[0038] Furthermore, in the said step 35, the specific process of constructing the test statistic AD(Z) according to the theoretical distribution function is:

[0039]

[0040] where AD(Z) represents the test statistic, and Z = [Z1, Z2,..., Z N .

[0041] Furthermore, in the said step 35, the specific process of determining the threshold value of the test statistic is:

[0042]

[0043] where P(AD(Z) ≤ λ) represents the probability of AD(Z) ≤ λ, and r j = (-1) j Γ(j + 0.5) / (Γ(0.5)j!), Γ is the Gamma function, and w represents the integration variable;

[0044] The significance level α = P(AD(Z) > λ) = 1 - P(AD(Z) ≤ λ), and calculate the threshold value λ corresponding to the significance level α

[0045] Furthermore, the specific process of the said step 38 is:

[0046]

[0047] where represents the detection probability of the waveform parameters of the a-th group

[0048] Further, the method further includes Step Four, which is: transmitting the signal to be transmitted by using the reserved waveform parameters.

[0049] Furthermore, the specific process of Step Four is as follows:

[0050] Step Four - 1: Performing QPSK mapping on the signal sequence to be transmitted at the sending end to obtain the mapped baseband signal sequence;

[0051] Step Four - 2: Selecting any group of waveform parameters from all the reserved group of waveform parameters, generating the weighting coefficients of the extended weighted fractional Fourier transform by using the selected waveform parameters, and then performing the extended weighted fractional Fourier transform on the mapped baseband signal sequence by using the generated weighting coefficients;

[0052] Step Four - 3: Modulating the signal after the extended weighted fractional Fourier transform to the carrier frequency to obtain the modulated data, and transmitting the modulated data through the antenna.

[0053] The beneficial effects of the present invention are as follows:

[0054] The present invention uses Gaussian white noise as the background noise model. By calculating the distance between the distribution function of the received sample signal and the distribution function of Gaussian white noise, comparing the distance with the threshold to judge the existence of the detection signal, and then selecting the waveform parameters according to the detection probability corresponding to each group of waveform parameters. The present invention gives full play to the diversity advantage of the waveform after the extended weighted fractional Fourier transform, completes the optimization selection of the transformation parameters with the advantage of low detection probability, makes the optimized waveform have the statistical characteristics of Gaussian-like distribution, increases the detection difficulty of the eavesdropper, reduces the probability of being detected, and greatly improves the security of communication behavior. Description of the Drawings

[0055] Figure 1 is a flowchart of a method for selecting fractional domain waveform parameters based on goodness-of-fit test of the present invention;

[0056] wherein: GWFRFT represents extended weighted fractional Fourier transform;

[0057] Figure 2 is a flowchart of signal transmission based on the selected parameters. Specific Embodiments

[0058] Specific Embodiment One: Combining Figure 1 to illustrate this embodiment. A method for selecting fractional domain waveform parameters based on goodness-of-fit test described in this embodiment, the method specifically includes the following steps:

[0059] Step One: Randomly generating M signal sequences with a length of 2N points, and denoting the d-th signal sequence as sd , s d = [s d,1 , s d,2 ,..., s d,2N , s d,1 , s d,2 ,..., s d,2N represents the 1st, 2nd,..., 2Nth elements in the signal sequence s d ;

[0060] Step 2: Perform QPSK mapping on the signal sequence s d to obtain the mapped N-point baseband signal sequence x d , x d = [x d,1 , x d,2 ,..., x d,N , x d,1 , x d,2 ,..., x d,N represents the 1st, 2nd,..., Nth elements in the baseband signal sequence x d ;

[0061] The present invention is compatible with various modulation methods, and here QPSK modulation method is taken as an example;

[0062] Step 3: Establish a waveform parameter database for the extended weighted fractional Fourier transform, and use the M baseband signal sequences obtained in Step 2 to select waveform parameters to obtain the selected waveform parameters.

[0063] Specific Embodiment Ⅱ: The difference between this embodiment and Specific Embodiment Ⅰ is that any set of waveform parameters in the waveform parameter database of the extended weighted fractional Fourier transform is denoted as The waveform parameters satisfy: and

[0064] Other steps and parameters are the same as those in Specific Embodiment Ⅰ.

[0065] In the present invention, a set of waveform parameters forms an arithmetic progression. By adjusting the value of and the interval between and , multiple sets of waveform parameters can be obtained, and a waveform parameter database is formed by using multiple sets of waveform parameters.

