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| A '''maximum length sequence''' ('''MLS''') is a type of [[pseudorandom binary sequence]].
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| They are bit sequences generated using maximal [[linear feedback shift register]]s and are so called because they are [[periodic function|periodic]] and reproduce every [[binary sequence]] that can be represented by the shift registers (i.e., for length-''m'' registers they produce a sequence of length 2<sup>''m''</sup> − 1). An MLS is also sometimes called an '''n-sequence''' or an '''m-sequence'''. MLSs are [[Frequency spectrum|spectrally flat]], with the exception of a near-zero DC term.
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| These sequences may be represented as coefficients of irreducible polynomials in a [[polynomial ring]] over [[Congruence_subgroup | Z/2Z]].
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| Practical applications for MLS include measuring [[impulse response]]s (e.g., of room [[reverberation]]). They are also used as a basis for deriving pseudo-random sequences in digital communication systems that employ [[direct-sequence spread spectrum]] and [[frequency-hopping spread spectrum]] [[transmission system]]s, and in the efficient design of some fMRI experiments<ref name="buracas">{{cite journal |author=Buracas GT, Boynton GM |title=Efficient design of event-related fMRI experiments using M-sequences |journal=Neuroimage |volume=16 |issue=3 Pt 1 |pages=801–13 |date=July 2002 |pmid=12169264 |doi=10.1006/nimg.2002.1116 |url=http://linkinghub.elsevier.com/retrieve/pii/S105381190291116X}}</ref>
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| ==Generation of maximum length sequences==
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| [[File:MLS shiftregisters L4.png|thumbnail|350px|right|Figure 1: The next value of register ''a''<sub>3</sub> in a feedback shift register of length 4 is determined by the modulo-2 sum of ''a''<sub>0</sub> and ''a''<sub>1</sub>.]]
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| MLS are generated using maximal linear feedback shift registers. An MLS-generating system with a shift register of length 4 is shown in Fig. 1. It can be expressed using the following recursive relation:
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| :<math>a_k[n+1] = \begin{cases}
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| a_0[n] + a_1[n], & k = 3 \\
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| \\
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| a_{k+1}[n], & \mbox{otherwise}
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| \end{cases}
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| </math>
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| where ''n'' is the time index, ''k'' is the bit register position, and <math>+</math> represents [[Modular arithmetic|modulo-2]] addition.
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| As MLS are periodic and shift registers cycle through every possible binary value (with the exception of the zero vector), registers can be initialized to any state, with the exception of the zero vector.
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| ===Polynomial interpretation===
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| A [[polynomial]] over [[Galois field|GF(2)]] can be associated with the linear feedback shift register. It has degree of the length of the shift register, and has coefficients that are either 0 or 1, corresponding to the taps of the register that feed the [[xor]] gate. For example, the polynomial corresponding to Figure 1 is ''x''<sup>4</sup> + ''x'' + 1.
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| A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be [[Primitive polynomial (field theory)|primitive]].<ref>"Linear Feedback Shift Registers-<small>
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| Implementation, M-Sequence Properties, Feedback Tables</small>"[http://www.newwaveinstruments.com/resources/articles/m_sequence_linear_feedback_shift_register_lfsr.htm],New Wave Intruments (NW), Retrieved 2013.12.03.</ref>
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| ===Implementation===
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| MLS are inexpensive to implement in hardware or software, and relatively low-order feedback shift registers can generate long sequences; a sequence generated using a shift register of length 20 is 2<sup>20</sup> − 1 samples long (1,048,575 samples).
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| ==Properties of maximum length sequences==
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| MLS have the following properties, as formulated by [[Solomon Golomb]]. <ref name="golumb">{{cite book |first=Solomon W. |last=Golomb |title=Shift register sequences |url=http://books.google.com/books?id=LqtMAAAAMAAJ |year=1967 |publisher=Holden-Day |isbn=0-89412-048-4}}</ref>
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| ===Balance property===
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| the occurrence of 0 and 1 in the sequence should be approximately the same
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| ===Run property===
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| Of all the "runs" in the sequence of each type (i.e. runs consisting of "1"s and runs consisting of "0"s):
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| * One half of the runs are of length 1.
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| * One quarter of the runs are of length 2.
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| * One eighth of the runs are of length 3.
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| * ... etc. ...
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| A "run" is a sub-sequence of "1"s or "0"s within the MLS concerned. The number of runs is the number of such sub-sequences.
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| ===Correlation property===
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| The [[autocorrelation]] function of an MLS is a very close approximation to a train of [[Kronecker delta]] function.
