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 $s$ as a sequence obtained by replacing every 0 with 1 and replacing every 1 with -1 as follows:

$$s_0,s_1,s_2,s_3.... = (-1)^{a_0},(-1)^{a_1},(-1)^{a_2},(-1)^{a_3}....$$ (7.4)

The auto-correlation function, $\rho$, is defined as [68]

$$\rho(i) = \frac{1}{n} \sum\limits_{j=0}^{n-1} s_j s_{i+j}^*.$$ (7.5)

where $n = 2^m-1$ is the length of the sequence. Note that the subscript $i+j$ can exceed $n$. When this happens the subscript is taken modulo $n$ (ie. $n$ is subtracted so that $s_{i+j}$ is a circularly shifted version of $s$). 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:

$$\rho(i) = \frac{A-D}{n},$$ (7.6)

where $A$ is the number of times the elements $s_j$ and $s_{i+j}$ agree and $D$ is the number of times they disagree. The first term in $\rho$ is given by $i=0$ so $s_j$ and $s_{i+j}$ agree for all values of $j$ giving $\rho(0) = 1$. When $1 \leq i \leq n-1$ 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 $A-D=-1$ for $1 \leq i \leq n-1$. The auto-correlation of the MLS is therefore

$$\rho(i) = \left\{ \begin{array} {r@{\quad:\quad}l} 1 & i=0, \\ -1/n & 1 \leq i \leq n-1, \end{array} \right.$$ (7.7)

which is distinguished from a perfect digital impulse by the presence of the small non-zero value when $i \neq 0$. While the frequency spectrum of an ideal digital impulse is equal for all frequencies, the frequency spectrum of $\rho$ is the same for all frequencies except for the zero frequency component.