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| : ''For complementary sequences in biology, see [[complementarity (molecular biology)]].''
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| In applied mathematics, '''complementary sequences (CS)''' are pairs of [[sequence]]s with the useful property that their out-of-phase aperiodic [[autocorrelation]] coefficients sum to zero. Binary complementary sequences were first introduced by [[Marcel J. E. Golay]] in 1949. In 1961–1962 Golay gave several methods for constructing sequences of length 2<sup>''N''</sup> and gave examples of complementary sequences of lengths 10 and 26. In 1974 R. J. Turyn gave a method for constructing sequences of length ''mn'' from sequences of lengths ''m'' and ''n'' which allows the construction of sequences of any length of the form 2<sup>''N''</sup>10<sup>''K''</sup>26<sup>''M''</sup>.
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| Later the theory of complementary sequences was generalized by other authors to polyphase complementary sequences, multilevel complementary sequences, and arbitrary complex complementary sequences. '''Complementary sets''' have also been considered; these can contain more than two sequences.
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| == Definition ==
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| Let (''a''<sub>0</sub>, ''a''<sub>1</sub>, ..., ''a''<sub>''N'' − 1</sub>) and (''b''<sub>0</sub>, ''b''<sub>1</sub>, ..., ''b''<sub>''N'' − 1</sub>) be a pair of bipolar sequences, meaning that ''a''(''k'') and ''b''(''k'') have values +1 or −1. Let the aperiodic [[autocorrelation function]] of the sequence '''x''' be defined by
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| :<math>R_x(k)=\sum_{j=0}^{N-k-1} x_jx_{j+k}.\,</math>
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| Then the pair of sequences ''a'' and ''b'' is complementary if:
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| :<math>R_a(k) + R_b(k) = 0,\, </math>
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| for ''k'' = 1, ..., ''N'' − 1.
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| Or using [[Kronecker delta]] we can write:
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| :<math>R_a(k) + R_b(k) = C\delta(k),\, </math>
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| where ''C'' is a constant.
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| So we can say that the sum of autocorrelation functions of complementary sequences is a delta function which is an ideal autocorrelations for many applications like [[radar]] [[pulse compression]] and [[spread spectrum]] [[telecommunications]].
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| == Examples ==
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| * As the simplest example we have sequences of length 2: (+1, +1) and (+1, −1). Their autocorrelation functions are (2, 1) and (2, −1), which add up to (4, 0).
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| * As the next example (sequences of length 4), we have (+1, +1, +1, −1) and (+1, +1, −1, +1). Their autocorrelation functions are (4, 1, 0, −1) and (4, −1, 0, 1), which add up to (8, 0, 0, 0).
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| * One example of length 8 is (+1, +1, +1, −1, +1, +1, −1, +1) and (+1, +1, +1, −1, −1, −1, +1, −1). Their autocorrelation functions are (8, −1, 0, 3, 0, 1, 0, 1) and (8, 1, 0, −3, 0, −1, 0, −1).
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| * An example of length 10 given by Golay is (+1, +1, −1, +1, −1, +1, −1, −1, +1, +1) and (+1, +1, −1, +1, +1, +1, +1, +1, −1, −1). Their autocorrelation functions are (10, −3, 0, −1, 0, 1,−2, −1, 2, 1) and (10, 3, 0, 1, 0, −1, 2, 1, −2, −1).
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| == Properties of complementary pairs of sequences ==
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| * Complementary [[sequences]] have complementary spectra. As the autocorrelation function and the power spectra form a Fourier pair, complementary sequences also have complementary spectra. But as the Fourier transform of a delta function is a constant, we can write
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| :: <math>S_a + S_b = C_S,\, </math>
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| : where ''C''<sub>''S''</sub> is a constant.
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| : ''S''<sub>''a''</sub> and ''S''<sub>''b''</sub> are defined as a squared magnitude of the [[Fourier transform]] of the sequences. The Fourier transform can be a direct DFT of the sequences, it can be a DFT of zero padded sequences or it can be a continuous Fourier transform of the sequences which is equivalent to the [[Z transform]] for ''Z'' = ''e''<sup>''j''ω</sup>.
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| * CS spectra is upper bounded. As ''S''<sub>''a''</sub> and ''S''<sub>''b''</sub> are non-negative values we can write
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| :: <math>S_a = C_S - S_b < C_S,\,</math>
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| : also
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| :: <math>S_b < C_S.\,</math>
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| * If any of the sequences of the CS pair is inverted (multiplied by −1) they remain complementary. More generally if any of the sequences is multiplied by ''e''<sup>''j''φ</sup> they remain complementary;
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| * If any of the sequences is reverted (inverted in time) they remain complementary;
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| * If any of the sequences is delayed they remain complementary;
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| * If the sequences are interchanged they remain complementary;
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| * If both sequences are multiplied by the same constant (real or complex) they remain complementary;
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| * If both sequences are decimated in time by ''K'' they remain complementary. More precisely if from a complementary pair ((''a''(''k''), ''b''(''k'')) we form a new pair (''a''(''Nk''), ''b''(''N''*''k'')) with zero samples in between then the new sequences are complementary.
