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Perfect Gaussian Integer Sequences of Arbitrary Length Soo-Chang Pei∗ and Kuo-Wei Chang† National Taiwan University∗ and Chunghwa Telecom†

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Perfect Gaussian Integer Sequences of Arbitrary Length Soo-Chang Pei∗ and Kuo-Wei Chang† National Taiwan University∗ and Chunghwa Telecom† Perfect Gaussian Integer Sequences of Arbitrary Length Soo-Chang Pei∗ and Kuo-Wei Chang† National Taiwan University∗ and Chunghwa Telecom† m Objectives N = p or N = p using Legendre N = pq where p and q are coprime Conclusion sequence and Gauss sum To construct perfect Gaussian integer sequences Simple zero padding method: We propose several methods to generate zero au- of arbitrary length N: Legendre symbol is defined as 1) Take a ZAC from p and q. tocorrelation sequences in Gaussian integer. If the 8 2 2) Interpolate q − 1 and p − 1 zeros to these signals sequence length is prime number, we can use Leg- • Perfect sequences are sequences with zero > 1, if ∃x, x ≡ n(mod N) n ! > autocorrelation. = < 0, n ≡ 0(mod N) to get two signals of length N. endre symbol and provide more degree of freedom N > 3) Convolution these two signals, then we get a than Yang’s method. If the sequence is composite, • Gaussian integer is a number in the form :> −1, otherwise. ZAC. we develop a general method to construct ZAC se- a + bi where a and b are integer. And the Gauss sum is defined as N−1 ! Using the idea of prime-factor algorithm: quences. Zero padding is one of the special cases of • A Perfect Gaussian sequence is a perfect X n −2πikn/N G(k) = e Recall that DFT of size N = N1N2 can be done by this method, and it is very easy to implement. sequence that each value in the sequence is a n=0 N taking DFT of size N1 and N2 seperately[14]. To Gaussian integer. A well known result[5] is that 8 √ construct perfect sequence, just References k • Considering the sequence length, there are <> N,N ≡ 1(mod 4) N √ 1) Divide n = 0, 1, 2, ..., N−1 into groups Sd, where G(k) = k three cases: N is prime, N is a power of prime > : − N i N,N ≡ 3(mod 4) m Sd = {n|gcd(n, N) = d} p or N = pq, where p and q are coprime but In other words, the Fourier transform of Legendre 2) For each group, use LS or GLS to ensure CA [1] Viterbi, A. J., CDMA. Addison Wesley, 1995 not necessary prime numbers. Sequences is almost CA, with the only exception on √ 3) Take DFT [2] Carni, E., and Spalvieri, A. IEEE Transactions on Wireless k = 0, the first point. The amplitude is N thus Communications, 2005 Example [3] Shah, S. F. A., and Tewfik, A. H,14th IEEE International Introduction our goal is to find a Gaussian integer a and some integers b, and c such that Conference on Electronics Circuits and Systems, 2007 The factors of 15 is 1,3,5,15. So we divide our [4] Benedetto, J.J., Konstantinidis, I. and Rangaswamy, M. IEEE Signal Perfect sequences are sequences with zero autocor- |a|2 = b2 + Nc2 (1) Processing Magazine 26 , 2009. signal into 4 groups [5] G. H. Hardy and E. M. Wright ,Oxford University Press, relation (ZAC). Then a sequence that 2 3 2 