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.

Figure 1: Magnitude Response of Rectangle Window (dB).

Now, how large should be for this approximation to be valid? Judging from Figure 2, the approximation is valid for .

Figure 2: Convergence of Peak Side-lobe Ratio with respect to the filter length M.

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## About sandeeppalakkal

I'm what the neural network in my brain feels what I am.

So, as long as the no. of taps is greater than 8, the peak to side lobe ratio is maintained @ 13.5dB. Thanks for sharing. I believed filter of any length had this ratio.

Similarly, for a linear antenna array, the main lobe to sidelobe difference is guaranteed to be 13.5dB only for more than 8 antenna elements. This is where 8 brings better understanding of the mathematical treatment of spatial filter.

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Thank you for the comment ARK:)

I don’t know much about antenna arrays but spatial filter is indeed an FIR filter. So, what you said — that if the number of taps is greater than 8, the main lobe to side-lobe difference is 13.5 dB — will be valid when the tap weights are same for every tap. Thus it is a filter with a rectangular window as the impulse response. But this is not in used in practice, no? Usually different weights are used which will improve the main-lobe to side-lobe ratio.

Can you give any antenna theory reference which says that the ratio is 13.5 dB if the number of taps are greater than 8?

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very good

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