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.

Live circuit register value 8

Maximal-Length LFSR

Change an input; step the timeline to watch the logic settle.

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

Notebook: 7 parts and 11 wires, modelled as clocked shift register over GF(2)

Recommended lab 10 of 18

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

Parts

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

Limits of the model

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

Diagnosis challenge The fault is hidden. Work it out from the changed outputs first.

Notebook: 7 parts and 11 wires, modelled as clocked shift register over GF(2)

Move one feedback tap away from the primitive polynomial. Must the period remain 151515?

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