Understanding Integer-Valued Zero Autocorrelation Sequences
Delve into the realm of integer-valued zero autocorrelation sequences, exploring concepts like periodic sequences, frequency domains, constant amplitudes, and more. Unravel the methods and techniques involved in creating these sequences and their significance in various applications.
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Integer-valued Zero Autocorrelation Sequences Presenter : Shiang-Chih Hua Advisor : Prof. Jian-jiun Ding
Outline Introduction Zero Autocorrelation Integer-valued Combination Method Recursive Method
Introduction ? = 6 ?1= 6 ?2= 6 ?,?1 2? 6 6 3 6? 6 3 2 6? 3 2 6 = 0, ?,?2 = 0
Zero Autocorrelation periodic sequence ? ? period = N ?=0 ?=0 ? 1? ? ? ? + ? = 0, ? = 1,2, ,? 1 ? 1? ? ? ? + ? = ???? ??? = 1, ? = 0, ?, 2?, 0, ?? ??????
Frequency domain ??= ??2? ? ? = ??? ? ? ? ? = ???? ? ? ? ? 1? ? ?? ?? = ?=0 1 ? ?=0 ? 1 ? ? ???? =
Constant Amplitude ? ? = ?, ? = 0,1, ,? 1
ZAC CA ZAC CA ? ? = ?=0 ? 1? ? ?? ?? ? ? ??+? ?+? ? 1 2= ? ? ? ? = ?=0 ?+?=0 ? 1? ? ? ? + ? ????= ?=0 ? 1???? ????= ? = ?2 ? 1 ?=0
ZAC CA ZAC CA ?=0 = = = 1 ? ?=0 1 ? ?=0 ? 1 ? ? ???? ? 1 ? ? ?? ? ?+? ? 1? ? ? ? + ? = ?=0 ? 1 1 ? 1 ? ? ? ? ?? ?? ?=0 ? ? ? ? 1?? ? 1 ?=0 ?2 ?=0 1 ? 1 ? ? ? ? ?? ?????? ? ? 1 ?=0 ?2 ?=0 1 ? ?=0 2?? ??= ?2??? = ???? ? 1 ? ?
Integer-valued ? ? = ? = ? 0 1 ? ?=0 ? 1 ? ? ???? , ? = 0,1, ,? 1, ?1 ? 1? ? = ?=0 ? ?? ? = ?=0 1 ? ? ? ???? , ? = 0,1, ,? 1, ? 1 ?2
Integer-valued ? ?,? ?|? ? ? ???? , ?|?, ? = 0,1, ,? 1, ? ?,? ??? ?,? =? ? ? ???? , ?|?, ? = 0,1, ,? 1, ?3 ?4
Example 1 1 1 1 1 1 1+ 3? 2 1+ 3? 2 1 1 3? 2 1 3? 2 1+ 3? 2 1 3? 2 1 1+ 3? 2 1 3? 2 1 3? 2 1+ 3? 2 1 1 3? 2 1+ 3? 2 1 3? 2 1 3? 2 1 1+ 3? 2 1+ 3? 2 ? 0 ? 1 ? 2 ? 3 ? 4 ? 5 1 1 1 1 6 1 1 1 1 1 1
Example 1 1 1 1+ 3? 2 1 3? 2 1 1+ 3? 2 1 3? 2 1 3? 2 1+ 3? 2 1 1 3? 2 1+ 3? 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ? 0 ? 2 ? 4 1 ? 0 ? 3 6 ? 0 , , 1 1 1
Example 1 1 1 1 1+ 3? 2 1 3? 2 1 1+ 3? 2 1 3? 2 1 3? 2 1+ 3? 2 1 1 3? 2 1+ 3? 2 1+ 3? 2 1+ 3? 2 1 1 3? 2 1 3? 2 1 3? 2 1 3? 2 1 1+ 3? 2 1+ 3? 2 1 1 1 1 1 1 1 1 1 1 1 1 ? 2 ? 4 ? 1 ? 5 6 ? 0 , ? 3 , ,
?4 ?2 ?3 ?4 ?=0 ? 1 ? ? ????= ?|? ??? ?,? =? ? ? ????= ?|?? ?,?
