LFSR / PRBS — Overview, Families & Architecture

Overview

Linear Feedback Shift Registers (LFSRs) and Pseudo‑Random Binary Sequences (PRBS) are compact, deterministic generators of pseudo‑random sequences with excellent statistical properties. They are used in test pattern generation, scramblers, whitening engines, BIST, spread‑spectrum systems, radar, correlators, and high‑speed serial link validation.
Their behavior is defined by polynomials over GF(2), making them mathematically related to CRCs but optimized for sequence generation.

1. Why LFSRs and PRBS Matter

test and validation of SERDES, PHYs, and datapaths
reproducible patterns with controlled statistics
scramblers and whitening engines
BER measurement and stress testing
BIST and pseudo‑random pattern generation
extremely low hardware cost
high speed (single XOR depth in Galois form)

2. PRBS Families

PRBS sequences use primitive polynomials to achieve maximal length 2^N-1.

Common families:

PRBS7 — low‑speed links
PRBS9 — legacy telecom
PRBS11 — optical/RF
PRBS15 — general‑purpose testing
PRBS23 — telecom, DSSS
PRBS31 — high‑speed SERDES (10G/25G/40G/100G)

Each family is defined by:

polynomial degree
tap positions
sequence length
statistical properties

PRBS31 is the de‑facto standard for stressing high‑speed CDRs.

3. Primitive Polynomials and Maximal Length

An LFSR is maximal‑length when its polynomial is primitive over GF(2).
Examples:

PRBS7: x^7+x^6+1
PRBS15: x^{15}+x^{14}+1
PRBS23: x^{23}+x^{18}+1
PRBS31: x^{31}+x^{28}+1

The all‑zero state is forbidden.

4. Fibonacci and Galois Architectures

Fibonacci LFSR

feedback XOR applied at the input
direct mapping from polynomial
higher XOR depth for large polynomials

Galois LFSR

feedback distributed across taps
lower XOR depth
better timing closure
preferred for high‑speed PRBS31 and scramblers

Both forms generate the same sequence when configured correctly.

5. Parallel LFSR / PRBS Generators

Parallel LFSRs unroll the recurrence to generate multiple bits per cycle.
Used in:

wide datapaths (16/32/64/128 bit)
scramblers
high‑speed serializers
protocol‑specific randomizers

Parallelization uses the same GF(2) matrix derivation as parallel CRCs.

6. Sequence Properties

Maximal‑length LFSRs exhibit:

uniform distribution of 0s and 1s
two‑level autocorrelation
flat spectrum except for DC notch
deterministic repeatability
sensitivity to initial seed

Ideal for stressing communication channels and validating signal integrity.

7. RTL Architecture

Typical signals:

lfsr_reg[N-1:0] — internal state
seed — non‑zero initialization
enable — update control
prbs_out — output bit or word

Implementation notes:

never allow all‑zero state
seed must be non‑zero
Galois preferred for timing
output can be MSB, LSB, or any tap

8. Primitive Polynomial Reference Table

PRBS Type	Polynomial (hex)	Expanded Polynomial	Length
PRBS7	0x48	x^7+x^6+1	2^7-1=127
PRBS15	0x6000	x^{15}+x^{14}+1	2^{15}-1=32767
PRBS23	0x00400020	x^{23}+x^{18}+1	2^{23}-1
PRBS31	0x10000008	x^{31}+x^{28}+1	2^{31}-1

9. Applications

BERTs and serial link testing
scramblers and descramblers
spread‑spectrum systems
radar and correlators
memory/storage randomization
BIST
pseudo‑random pattern generation

10. Use in TRNG Conditioning

LFSRs are deterministic and cannot generate entropy, but they are widely used as whitening and decorrelation stages in TRNGs.
They remove bias and spread entropy across bits after physical noise extraction.

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