Maximum length 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 (except the zero vector) that can be represented by the shift registers (i.e., for length-m registers they produce a sequence of length 2m − 1).
MLSs are spectrally flat, with the exception of a near-zero DC term.
Practical applications for MLS include measuring impulse responses (e.g., of room reverberation or arrival times from towed sources in the ocean[1]).
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.
[2] MLS are generated using maximal linear-feedback shift registers.
An MLS-generating system with a shift register of length 4 is shown in Fig.
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.
A necessary and sufficient condition for the sequence generated by a LFSR to be maximal length is that its corresponding polynomial be primitive.
[3] 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).
[vague] Of all the "runs" (consisting of "1"s or "0"s) in the sequence : The circular autocorrelation of an MLS is a Kronecker delta function[5][6] (with DC offset and time delay, depending on implementation).
The linear autocorrelation of an MLS approximates a Kronecker delta.
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.
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.
[7][8] However, after analog 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.
[9] Other signals have been designed with minimal crest factor, though it is unknown if it can be improved beyond 3 dB.
[10] Cohn and Lempel[11] 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.
