# nuttallwin

Nuttall-defined minimum 4-term Blackman-Harris window

## Syntax

``w = nuttallwin(L)``
``w = nuttallwin(L,sflag)``

## Description

example

````w = nuttallwin(L)` returns a Nuttall defined `L`-point, four-term symmetric Blackman-Harris window. The coefficients for this window differ from the Blackman-Harris window coefficients computed with `blackmanharris` and produce slightly lower sidelobes.```
````w = nuttallwin(L,sflag)` uses `sflag` window sampling.```

## Examples

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Compare 64-point Nuttall and Blackman-Harris windows. Plot them using `wvtool`.

```L = 64; w = blackmanharris(L); y = nuttallwin(L); wvtool(w,y)```

Compute the maximum difference between the two windows.

`max(abs(y-w))`
```ans = 0.0099 ```

## Input Arguments

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Window length, specified as a real positive scalar.

Data Types: `single` | `double` | `int8` | `int16` | `int32` | `int64` | `uint8` | `uint16` | `uint32` | `uint64`

Window sampling, specified as `"symmetric"` or `"periodic"`. For the equations that define the symmetric and periodic windows, see Algorithms.

## Output Arguments

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Nuttall defined Blackman-Harris window, returned as a column vector of length `L`. The function minimizes the maximum sidelobes of the window.

## Algorithms

The equation for the symmetric Nuttall defined four-term Blackman-Harris window is

`$w\left(n\right)={a}_{0}-{a}_{1}\mathrm{cos}\left(2\pi \frac{n}{N-1}\right)+{a}_{2}\mathrm{cos}\left(4\pi \frac{n}{N-1}\right)-{a}_{3}\mathrm{cos}\left(6\pi \frac{n}{N-1}\right)$`

where n= 0,1,2, ... N-1.

The equation for the periodic Nuttall defined four-term Blackman-Harris window is

`$w\left(n\right)={a}_{0}-{a}_{1}\mathrm{cos}\left(2\pi \frac{n}{N}\right)+{a}_{2}\mathrm{cos}\left(4\pi \frac{n}{N}\right)-{a}_{3}\mathrm{cos}\left(6\pi \frac{n}{N}\right)$`

where n= 0,1,2, ... N-1. The periodic window is N-periodic.

The coefficients for this window are

a0 = 0.3635819

a1 = 0.4891775

a2 = 0.1365995

a3 = 0.0106411

## References

[1] Nuttall, Albert H. “Some Windows with Very Good Sidelobe Behavior.” IEEE® Transactions on Acoustics, Speech, and Signal Processing. Vol. ASSP-29, February 1981, pp. 84–91.

## Version History

Introduced before R2006a