Exploring equivalence of gold code sequence matrix with Hadamard matrix

Raza, Ammar
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The purpose of this study is to explore how to find equivalence of Gold code sequence matrices to Walsh-Hadamard matrices. The aim is to use Gold code sequence in place of m-sequence in GPS signal acquisition, so that we can take advantage of the large number of Gold code sequences available to us for a particular pair of LFSR generator. Different approach used to figure out such an equivalence are the method used for m-sequence matrices equivalence with Walsh-Hadamard matrices, optimization equations consisting of Gold sequence matrix, Walsh-Hadamard matrix and a permutation matrix; and applying a few graph matching algorithms on our matrices of Gold code sequence and Walsh-Hadamard. The optimization formulation that we derive in chapter three is already in use in graph matching problems. Appropriate existing Matlab libraries have been utilized with required modifications to run the Matlab functions as per our needs and requirements. The result has been very encouraging for the tests conducted with matrices of size 8x8, 32x32, 64x64 and 128x128. Although we do not see exact equivalence between Gold sequence matrices and Walsh-Hadamard matrices, we do find a method which efficiently produces a permutation matrix between the two matrices under consideration. This permutation matrix, when applied on Walsh-Hadamard matrix, gives us a small Frobenius norm and a fairly recognizable cross-correlation peak. After having acquired the best fit permutation matrix, we can then use the Fast Walsh-Hadamard transform and take advantage of the vastly reduced correlation cost.

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Gold code, Gold sequence, GPS, Hadamard matrix, m-sequence, Walsh-Hadamard matrix
Electrical and Computer Engineering