The window is a weight function that is used to control the effects caused by the presence of side lobes in the spectral estimates (spreading of the spectrum). It is convenient to consider the available final data record or the existing final correlation sequence as some part of the corresponding infinite sequence, visible through the applied window. For example, the sequence of observed data of Mathematics can be written as a product of a rectangular function of unit amplitude
![{\ displaystyle x_ {0} [n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e058f1c5c690ed85590a55c093256c2af99c2948)
![{\ displaystyle \ operatorname {rect} [n] = \ left \ {{\ begin {array} {* {40} l} 1, & 0 \ leq n \ leq N-1; \\ 0, \\\ end { array}} \ right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eaf4c608f5e34e1232da9229c7e36e41f4ac6b1)
and endless sequence
![x [n]](https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d)
![{\ displaystyle x_ {0} [n] = x [n] \ cdot \ operatorname {rect} [n].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16ea9503f59991c405b3b44e4458e8517a25bc58)
In this case, an obvious assumption is made that all unobservable readings are zero, regardless of whether this is in fact true or not. Discrete Fourier transform (DFT) weighted by a sequence window, expressed through a sequence transformation and rectangular window , equal to convolution of these transformations
![{\ displaystyle \ operatorname {rect} [n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5983f1917f8dfa133e4b905554d406719e8a5e0b)
Where is a discrete sinc function , or Dirichlet kernel , which is a DFT of a rectangular function.
