Feedback Shift Register Sequences

Feedback Shift Register (FSR) sequences have been successfully implemented in many communication systems for their randomness properties and ease of implementation. These include ranging and navigation systems, spread spectrum communication systems, CDMA mobile communication systems, and crypto systems such as streamciphers. This article gives a brief overview of FSR sequences, both linear and non-linear. Two conditions on the connection logic of FSRs for better output sequences are described, which are the branchless condition and the balanced logic condition. We use mostly the state transition diagram of an FSR to describe the property of its output sequences. For linear FSR sequences, we describe the relation between the connection polynomials and the structure of the cycle decomposition in the state diagram, and hence the periodicity of the output sequences. Various randomness properties of the maximal length linear FSR sequences, known as m-sequences, are described: balance, run-distribution, span, ideal autocorrelation, constant-on-the-coset, and cycle-and-add properties. Two properties, the ideal autocorrelation function and the smallest linear complexity, of m-sequences are described in detail. Finally, a complete analysis of 4-stage FSRs is provided.