Let {a t } be a sequence of period n (so a t = a t+n for all values of t) with symbols being the integers mod q (see modular arithmetic). The periodic auto-correlation of the sequence {a t } at shift τ is defined as
where ω is a complex qth root of unity.
In most applications one considers binary sequences when q=2 and ω=-1. Then the autocorrelation at shift τ equals the number of agreements minus the number of disagreements between the sequence {at} and its cyclic shift {at+τ}. Note that in most applications one wants the autocorrelation for all nonzero shifts τ ≠ 0 (mod n) (the out-of-phase autocorrelation) to be low in absolute value. For example, this property of a sequence is extremely useful for synchronization purposes.
This is a preview of subscription content, log in via an institution to check access.
References
Golomb, S.W. (1982). Shift Register Sequences. Aegean Park Press, Laguna Hills, CA.
Helleseth, T. and P.V. Kumar (1998). “Sequences with low correlation.” Handbook of Coding Theory, eds. V.S. Pless and W.C. Huffman. Elsevier, Amsterdam.
Helleseth, T. and P.V. Kumar (1999). “Pseudonoise sequences.” The Mobile Communications Handbook, ed. J.D. Gibson. CRC Press, Boca Raton, FL, Chapter 8.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 International Federation for Information Processing
About this entry
Cite this entry
Helleseth, T. (2005). Autocorrelation. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_21
Download citation
DOI: https://doi.org/10.1007/0-387-23483-7_21
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-23473-1
Online ISBN: 978-0-387-23483-0
eBook Packages: Computer ScienceReference Module Computer Science and Engineering