[0066] Specific Embodiment Ⅲ: The difference between this embodiment and Specific Embodiment Ⅰ or Ⅱ is that the process of using the M baseband signal sequences obtained in Step 2 to select waveform parameters to obtain the selected waveform parameters is as follows:

[0067] Step 3.1: Initialize the number of waveform parameter groups \(a = 1\);

[0068] Step 3.2: Initialize the number of baseband signal sequences \(d = 1\);

[0069] Step 3.3: Obtain the weighting coefficients of the extended weighted fractional Fourier transform according to the \(a\)-th group of waveform parameters Then perform the extended weighted fractional Fourier transform on the \(d\)-th baseband signal sequence using the weighting coefficients to obtain the transformed signal

[0070]

[0071] where is the \(a\)-th group of waveform parameters, \(f(t)=x\) d , the frequency domain component \(F(t)\) and the time domain component \(f(t)\) are a Fourier transform pair, that is, \(F(t)\) is the Fourier transform result of \(f(t)\), \(f(-t)\) is the inversion function of \(f(t)\) centered at the origin, and \(F(-t)\) is the inversion function of \(F(t)\) centered at the origin;

[0072] Step 3.4: For each element \(X_1, X_2,\cdots,X\) in the transformed signal N , take the square of the modulus respectively, then sort the obtained \(N\) results from small to large, and obtain the theoretical distribution function according to the sorting result;

[0073] Step 3.5: Construct the test statistic \(AD(Z)\) according to the theoretical distribution function, and then determine the threshold value \(\lambda\) of the test statistic;

[0074] Step 3.6: Compare the test statistic \(AD(Z)\) with the threshold \(\lambda\) to obtain the decision result; specifically:

[0075] If the test statistic is greater than or equal to the threshold value, the decision is \(H_1\), that is

[0076] If the test statistic is less than the threshold value, the decision is \(H_0\), that is

[0077] Step 3.7: Determine whether \(d = M\) is satisfied, where \(M\) represents the total number of baseband signal sequences;

[0078] If \(d = M\) is satisfied, execute Step 3.8;

[0079] If \(d = M\) is not satisfied, let \(d = d + 1\), and return to execute Step 3.3;

[0080] Step 3.8: Calculate the probability of accepting the alternative hypothesis \(H_1\) for the \(a\)-th group of waveform parameters, that is, calculate the detection probability of the \(a\)-th group of waveform parameters

[0081] Compare the detection probability according to the security requirements of the system with the threshold t0. If it is less than t0, then retain the waveform parameters of the a-th group. If it is greater than or equal to t0, then discard the waveform parameters of the a-th group; the threshold t0 can be set according to the actual security requirements during the communication process;

[0082] Then determine whether a = K is satisfied, where K represents the total number of groups of waveform parameters in the waveform parameter database;

[0083] If a = K is satisfied, then execute Step 39;

[0084] If a = K is not satisfied, then set a = a + 1 and return to execute Step 32;

[0085] Step 39: Obtain the set composed of all the retained groups of waveform parameters.

[0086] Other steps and parameters are the same as those in the First or Second Specific Embodiment.

[0087] Specific Embodiment 4: The difference between this embodiment and any one of the First to Third Specific Embodiments is that the method for obtaining the weighting coefficient of the extended weighted fractional Fourier transform according to the waveform parameters of the a-th group is specifically:

[0088]

[0089] wherein, represents the weighting coefficient of the extended weighted fractional Fourier transform, and i represents the imaginary unit.

[0090] Other steps and parameters are the same as those in any one of the First to Third Specific Embodiments.

[0091] Specific Embodiment 5: The difference between this embodiment and any one of the First to Fourth Specific Embodiments is that the specific process of Step 34 is:

[0092] Step 341: Assume that X1, X2,... X N are all complex Gaussian distribution variables, and X n = b n + ic n , n = 1, 2,..., N. The real part b n and the imaginary part c n are independent and identically distributed real Gaussian random variables. The real part b n and the imaginary part c n obey the Gaussian distribution N(0, σ 2 ), and σ 2 is the variance of the Gaussian distribution. Take the square of the modulus of the variable X n [[ID=6):

[0093]

[0094] Step 342, right | X n | 2 Perform ascending sorting and record each element in the sorting result as X1′, X2′, …, X′ N , and then calculate the variable U according to each element in the sorting result n :

[0095]

[0096] Step 343. Variable U n ,n=1,2,…,N obeys the chi-square distribution with 2 degrees of freedom, then the variable U n The theoretical distribution function of is:

[0097]

[0098] Where F(·) represents the theoretical distribution function, e represents the base of the natural logarithm, and Z n Indicates U n The corresponding theoretical distribution function value.

[0099] The other steps and parameters are the same as those in the first to fourth embodiments.

[0100] Specific embodiment 6: This embodiment differs from any one of specific embodiments 1 to 5 in that, in steps 3 and 5, the specific process of constructing the test statistic AD(Z) according to the theoretical distribution function is as follows:

[0101]

[0102] Where AD(Z) represents the test statistic, Z=[Z1,Z2,...,Z N ].