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| * If alternating bits of both sequences are inverted they remain complementary. In general for arbitrary complex sequences if both sequences are multiplied by ''e''<sup>''j''π''kn''/''N''</sup> (where ''k'' is a constant and ''n'' is the time index) they remain complementary
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| * A new pair of complementary sequences can be formed as [''a'' ''b''] and [''a'' −''b''] where [..] denotes concatenation and ''a'' and ''b'' are a pair of CS;
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| * A new pair of sequences can be formed as {''a'' ''b''} and {''a'' −''b''} where {..} denotes interleaving of sequences.
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| * A new pair of sequences can be formed as ''a'' + ''b'' and ''a'' − ''b''.
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| ==Golay pair==
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| A complementary pair ''a'', ''b'' may be encoded as polynomials ''A''(''z'') = ''a''(0) + ''a''(1)''z'' + ... + ''a''(''N'' − 1)''z''<sup>''N''−1</sup> and similarly for ''B''(''z''). The complementarity property of the sequences is equivalent to the condition
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| :<math>\vert A(z) \vert^2 + \vert B(z) \vert^2 = 2N \, </math>
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| for all ''z'' on the unit circle, that is, |''z''| = 1. If so, ''A'' and ''B'' form a '''Golay pair''' of polynomials. Examples include the [[Shapiro polynomials]], which give rise to complementary sequences of length a power of 2.
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| == Applications of complementary sequences ==
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| * Multislit spectrometry
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| * Ultrasound measurements
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| * Acoustic measurements
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| * [[radar]] [[pulse compression]]
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| * [[Wi-Fi]] networks,
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| * [[3G]] [[CDMA]] wireless networks
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| * [[OFDM]] communication systems
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| * Train wheel detection systems
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| * Non-destructive tests (NDT)
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| * Communications
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| ==See also==
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| * [[Pseudorandom binary sequence]]s (also called [[maximum length sequence]]s or M-sequences)
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| * [[Gold code|Gold sequences]]
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| * [[Kasami code|Kasami sequences]]
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| * [[Walsh matrix|Walsh–Hadamard sequences]]
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| * [[Binary Golay code]] ([[Error-correcting code]])
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| * [[Ternary Golay code]] ([[Error-correcting code]])
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| == References ==
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| {{refbegin}}
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| *{{cite journal |first=Marcel J.E. |last=Golay |authorlink=Marcel J. E. Golay |title=Multislit spectroscopy |journal=J. Opt. Soc. Am. |volume=39 |pages=437–444 |year=1949 |doi=10.1364/JOSA.39.000437 |url=http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-39-6-437}}
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| *{{cite journal |first=Marcel J.E. |last=Golay |title=Complementary series |journal=IRE Trans. Inform. Theory |volume=7 |issue=2 |pages=82–87 |date=April 1961 |doi=10.1109/TIT.1961.1057620 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1057620}}
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| *{{cite journal |first=Marcel J.E. |last=Golay |title=Note on “Complementary series″ |journal=[[Proc. IRE]] |volume=50 |pages=84 |year=1962 |doi=10.1109/JRPROC.1962.288278 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=4066535}}
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| *{{cite journal |first=R.J. |last=Turyn |title=Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings |journal=J. Combin. Theory (A) |volume=16 |pages=313–333 |year=1974 |doi=10.1016/0097-3165(74)90056-9 |issue=3 }}
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| *{{cite book |authorlink=Peter Borwein |first=Peter |last=Borwein |title=Computational Excursions in Analysis and Number Theory |url=http://books.google.com/books?id=A_ITwN13J6YC&pg=PA110 |year=2002 |publisher=Springer |isbn=978-0-387-95444-8 |pages=110–9}}
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| *{{cite journal |first1=P.G. |last1=Donato |first2=J. |last2=Ureña |first3=M. |last3=Mazo |first4=C. |last4=De Marziani |first5=A. |last5=Ochoa |title=Design and signal processing of a magnetic sensor array for train wheel detection |journal=Sensors and Actuators A: Physical |volume=132 |pages=516–525 |year=2006 |doi=10.1016/j.sna.2006.02.043 |issue=2 }}
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| {{refend}}
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| [[Category:Sequences and series]]
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| [[Category:Signal processing]]
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| [[Category:Pseudorandom number generators]]
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