Once I was asked, “Do you know the peak side-lobe ratio of the frequency response of the rectangular window?”
It was embarrassment for me to reply, “No, I don’t remember.” I was not smart!
A rectangular window is defined as
and its frequency response is
The peak side-lobe ratio depends only on the magnitude response and hence the factor above is inconsequential to our discussion. The remaining part of the expression is very similar to a sinc function. Particularly, the it peaks at , where
and reaches its minimum at or , where
The maximum side-lobe amplitude of the magnitude response coincides with the minimum value. Therefore, the peak side-lobe ratio is given by
Now, for large values of , the argument of the sin function in the above equation becomes close to zero. Using the Taylor approximation of the function near , we have
for large values of .
That’s it! Most of the books and references say 13 dB, perhaps an integer approximation.
The magnitude response of the rectangle window in dB is plotted in Figure 1.
Now, how large should be for this approximation to be valid? Judging from Figure 2, the approximation is valid for .