?4 ?2 ?3 ?4 ? ?+?? ?=0 = ?=0 = ? ?|? ? ? ????= ?? ?,? ? ?,? = ? 1? ? + ?? ? 1 ?=0 ? 1 ? ? ?? ?= ?=0 ? ? 1 ? ? ???? ?=0 ? 1????= ?=0 ? 1 ? ? ???????? 1 ? ?=0 ? 1? ? + ?? ?
?4 ?2 ?3 ?4 ? ?,? = ?|? ? ? ????= ?|?|? ??? ?,? =? ? ? ???? = ?|?|?? ?,? = ? ?|? ? ? ????? ????????? ??????? ? ?,? = ? ?|? ? ? ?,? ? ?? ?,? = ?|?|?? ? ?? ?,? ?
? = 12 ? 1,? = ? 1,? + ? 2,? + ? 3,? + ? 4,? + ? 6,? + ? 12,? ? 2,? = ? 2,? + ? 4,? + ? 6,? + ? 12,? ? 3,? = ? 3,? + ? 6,? + ? 12,? ? 4,? = ? 4,? + ? 12,? ? 6,? = ? 6,? + ? 12,? ? 12,? = ? 12,?
? = 12 ? 12,? = ? 12,? ? 6,? = ? 6,? ? 12,? ? 4,? = ? 4,? ? 12,? ? 3,? = ? 3,? ? 6,? ? 2,? = ? 2,? ? 4,? ? 6,? + ? 12,? ? 1,? = ? 1,? ? 2,? ? 3,? + ? 6,?
Example 1 1 1 1 1+ 3? 2 1 3? 2 1 1+ 3? 2 1 3? 2 1 3? 2 1+ 3? 2 1 1 3? 2 1+ 3? 2 1+ 3? 2 1+ 3? 2 1 1 3? 2 1 3? 2 1 3? 2 1 3? 2 1 1+ 3? 2 1+ 3? 2 1 1 1 1 1 1 1 1 1 1 1 1 ? 2 ? 4 ? 1 ? 5 6 ? 0 , ? 3 , ,
Example ? 2 ? 4 ? 1 ? 5 1 3? 1 1 3? 1 ? 0 , ? 3 , , 3? 3? ? 0 = 1, ? 2 = ? + 1 1 3??, ? 4 = ? 3??, ? 3 = 1, ? 1 = ? + 1 1 3??, ? 5 = ? 3?? ?,?,?,? , ?2+1 3?2= 1, ?2+1 3?2= 1
Example 2? 2? 1 1 1 1 1 1 1 ? ? ? ? 2? ? + ? ? + ? ? ? ? + ? 2? ? ? ? + ? 1 1 1 1 1 6? = + + ?,?,?,? , ?2+1 3?2= 1, ?2+1 3?2= 1
Initial solution General solution ?2+1 ? = 1 + ?1?, ? = 0 + ?2? 1 + 2?1? + ?12?2+1 3?2= 1,??????? ???????? ?,? = 1,0 2?1 ?12+1 3?22?2= 1 ? = 3?22 ?12+1 ?12+1 3?22 3?22, ? = ? = 2?1?2 ?12+1 3?22
Combination ??? ? = ??? ? = ? 0 ??? ? = ? 0 ??? ? = ? 0 ? = ???? ? ? 0 ? 1 ? 2 ? 3 ? 4 ? 5 ? ? 1 ? 2 ? 3 ? 4 ? 5 ? ? 1 ? 2 ? 3 ? 4 ? 5 ? ? 1 ? 2 ? 3 ? 4 ? 5 ? ? 0 ? 1 ? 2 ? 3 ? 4 ? 5
Example ??? 1 ??? 1 ??? 0 ??? 0 ? =1 0?= 1 0?= 1 1? 0 0 0 0 1 1 1 1 1? 0 0 0 0 1 1 1 1 ?61 ?62 ?64 ?61 ? ?65 ? 1?= 1 0 0 0 0 1 ?65 ?64 ?62 0?= 1 1 0 0 0 1 5? 62 1 1 2 1
