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## Auto-correlation property of MLS

In order to use an MLS in a measurement, it is preferable to convert the signal to one which oscillates around zero rather than above zero because this reduces the dc offset and therefore improves the efficiency of loudspeakers and measurement systems. This can be done by defining as a sequence obtained by replacing every 0 with 1 and replacing every 1 with -1 as follows: (7.4)

The auto-correlation function, , is defined as (7.5)

where is the length of the sequence. Note that the subscript can exceed . When this happens the subscript is taken modulo (ie. is subtracted so that is a circularly shifted version of ). The symbol * denotes complex conjugation which may be dropped in the current application because all entries in the sequence are real.

For the MLS defined in equation (7.4), the auto-correlation becomes: (7.6)

where is the number of times the elements and agree and is the number of times they disagree. The first term in is given by so and agree for all values of giving . When we are calculating the agreement between the signal and a circularly shifted version. Since the signals appear random, it is intuitive that the agreement and disagreement should be almost equal. The proof in  shows that for . The auto-correlation of the MLS is therefore (7.7)

which is distinguished from a perfect digital impulse by the presence of the small non-zero value when . While the frequency spectrum of an ideal digital impulse is equal for all frequencies, the frequency spectrum of is the same for all frequencies except for the zero frequency component.

Back to Kemp Acoustics Home    Next: Extracting the system impulse Up: Maximum length sequences Previous: Generating an MLS sequence   Contents
Jonathan Kemp 2003-03-24