# polyphase

Polyphase components of Laurent polynomial

## Syntax

``[E,O] = polyphase(P)``

## Description

example

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

## Examples

collapse all

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 $b\left(z\right)$.

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

## Input Arguments

collapse all

Laurent polynomial, specified as a `laurentPolynomial` object.

## Output Arguments

collapse all

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