Method for calculating the PN generator mask to obtain a desired shift of the PN code

A method, implemented by a processor, for determining a plurality n of mask vector components to control a phase shift network by combining the outputs of the n stages of an MSRG-type PN code generator, for any initial state of the PN generator, said method comprising the steps of:

• a) determining the value of q which is the number of bits that the MSRG output is shifted relative to the reference sequence, 1/f(x) wherein f(x) is the characteristic polynomial of the sequence, specified by a plurality of n−1 coefficients;
• b) determining the mask vector as a plurality n of the coefficients (m0, m1, . . . , mn−1) of the mask polynomial using a mathematical relationship:
• mask polynomial=m(x)=terms of [xr modulo f(x)]/f(x) having degree less than n,
• wherein
• r=mask parameter=k−q+n−1,
• wherein n is the number of stages of the PN generator circuit, which is the desired sequence shift by which the phase shift network output is to be delayed by k bits from the reference sequence 1/f(x) using the mask, and the calculation is taken modulo the period of the sequence;
• and
• c) controlling the phase shift network, utilizing said mask, by combining a plurality of the outputs of the MSRG shift register stages to implement the desired delay.

A computer-readable medium for determining a mask vector to control a phase shift network by combining a plurality of the stages of an SSRG-type PN code generator, said computer-readable medium implementing the steps of:

• a) determining a mask parameter r by a mathematical relationship:
• mask polynomial=m(x)=xr modulo f(x);
• wherein
• r=Mask parameter=k−q+n−1,
• b) constructing an n by n transition matrix, denoted by T, as follows:
• T = [ c n - 1 1 0 0 ⋯ 0 c n - 2 0 1 0 ⋯ 0 c n - 3 0 0 1 ⋯ 0 ⋮ ⋮ c 1 0 0 0 1 1 0 0 0 0 0 ]
• c) calculating the coefficients (g0, g1, . . . , gn−1), utilizing said transition matrix, of the polynomial gr(x)=xr modulo f(x) by implementing the equation [ g n - 1 g n - 2 g n - 3 ⋮ g 1 g 0 ] = [ c n - 1 1 0 0 ⋯ 0 c n - 2 0 1 1 ⋯ 0 c n - 3 0 0 1 ⋯ 0 ⋮ ⋮ c 1 0 0 0 1 1 0 0 0 0 0 ] r ⁡ [ 0 0 0 ⋮ 0 1 ]
• d) equating the mask vector (m0, m1, . . . , mn−1) with the coefficients (g0, g1, . . . , gn−1) of the polynomial gr(x) from said step c.

A computer-readable medium for determining a mask vector to control a phase shift network by combining a plurality of the stages of an MSRG-type PN code generator, said computer-readable medium implementing the steps of:

• a) determining the mask parameter r by a mathematical relationship:
• r=mask parameter=k−q+n−1,
• wherein n is the number of stages of the PN generator circuit, which is the desired sequence shift at the phase shift network output is to be delayed by k bits from the reference sequence 1/f(x) using the mask, and the calculation is taken modulo the period of the sequence;
• b) constructing an n by n transition matrix, denoted T, as:
• T = [ c n - 1 1 0 0 ⋯ 0 c n - 2 0 1 0 ⋯ 0 c n - 3 0 0 1 ⋯ 0 ⋮ ⋮ c 1 0 0 0 1 1 0 0 0 0 0 ] ;
• c) performing, utilizing said transition matrix, the modular calculation of the polynomial
• gr(x)=xr modulo f(x) by implementing the equation g=Tr s0, wherein the vectors g and s0 are as:
• [ g n - 1 g n - 2 g n - 3 ⋮ g 1 g 0 ] = [ c n - 1 1 0 0 ⋯ 0 c n - 2 0 1 1 ⋯ 0 c n - 3 0 0 1 ⋯ 0 ⋮ ⋮ c 1 0 0 0 1 1 0 0 0 0 0 ] r ⁡ [ 0 0 0 ⋮ 0 1 ] ;
• d) equating the mask polynomial with the terms of degree less than n found by dividing the polynomial gr(x) found in step 4C by the polynomial, f(x), that is, solve the following set of equations for the mask vector as the plurality of coefficients (m0, m1, . . . , mn−1) of the mask polynomial:
• g0=m0
• g1=m1+c1m0
• g2=m2+c1m1+c2m0
• g3=m3+c1m2+c2m1+c3m0
• :
• gn−1=mn−1+c1mn−2+ . . . +cn−2m1+cn−1m0.

A computer-readable medium for determining a mask vector to control a phase shift network by combining a plurality of the stages of an SSRG-type PN code generator, said computer-readable medium implementing the steps of:

• a) determining the mask parameter r by a mathematical relationship:
• mask polynomial=m(x)=xr modulo f(x);
• wherein r=Mask parameter=k−q+n −1,;
• b) performing, utilizing a factoring technique, a modular calculation of the polynomial gr(x)=xr modulo f(x), wherein the factoring technique is implemented using the sub-steps of:
• i) if r is less than n, setting gr(x)=xr;
• ii) if r is not less than n, that is, xr=xdxn, where D is greater than or equal to zero, setting a temporary polynomial b(x) equal to xn;
• iii) substituting for xn the polynomial of lesser degree found from the equation f(x)=0;
• iv) if the degree of xDb(x) is less than n, setting gr(x)=xDb(x);
• otherwise, factoring xDb(x) as (xD−p)(xpb(x)) so that the degree of the new b(x)=xpb(x) is equal to n, and proceeding to said step iii), using D−p as a new value of D;
• c) equating the mask vector (m0, m1, . . . , mn−1) with the coefficients of the mask polynomial given by the polynomial gr(x) found in said step b.

A computer-readable medium for determining a mask vector to control a phase shift network by combining a plurality of the stages of an MSRG-type PN code generator, said computer-readable medium implementing the steps of:

• a) determining the mask parameter r by a mathematical relationship:
• mask polynomial=m(x)=terms of [xr modulo f(x)]/f(x) having degree less than n
• wherein
• r=mask parameter=k−q+n−1,
• wherein n is the number of stages of the PN generator circuit, which is the desired sequence shift at the phase shift network output is to be delayed by k bits from the reference sequence 1/f(x) using the mask, and the calculation is taken modulo the period of the sequence;
• b) performing, utilizing a factoring technique, a modular calculation of the polynomial gr(x)=xr modulo f(x), wherein the factoring technique is implemented using the sub-steps of:
• i) if r is less than n, then setting gr(x)=x′;
• ii) if r is not less than n, that is, xr=xDxn, where D is greater than or equal to zero, then setting a temporary polynomial b(x) equal to xn;
• iii) substituting for xn the polynomial of lesser degree found from the equation f(x)=0;
• iv) if the degree of xDb(x) is less than n, setting gr(x)=xDb(x);
• otherwise, factoring xDb(x) as (xD−p)(xpb(x)) so that the degree of the new b(x)=xpb(x) is equal to n, and proceeding to said step iii), using D−p as a new value of D;
• c) equating the mask vector with the coefficients of the mask polynomial expressed as the terms of degree less than n found by dividing the polynomial gr(x) found in step 6B by the characteristic polynomial f(x), that is, solving the following set of equations for the plurality of mask vector components (m0, m1, . . . , mn−1):
• g0=m0
• g1=m1+c1m0
• g2=m2+c1m1+c2m0
• g3=m3+c1m2+c2m1+c3m0
• :
• gn−1=mn−1+c1mn−2+ . . . +cn−2m1+cn−1m0.