CN104158557B - Gold sequence method for parameter estimation - Google Patents

Abstract

The present invention is claimed a kind of Gold sequence method for parameter estimation.Gold sequence is a kind of pseudo-random sequence, and its cross correlation is better than m-sequence, is widely used in modern communications.The parameter of Gold sequence is the important component part of Gold sequence, but its parameter estimation correlational study is the fewest.For Gold sequence Parameter Estimation Problem, the present invention proposes Gold sequence method for parameter estimation based on the western iterative algorithm of prunus mume (sieb.) sieb.et zucc..This method, first according to the partial sequence of the Gold sequence intercepted, utilizes the western iterative algorithm of prunus mume (sieb.) sieb.et zucc. to estimate its generator polynomial, then by polynomial division estimate to produce the m-sequence of this Gold sequence preferably to primitive polynomial.Intercepting, sequence length is shorter and containing in the case of error code, and the method can estimate the parameter of Gold sequence effectively.

Description

Gold sequence method for parameter estimation

Technical field

The present invention relates to a kind of Gold sequence method for parameter estimation, it is adaptable to spread spectrum communication, pseudo-code scrambling, pseudo-random code ranging etc. The parameter estimation of the pseudo-random sequence in field.

Background technology

Pseudo-random sequence suffers from widely should in fields such as spread spectrum communication, pseudo-code scrambling, pseudo-random code ranging and cryptographies With.In spread spectrum communication, frequency expansion sequence plays a very important role, and applying most frequency expansion sequences in reality is m-sequence, Gold sequence.M-sequence and Gold sequence are obtained by linear feedback shift register, have and realize simple, pseudo-randomness Good and that dependency is good advantage, the most generally replaces random sequence to use as frequency expansion sequence.

M-sequence is the most complete theoretical a kind of pseudo-random sequence in current sequence research, and it is research and constructs it The basis of its sequence.M-sequence has good randomness and balance, but the m-sequence number of same length is few, and m-sequence Cross correlation value undesirable.Compared with m-sequence, the sequence quantity that Gold sequence can be used as spectrum-spreading address code is many, has good Cross correlation, is widely used in modern communications.

In a Gold sequence race, the m-sequence both having comprised this race's Gold sequence of generation is the most right, also includes that both shift Mould two adds the new sequence of generation.The parameter of Gold sequence, including its generator polynomial and this Gold sequence m-sequence preferably to Primitive polynomial, has been the basis of the work such as frequency expansion sequence recovery, information deciphering.

Summary of the invention

The technical problem to be solved is, to Gold sequence, the research related at present is mainly performance evaluation, phase Close Journal of Sex Research and the application in spread spectrum communication system thereof, and the correlational study of its parameter estimation is the fewest.The parameter of Gold sequence Problem is the important research link in sequence analysis, and therefore the parameter estimation of Gold sequence gets a good eye value.

The present invention solves the technical scheme of the problems referred to above, for solving Gold sequence Parameter Estimation Problem, it is proposed that a kind of Method for parameter estimation based on the western iterative algorithm of prunus mume (sieb.) sieb.et zucc..The present invention first according to the part Gold sequence Han error code intercepted, utilizes Mei Xi Iterative algorithm estimates its generator polynomial, then by two element field polynomial division estimate the m-sequence of Gold sequence preferably to basis Former multinomial.

The Gold sequence method for parameter estimation that the present invention provides, comprises the steps:

Step 1: intercepted length is the n rank to be estimated Gold sequence of L, L > 8n；

Step 2: the m position of the intercepted sequence of selecting step 1 is to m+4n-1 position, and m≤L-8n+1, m initial value is 1, utilizes The western iterative algorithm of prunus mume (sieb.) sieb.et zucc. seeks the Gonjuctive polynomial of this intercepting sequence；

Step 3: produce one according to the m position of the intercepted sequence of Gonjuctive polynomial described in step 2 and step 1 to m+2n-1 position Individual new sequence；

Step 4: by the 4n+m position of the 4n+1 position of sequence new described in step 3 to L-m+1 position Yu the intercepted sequence of step 1 Compare to L position, if the probability that in these two sections of sequences, correspondence position is identical is more than 80%, then this Gonjuctive polynomial be described in treat Estimate the generator polynomial of n rank Gold sequence；

Step 5: by generator polynomial divided by the primitive polynomial of n rank m-sequence, if residue is zero, except formula and business's formula be then For the m-sequence corresponding with this n rank to be estimated Gold sequence preferably to primitive polynomial.

The Gold sequence method for parameter estimation that the present invention provides, in the case of intercepting sequence length is shorter, can preferably be estimated Count out the parameter of Gold sequence and have certain fault-tolerance.When there is continuous error code in intercepting sequence, the method that the present invention provides Still it is suitable for.

