Computation of Autocorrelation Coefficients for m-Sequences Defined by Characteristic Polynomials

To obtain a comprehensive understanding of m-sequences utilized in the context of characteristic polynomials, analyzing their autocorrelation coefficients is essential. These coefficients quantify the self-similarity of the sequence over different shifts, which is crucial for evaluating properties like randomness, periodicity, and applicability in communication systems, cryptography, and signal processing. In code implementations, autocorrelation coefficients for an m-sequence can be computed by comparing the sequence with its shifted versions. For a binary m-sequence of length N, the autocorrelation function R(τ) at lag τ is typically calculated as: R(τ) = (Number of agreements - Number of disagreements) / N, where agreements and disagreements are counted between the original sequence and its τ-shifted version. The autocorrelation function of an m-sequence exhibits a distinct peak at zero lag and low values for all non-zero lags, a property that makes m-sequences ideal for synchronization and spreading codes. By analyzing these coefficients, one can evaluate the sequence’s repetition characteristics, noise immunity, and suitability for applications such as CDMA systems, radar signal design, and error detection. Understanding the autocorrelation behavior thus provides deeper insights into both the theoretical properties and practical utility of m-sequences derived from characteristic polynomials.