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南航矩阵论英文小论文_最小二乘法

Key words: Wireless communication; Pseudo Noise (PN) phase estimate; Precision synchronization;Anti-frequency offset; Highly dynamic condition; Least Squares (LS) 1Introduction

With the continuous development of wireless communication and network, the precise synchronization requirement of the system is higher and higher, such as the American GPS , Chinese Beidou Positioning System, American Tactical Tatgeting Network Technology and Intra Flight Data Link Network Technology put forward new requirements on precise synchronization technology, and often need to work under the condition of high dynamic and low noise ratio. Direct Sequence Spread Spectrum, because of its good correlation and good anti-noise ability, is often used in the process of synchronous communication system, especially in the field of aerospace and military communications, Direct Spread Spectrum signal usually applied to complete the range and positioning work [1, 2]. Both synchronous and ranging, the phase measuring accuracy of PN code sequence is the underlying factors of affecting the performance, so a lot of literatures do many researches on the phase measurement of the PN sequence [3-6].For example, literature[3] analysis of the effect of deviation of phase measurement in the time domain, and subsection cross-correlation method is proposed to improve the ability to resist frequency deviation of the system, but the article didn't mention the error range of system accurate synchronization in the end.The literature[4], with fast Fourier transform method, by the process of calculating reference signal and the input signal correlation spectrum achieves DSSS signal PN code phase measurement, this method is of high efficiency and widely used. Defects of the method is, however, that it must increase

the sampling frequency in order to further improve the accuracy of phase measurement, leading to increase the points of FFT, and increase the amount of calculation and implementation difficulty. The literature[5] with the two-point linear interpolation deals with datas of the peak and the peak times of the reference signal and the input signal correlation spectrum in order to improve the measurement precision, but the algorithm is still not high precision in practical application. The literature[6] uses the correlation spectrum and its neighboring two data points to determine quadratic interpolation polynomial, then calculate the maximum point of interpolation polynomial to determine precise location related to spectral spectrum peak, the effect of improving the measuring accuracy of DSSS signal PN code phase by the method is not obvious, but not the analysis of the effect of deviation, according to the analysis of the literature[3], however, this method is largely affected by the frequency deviation, this article also shows that it is not suitable for high dynamic environment through simulation.The literature[7] proposes that it uses Delay-Locked Loop S phase curve linear characteristics of the middle section, and uses the least square method to get precise pseudo-code phase difference, this method is simple, high precision, can yet be regarded as a kind of good accurate synchronization method, but it also did not consider the effects of deviation in its article. The literature[8] quantitatively analyses the effect of the arbitrary channel non-ideal features for pseudo code ranging zero value, the receiver of the software has carried on the simulation to the analysis result, but it has nothing about how to solve the pseudo code ranging in the condition of high dynamic problems. The literature[9] solves the problem of algorithm complexity increases due to carrier phase measurement uncertainty by using the resampling, but it does not relate to how to improve the PN phase measurement accuracy.

In high dynamic condition, for example, satellite communication, space combat and so on, synchronization accuracy will be affected by the frequency deviation and the rate of migration.Considering the two factors, this paper puts forward an precise measurement PN phase method based on the improved least squares, and shows the measuring results under the condition of high dynamic using the improved method through theoretical derivation and numerical simulation.

2PN phase measurement method and the discrete expression of the cross-correlation function under the condition of high dynamic

2.1 The PN phase measurement method of the least squares

The literature[7] puts forward using the least square method to measure the phase of the received PN sequence, the basic idea is to use the PN code related symmetry, in order to achieve phase curves. As shown in figure 1 and figure 2.

figure 1 relevant curve under ideal conditions

figure 2 phase curve under ideal conditions

The mathematic expression of the phase curve is as follows:

（1）

in the equation expresses the phase difference of local PN and received PN, is for code cycle.

Under ideal conditions, the local pseudo-code shift around N sampling points respectively, input phase detecting unit, and output phase error; If phase discriminator of PN sequence and the input of the PN sequence of phase difference is less than , then the output value must form a straight line, the corresponding in figure 2

scope of the straight line.Make these points by using the least squares fitting of this line, set curve equation is y = bx + a, then under the minimum mean square error criterion of optimal estimation is for the expression form of the equation(2) [7,10].

（2）

Due to is symmetrical, so ，plug in the equation(2) to get the phase difference of the local PN and the accepted PN

（3）The unit of is in the equation.

