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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]]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. | |||
These sequences may be represented as coefficients of irreducible polynomials in a [[polynomial ring]] over [[Congruence_subgroup | Z/2Z]]. | |||
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> | |||
==Generation of maximum length sequences== | |||
[[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>.]] | |||
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: | |||
:<math>a_k[n+1] = \begin{cases} | |||
a_0[n] + a_1[n], & k = 3 \\ | |||
\\ | |||
a_{k+1}[n], & \mbox{otherwise} | |||
\end{cases} | |||
</math> | |||
where ''n'' is the time index, ''k'' is the bit register position, and <math>+</math> represents [[Modular arithmetic|modulo-2]] 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|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. | |||
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> | |||
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> | |||
===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 2<sup>20</sup> − 1 samples long (1,048,575 samples). | |||
==Properties of maximum length sequences== | |||
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> | |||
===Balance property=== | |||
the occurrence of 0 and 1 in the sequence should be approximately the same | |||
===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|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. | |||
If the impulse response of a system is ''h''[''n''] and the MLS is ''s''[''n''], then | |||
:<math>y[n] = (h*s)[n].\,</math> | |||
Taking the cross-correlation with respect to ''s''[''n''] of both sides, | |||
:<math>{\phi}_{sy} = h[n]*{\phi}_{ss}\,</math> | |||
and assuming that φ<sub>''ss''</sub> is an impulse (valid for long sequences) | |||
:<math>h[n] = {\phi}_{sy}.\,</math> | |||
<!-- had to do the above in a rush. not entirely accurate!--> | |||
==Relationship to Hadamard transform== | |||
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]]. | |||
==See also== | |||
* [[Impulse response]] | |||
* [[Frequency response]] | |||
* [[Polynomial ring]] | |||
* [[Federal Standard 1037C]] | |||
* [[Gold code]] | |||
* [[Complementary sequences]] | |||
==References== | |||
*{{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}} | |||
{{reflist}} | |||
==External links== | |||
* {{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. | |||
* {{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. | |||
* {{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}} ] | |||
* {{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). | |||
*{{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] | |||
*{{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 | |||
[[Category:Pseudorandomness]] | |||
[[Category:Polynomials]] | |||
[[Category:Binary sequences]] |
Revision as of 20:19, 28 January 2014
A maximum length sequence (MLS) is a type of pseudorandom binary sequence.
They are bit sequences generated using maximal linear feedback shift registers and are so called because they are 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 2m − 1). An MLS is also sometimes called an n-sequence or an m-sequence. MLSs are spectrally flat, with the exception of a near-zero DC term.
These sequences may be represented as coefficients of irreducible polynomials in a polynomial ring over Z/2Z.
Practical applications for MLS include measuring impulse responses (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 systems, and in the efficient design of some fMRI experiments[1]
Generation of maximum length sequences
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 n is the time index, k is the bit register position, and represents modulo-2 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 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 x4 + x + 1.
A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be primitive.[2]
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. [3]
Balance property
the occurrence of 0 and 1 in the sequence should be approximately the same
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 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.
If the impulse response of a system is h[n] and the MLS is s[n], then
Taking the cross-correlation with respect to s[n] of both sides,
and assuming that φss is an impulse (valid for long sequences)
Relationship to Hadamard transform
Cohn and Lempel [4] 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
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
External links
- Template:Cite web — 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.
- Template:Cite web — Paper describing MLS generation. Contains C-code for MLS generation using up to 18-tap-LFSRs and matching Hadamard transform for impulse response extraction.
- Template:Cite web ]
- Template:Cite web — 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).
- Template:Cite web A (binaural) room impulse response database generated by means of maximum length sequences]
- Template:Cite web — Implementing lfsr's in FPGAs includes listing of taps for 3 to 168 bits
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ "Linear Feedback Shift Registers- Implementation, M-Sequence Properties, Feedback Tables"[1],New Wave Intruments (NW), Retrieved 2013.12.03.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang