This is not possible with truly random events such as nuclear decay. If you know the present state, you can predict the next state. This same property is statistically expected in a truly random sequence. Exactly 1/2 of these runs will be one bit long, 1/4 will be two bits long, up to a single run of zeroes n-1 bits long, and a single run of ones n bits long. For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is::: x^ runs occur (for example, a six bit LFSR will have 32 runs). This is called the feedback polynomial or characteristic polynomial. This means that the coefficients of the polynomial must be 1's or 0's. The tap sequence of an LFSR can be represented as a polynomial mod 2. The sequence of numbers generated by an LFSR can be considered a binary numeral system just as valid as Gray code or the natural binary code. cycles through all possible 2^n-1 states within the shift register except the state where all bits are zero), unless it contains all zeros, in which case it will never change. * A maximal LFSR produces an m-sequence (i.e. * The outputs that influence the input are called 'taps' (white in the diagram). The taps are XOR'd sequentially with the output and then fed back into the leftmost bit. The list of the bits' positions that affect the next state is called the tap sequence. Both hardware and software implementations of LFSRs are common. However, an LFSR with a well-chosen feedback function can produce a sequence of bits which appears random and which has a very long cycle.Īpplications of LFSRs include generating pseudo-random numbers, pseudo-noise sequences, fast digital counters, and whitening sequences. Likewise, because the register has a finite number of possible states, it must eventually enter a repeating cycle. The initial value of the LFSR is called the seed, and because the operation of the register is deterministic, the sequence of values produced by the register is completely determined by its current (or previous) state. The only linear functions of single bits are xor and inverse-xor thus it is a shift register whose input bit is driven by the exclusive-or (xor) of some bits of the overall shift register value. The maximal sequence consists of every possible state except the '0000' state.]Ī linear feedback shift register (LFSR) is a shift register whose input bit is a linear function of its previous state.

Xor gate provides feedback to the register that shifts bits from left to right.

Linear feedback shift register