2.2 Correlation function of PN sequence under the condition of high dynamic

Generally defined as the condition of high dynamic carrier communication with high speed, acceleration and acceleration.Typical high dynamic model of the Jet Propulsion Laboratory given in the literature[11] in 1988, usually a data frame transmission time is relatively short, the relative speed at this stage is constant,thus first-order dynamic condition is more commonly used[12].However, because of the influence of the bit rate offset and carrier frequency offset under the condition of the first order dynamic, phase discriminator curve will change, the method of the literature[7] is bound to be deteriorating in the precision.

Assumes that the pseudo random code bit rate throughout the integration period of migration is a fixed value, set the pseudo-random code signal to C(t) =

,of it is bipolar pseudo

random sequence, values of . Based on the literature[13] can define pseudo random codes with deviation signal to C'(t) = C(t + dt - qTc), including dt says rate deviation caused by relative motion, qTc said receiving the initial phase of pseudo random code.So the correlation function can be expressed as

（4）In this paper, make the time discretization, namely achieve the signal sampling.Set the sample rate is fs, thus the expression of correlation function can be expressed as

（5）Rate of migration into on the expression of relative motion is as follows:

（6）

First cosmic speed, maximum rate deviation is . Therefore, in most cases of applications, bit rate deviation can be limited in the range of .

3Deviation factor and frequency folk prescription method

Suppose that send signals is, normalized residual angular

frequency is, is the residual deviation,is the sampling

frequency, and initial assumptions carrier phase is 0, rate is 1, is the channel noise, assume that as a gaussian white noise.Then received signal is

So the cross-correlation curve is

（7）set

（8）D(q) is the phase values, is as a known quantity through the phase detecting unit, N is an integer.The qTc is the initial phase of measuring PN sequence.

Let received PN sequence to get N sampling point in the left and right, its phase value of the phase detector unit output value is the expression that:

（9）

Among them.

Make plug in the equation(3), so

（10）Combinate (7) - (10) to get it that the measured value of the phase will include frequency deviation, the accuracy of measurement will be affected by the deviation, and this effect will gradually increase with the accumulation of time.As far as possible in order to eliminate the influence of the residual deviation, this paper puts forward the data points of each receives the section modulus square method, namely to I, Q two-way signal segmentation related square sum, in piecewise interval accumulation of doppler frequency can be thought very small, namely think that each sample point carrier offset is consistent, so relevant expressions can be expressed as follows:

（11）

（12）Among them

is the phase of the carrier frequency offset of n period relatived starting position.

Then the section modulus summing

（13）According to the relevant characteristics of pseudorandom code, when the signal-to-noise ratio is above a certain threshold,

So get

（14）

Plug the equation(14) into the equation(10) , we know that measured value has nothing to do with the frequency offset, so the improved correlation method can largely eliminate the influence of carrier frequency offset.

4 The analysis of system anti-noise and anti-interference ability

The related process after discretization is equivalent to local pseudo random code multiplication accumulation process.Assume that noise component is n(i), human

disturbance component is J(i), different communication network interference is .

This can be expanded into a received signal

（15）The relevant operation according to the decomposition process, first of all, with the multiplication of local pseudo random code

（16）This process is essentially spread spectrum communication algorithm to process.In this process, the noise component of the power spectral density has not changed, but the effective width is narrower.And human disturbance after and local pseudo random code of multiplication, power spectrum is broadened, power spectral density is lower, after the relevant filter can significantly reduce the noise power.And the interference between different communication network, because used pseudo random code is not relevant, jamming signal and local random codes can be multiplied as once again by spread spectrum, and under the condition that the communication energy are basically identical, the interference can be neglected. We can express entering the accumulator of signal and noise interference signal as follows:

（17）

（18）K is parameters related to the modulation mode among them, for PSK, K = 0.903.The Gp is the processing gain of spread spectrum communication, related to the related

length, set the related degree as H, so .

5Conclusion

The least squares makes clever use of the linear characteristic of the phase curve in the middle part, use the discrete points to continuous linear fitting, and sampling rate of the phase measuring accuracy broke through the system constraints, gained great improvement.Under the condition of high dynamic, such as satellite communications, missile, rocket, fighter aircraft of telemetering and communication fields, carrier frequency offset and bit rate offset become phase measurement accuracy of the two factors which cannot be ignored.This article through the analysis and simulation of the two factors on the least squares method to measure the PN phase method, proved that in first order dynamic conditions whose code length is not long, least-square method is affected by the rate of migration which is not very big, but the influence of carrier frequency offset is very serious.After it,we put forward an improved method which can effectively resist deviation of phase measurement scheme based on least square method, through theoretical analysis and numerical simulation, compared with other methods show that the method not only high precision, strong ability to resist noise, and influence of the frequency deviation is small, can satisfy the requirement of military communications and satellite communications, and other applications.So the improvement of phase measurement method has the very broad application prospects.

