Pappus's hexagon theorem

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 2^m − 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

Generation of maximum length sequences

Figure 1: The next value of register a₃ in a feedback shift register of length 4 is determined by the modulo-2 sum of a₀ and a₁.

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:

$a_{k}[n+1]={\begin{cases}a_{0}[n]+a_{1}[n],&k=3\\\\a_{k+1}[n],&{\mbox{otherwise}}\end{cases}}$

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.

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.

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

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

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

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

and assuming that φ_ss is an impulse (valid for long sequences)

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

Relationship to Hadamard transform

Cohn and Lempel 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

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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