# 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).

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