L = 2N – 1
Feedback taps configuration of the LFSR can be represented in finite field arithmetic as a polynomial modulo 2 . Well-known LFRS polynomials can be found in many publications . For example, PBRS9 has two taps at 9th and 5th can be represented as the following polynomial:
x9+ x5 + 1
A feedback polynomial can be configured in either a Fibonacci or Galois style , as show in Figure 1and Figure 2, respectively.
Figure 1: PBRS9 with Fibonacci configuration
Figure 2: PBRS9 with Galois configuration
Using PBRS in bit errors measurement
Figure 3 shows a simple digital communication chain with PBRS9 as a random bit generator with a maximum length of 511 bits. Initial state of the LFSR of the transmit PBRS9 can be an arbitrary value but must be different from zero.
At the receiver, once the first bit is detected and demodulated by the demodulator, the receive PBRS9 generator uses the first 9 bits as an initial state for its LFSR. As a result, the receive PBRS9 generates the same sequence delayed by 9 bits to that generated by the transmitter. An alignment of the demodulated bits from the demodulator and the receive PRRS9 generator is required to perform a measurement of error bits.
Figure 3: BER measurement using PBRS9
In order to avoid an invalid initial state of the receive PBRS9, an additional mechanism is required to detect burst errors between the demodulated bits and the PBRS9 generated bits. If there are too many consecutive bits corrupted by the channel or bad synchronization, the receive PBRS9 is required to re-initialize its initial state before generating a new bit sequence.
This blog post showed a simple method to measure BER in digital communication systems by using PBRS sequences. By using demodulated bits at the receiver as the initial state of the LFSR inside the PBRS generator, the receive PBRS can generate a delayed copy version of the PBRS sequence in the transmitter for BER measurement. In upcoming blog posts, I will demonstrate how to implement a PBRS generator for BER measurement of a simple digital communication chain using Xilinx System Generator  in conjunction with Nutaq’s Model-Based Design Kit (MBDK) .