Method and apparatus for generating a large number of codes having desirable correlation properties
First Claim
1. A method of generating a family of number sequences, comprising the steps of:
- specifying a prime number, p, having magnitude greater than 3 and an associated Galois field, GF(p);
selecting a first primitive element of GF(p), α
, and a second primitive element of GF(p), β
, wherein α
=β
or a α
≠
β
;
defining a family of p+1 polynomials, wherein k is an integer and 0≦
k≦
p, wherein fk(x)=modp(x2+x+k) when k<
p and fk(x)=modp(x+1) when k=p;
defining a family of p+1 matrices, wherein each matrix, A(k), has dimension (p−
1)×
(p−
1) and the element of row i in column j of matrix A(k), A(k)ij, is defined to have a first state when fk(α
j)=modp(β
i) and is otherwise defined to have a second state; and
producing a family of p+1 number sequences, wherein the values of members of each number sequence, Sk, are determined in accordance with the elements of a corresponding matrix, A(k), of said family of p+1 matrices.
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Abstract
The present invention relates to a method for generating and apparatus for employing code families having desirable correlation properties. Regardless of code length, maximum autocorrelation of the codes is 4 for any nonzero offset and maximum cross-correlation of any two codes from a code family is 4 for any offset. The codes can be used in impulse radio systems and non-impulse radio systems including CDMA, TDMA, FDMA, OFDM and various other frequency hopping and direct sequence systems. The codes can be used to specify various impulse radio and non-impulse radio signal characteristics including pulse position in time, amplitude, width, type, phase, phase difference, frequency, spreading code, etc. The codes have the unique property that they can specify as many as two components in which a signal is not present. A method of code compression is provided.
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Citations
20 Claims
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1. A method of generating a family of number sequences, comprising the steps of:
-
specifying a prime number, p, having magnitude greater than 3 and an associated Galois field, GF(p);
selecting a first primitive element of GF(p), α
, and a second primitive element of GF(p), β
, wherein α
=β
or a α
≠
β
;
defining a family of p+1 polynomials, wherein k is an integer and 0≦
k≦
p, wherein fk(x)=modp(x2+x+k) when k<
p and fk(x)=modp(x+1) when k=p;
defining a family of p+1 matrices, wherein each matrix, A(k), has dimension (p−
1)×
(p−
1) and the element of row i in column j of matrix A(k), A(k)ij, is defined to have a first state when fk(α
j)=modp(β
i) and is otherwise defined to have a second state; and
producing a family of p+1 number sequences, wherein the values of members of each number sequence, Sk, are determined in accordance with the elements of a corresponding matrix, A(k), of said family of p+1 matrices. - View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12)
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13. A method of generating a family of codes that satisfy defined correlation properties, comprising the steps of:
-
defining a signal characteristic value layout having n components;
subdividing each of said n components into m subcomponents;
generating said family of codes, wherein at least one code of said family of codes specifies that a signal is not present in at least one of said components but no more than two of said components when mapped to said signal characteristic value layout. - View Dependent Claims (14, 15, 16)
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17. A method of code compression, comprising the steps of:
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enumerating the position of each element of a matrix corresponding to a code having defined correlation properties, wherein said code specifies a signal is not present in one or more components of a signal characteristic value layout; and
storing in an array the assigned position number of each said element having a value that corresponds to a subcomponent within a component of said signal characteristic value layout. - View Dependent Claims (18, 19, 20)
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Specification