Pseudorandom binary sequence

A binary sequence (BS) is a sequence of bits, i.e.

for .

A BS consists of ones and zeros.

A BS is a pseudorandom binary sequence (PRBS) if its autocorrelation function, given by

has only two values:

where

is called the duty cycle of the PRBS, similar to the duty cycle of a continuous time signal. For a maximum length sequence, where , the duty cycle is 1/2.

A PRBS is 'pseudorandom', because, although it is in fact deterministic, it seems to be random in a sense that the value of an element is independent of the values of any of the other elements, similar to real random sequences.

A PRBS can be stretched to infinity by repeating it after elements, but it will then be cyclical and thus non-random. In contrast, truly random sequence sources, such as sequences generated by radioactive decay or by white noise, are infinite (no pre-determined end or cycle-period). However, as a result of this predictability, PRBS signals can be used as reproducible patterns (for example, signals used in testing telecommunications signal paths).

Pseudorandom binary sequences can be generated using linear-feedback shift registers.

Some common sequence generating monic polynomials are

PRBS7 =

PRBS9 =

PRBS11 =

PRBS13 =

PRBS15 =

PRBS20 =

PRBS23 =

PRBS31 =

An example of generating a "PRBS-7" sequence can be expressed in C as

#include <stdio.h> #include <stdint.h> #include <stdlib.h> int main(int argc, char* argv[]) { uint8_t start = 0x02; uint8_t a = start; int i; for (i = 1;; i++) { int newbit = (((a >> 6) ^ (a >> 5)) & 1); a = ((a << 1) | newbit) & 0x7f; printf("%x\n", a); if (a == start) { printf("repetition period is %d\n", i); break; } } }

In this particular case, "PRBS-7" has a repetition period of 127 values.

The PRBSk or PRBS-k notation (such as "PRBS7" or "PRBS-7") gives an indication of the size of the sequence. is the maximum number of bits that are in the sequence. The k indicates the size of a unique word of data in the sequence. If you segment the N bits of data into every possible word of length k, you will be able to list every possible combination of 0s and 1s for a k-bit binary word, with the exception of the all-0s word. For example, PRBS3 = "1011100" could be generated from . If you take every sequential group of three bit words in the PRBS3 sequence (wrapping around to the beginning for the last few three-bit words), you will find the following 7 word arrangements:

"1011100" → 101 "1011100" → 011 "1011100" → 111 "1011100" → 110 "1011100" → 100 "1011100" → 001 (requires wrap) "1011100" → 010 (requires wrap)

Those 7 words are all of the possible non-zero 3-bit binary words, not in numeric order. The same holds true for any PRBSk, not just PRBS3.