# polyphase

Polyphase components of Laurent polynomial

## Description

example

[E,O] = polyphase(P) returns the even part E and odd part O of the Laurent polynomial P.

## Examples

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Create the Laurent polynomial $b\left(z\right)={z}^{3}+3{z}^{2}-1+2{z}^{-1}$.

b = laurentPolynomial(Coefficients=[1 3 0 -1 0 2],MaxOrder=3);

Use the polyphase function to obtain the even and odd parts of $b\left(z\right)$. Use the helper function helperPrintLaurent to print the Laurent polynomials in algebraic form.

[evenP,oddP] = polyphase(b);
resE = helperPrintLaurent(evenP);
disp(resE)
3*z - 1 + 2*z^(-1)

resO = helperPrintLaurent(oddP);
disp(resO)
z^(2)

Confirm the identity $E\left({z}^{2}\right)+{z}^{-1}O\left({z}^{2}\right)==b\left(z\right)$, where $E\left(z\right)$ and $O\left(z\right)$ are the even and odd parts, respectively, of $b\left(z\right)$.

lpz = laurentPolynomial(Coefficients=1,MaxOrder=-1);
leftSide = evenPz2+(lpz*oddPz2);
areEqual = (leftSide == b)
areEqual = logical
1

## Input Arguments

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Laurent polynomial, specified as a laurentPolynomial object.

## Output Arguments

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Even part of the Laurent polynomial P, returned as a laurentPolynomial object. The polynomial E is such that:

E(z2) = [P(z) + P(-z)]/2.

Odd part of the Laurent polynomial P, returned as a laurentPolynomial object. The polynomial O is such that:

O(z2) = [P(z) - P(-z)]/ [ 2 z-1].

## Version History

Introduced in R2021b