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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 autocorrelation function, , is defined as [68]

(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 autocorrelation 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 [68] shows that
for
. The autocorrelation of the MLS is therefore

(7.7) 
which is distinguished from a perfect digital impulse by the presence of
the small nonzero 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
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Previous: Generating an MLS sequence
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Jonathan Kemp
20030324