Accompanying drawing explanation

Fig. 1 is linear feedback shift register schematic diagram；

Fig. 2 is Gold sequence schematic diagram；

Fig. 3 is containing error code Gold sequence parameter estimation flow chart；

Fig. 4 is different rank Gold sequence fault freedom comparison diagram；

Fig. 5 is the identical Gold sequence fault freedom comparison diagram that intercepted length is different.

Detailed description of the invention

M-sequence is the abbreviation of longest linear feedback shift register sequence, and Fig. 1 show a linear feedback shift register The schematic diagram of device.a_n(n=0,1,2 ...) is represented by:

a_n=c₁a_n-1⊕c₂a_n-2⊕…⊕c_n-1a₁⊕c_na₀ (1)

Wherein c_i∈ { 0,1} (i=1,2 ..., n).

The feedback link of linear shift register and the structure of sequence, be expressed as with p (x):

$p\left(x\right)=1+\underset{i=1}{\overset{S}{\mathrm{\Σ}}}{c}_i{x}^i---\left(2\right)$

In formula, c_iRepresent the on-off of feedback link, wherein c_i=1 represents participation feedback, c_iFeedback is not participated in in=0 expression. xⁱRepresent the position of feedback link.

When p (x) is primitive polynomial, linear feedback shift register can produce the sequence of maximum cycle and be m sequence Row, the cycle greatest length of the m-sequence that n level linear feedback shift register can be generated by is 2ⁿ-1。

Gold sequence based on m-sequence preferably to produce.M-sequence, preferably to referring to, is concentrated at m-sequence, cross-correlation letter The absolute value of number is less than two m-sequence of certain value.If { a} is by n primitive polynomial m to sequence_aX m-sequence that () produces, sequence { b} is by n primitive polynomial m for row_bX m-sequence that () produces.If their cross-correlation function value R_ab(τ) inequality is met:

So { a} is with { it is the most right that b} is constituted m-sequence.M-sequence preferably to cross-correlation function have 3 values-1 ,-t (n), t (n)-2}, wherein(Represent the integer part of real number a).

Assuming that X and Y is is P=2 in the cycleⁿThe m-sequence of-1 is the most right, to wherein any one, such as X, carry out arbitrary Cyclic shift obtains TⁱX (i=0,1,2 ..., P-1), then by TⁱX with Y carries out mould two and adds, i.e. available preferred based on m-sequence Gold sequence G to X and Y_i, its expression formula is:

G_i=TⁱX Y (i=0,1,2 ..., P-1) (4)

Based on m-sequence preferably to generating the schematic diagram of Gold sequence as shown in Figure 2.

Gold sequence collection G (X, Y) preferably generated X and Y by m-sequence is:

G (X, Y)={ X, Y, X Y, TX Y, T²X⊕Y,…,T^P-1X⊕Y} (5)

The method of another kind of structure Gold sequence is directly to be produced by multinomial.Assume m_a(x) and m_bX () is n rank m-sequence The preferably primitive polynomial to X and Y, makees m (x)=m_a(x)m_b(x), and gcd (m_a(x),m_b(x))=1, then the sequence that m (x) produces Row collection is identical with the Gold sequence collection that formula (4) describes, and m (x) now is called the generator polynomial of this Gold sequence.

Assume a₀,a₁,a₂,…,a_B-1It is two element field F₂On constituted orderly group of B element, then claim this sequence to be The binary sequence of a length of B.For a F₂On multinomial:

F (x)=c₀+c₁x+c₂x²+…+c_lx^l (6)

Wherein c₀=1, but do not limit c_l=1.We are with the f (x) the l level linear shift register as Gonjuctive polynomial It is abbreviated as<f (x), l>.If recurrence relation:

a_k=c₁a_k-1+c₂a_k-2+…c_la_k-l(k=l, l+1 ..., B-1) (7)

Set up, then<f (x), l>produces this binary sequence.

An any given long binary sequence of B.N is defined a series of < f by inductive method_n(x),l_n>, n=1,2 ..., B.

