sdr.preferred_pairs

sdr.preferred_pairs(degree: int, poly: PolyLike | None = None) → Iterator[tuple[Poly, Poly]]

Generates primitive polynomials of degree $m$ that produce preferred pair $m$ -sequences.

Parameters:¶

degree: int¶: The degree $m$ of the $m$ -sequences.
poly: PolyLike | None = None¶: The first polynomial $f (x)$ in the preferred pair. If None, all primitive polynomials of degree $m$ that yield preferred pair $m$ -sequences are returned.

Returns:¶

An iterator of primitive polynomials $(f (x), g (x))$ of degree $m$ that produce preferred pair $m$ -sequences.

Notes¶

A preferred pair of primitive polynomials of degree $m$ are two polynomials $f (x)$ and $g (x)$ such that the periodic cross-correlation of the two $m$ -sequences generated by $f (x)$ and $g (x)$ have only the values in ${- 1, - t (m), t (m) - 2}$ .

\begin{array}{r} t (m) = {\begin{cases} 2^{(m + 1) / 2} + 1 & if m is odd \\ 2^{(m + 2) / 2} + 1 & if m is even \end{cases} \end{array}

There are no preferred pairs for degrees divisible by 4.

References¶

John Proakis, Digital Communications, Chapter 12.2-5: Generation of PN Sequences.

Examples¶

Generate one preferred pair with $f (x) = x^{5} + x^{3} + 1$ .

In [1]: next(sdr.preferred_pairs(5, poly="x^5 + x^3 + 1"))
Out[1]: (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^3 + x^2 + x + 1, GF(2)))

Generate all preferred pairs with $f (x) = x^{5} + x^{3} + 1$ .

In [2]: list(sdr.preferred_pairs(5, poly="x^5 + x^3 + 1"))
Out[2]: 
[(Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^3 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2)))]

Generate all preferred pairs with degree 5.

In [3]: list(sdr.preferred_pairs(5))
Out[3]: 
[(Poly(x^5 + x^2 + 1, GF(2)), Poly(x^5 + x^3 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^2 + 1, GF(2)), Poly(x^5 + x^4 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^2 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x + 1, GF(2))),
 (Poly(x^5 + x^2 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^3 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2))),
 (Poly(x^5 + x^3 + x^2 + x + 1, GF(2)), Poly(x^5 + x^4 + x^2 + x + 1, GF(2))),
 (Poly(x^5 + x^3 + x^2 + x + 1, GF(2)), Poly(x^5 + x^4 + x^3 + x + 1, GF(2))),
 (Poly(x^5 + x^4 + x^2 + x + 1, GF(2)),
  Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2))),
 (Poly(x^5 + x^4 + x^3 + x + 1, GF(2)),
  Poly(x^5 + x^4 + x^3 + x^2 + 1, GF(2)))]

Note, there are no preferred pairs for degrees divisible by 4.

In [4]: list(sdr.preferred_pairs(4))
Out[4]: []

In [5]: list(sdr.preferred_pairs(8))
Out[5]: []