# Discrete-time Fourier transform of a window function

Used in ECE438.

This page can be used to study the frequency-domain behavior of a discrete-time window function, as its length increases.

Consider the perfect discrete-time window function

$ w[n]= \left\{ \begin{array}{ll} 1,&\text{ if }0 \leq n < N \\ 0, & \text{ else}. \end{array} \right., $

for any integer values of n. Note that N represents the length of the window. The DTFT of this window function is

$ W(\omega) =\frac{e^{\frac{-j \omega (N-1)}{2}} \sin\left( \frac{\omega N}{2}\right)}{\sin \left( \frac{\omega}{2} \right)} $.

Below is the graph of the magniture of $ W(\omega) $ for $ N=15 $.

Below is the graph of the magniture of $ W(\omega) $ for $ N=100 $. Observe that the ripples are "thinner" and more numerous than in the previous case of $ N=10 $.

Below is the graph of the magniture of $ W(\omega) $ for $ N=10000 $.

Observe the close resemblance of this graph to that of the magnitude of the Fourier transform of the signal $ x[n]=1 $, for any n integer (in other words, an "infinite-length" window).