Building an LFSR logical bitmap using characteristic polynomials

Question

How would one build an LFSR logical map, with a characteristic polynomial such as

$$P(x) = x^4 + x^3 + 1$$?

Also, how would you interpret it, and what would you do to do the reverse it in order to create a characteristic polynomial from an LFSR map?

Answer 1

The recipocal $P^*(x)$ of polynomial $$P(x)= a_0 + a_1x + a_2x^2 + \cdots + a_nx^n$$ of degree $n = deg(P(x))$ is given by;

$$P^*(x) = a_n + a_{n-1}x + \cdots + a_0x^n = x^n P(x^{-1})$$

The feedback polynomial defines the tap points of LFSR. The characteristic polynomial and is the reciprocal of feedback. Therefore;

Let $P^*(x)= x^4 \cdot P(1/x) = 1 + x + x^4 $ be the feedback polynomial LFSR calculated by the recipocal of characteristic polynomial $P(x)$. Then the taps of the LFSR as follows;

For further reading in this subject, Golomb's classic book is advised.

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_{The image is produced with the LFSR drawing library.}