Use of this web site signifies your agreement to the terms and conditions. Efficient Composited de Bruijn Sequence Generators Abstract: A binary de Bruijn sequence with period 2 n is a sequence in which every tuple of n bits occurs exactly once. De Bruijn sequence generators have randomness properties that make them attractive for pseudorandom number generators and as building blocks for stream ciphers. Unfortunately, it is very difficult to find de Bruijn sequence generators with long periods e.
OcDeb-k-n efficiently computes a composited de Bruijn sequence where k levels of composition are added to a de Bruijn sequence of period 2 n. Furthermore, it enables efficient parallelization and hardware retiming. Comprehensive result analysis is conducted for 65 nm ASIC technology. In section "Evaluation of binary De Bruijn sequences in DS-CDMA systems," a comparative evaluation of the Welch bound for different families of binary spreading codes will be also presented.
In order to test the performance obtainable by the application of De Bruijn sequences as spreading codes in a classical DS-CDMA system, we assume a gaussian channel affected by multipath that is described by means of either the indoor office test environment and the outdoor to indoor and pedestrian test environment described in [ 15 ]. In both the cases, the so-called Channel A specified by the Recommendation has been considered. Both the channel configurations are simulated by means of a tapped-delay-line model, with different values assigned to relative delay in ns and average power in dB of each path: there are five secondary paths in the indoor test environment, and three secondary paths in the outdoor model.
A detailed description of each model may be found in the related reference. Being maximal period binary sequences, the length of a De Bruijn sequence is always an even number. When comparing the total number of De Bruijn sequences of length L to the total number of available m -sequences, Gold, or Kasami sequences, similar but not identical length values shall be considered, as reported in Table 1.
The table confirms the double exponential growth in the cardinality of De Bruijn sequences, at a parity of the span n , with respect to the other sequences. Of course, not all the De Bruijn sequences of span n may be suitable for application in a multi-user system; anyway, even if strict selection criteria are applied, it is reasonable to expect that a quite extended subset of sequences may be extracted from the entire family. Obviously, at a parity of the chip time, the time duration of each null sample is reduces.
These null values are adjacent to the auto-correlation peak value, and contribute to provide resistance against possible multipath effects. As any binary De Bruijn sequence c comprises the same number of 1's and 0's, when converted into a bipolar form, the following holds:.
So, when n increases, the auto-correlation profiles of the De Bruijn sequences will show many samples equal to 0, a symmetric distribution of the samples, and a reduced number of different positive and negative samples, as to give an average auto-correlation equal to 0. Figure 1 shows the average auto-correlation profile of the set of span 5 De Bruijn sequences that confirms the previous properties.
A simple bound may be defined for the positive values of the correlation functions sidelobes in De Bruijn sequences [ 16 ]:. The left inequality follows from the second and the third properties in 6 ; the right inequality is due to the peculiar features of De Bruijn sequences that are full-length sequences, a period of which includes all the possible binary n -tuples.
All the possible cross-correlation values are integer multiple of 4. Figure 2 shows the average cross-correlation profile of binary De Bruijn sequences of span 5. This also motivates the need for a proper selection criterion to be applied on the whole set of sequences, to extract the most suitable spreading codes to use in the DS-CDMA system. As previously stated in the "Introduction," we can provide a comprehensive evaluation of binary De Bruijn sequences of length 32, i.
For increasing values of n , the brute force generation process becomes unfeasible, and more sophisticated techniques shall be applied [ 13 ]. A useful overview of possible alternative approaches suggested in the literature may be found in [ 17 ]. However, the main limitation of such solutions is related to the reduced number of sequences they allow to obtain by a single generation step. As a consequence, in this article, we opted for a generation strategy that we named "tree approach".
Basically, sequence generation starts with n zeros the all-zero n -tuple shall be always included in a period of a span n De Bruijn sequence and appends a one or a zero, as the next bit of the sequence, thus originating two branches.
As long as the last n -tuple in the partial sequence obtained has not yet appeared before, generation goes on by iterating the process; otherwise the generation path is discarded. This generation scheme that proceeds by parallel branches is fast to execute, and has the advantage of providing the whole set of sequences that we need to perform our correlation-related evaluations. However, the approach suggested suffers for memory limitations, because all the sequences having the same span n must be generated at the same time.