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| ==Extraction of impulse responses==
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| If a [[LTI system theory|linear time invariant]] (LTI) system's impulse response is to be measured using a MLS, the response can be extracted from the measured system output ''y''[''n''] by taking its circular cross-correlation with the MLS. This is because the [[autocorrelation]] of a MLS is 1 for zero-lag, and nearly zero (−1/''N'' where ''N'' is the sequence length) for all other lags; in other words, the autocorrelation of the MLS can be said to approach unit impulse function as MLS length increases.
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| If the impulse response of a system is ''h''[''n''] and the MLS is ''s''[''n''], then
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| :<math>y[n] = (h*s)[n].\,</math>
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| Taking the cross-correlation with respect to ''s''[''n''] of both sides,
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| :<math>{\phi}_{sy} = h[n]*{\phi}_{ss}\,</math>
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| and assuming that φ<sub>''ss''</sub> is an impulse (valid for long sequences)
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| :<math>h[n] = {\phi}_{sy}.\,</math>
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| <!-- had to do the above in a rush. not entirely accurate!-->
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| ==Relationship to Hadamard transform==
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| Cohn and Lempel <ref name="cohn">{{cite journal |last1=Cohn |first1=M. |last2=Lempel |first2=A. |title=On Fast M-Sequence Transforms |journal=IEEE Trans. Information Theory |volume=23 |issue=1 |pages=135–7 |date=January 1977 |doi=10.1109/TIT.1977.1055666 |url=http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1055666}}</ref> showed the relationship of the MLS to the [[Hadamard transform]]. This relationship allows the [[correlation]] of an MLS to be computed in a fast algorithm similar to the [[Fast Fourier transform|FFT]].
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| ==See also==
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| * [[Impulse response]]
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| * [[Frequency response]]
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| * [[Polynomial ring]]
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| * [[Federal Standard 1037C]]
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| * [[Gold code]]
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| * [[Complementary sequences]]
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| ==References==
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| *{{cite book |first1=Solomon W. |last1=Golomb |author2=Guang Gong |title=Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar |url=http://books.google.com/books?id=DhYXL4miZj4C |year=2005 |publisher=[[Cambridge University Press]] |isbn=978-0-521-82104-9}}
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| {{reflist}}
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| ==External links==
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| * {{cite web |last=Bristow-Johnson |first=Robert |title=A Little MLS Tutorial |url=http://www.dspguru.com/dsp/tutorials/a-little-mls-tutorial }} — Short on-line tutorial describing how MLS is used to obtain the [[impulse response]] of a [[linear time-invariant system]]. Also describes how nonlinearities in the system can show up as spurious spikes in the apparent impulse response.
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| * {{cite web |first=Jens |last=Hee |title=Impulse response measurement using MLS |format=PDF |url=http://jenshee.dk/signalprocessing/mls.pdf}} — Paper describing MLS generation. Contains C-code for MLS generation using up to 18-tap-LFSRs and matching Hadamard transform for impulse response extraction.
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| * {{cite web |first=Wesley |last=Kerr |title=Creation of M-Sequences |work=Geoffrey Aguirre Lab |publisher=University of Pennsylvania |url=http://www.cfn.upenn.edu/aguirre/wiki/public:m_sequences}} ]
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| * {{cite web |title=Linear Feedback Shift Registers |year=2005 |publisher=New Wave Instruments |url=http://www.newwaveinstruments.com/resources/articles/m_sequence_linear_feedback_shift_register_lfsr.htm}} — Properties of maximal length sequences, and comprehensive feedback tables for maximal lengths from 7 to 16,777,215 (3 to 24 stages), and partial tables for lengths up to 4,294,967,295 (25 to 32 stages).
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| *{{cite web |first=Magnus |last=Schäfer |title=Aachen Impulse Response Database |date=October 2012 |publisher=Institute of Communication Systems and Data Processing, RWTH Aachen University |url=http://www.ind.rwth-aachen.de/en/research/tools-downloads/aachen-impulse-response-database/ |id=V1.4}} A (binaural) room impulse response database generated by means of maximum length sequences]
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| *{{cite web |id=XAPP052 v1.1 |title=Efficient Shift Registers, LFSR Counters, and Long Pseudo-Random Sequence Generators — Obsolete |date=July 1996 |publisher=Xilinx |url=http://www.xilinx.com/support/documentation/application_notes/xapp052.pdf}} — Implementing lfsr's in FPGAs includes listing of taps for 3 to 168 bits
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| [[Category:Pseudorandomness]]
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| [[Category:Polynomials]]
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| [[Category:Binary sequences]]
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