Editing Maximum length sequence (section)

==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 (&minus;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!-->Any signal with an impulsive autocorrelation can be used for this purpose, but signals with high [[crest factor]], such as the impulse itself, produce impulse responses with poor [[signal-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&nbsp;dB.<ref>{{Cite web|url=http://dspguru.com/dsp/tutorials/a-little-mls-tutorial|title=A Little MLS (Maximum-Length Sequence) Tutorial {{!}} dspGuru.com|website=dspguru.com|access-date=2016-05-19|quote=its RMS and peak values are both X, making its crest factor (peak/RMS) equal to 1, the lowest it can get.}}</ref><ref>{{Cite web|url=https://www.clear.rice.edu/elec301/Projects00/elec301/OtherTechniques/othertechniques.html|title=Other Electro-Acoustical Measurement Techniques|website=www.clear.rice.edu|access-date=2016-05-19|quote=The crest factor for MLS is very close to 1, so it makes sense to use this kind of input signal when we need a high signal-to-noise ratio for our measurement}}</ref>  However, after [[Digital-to-analog converter|analog reconstruction]], the sharp discontinuities in the signal produce strong intersample peaks, degrading the crest factor by 4-8&nbsp;dB or more, increasing with signal length, making it worse than a sine sweep.<ref>{{Cite web |last=Chan |first=Ian H. |title=Swept Sine Chirps for Measuring Impulse Response |url=http://www.thinksrs.com/downloads/PDFs/ApplicationNotes/SR1_SweptSine.pdf |access-date=2016-05-19 |website=thinksrs.com |quote=Maximum-length sequence (MLS) theoretically fits the bill because it has a mathematical crest factor of 0dB, the lowest crest factor possible. However, in practice, the sharp transitions and bandwidth-limited reproduction of the signal result in a crest factor of about 8dB}}</ref>  Other signals have been designed with minimal crest factor, though it is unknown if it can be improved beyond 3&nbsp;dB.<ref>{{Cite journal|last=Friese|first=M.|date=1997-10-01|title=Multitone signals with low crest factor|url=https://stanford.edu/~boyd/papers/pdf/multitone_low_crest.pdf|journal=IEEE Transactions on Communications|volume=45|issue=10|pages=1338–1344|doi=10.1109/26.634697|issn=0090-6778}}</ref>