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# nuttallwin

Nuttall-defined minimum 4-term Blackman-Harris window

## Syntax

w = nuttallwin(N)
w = nuttalwin(N,SFLAG)

## Description

w = nuttallwin(N) returns a Nuttall defined N-point, 4-term symmetric Blackman-Harris window in the column vector w. The window is minimum in the sense that its maximum sidelobes are minimized. The coefficients for this window differ from the Blackman-Harris window coefficients computed with blackmanharris and produce slightly lower sidelobes.

w = nuttalwin(N,SFLAG) uses SFLAG window sampling. SFLAG can be 'symmetric' or 'periodic'. The default is 'symmetric'. You can find the equations defining the symmetric and periodic windows in Definitions.

## Examples

expand all

### Nuttall and Blackman-Harris Windows

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

```

## Definitions

The equation for the symmetric Nuttall defined 4-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 4-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 = .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.