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# tukeywin

Tukey (tapered cosine) window

## Syntax

w = tukeywin(L,r)

## Description

w = tukeywin(L,r) returns an L-point Tukey window in the column vector, w. A Tukey window is a rectangular window with the first and last r/2 percent of the samples equal to parts of a cosine. See Definitions for the equation that defines the Tukey window. r is a real number between 0 and 1. If you input r ≤ 0, you obtain a rectwin window. If you input r ≥ 1, you obtain a hann window. r defaults to 0.5.

## Examples

expand all

### Tukey Windows

Compute 128-point Tukey windows with five different values of r, or "tapers." Display the results using wvtool.

```L = 128;
t0 = tukeywin(L,0);        % Equivalent to a rectangular window
t25 = tukeywin(L,0.25);
t5 = tukeywin(L);          % r = 0.5
t75 = tukeywin(L,0.75);
t1 = tukeywin(L,1);        % Equivalent to a Hann window
wvtool(t0,t25,t5,t75,t1)
```

## Definitions

The following equation defines the L–point Tukey window:

$w\left(x\right)=\left\{\begin{array}{ll}\frac{1}{2}\left\{1+\mathrm{cos}\left(\frac{2\pi }{r}\left[x-r/2\right]\right)\right\}\hfill & 0\le x<\frac{r}{2}\hfill \\ 1\hfill & \frac{r}{2}\le x<1-\frac{r}{2}\hfill \\ \frac{1}{2}\left\{1+\mathrm{cos}\left(\frac{2\pi }{r}\left[x-1+r/2\right]\right)\right\}\hfill & 1-\frac{r}{2}\le x\le 1\hfill \end{array}$

where x is an L-point linearly spaced vector generated using linspace. The parameter r is the ratio of cosine-tapered section length to the entire window length with 0 ≤ r ≤ 1. For example, setting r = 0.5 produces a Tukey window where 1/2 of the entire window length consists of segments of a phase-shifted cosine with period 2r = 1. If you specify r ≤ 0, an L-point rectangular window is returned. If you specify r ≥ 1, an L-point von Hann window is returned.

## References

[1] Bloomfield, P. Fourier Analysis of Time Series: An Introduction. New York: Wiley-Interscience, 2000.