Construction Of Perfect And Nearly Perfect Gaussian Integer Sequences

A Gaussian integer sequence is a sequence in which all of the elements have the form of complex numbers that the real and imaginary pars are both integers.The GIS is different from traditional sequence,and it is a kind of non-equal amplitude sequence.This kind of sequence is the generalization of orthogonal amplitude modulation sequence,and it does not need to satisfy the condition that both real and imaginary parts are odd numbers.The GIS with ideal autocorrelation performance have important applications in wireless communication systems.For example,they can be used as spread spectrum sequences in code division multiple access wireless communication systems to suppress multipath interference and as precoding matrices in orthogonal frequency division multiplexing communication systems to reduce peak-to-average power ratio.Therefore,this paper takes the autocorrelation function of the sequence as the key element to study the construction methods of perfect and nearly perfect Gaussian integer sequences.Firstly,constructions of perfect Gaussian integer sequence based on the classical cyclotomic classes were proposed.The PGIS with degree 3 and 5 were constructed respectively from the cyclotomic classes of order 2 and 4.The presented sequences with odd prime length have ideal autocorrelations.This method is combined the analysis in frequency domain with the classical cyclotomic classes.The proposed method can solve the problem that constructing the PGIS by the cyclotomic classes has high computational complexity.Secondly,a construction of Perfect Gaussian Integer Sequences based on d-form function was proposed.The pseudorandom sequence with length p~n-1 is constructed by the d-form function,where p?1(mod 4)and p is odd number.A mapping relation between the finite fieldGF(p~n)and GF(p)is generated based on the d-form sequence which satisfies the balance property and 2-tuple balance property.As a result of the Gaussian prime residue classes G_?and the finite field GF(p)are mathematically equivalent,where p=??~*,the PGIS of length N=(p~n-1)(p-1) with degree-(p-1) can be constructed over G_?.Finally,a construction of Gaussian integer sequences based on pseudo-random sequences.Gaussian integer sequences with period p~m-1 whose degree p-1 was constructed from p-ary pseudo-random sequences with period p~m-1.The presented sequences are nearly perfect Gaussian integer sequences with p-2 non-zero out-of-phase autocorrelation values.Moreover,these Gaussian integer sequences have balance property,...