[0103] The other steps and parameters are the same as those in the first to fifth embodiments.

[0104] The distribution of AD(Z) is independent of the distribution of the null hypothesis H0, and when N≥5, the distribution function of AD(Z) converges.

[0105] Specific embodiment seven: This embodiment differs from any one of specific embodiments one to six in that, in step three and five, the specific process of determining the threshold value of the test statistic is as follows:

[0106]

[0107] Among them, P(AD(Z)≤λ) represents the probability that AD(Z)≤λ, r j =(-1) jΓ(j + 0.5) / (Γ(0.5)j!), where Γ is the Gamma function and w represents the integration variable;

[0108] The significance level α = P(AD(Z) > λ) = 1 - P(AD(Z) ≤ λ). Calculate the threshold value λ corresponding to the significance level α. α is set according to the actual requirements during the communication process and is set to 0.05 in the present invention.

[0109] Other steps and parameters are the same as those in any one of the first to sixth specific embodiments.

[0110] Specific embodiment eight: The difference between this embodiment and any one of the first to seventh specific embodiments is that the specific process of step three eight is as follows:

[0111]

[0112] Among them, represents the detection probability of the waveform parameters of the a-th group.

[0113] Other steps and parameters are the same as those in any one of the first to seventh specific embodiments.

[0114] Specific embodiment nine: The difference between this embodiment and any one of the first to eighth specific embodiments is that the method further includes step four, and step four is: Transmit the signal to be transmitted by using the reserved waveform parameters.

[0115] Other steps and parameters are the same as those in any one of the first to eighth specific embodiments.

[0116] Specific embodiment ten: Combine Figure 2 to illustrate this embodiment. The difference between this embodiment and any one of the first to ninth specific embodiments is that the specific process of step four is as follows:

[0117] Step four one: Perform QPSK mapping on the signal sequence to be transmitted at the sending end to obtain the mapped baseband signal sequence;

[0118] Step four two: Select any group of waveform parameters from all the reserved groups of waveform parameters, generate the weighting coefficients of the extended weighted fractional Fourier transform by using the selected waveform parameters, and then perform the extended weighted fractional Fourier transform on the mapped baseband signal sequence by using the generated weighting coefficients;

[0119] Step four three: Modulate the signal after the extended weighted fractional Fourier transform to the carrier frequency to obtain the modulated data, and transmit the modulated data through the antenna.

[0120] Other steps and parameters are the same as those in any one of the first to ninth specific embodiments.

[0121] The above calculation examples of the present invention are only for illustrating in detail the calculation model and calculation process of the present invention, rather than limiting the implementation manner of the present invention. For those of ordinary skill in the art, other different forms of changes or variations can be made on the basis of the above description. It is impossible to list all the implementation manners here. Any obvious changes or variations derived from the technical solution of the present invention still fall within the protection scope of the present invention.

Claims

1. A method for selecting waveform parameters in the fractional domain based on goodness-of-fit test, characterized in that The method specifically includes the following steps: Step 1: Randomly generate M signal sequences each with a length of 2N points, and denote the d-th signal sequence as s d , s d = [s d,1 , s d,2 ,..., s d,2N , s d,1 , s d,2 ,…, s d,2N represents the 1st, 2nd,..., 2N-th elements in the signal sequence s d ; Step 2: Perform QPSK mapping on the signal sequence s d to obtain the mapped N-point baseband signal sequence x d , where x d = [x d,1 , x d,2 , …, x d,N , and x d,1 , x d,2 , …, x d,N represent the 1st, 2nd, …, Nth elements in the baseband signal sequence x d ; Step 3: Establish a waveform parameter database for the extended weighted fractional Fourier transform, and select waveform parameters using the M baseband signal sequences obtained in Step 2 to obtain the selected waveform parameters.

