Maximum length sequence

Amaximum length sequence(MLS) is a type ofpseudorandom binary sequence.

They are bit sequences generated using maximallinear-feedback shift registersand are so called because they areperiodicand reproduce everybinary sequence(except the zero vector) that can be represented by the shift registers (i.e., for length-mregisters they produce a sequence of length 2^m− 1). An MLS is also sometimes called ann-sequenceor anm-sequence.MLSs arespectrally flat,with the exception of a near-zero DC term.

These sequences may be represented as coefficients of irreducible polynomials in apolynomial ringoverZ/2Z.

Practical applications for MLS include measuringimpulse responses(e.g., of roomreverberationor arrival times from towed sources in the ocean). They are also used as a basis for deriving pseudo-random sequences in digital communication systems that employdirect-sequence spread spectrumandfrequency-hopping spread spectrumtransmission systems,and in the efficient design of somefMRIexperiments.

Generation

Figure 1: The next value of registera₃in a feedback shift register of length 4 is determined by the modulo-2 sum ofa₀anda₁.

MLS are generated using maximallinear-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:

${\begin{cases}a_{3}[n+1]=a_{0}[n]+a_{1}[n]\\a_{2}[n+1]=a_{3}[n]\\a_{1}[n+1]=a_{2}[n]\\a_{0}[n+1]=a_{1}[n]\\\end{cases}}$

wherenis the time index and$+$representsmodulo-2addition. For bit values 0 = FALSE or 1 = TRUE, this is equivalent to the XOR operation.

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

ApolynomialoverGF(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 thexorgate. For example, the polynomial corresponding to Figure 1 is$x^{4}+x+1$.

A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial beprimitive.

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 bySolomon Golomb.

Balance property

The occurrence of 0 and 1 in the sequence should be approximately the same. More precisely, in a maximum length sequence of length$2^{n}-1$there are$2^{n-1}$ones and$2^{n-1}-1$zeros. The number of ones equals the number of zeros plus one, since the state containing only zeros cannot occur.

Run property

A "run" is a sub-sequence of consecutive "1" s or consecutive "0" s within the MLS concerned. The number of runs is the number of such sub-sequences.

Of all the "runs" (consisting of "1" s or "0" s) in the sequence:

• 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....

Correlation property

The circularautocorrelationof an MLS is aKronecker deltafunction(with DC offset and time delay, depending on implementation). For the ±1 convention, i.e., bit value 1 is assigned$s=+1$and bit value 0$s=-1$,mapping XOR to the negative of the product:

$R(n)={\frac {1}{N}}\sum _{m=1}^{N}s[m]\,s^{*}[m+n]_{N}={\begin{cases}1&{\text{if }}n=0,\\-{\frac {1}{N}}&{\text{if }}0<n<N.\end{cases}}$

where$s^{*}$represents the complex conjugate and$[m+n]_{N}$represents acircular shift.

The linear autocorrelation of an MLS approximates a Kronecker delta.

Extraction of impulse responses

If alinear time invariant(LTI) system's impulse response is to be measured using a MLS, the response can be extracted from the measured system outputy[n] by taking its circular cross-correlation with the MLS. This is because theautocorrelationof a MLS is 1 for zero-lag, and nearly zero (−1/NwhereNis 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 ish[n] and the MLS iss[n], then

$y[n]=(h*s)[n].\,$

Taking the cross-correlation with respect tos[n] of both sides,

${\phi }_{sy}=h[n]*{\phi }_{ss}\,$

and assuming that φ_ssis an impulse (valid for long sequences)

$h[n]={\phi }_{sy}.\,$

Any signal with an impulsive autocorrelation can be used for this purpose, but signals with highcrest factor,such as the impulse itself, produce impulse responses with poorsignal-to-noise ratio.It is commonly assumed that the MLS would then be the ideal signal, as it consists of only full-scale values and its digital crest factor is the minimum, 0 dB.However, afteranalog reconstruction,the sharp discontinuities in the signal produce strong intersample peaks, degrading the crest factor by 4-8 dB or more, increasing with signal length, making it worse than a sine sweep.Other signals have been designed with minimal crest factor, though it is unknown if it can be improved beyond 3 dB.

Relationship to Hadamard transform

Cohn and Lempelshowed the relationship of the MLS to theHadamard transform.This relationship allows thecorrelationof an MLS to be computed in a fast algorithm similar to theFFT.

See also

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

References

• Golomb, Solomon W.;Guang Gong(2005).Signal Design for Good Correlation: For Wireless Communication, Cryptography, and Radar.Cambridge University Press.ISBN978-0-521-82104-9.

External links

• Bristow-Johnson, Robert."A Little MLS Tutorial".— Short on-line tutorial describing how MLS is used to obtain theimpulse responseof alinear time-invariant system.Also describes how nonlinearities in the system can show up as spurious spikes in the apparent impulse response.
• Hee, Jens."Impulse response measurement using 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.
• Schäfer, Magnus (October 2012)."Aachen Impulse Response Database".Institute of Communication Systems and Data Processing, RWTH Aachen University. V1.4.A (binaural) room impulse response database generated by means of maximum length sequences.
• "Efficient Shift Registers, LFSR Counters, and Long Pseudo-Random Sequence Generators — Obsolete"(PDF).Xilinx. July 1996. XAPP052 v1.1.— Implementing lfsr's in FPGAs includes listing of taps for 3 to 168 bits