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Maximal-Length LFSR

Sequential digital

A 4-bit Fibonacci linear-feedback shift register with taps at stages 3 and 4 — the primitive polynomial $x^4+x^3+1$ . Each clock edge shifts the register and feeds back the XOR of the tapped bits. The result cycles through all 15 non-zero states before repeating: a deterministic sequence wearing the disguise of randomness.

7 components · 11 connections · clocked shift register over GF(2)

recommended lab 10 of 11

Connect primitive feedback taps to the period of an LFSR sequence.

Builds on: 8-Bit Digital Adder

predict · then test

Move one feedback tap away from the primitive polynomial. Must the period remain $15$ ?

Commit before the explanation appears. This is a prediction, not a quiz score.

governing model

x^4 + x^3 + 1 \;\cdot\; \text{period} = 2^n - 1 = 15

structural inventory

shared clock
$4\times$ D flip-flop chain
XOR feedback taps $3,4$
serial m-sequence output

boundaries

period $2^4-1=15$
all-zero state forbidden
primitive polynomial taps

diagnosis challenge Fault identity withheld: infer it from the changed outputs first.

causal signal trace paused

Form the feedback bit Before the edge, the selected Q3 and Q4 taps XOR to establish the next D input.

t_{\mathrm{stage}}=\htmlData{math-slot=value}{\mathtt{0.00}}\,\mathrm{s}

step 1 / 4 Form the feedback bit

control console

seed (1–15)

inject hidden fault

test bed · mode=full

Maximal-Length LFSR

instrumentation rack

sequence period

output bit

scope

Primitive taps — all 15 non-zero states before repeating.