1) initial value is taken:

f₀(x)=1, l₀=0

2) < f is set_l(x),l_l>, l=0,1 ..., n (0≤n<N) tries to achieve the most, and has:

l₀≤l₁≤…≤l_n

Note: $f_n\left(x\right){=c}_0^{\left(n\right)}+c_1^{\left(n\right)}x+\&CenterDot;\&CenterDot;\&CenterDot;+c_{l_n}^{\left(n\right)}{x}^{l_n},c_0^{\left(c\right)}=1$

Calculate again:

$d_n=c_0^{\left(n\right)}{a}_n+c_1^{\left(n\right)}{a}_{n-1}+\&CenterDot;\&CenterDot;\&CenterDot;+c_{l_n}^{\left(n\right)}{a}_{n-l_n}$

Claim d_nIt it is the n-th step difference.Then distinguish two following situations then:

If a) d_n=0, then order:

f_n+1(x)=f_n(x),l_n+1=l_n

If b) d_n=1, then need distinguish two following situations:

I. l is worked as₀=l₁=...=l_nWhen=0, take:

f_n+1(x)=1+xⁿ⁺¹,l_n+1=n+1

Ii. when there being m (0≤m < n), make:

l_m<l_m+1=l_m+2=...=l_n,

Just put:

f_n+1(x)=f_n(x)+x^n-mf_m(x),

l_n+1=max{l_n,n+1-l_n}。

< the f finally obtained_B(x),l_B> it is the line of shortest length shift register producing sequence.

Gold sequence is periodic sequence, and the cycle of n rank Gold sequence is N=2ⁿ-1, the exponent number of its generator polynomial is 2n. To n rank Gold sequence, there is a_k=c₁a_k-1+c₂a_k-2+…+c_2na_k-2n, wherein k=2n, 2n+1 ..., 2ⁿ-2, i.e. as k >=2n, Every of Gold sequence is to be determined by the continuous 2n position before it, and such iteration can get Gold sequence, namely Gold sequence Arrange the partial sequence by continuous 2n position and generator polynomial determines.

Respectively with x¹⁴+x⁹+x⁸+x⁶+x⁵+x⁴+x²+x+1、x²⁶+x²⁵+x²⁴+x²²+x²¹+x¹⁸+x¹⁷+x¹¹+x¹⁰+x⁹+x⁵+x⁴+x³ + x+1 and x³⁶+x²⁸+x²³+x¹⁷+x¹⁴+x¹²+x¹⁰+x⁵+ 1 is generator polynomial, uses MATLAB software to randomly generate 7 rank, 13 rank And 18 rank Gold sequence.

One section of sequence of random intercepting, the length wherein intercepting and capturing sequence is generally less than the Cycle Length of Gold sequence, uses The western iterative algorithm of prunus mume (sieb.) sieb.et zucc. seeks its Gonjuctive polynomial.Carry out 1000 emulation experiments respectively, when Gonjuctive polynomial should with above generation When the generator polynomial of Gold sequence is identical, then it is assumed that this time emulation experiment can correctly identify the generator polynomial of this sequence.Nothing During error code, with correct, the sequence length of intercepting identifies that the probability of generator polynomial is as shown in the table.

Table 1 n=7, intercepted length and the correct estimated probability of generator polynomial

Intercepted length	25	26	27	28	29	30
							Correct probability	2.7%	11.4%	48.7%	100%	100%	100%

Table 2 n=13, intercepted length and the correct estimated probability of generator polynomial

Intercepted length	49	50	51	52	53	54
							Correct probability	2.5%	14.8%	50.2%	100%	100%	100%

Table 3 n=18, intercepted length and the correct estimated probability of generator polynomial

Intercepted length

69

70

71

72

73

74

Correct probability

3.3%

12.9%

47.8%

100%

100%

100%

From above table, when seeking the generator polynomial without error code n rank Gold sequence with the western iterative algorithm of prunus mume (sieb.) sieb.et zucc., it is not necessary to The Gold sequence of one complete cycle, only needs the most correct 4n position.To without error code n rank Gold sequence, work as intercepting When the length of sequence is more than or equal to 4n position, all can 100% to estimate the generation of this Gold sequence multinomial with the western iterative algorithm of prunus mume (sieb.) sieb.et zucc. Formula；When the length intercepting sequence is less than 4n position, this algorithm can partly estimate generator polynomial, and intercept the length of sequence The shortest, its correct estimated probability is the least.

Parameter containing error code Gold sequence is estimated by the present invention on this basis, is illustrated in figure 3 the stream of the present invention Cheng Tu.The present invention estimates the generator polynomial containing error code Gold sequence according to the western iterative algorithm of prunus mume (sieb.) sieb.et zucc., recycles two element field multinomial Division estimate the m-sequence of this Gold sequence preferably to primitive polynomial.

Concretely comprise the following steps:

Step 1: intercepted length is part Gold sequence on error code n rank of L.

The effectiveness estimated for continuous generator polynomial under ensureing, intercepts sequence length and should be greater than 8n.