2. The method for selecting fractional-domain waveform parameters based on goodness-of-fit test according to claim 1, wherein Any set of waveform parameters in the waveform parameter database of the extended weighted fractional Fourier transform is denoted as k = 0, 1, 2, 3, and the waveform parameters satisfy: and 3. The method for selecting waveform parameters in the fractional domain based on the goodness-of-fit test according to claim 2, wherein The process of selecting waveform parameters using the M baseband signal sequences obtained in Step 2 to obtain the selected waveform parameters is as follows: Step 3-1: Initialize the number of waveform parameter groups a = 1; Step 3-2: Initialize the number of baseband signal sequences d = 1; Step 33: Obtain the weighting coefficient of the extended weighted fractional Fourier transform according to the waveform parameters of Group a When l = 0, 1, 2, 3, perform the extended weighted fractional Fourier transform on the d-th baseband signal sequence using the weighting coefficient to obtain the transformed signal Steps three and four: For the transformed signal with each element X1, X2,... X N take the square of the modulus respectively, then sort the N obtained results from small to large, and obtain the theoretical distribution function according to the sorting results; Step 3-5: Construct a test statistic AD(Z) according to the theoretical distribution function, and then determine the threshold λ of the test statistic; Step 3-6: Compare the test statistic AD(Z) with the threshold λ to obtain a decision result; Specifically: If the test statistic is greater than or equal to the threshold value, then If the test statistic is less than the threshold value, then Step 3-7: Determine whether d = M is satisfied, where M represents the total number of baseband signal sequences; If d = M is satisfied, then execute Step 3-8; If d = M is not satisfied, then set d = d + 1 and return to execute Step 3-3; Step 38. Calculate the detection probability of the waveform parameters of Group a Compare the detection probability with the threshold t0. If it is less than t0, retain the waveform parameters of the a-th group. If it is greater than or equal to t0, discard the waveform parameters of the a-th group; Then determine whether a = K is satisfied, where K represents the total number of waveform parameter groups in the waveform parameter database; If a = K is satisfied, then execute Step 3-9; If a = K is not satisfied, then set a = a + 1 and return to execute Step 3-2; Step 3-9: Obtain a set composed of all the retained groups of waveform parameters.

4. A method for selecting waveform parameters in the fractional domain based on goodness-of-fit test according to claim 3, characterized in that The process of obtaining the weighting coefficients of the extended weighted fractional Fourier transform according to the a-th group of waveform parameters is specifically as follows: Among them, represents the weighting coefficient of the extended weighted fractional Fourier transform, and i represents the imaginary unit.

5. A method for selecting fractional domain waveform parameters based on goodness-of-fit test according to claim 4, characterized in that The specific process of Step 3-4 is as follows: Step 341. Let X1, X2, ..., X N be complex Gaussian distributed variables, X n = b n + ic n , n = 1, 2, ..., N. The real part b n and the imaginary part c n are independent and identically distributed real Gaussian random variables. The real part b n and the imaginary part c n follow the Gaussian distribution N(0, σ 2 ), where σ 2 is the variance of the Gaussian distribution. Take the square of the modulus of the variable X n : Step 342: Perform an ascending sort on |X n | 2 and denote each element in the sorting result as X1′, X2′, …, X′ N . Then calculate the variable U respectively according to each element in the sorting result n : Step Three Four Three, Variable U n , where \(n = 1, 2, \ldots, N\) follows a chi-square distribution with 2 degrees of freedom, then the variable U n has the following theoretical distribution function: Among them, F(·) represents the theoretical distribution function, e represents the base of the natural logarithm, and Z n represents U n corresponding theoretical distribution function value.

6. The method for selecting fractional domain waveform parameters based on goodness-of-fit test according to claim 5, characterized in that In Step 3-5, the specific process of constructing the test statistic AD(Z) according to the theoretical distribution function is as follows: Among them, AD(Z) represents the test statistic, where Z = [Z1, Z2, …, Z N .

7. A method for selecting waveform parameters in the fractional domain based on goodness-of-fit test according to claim 6, characterized in that In Step 3-5, the specific process of determining the threshold of the test statistic is as follows: where P(AD(Z) ≤ λ) represents the probability that AD(Z) ≤ λ, r j = (-1) j Γ(j + 0.5) / (Γ(0.5)j!), Γ is the Gamma function, and w represents the integration variable; The significance level α = P(AD(Z)>λ) = 1 - P(AD(Z)≤λ), and calculate the threshold λ corresponding to the significance level α.

8. A method for selecting waveform parameters in the fractional domain based on goodness-of-fit test according to claim 7, characterized in that The specific process of Step 3-8 is as follows: Among them, represents the detection probability of the waveform parameters of the a-th group.

9. A method for selecting waveform parameters in the fractional domain based on goodness-of-fit test according to claim 8, characterized in that The method further includes Step 4, and Step 4 is: Transmit the signal to be transmitted using the retained waveform parameters.

10. A method for selecting fractional-domain waveform parameters based on goodness-of-fit test according to claim 9, characterized in that The specific process of Step 4 is as follows: Step 4-1: Perform QPSK mapping on the signal sequence to be transmitted at the sending end to obtain the mapped baseband signal sequence; Step 4-2: Select any group of waveform parameters from all the retained groups of waveform parameters, generate the weighting coefficients of the extended weighted fractional Fourier transform using the selected waveform parameters, and then perform the extended weighted fractional Fourier transform on the mapped baseband signal sequence using the generated weighting coefficients; Step 4-3: Modulate the signal after the extended weighted fractional Fourier transform to the carrier frequency to obtain the modulated data, and transmit the modulated data through the antenna.

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