As a consequence, taking into account our focus on the correlation properties of the sequences, we introduce in the generation process a constraint related to cross-correlation: when two generation paths share a common pattern of bits in their initial root, one of them is pruned, in order to reduce a priori the number of sequences that will provide high cross-correlation, due to the presence of common bit patterns. As previously stated, the Welch bound allows to evaluate a family of binary spreading codes in terms of its cross-correlation performance.
The bound is a lower one, as a consequence, by evaluating such bound over different code sets we can draw conclusions about the one providing the worst performance, i. According to this statement, we can compare the Welch bound profile of different sets of spreading codes, namely m -sequences, Gold, OVSF, Kasami, and De Bruijn sequences, at a parity of the span n.
To such an aim, we first compute the expression of the Welch bound for each set of spreading codes, starting from the general definition of Equation 5. However, the orthogonality is ensured in the synchronous case, whereas it is usually lost when OVSF codes are applied asynchronously. In the case of Kasami sequences that are generated from m-sequences as well, we have to distinguish between the so-called small set and the large set of sequences.
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 In evaluating the asymptotic curve, we assume. Welch bound curves for different families of spreading codes. The curves corresponding to Kasami sequences are interpolated for the values of n for which they are not defined, in order to allow an easy comparison with the other curves.
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.
As shown, the asymptotic curve is well approached by the De Bruijn sequences, even for small values of n , thanks to the double exponential growth of M with n. 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. Table 3 provides a description of the statistical properties of the auto-correlation functions for the sequences included in this set; as shown, from the whole family of sequences, two subsets are extracted, corresponding to different thresholds on the maximum absolute value of the auto-correlation sidelobes i.
As expected, all the sequences in any set have an average auto-correlation equal to 0. As a consequence, given the DS-CDMA context of application, it is necessary to avoid the presence of complementary sequences in the set from which spreading codes are chosen.
Table 4 describes the statistical properties of the cross-correlation functions computed over 1, De Bruijn sequences of span 5 that are divided into different subsets by setting different thresholds on the maximum absolute value of the cross-correlation peak. The analysis performed on the cross-correlation properties shows that the two sequences extracted from the half set, for which the cross-correlation absolute peak value is 8, are also the two optimum sequences for auto-correlation.
If we want a limited cross-correlation peak, we must accept higher sidelobes, and viceversa. As a further remark, we may say that high values of the cross-correlation functions i. Results similar to those presented in Table 3 have been derived also for a partial set of De Bruijn sequences of span 6. The generation of span 6 De Bruijn sequences is performed by resorting to the "tree approach" under development. In a first round, the generated paths are pruned every 8 steps; by this way, we limit the generation to a partial set of , sequences.
Among them, we select those sequences for which the maximum absolute value of the auto-correlation sidelobes does not exceed 8, and we obtain sequences. These are further selected into a subset of 15 sequences, for which the maximum cross-correlation equals 24, and into a subset of 34 sequences, for which the maximum cross-correlation equals It is worth noting that even when limiting the subset of sequences to those having a maximum absolute value of the auto-correlation sidelobes equal to 8, we still get different sequences among which we can select the required spreading codes for the DS-CDMA system.
A similar approach is applied to the sequences generated by pruning the partial paths every 6 steps. A smaller set is obtained, including 4, sequences, among which we select sequences having a maximum absolute value of the auto-correlation sidelobes equal to From this subset, we further select 7 sequences with a maximum cross-correlation peak equal to 24, and 18 sequences with a maximum cross-correlation peak of The properties of the sequences obtained are described in Tables 5 and 6.
We computed the average error probability at the output of a correlator receiver of the i th user, in a gaussian channel affected by multipath, according to the Channel A indoor and outdoor-to-indoor test environments specified in [ 15 ]. The performance provided by the adoption of De Bruijn sequences are compared to those obtainable by adopting OVSF sequences in the dowlink section, Gold sequences in the uplink section, and to the ideal behavior of the system no interference.
