Impact of truncation on the statistical properties of LFSR sequences Marco Baldi, Franco Chiaraluce Universit` a Politecnica delle Marche, Ancona, Italy {m.baldi, f.chiaraluce}@univpm.it Noureddine Boujnah, Roberto Garello Politecnico di Torino, Italy {noureddine.boujnah, roberto.garello}@polito.it Abstract—This paper investigates some theoretical issues re- lated with the truncation of maximum length sequences. It is shown that truncation can have a significant impact on the auto- correlation properties, mining the applicability of these sequences in practical applications. First and second order statistics for the autocorrelation function are considered, and some new relations are presented that simplify computation. As an example of practical impairment, we focus on space communication links and we consider the changes that occur in the transmitted waveform power spectral density when truncated sequences are used for data randomization. Index Terms—Maximum-length sequences, Linear Feedback Shift Registers, autocorrelation function, truncation noise. I. I NTRODUCTION Linear feedback shift registers (LFSRs) are widely used for generating maximum-length sequences (in short, m-sequences) with very good correlation properties. It is known that, through a suitable choice of the feedback coefficients [1], an LFSR with L cells produces a sequence with length N =2 L − 1 with out-of-phase autocorrelation values R(τ )= −1 for any non-zero shift 1 ≤ τ ≤ N − 1. Taking into account that the maximum autocorrelation value is R(0) = N , the behavior of these sequences, for large N , is very similar to those of ideal sequences (R(τ )=0 for any non-zero shift). Since a long time, m-sequences have been applied in several telecommunication applications, e.g., for rapid acquisition in radar ranging systems or as spreading sequences in Code Di- vision Multiple Access (CDMA) systems. As another relevant application, they can be used to increase the randomness of transmitted data, in such a way to approach the power spectrum of ideal uncorrelated data. As an example, this target is very important in space missions, where the power flux density at the Earth’s surface must not exceed prefixed values, for all operation conditions and methods of modulation [2], [3]. However, the nearly-optimal correlation properties of m- sequences are strictly dependent on the sequence length. In fact, by a mechanism that recalls prime numbers theory, just cutting out one bit in the designed length N can significantly modify the correlation properties. Clear examples of this behavior will be given in the following. On the other hand, the need to shorten the m-sequence appears many times in practice. In synchronization problems, for example, in order to decrease the acquisition time and still preserve the advantage of a long m-sequence, one can use a relatively short segment of the long m-sequence as the correlator reference signal. Even more commonly, the m- sequence could be shortened to match the data field size in frame structures wherein the sequence must be embedded [3], [4]. The resulting sequences, also called truncated m- sequences in the sequel, are segments of m-sequences, where the first (or, equivalently, the last) M bits have been cut. As we will show, pruning causes the appearance of a truncation noise in the autocorrelation function. In this paper, we focus on the properties of truncated m- sequences, with special emphasis on the mean and variance of their autocorrelation function. It is interesting to note that this topic was widely investigated in the 70’s (see [5] and [6] for example), yielding a number of theoretical results that were also the starting point of our study. In spite of its importance, however, the problem was nearly abandoned in the subsequent years, though many issues are still open and deserve further deepening. Moving from these premises, we present some new theoretical results and numerical examples that permit to disclose more in depth the autocorrelation properties of truncated m-sequences. These results will be applied to investigate the sub-optimality of communication systems employing truncated m-sequences. The organization of the paper is as follows. In Section II we remind the definition of m-sequences and the properties of their autocorrelation function. In Section III we introduce truncated m-sequences and the statistical issues for studying their truncation noise. In Section IV we derive a number of recurrence relations for computing their autocorrelation function. In Section V we discuss the impact of truncation on transmitted power spectral density, with focus on real space links. Section VI concludes the paper. II. MAXIMUM- LENGTH SEQUENCES A Linear Feedback Shift Register (LFSR) is shown in Fig.1 and consists of: • L binary cells (binary flip-flops, that can contain 0 or 1) s(1),s(2), ...s(L) ∈ GF (2) (Galois Field of size 2); • L potential feedback connections c(1),c(2), ...c(L) ∈ GF (2). At time t =0 the LFSR contains the starting non-zero seed: s 0 (1),s 0 (2), ...s 0 (L). The i-th connection exists if c(i)=1 and does not exist if c(i)=0: they define the coefficients of the characteristic polynomial g(D)= D L + ∑ L i=1 c(i)D L−i