On the Statistical Independency of Random Signal Based on M-Sequences

Abstract

The M-sequence, which is generated by connecting a feedback loop on an n-stage shift register with mod. 2 adders, is frequently used for a binary random signal. But the random signal x(t) obtained from the M-sequences with a weight adder is not statistically independent because its randomness is not complete.
In this paper, a statistical dependency d of the x(t) is defined by
d=∫∞-∞…∫∞-∞{F(x₁, …, x_m)
-F₁(x₁)…F_m(x_m)}²dx₁…dx_m,
where F_i(x_i) is a cumulative distribution function of the random variable x_i=x(t_i), F(x₁, …, x_m) is a jointly cumulative distribution function of x₁, …, x_m. The statistical independency of the random signal based on the M-sequences is estimated by this definition. As a result, it is shown that d depends on the feedback loop and also on the weight arrangement of the weight adder even if the feedback loop is the same, and that d decreases if the weight elements are arranged according to the linearly dependent structure of the feedback loop.