Determining Linear Shift Register Generating Polynomials Using Fourier Transform

Using Fourier Transform to determine linear shift register generating polynomials represents a widely adopted methodology in digital signal processing. Fourier Transform serves as a fundamental mathematical tool that decomposes functions or signals into sums of sinusoidal components (sine and cosine waves), facilitating comprehensive signal analysis and manipulation. Linear shift registers are electronic circuits designed for binary data storage and sequential shifting operations, while generating polynomials define the characteristic equations governing output sequence generation. The Fourier Transform approach enables polynomial determination by converting time-domain register sequences into frequency-domain representations, where polynomial roots correspond to specific spectral components. From an implementation perspective, this typically involves applying Fast Fourier Transform (FFT) algorithms to captured output sequences, followed by root-finding procedures to identify polynomial coefficients. This method provides significant advantages for circuit analysis and optimization, allowing engineers to characterize register behavior through spectral decomposition and verify polynomial properties using frequency-domain insights. The technique proves particularly valuable for testing maximal-length sequences and analyzing register performance in communication systems and cryptography applications.