Maximum length sequence
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A maximum length sequence (MLS is a type of pseudorandom binary sequence They are bit sequences generated using maximal linear feedback shift register and are so called because they are periodic function and reproduce every binary sequence that can be reproduced by the shift registers (i.e., for length-mregisters they produce a sequence of length 2mlt;/sup> − 1). A MLS is also sometimes called a n-sequence or a m-sequence MLSs are Frequency spectrum with the exception of a near-zero DC term. These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Congruence_subgroup Practical applications for MLS include measuring impulse response (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 , and in the efficient design of some fMRI experimentshttp://www.ncbi.nlm.nih.gov/pubmed/12169264?doptAbstract Buracas GT, Boynton GM. Efficient design of event-related fMRI experiments using M-sequences. Neuroimage. 2002 Jul;16(3 Pt 1):801-13.]

Generation of maximum length sequences

Image:MLS shiftregisters L4.png 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: : where nis the time index, kis the bit register position, and $+$ represents Modular arithmetic addition. 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.

Polynomial interpretation

A polynomial over Galois field 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 Fig. 1 is xlt;sup>4 + xlt;sup>1 + 1. A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be primitive polynomial

Implementation

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 2²⁰ − 1 samples long (1,048,575 samples).

Properties of maximum length sequences

MLS have the following properties, as formulated by Solomon Golomb Golomb, S. Shift Register Sequences San Francisco, Holden–Day, 1967. ISBN 0894120484

Balance property

The number of "1"s in the sequence is one greater than the number of "0"s.

Run property

Of all the "runs" in the sequence of each type (i.e. runs consisting of "1"s and runs consisting of "0"s): * One half of the runs are of length 1. * One quarter of the runs are of length 2. * One eighth of the runs are of length 3. * ... etc. ... 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.

Correlation property

The autocorrelation function of an MLS is a very close approximation to a train of Kronecker delta function.

Extraction of impulse responses

If a LTI system theory (LTI) systems impulse response is to be measured using a MLS, the response can be extracted from the measured system output yn 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/Nwhere Nis 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. If the impulse response of a system is hn and the MLS is sn, then :$yn]\; (h*s)n].\backslash ,$ Taking the cross-correlation with respect to sn of both sides, :$\backslash phi\}\_sy\}\; hn]*\backslash phi\}\_ss\}\backslash ,$ and assuming that φsslt;/sub> is an impulse (valid for long sequences) :$hn]\; \backslash phi\}\_sy\}.\backslash ,$

Relationship to Hadamard transform

Cohn and Lempel Cohn, M. and Lempel, A. On Fast M-Sequence Transforms IEEE Trans. Information Theory, vol. IT-23, pp. 135–137, January, 1977. 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 FFT

See also

* Impulse response * Frequency response * Polynomial ring * Federal Standard 1037C * Gold code * Complementary sequences

References

*Solomon W. Golomb and Guang Gong Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar 2005. ISBN 0521821045

External links

* http://jenshee.dk/signalprocessing/mls.pdf Impulse response measurement using MLS] — Paper by Jens Hee describing MLS generation. Contains C-code for MLS generation using up to 18-tap-LFSRs and matching Hadamard transform for impulse response extraction. * http://www.cfn.upenn.edu/aguirre/wiki/public:m_sequences Guide to Creation of M-Sequences] * http://www.newwaveinstruments.com/resources/articles/m_sequence_linear_feedback_shift_register_lfsr.htm LFSR Reference] — 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). *http://www.ind.rwth-aachen.de/de/veroeffentlichungen/details/publabel/jeub09a/# A (binaural) room impulse response database generated by means of maximum length sequences] Category:Pseudorandomness Category:Polynomials Category:Binary sequences de:Maximum Length Sequence it:Sequenza di lunghezza massima ja:M系列 ru:М-последовательность zh:M-sequence