### 21.5.14 Tukey window

The tukey_window command applies the Tukey window function to a sequence or its segment.

- tukey_window takes one mandatory argument and up to three optional arguments:
  - v, a real vector with length n.
  - Optionally, one of:
    - α, a real number in [0,1] (by default, α=0.5).
    - m,N⟨,α⟩, a sequence of two integers and optionally a real number α∈[0,1] (by default, m=0 and N=n).
- tukey_window returns the elementwise product of [vm,vm+1,…,vm+N−1] and the vector w of length N 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

listplot(tukey_window([1$1000],0.4))