Window function: Difference between revisions

Revision as of 05:41, 23 November 2024

Revision as of 05:44, 23 November 2024

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=== ''B''-spline windows ===

''B''-spline windows can be obtained as ''k''-fold convolutions of the rectangular window. They include the rectangular window itself (''k''&nbsp;=&nbsp;1), the {{slink|#Triangular window}} (''k''&nbsp;=&nbsp;2) and the {{slink|#Parzen window}} (''k''&nbsp;=&nbsp;4).<ref name=Toraichi89/> Alternative definitions sample the appropriate normalized [[B-spline|''B''-spline]] [[basis function]]s instead of convolving discrete-time windows. A ''k''^th-order ''B''-spline basis function is a piece-wise polynomial function of degree ''k''−1&nbsp;−&nbsp;1 that is obtained by ''k''-fold self-convolution of the [[rectangular function]].

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[[File:Window function and its Fourier transform – Triangular (n = 0...N).svg|thumb|480px|right|Triangular window (with ''L''&nbsp;=&nbsp;''N''&nbsp;+&nbsp;1)]]

Triangular windows are given by:

:<math>w[n] = 1 - \left|\frac{n - \frac{N}{2}}{\frac{L}{2}}\right|,\quad 0\le n \le N,</math>

where ''L'' can be ''N'',<ref name=Bartlett/> ''N''&nbsp;+&nbsp;1,<ref name=Harris/><ref name=Tukey/><ref name=MWtriang/> or ''N''&nbsp;+&nbsp;2.<ref name=Welch1967/> The first one is also known as '''[[M. S. Bartlett|Bartlett]] window''' or '''[[Lipót Fejér|Fejér]] window'''. All three definitions converge at large&nbsp;''N''.

<ref name=Tukey/><ref name=MWtriang/> or ''N''&nbsp;+&nbsp;2.<ref name=Welch1967/> The first one is also known as '''[[M. S. Bartlett|Bartlett]] window''' or '''[[Lipót Fejér|Fejér]] window'''. All three definitions converge at large&nbsp;''N''.

The triangular window is the 2{{Superscript|nd}}2nd-order ''B''-spline window. The ''L''&nbsp;=&nbsp;''N'' form can be seen as the convolution of two {{Fraction|N|2}}-width rectangular windows. The Fourier transform of the result is the squared values of the transform of the half-width rectangular window.

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{{Distinguish|Kernel density estimation}}

Defining {{math|''L'' ≜ ''N'' + 1}}, the Parzen window, also known as the '''de la Vallée Poussin window''',<ref name=Harris/> is the 4{{Sup|th}}4th-order ''B''-spline window given by:

:<math>