3 f(0) 0 0 0 f(5) f(10) [6] Hu, Wei-Wen and Wang, Sen-Hung and Li, Chih-Peng, ICC, 2011 N−1 8 6 7 6 7 X ∗ < a, n = 0 6 7 6 7 x (n − m)x(n) = Cδ(m) √ 6 0 0 0 7 6 0 0 0 7 [7] B. C. Berndt , R. J. Evans and K. S. Williams ,A Wiley-Interscience f(n) = n (2) 6 7 6 7 n=0 : Nc + bi, n =6 0 6 7 6 7 publication,1998 N 6 0 0 0 7 + 6 0 0 0 7 + 6 7 6 7 for some constant C. 6 7 6 7 [8] Yang Yang, Xiaohu Tang, Zhengchun Zhou, IEEE Signal Processing is CA, and the DFT of f(n) is ZAC in Gaussian 6 7 6 7 6 0 0 0 7 6 0 0 0 7 Letters , 2012 When considering whether a sequence if ZAC or 4 5 4 5 integer. 0 0 0 0 0 0 [9] Soo-Chang Pei and Kuo-Wei Chang, APSIPA, 2013 not, a smart way is to calculate its discrete Fourier 2 3 2 3 Example 0 0 0 0 0 0 [10] Xiuwen MA, Qiaoyan WEN, Jie ZHANG and Huijuan ZUO,‘IEICE, transform (DFT). Benedetto[4] has proven that a 6 7 6 7 Vol.E96-A, No.11, pp.2290-2293 6 7 6 7 6 f(3) 0 0 7 6 0 f(8) f(13) 7 sequence is ZAC if and only if its DFT is constant 6 7 6 7 [11] L. G. Hua, Introduction to Number Theory, 1997 6 7 6 7 √ √ 6 f(6) 0 0 7 + 6 0 f(11) f(1) 7 amplitude (CA). 6 7 6 7 [12] M. R. Schroeder,Number Theory in Science and Communications, f(n) = 6 + 1i, 5i + 2 3, 5i − 2 3 6 7 6 7 6 f(9) 0 0 7 6 0 f(14) f(4) 7 1997 :Springer-Verlag |X(m)| = A 6 7 6 7 F (k) = {6 + 11i, 6 − 10i, 6 + 2i} 4 f(12) 0 0 5 4 0 f(2) f(7) 5 [13] Pei, Soo-chang and Chia-Chang Wen and Jian-Jiun Ding, IEEE for some constant A. Transactions on Circuits and Systems, 2008 One possible outcome is by When the sequence length is prime or power of [14] Good, I. J., Journal of the Royal Statistical Society Example 61 = 62 +52 = 1+22 ×15 = 72 +22 ×3 = 42 +32 ×5 prime, we can use Legendre symbol to construct per- 2 √ √ 3 6 + 5i 7i + 2 3 7i − 2 3 fect Gaussian integer sequences. Legendre symbol 2 6 √ √ √ 7 Contact Information Let N = 3 , and c=2,b=5,a=6+1i, since d(n) is 6 7 6 4i + 3 5 1i + 2 15 1i − 2 15 7 or sequence(LS)[11, 12] has a strong connection to 6 √ √ √ 7 6 7 d(n) = {0, 1, −1, 0, 1, −1, 0, 1, −1} 6 4i − 3 5 1i − 2 15 1i + 2 15 7 • Email: [email protected] quadratic Gauss sum, which is also a summation of 6 7 6 √ √ √ 7 6 7 Email: [email protected] complex roots of unity, like Ramanujan’s sum. The first iteration gives us 6 4i − 3 5 1i − 2 15 1i + 2 15 7 • √ √ √ √ 4 √ √ √ 5 For composite number, we propose some novel meth- {0, 2 3 + 5i, −2 3 + 5i, 0, 2 3 + 5i, −2 3 + 5i 4i + 3 5 1i + 2 15 1i − 2 15 ods to construct perfect Gaussian integer sequences, √ √ 0, 2 3 + 5i, −2 3 + 5i} which is CA, and its Fourier transform from the naïve zero padding and convolution, to de- 2 3 Now second eliminates the remaining zeros. 6 + 43i 6 + 4i 6 + 16i composing N into different groups. 6 7 √ √ 6 7 6 21 + 13i 21 − 41i 21 + 31i 7 f(n) ={6 + 1i, 2 3 + 5i, −2 3 + 5i 6 7 √ √ 6 7 6 −9 + 13i −9 + 19i −9 − 29i 7 1 − 6i, 2 3 + 5i, −2 3 + 5i 6 7 6 7 √ √ 6 −9 + 13i −9 + 19i −9 − 29i 7 1 − 6i, 2 3 + 5i, −2 3 + 5i} 6 7 4 21 + 13i 21 − 41i 21 + 31i 5 F (k) ={8 + 19i, 5 + 7i, 5 + 7i, 8 − 44i, 5 + 7i, 5 + 7i 8 − 8i, 5 + 7i, 5 + 7i} is a perfect Gaussian integer sequence..
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