Step 2: (m+4n-1) position is arrived in the m position (m initial value is 1, m≤L-8n+1) choosing intercepting sequence, namely cuts Take continuous print 4n position in sequence, utilize the western iterative algorithm of prunus mume (sieb.) sieb.et zucc. to seek its Gonjuctive polynomial.

Step 3: according to this Gonjuctive polynomial and intercept sequence m position to (m+2n-1) position, i.e. continuous print 2n position, generation One new sequence.

Step 4: (4n+1) position of new sequence to (L-m+1) position and (4n+m) position intercepting sequence are carried out to L position Relatively.If the probability that wherein correspondence position is identical is more than 80%, then this Gonjuctive polynomial is that the generation of Gold sequence to be estimated is many Item formula.

If the probability that correspondence position is identical is less than 80%, then the value of m is added 1, repeat step 2 and 3.If it is now corresponding The value of m, still less than 80%, is added 1 by position identical probability the most again, continues to repeat step 2 and 3, until m > L-8n+1.

Step 5: by this generator polynomial m (x) divided by the primitive polynomial of n rank m-sequence (by primpoly letter in MATLAB Number understands all primitive polynomials of n rank m-sequence), if residue is zero, such as m (x)/m_a(x)=m_b(x), the most now except formula m_a (x) and business's formula m_b(x) be the m-sequence of this Gold sequence preferably to primitive polynomial.

Parameter without error code Gold sequence is estimated by said method, is also applicable.

Parameter containing error code Gold sequence is estimated by the method utilizing the present invention to propose.Fig. 3 is containing error code Gold sequence Row parameter estimation flow chart, the present invention utilizes the western iterative algorithm of prunus mume (sieb.) sieb.et zucc. and two element field polynomial division to estimate containing error code Gold sequence Parameter.

Fig. 4 is 9 rank, 11 rank and 15 rank Gold sequence fault freedom comparison diagrams, the partial order that wherein 9 rank Gold sequence intercept Arrange a length of 200, a length of 300 of the partial sequence that 11 rank Gold sequence intercept, the partial order that 15 rank Gold sequence intercept Arrange a length of 500.By 1000 random experiments, the correct estimated probability of available generator polynomial.For Gold sequence, Exponent number is the highest, and the correct figure place estimating that during its generator polynomial, needs are the most correct is the most.From fig. 4, it can be seen that bit error rate phase Meanwhile, Gold sequence exponent number is the highest, and correct estimation generator polynomial probability is the least, estimates the most difficult.

Fig. 5 is the 13 rank Gold sequence fault freedom comparison diagrams that intercepted length is respectively 300,400 and 500.Logical Cross 1000 random experiments, the correct estimated probability of available generator polynomial.As shown in Figure 5, when the bit error rate is identical, intercept long Spending the longest, correct estimation generator polynomial probability is the biggest；When intercepted length is identical, the bit error rate is the highest, and correct estimation generates multinomial Formula probability is the least.

Claims (3)

1.Gold sequential parameter method of estimation, it is characterised in that comprise the steps:

Step 1: intercepted length is the n rank to be estimated Gold sequence of L, L > 8n；

Step 2: the m position of the intercepted sequence of selecting step 1 is to m+4n-1 position, and m≤L-8n+1, m initial value is 1, utilizes Mei Xi Iterative algorithm seeks the Gonjuctive polynomial of this intercepting sequence；

Step 3: produce one newly to m+2n-1 position according to the m position of the intercepted sequence of Gonjuctive polynomial described in step 2 and step 1 Sequence；

Step 4: by the 4n+m position of the 4n+1 position of sequence new described in step 3 to L-m+1 position and the intercepted sequence of step 1 to L Position compares, if the probability that in these two sections of sequences, correspondence position is identical is more than 80%, then this Gonjuctive polynomial be described in wait to estimate The generator polynomial of meter n rank Gold sequence；

Step 5: by generator polynomial divided by the primitive polynomial of n rank m-sequence, if residue is zero, then except formula and business's formula be with M-sequence corresponding to this n rank to be estimated Gold sequence preferably to primitive polynomial.

Gold sequence method for parameter estimation the most according to claim 1, it is characterised in that: if new sequence in described step 4 4n+1 position is to the 4n+m position of L-m+1 position and the intercepted sequence of step 1 to L position, and the probability that correspondence position is identical is less than 80%, Then the value of m is added 1, repeat step 2 and 3, if the probability that now correspondence position is identical is still less than 80%, the most again the value of m is added 1, continue to repeat step 2 and 3, until m > L-8n+1.

Gold sequence method for parameter estimation the most according to claim 1 or claim 2, it is characterised in that: described n rank to be estimated Gold sequence Row include the Gold sequence without error code and the Gold sequence containing error code.