# Tukey window function: tukey_window

### 15.2.26 Tukey window function: tukey_window

The tukey_window command finds a Tukey window of a sequence.

- tukey_window takes one mandatory argument and two optional arguments:
  - v, a real vector with length n.
  - Optionally, α, a real number in [0,1] (by default 0.5).
  - Optionally, an interval n0..n1 (by default 0..(n−1)).
- tukey_window(v ⟨,α,n1..n2⟩) returns the elementwise product of [vn1,…,vn2] and the vector w of length N=n2−n1+1 defined by

  wk=

  ⎧
  ⎪
  ⎪
  ⎪
  ⎪
  ⎪
  ⎨
  ⎪
  ⎪
  ⎪
  ⎪
  ⎪
  ⎩

  1
  2

  ⎛
  ⎜
  ⎜
  ⎝

  1+cos

  ⎛
  ⎜
  ⎜
  ⎝

  π

  ⎛
  ⎜
  ⎜
  ⎝

  k
  β

  −1

  ⎞
  ⎟
  ⎟
  ⎠

  ⎞
  ⎟
  ⎟
  ⎠

  ⎞
  ⎟
  ⎟
  ⎠

  ,

  k<β,

  1,

  β≤ k≤(N−1)

  ⎛
  ⎜
  ⎜
  ⎝

  1−

  α
  2

  ⎞
  ⎟
  ⎟
  ⎠

  ,

  1
  2

  ⎛
  ⎜
  ⎜
  ⎝

  1+cos

  ⎛
  ⎜
  ⎜
  ⎝

  π

  ⎛
  ⎜
  ⎜
  ⎝

  k
  β

  −

  2
  α

  +1

  ⎞
  ⎟
  ⎟
  ⎠

  ⎞
  ⎟
  ⎟
  ⎠

  ⎞
  ⎟
  ⎟
  ⎠

  ,

  otherwise,

  where β=α (N−1)/2, for k=0,1,…,N−1.

#### Example.

Input:

L0:=randvector(1000,0..1):;
scatterplot(L0);

Output:

Input:

L:=tukey_window(L0,0.4):;
scatterplot(L);

Output: