The relation between the geometrical parameters, the load and the material characteristics is first discussed in order to find out the limit value of the span length.
In the case of binary De Bruijn sequence, for any value of the span n we have: and L = 2 n, so that: (16).
As long as the value of the span n increases, the De Bruijn sequences show a better adherence to the Welch bound than the other families of spreading codes considered for comparison.
where a and b are two binary sequences in the set having the same period L, and l denotes any possible value of the shift among the sequences (0 ≤ l ≤ L - 1); the approximation holds when M ≫ L (increasing value of the span n).
The dimensionless values of the span limits are 0.873 and 1 with value of (d(bar{z})) as 1.81 and 2.2, respectively, based on the selected graph.
Given the amount of binary De Bruijn sequences obtainable, even for small values of the span parameter, and considering the great complexity of the generation process [13], we can provide an exhaustive analysis of binary sequences of length 32 (i.e., span 5) that form a set of 2,048 different sequences, and partial results for sequences generated by increasing values of the span.
For the smallest values of the span n, m-sequences and De Bruijn sequences show the lowest values of the bound; when n increases, De Bruijn sequences exhibit performance comparable to Gold and Kasami large set sequences.
Gold sequences are generated from the so-called preferred pairs of m-sequences, for values of the span n that satisfy the conditions: n ≠ 0 (mod 4) or n = 2 (mod 4).
Binary De Bruijn sequences feature great cardinality of the available sequence sets, even for small values of the span parameter, and may consequently allow the definition of proper selection criteria, based on thresholds applied on the auto- and cross-correlation profiles, though preserving a great number of available codes.
Once derived the expression of the Welch bound specific for each code set, it is possible to compare the sequences' behaviors by evaluating each bound equation for different values of the span n, ranging from 3 to 10. Figure 3 shows the resulting performance, together with the asymptotic curve, corresponding to that holds when M ≫ L. In evaluating the asymptotic curve, we assume.
We favored smaller values of the span when the difference in AIC for a local minimum and a global minimum was not meaningful to identify important features of the disease point process.