At the same time, 32 OVSF sequences are generated, and the average performance computed over all the possible subsets of 4 sequences obtainable from the whole set. Simulation results are shown in Figures 4 and 5 , for the indoor and outdoor Channel A test environments , respectively. As a general remark, we may observe that De Bruijn sequences generally perform slightly better than OVSF sequences, thanks to their more favorable autocorrelation profiles, with respect to OVSF codes. In the uplink section of the CDMA system, we compare De Bruijn sequences of length 32 and Gold sequences of length 31, in the case of 2, 3, and 4 active users.
The performance is averaged over all the possible selections of 2, 3, and 4 sequences in the whole set. In a similar way, we also test the performance provided by the set of 33 Gold sequences, by averaging the results obtained by different choices of 4, 3, and 2 spreading codes. Figures 6 and 7 show the estimated behavior, in the indoor and outdoor Channel A test environments , respectively. It is evident that in all the situations considered, Gold codes perform better than De Bruijn ones, even if the differences in the average probability of error are not so significant.
As a final evaluation, we consider span 6 sequences, i. We test their performance in the outdoor test environment only, either in the downlink or in the uplink sections. Similar to the previous test, we compare De Bruijn sequences to Gold codes in the uplink section, and to the OVSF codes in the downlink section, and consider the case of four users active in the system. It is confirmed that Gold codes perform better than De Bruijn ones, even for increased span, whereas De Bruijn sequences are better than OVSF codes in the downlink section.
Average probability of error for users adopting De Bruijn spreading codes of span 6, compared to Gold sequences in the uplink section, and to OVSF codes in the downlink section, in the outdoor test environment. 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.
The performance provided by De Bruijn sequences have been compared to those obtained by more consolidated solutions, relying on the use of m -sequences, Gold, and OVSF sequences as spreading codes. From simulations, it is evident that De Bruijn codes show a rather similar behavior to the code sets traditionally considered, and designed ad hoc to provide good CDMA performance. Consequently, the results discussed in this article encourage further studies and analyses, to extensively test the applicability of De Bruijn sequences in multi-user contexts, even by resorting to longer codes, that, however, require more sophisticated generation techniques.
At the same time, a thorough investigation of the sequences correlation properties is fundamental, to design suitable selection criteria for each specific application scenario. Pursley MB: Performance evaluation for phase-coded spread spectrum multiple-access communication--part I: system analysis. Haykin S: Communication Systems. Wiley, New York; Google Scholar. Proc IEEE , 68 05 Pursley MB, Sarwate DV: Performance evaluation for phase-coded spread spectrum multiple-access communication-part ii: code sequence analysis.
De Bruijn N: A combinatorial problem. Proc Ned Akad Wet , Mayhew GL: Clues to the hidden nature of de Bruijn sequences. Comput Math Appl ,
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Abstract A binary de Bruijn sequence has the property that every n-tuple is distinct on a given period of length 2". An efficient algorithm to generate a class of classi- cal de Bruijn sequences is given based upon the distance between cycles within the Good - de Bruijn digraph. Utilizing this randomness we find additional new structure in de Bruijn sequences. We analyze binary sequences that are not de Bruijn but instead possess the sufficient structure so that every distinct binary n-tuple can be systematically "combed" out of the se- quence.
These complete or nonclassical de Bruijn sequences are a generalization of the well-known de Bruijn cycle. Our investigation focuses on binary sequences, called double Eulerian cycles, that define a cycle along a graph digraph visiting each edge arc exactly twice. A new algorithm to generate a class of double Eulerian cycles on graphs and digraphs is found. Double Eulerian cycles along the binary Good - de Bruijn digraph are partitioned by the run structure of their defining sequences.
This partition allows for a statistical analysis to determine the relative size of the set of complete cycles defined by the sequences we study. A measure that categorizes double Eulerian cycles along graphs digraphs by the distance between the two visitations of each edge arc is provided.
OcDeb-k-n efficiently computes a composited de Bruijn sequence where k levels of composition are added to a de Bruijn sequence of period 2 n. Furthermore, it enables efficient parallelization and hardware retiming.
Comprehensive result analysis is conducted for 65 nm ASIC technology. For example, OcDeb has an area of GE with 1. The area of OcDeb generating a de Bruijn sequence of period 2 is 7, GE and the performance is 1. Article :. Date of Publication: 02 March