Reference documentation

[1] Bao J and Tsui Y Fundamentals of Global Positioning System Receivers: A Software Approach [M]. New York: Wiley Inter-Science, 2000: 73-74.

[2] Ma Hong and Wang Yuan-qin. Research on relative autonomous ranging using spread-spectrum technology[J]. Journal of Astronautics, 2005, 26(1): 29-33.

[3] Wang Lei and Xu Da-zhuan. An anti frequency offset fine time synchronization mehod and its performance analysis[J]. Journal of Electronics & Information Technology, 2011, 33(2): 300-303.

[4] Van Nee D J R and Coemen A. New fast GPS codeacquisition technique using FFT[J]. Electronics Letters, 1991, 27(2): 158-160.

[5] Krasner N F. GPS receiver and method for processing GPS signals[P]. United States Patent, 6725159, 2004.

[6] Gong Guo-hui and Li Si-kun. Improving DSSS signal PN code phase measurement precision by 3-points quadratic interpolation[J]. Journal on Communications, 2007, 28(2): 130-133.

[7] Hu Xiu-lin, Zeng Zhen, Zhang Jun, et al.. Synchronization of psceudorandom code and its implementation on FPGA in DS/SS[J]. Huazhong University of Science & Technology. (Nature Science Edition), 2005, 3(6): 44-46.

[8] Li Bai-yu, Chen Lei, Li Cai-hua, et al.. The imact non-ideal-front-end characteristic on PN zero value measurement of navigation receivers[J]. Journal of Electronics & Information Technology, 2011, 33(9): 2138-2143.

[9] Li Li-min, Ma Lu, Ren Qian-yi, et al.. Precise intersatellite ranging-technique based on fading memorey Gaussian sum filtering[J]. Journal of Electronics & Information Technology, 2011, 33(2): 295-299.

[10] Li Chun, Liu Cong-feng, Liao Gui-sheng, et al.. Solution and analysis of constrained least square passive location algorithm[J]. Systems Engineering and Electronics, 2012, 34(2): 221-226.

[11] Hinedi S and Statman J L. High-dynamic GPS tracking final report[R]. Jet Propulsion Laboratory, 1988.

[12] Li Xiao-min, Liu Hui, et al.. GPS signal parameters estimation algorithm based on data transition detection in high dynamic circustances[J]. Acta Aeronautica et Astronautica Sinica, 1999, 20(5): 430-434.

[13] Li Chun-xia. The Characteristics of PN code correlation and its applications under high dynamics[D]. [Ph.D. dissertation], National University of Defense Technology, 2005.

南航矩阵论等价关系

Student’s Name: Student’s ID No.: College Name: The study of Equivalence Relations Abstract According to some relative definitions and properties, to proof that if B can be obtained from A by performing elementary row operations on A, ~ is an equivalence relation, and to find the properties that are shared by all the elements in the same equivalence class. To proof that if B is can be obtained from A by performing elementary operations, Matrix S A ∈ is said to be equivalent to matrix S B ∈, and ~A B means that matrix S A ∈ is similar to S B ∈, if let S be the set of m m ? real matrices. Introduction The equivalence relations are used in the matrix theory in a very wide field. An equivalence relation on a set S divides S into equivalence classes. Equivalence classes are pair-wise disjoint subsets of S . a ~ b if and only if a and b are in the same equivalence class.This paper will introduce some definitions and properties of equivalence relations and proof some discussions. Main Results Answers of Q1 (a) The process of the proof is as following,obviously IA=A,therefore ~ is reflexive;we know B can be obtained from A by performing elementary row operations on A,we assume P is a matrix which denote a series of elementary row operations on A.Then ,we have PA=B,(A~B),and P is inverse,obviously we have A=P -1B,(B~A).So ~ is symmetric.We have another matrix Q which denote a series of elementary row operations on B,and the result is C,so we have QB=C.And we can obtain QB=Q(PA)=QPA=C,so A~C.Therefore,~ is transitive. Hence, ~ is an equivalence relation on S . (b) The properties that are shared by all the elements in the same equivalence class are as followings: firstly,the rank is the same;secondly,the relation of column is not changed;thirdly,two random matrices are row equivalent;fourthly,all of the matrices

2016矩阵论试题

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