Shapiro Polynomials

In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are:

$\begin{align} P_1(x) & {} =1 + x \\ P_2(x) & {} =1 + x + x^2 - x^3 \\ P_3(x) & {} =1 + x + x^2 - x^3 + x^4 + x^5 - x^6 + x^7 \\ ... \\ Q_1(x) & {} =1 - x \\ Q_2(x) & {} =1 + x - x^2 + x^3 \\ Q_3(x) & {} =1 + x + x^2 - x^3 - x^4 - x^5 + x^6 - x^7 \\ ... \\ \end{align}$

where the second sequence, indicated by Q, is said to be complementary to the first sequence, indicated by P.

Read more about Shapiro Polynomials: Construction, Properties, See Also

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