? = prime ? 1,? ? 0 = 1, ? 0 = 1, ? 1 , , ? ? 1 = 1 ? =1 ? ? 2 ? 1 , , ? ? 1 2? 2 2 2
? = 6 2? 2? 1 1 1 1 1 1 1 ? ? ? ? 2? ? + ? ? + ? ? ? ? + ? 2? ? ? ? + ? 1 1 1 1 1 6? = + + ?,?,?,? , ?2+1 3?2= 1, ?2+1 3?2= 1
? = 5 4? 1 1 1 1 1 ? + 5? + 10? ? 5? + 10? ? 5? 10? ? + 5? 10? 5? = + ?,?,?,? , ?2+ 5?2+ 10?2+ 10?2= 1, ?? + 4?? + ?2 ?2= 0
? = 5 ?,?,?,? , ?2+ 5?2+ 10?2+ 10?2= 1, ?? + 4?? + ?2 ?2= 0 ?? ? = 0 ?,? = 0 ? = 1 ? = 1 + ?1?, ? = ?, ? = ?3?, ? = ?4? ? = ?12+5+10?32+10?42= ?1+4?3?4+?32 ?42 2?1 1
? = 5 ??????? ???????? ?1,?3,?4 = ?12+ 2 4?3?4+ ?32 ?42?1= 5 + 10?32+ 10?42 ?1 ?3 5 ?1?12+ 5+?1?22 4?1?1?2 ?4 5 ?1?12+ 5+?1?22 4?1?1?2 5 2,1 2,0 5 ?1?32+ 5 + ?1?42 4?1?3?4=?12 5 2 ???= ?1 2 4?3?4+ ?32 ?42 5 ?1?3+4?1?4?12+ 5+?1?3?22 2 5+?1?4?1?2 ???= 5 ?1?4?12+ 4?1?3 5+?1?4?22 2 5 ?1?3?1?2 ???=
Result 2? 1/11 6 6 6 3 9? 1/31 22 18 6 6 4? 1/41 36 12 12 9 3? 1/61 48 24 24 16 52? 1/101 69 36 36 12 57? 1/121 96 24 24 32 39? 1/121 108 36 12 4
Special cases ? = 4? + 2 = 2? + 1 + 2? + 1 ? 0 ? 1 ? 2 ? = ? 0 ? 3 ? = 2? + 2? + 4 ? = 1 0 ? = 1 0 ? ?& ? 0 ? ? 3 ? 4 ? 1 ? 2 ? 3 ? 4 ? ? 1 ? 4 ? 2 ? 0 ? 3 ? 1 ? 4 ? 2 0? 0? 0 1 0 0 1 0 0 1 0 0 1 0 ? 0 1 0 ? 0 1 0 ?
Reference [1] Pei, Soo-Chang, and Kuo-Wei Chang. "On integer-valued zero autocorrelation sequences." Signal and Information Processing Association Annual Summit and Conference (APSIPA), 2013 Asia-Pacific. IEEE, 2013. [2] Carni, E., and Spalvieri, A. Synchronous CDMA based on the cyclical translations of a CAZAC sequence , IEEE Transactions on Wireless Communications, 2005 [3] Hu, Wei-Wen and Wang, Sen-Hung and Li, Chih-Peng, Gaussian Integer Sequences with Ideal Periodic Autocorrelation Functions , 2011 IEEE International Conference on Communications ICC, pp. 1-5
