## Peak Side-lobe Ratio of the Frequency Response of the Rectangular Window

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

$\displaystyle h[n] = \left\{\begin{array}{cc} 1/M, & n = 0, 1, \ldots M-1\\ 0, & \text{ otherwise}. \end{array}\right.$

and its frequency response is

$\displaystyle H(e^{j\omega}) = e^{-j\omega(M-1)/2} \frac{\sin \left(\frac{\omega M}{2}\right)}{M\sin \left(\frac{\omega}{2}\right)}$

The peak side-lobe ratio depends only on the magnitude response and hence the ${e^{-j\omega(M-1)/2}}$ 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 ${\omega =0}$, where

$\displaystyle \left.H(e^{j\omega})\right|_{\omega =0} = 1,$

and reaches its minimum at ${\frac{\omega M}{2} = \frac{3\pi}{2}}$ or ${\omega = 3\pi/M}$, where

$\displaystyle \left.H(e^{j\omega})\right|_{\omega = \frac{3\pi}{M}} = \frac{-1}{M\sin\left(\frac{3\pi}{2M}\right)}.$

The maximum side-lobe amplitude of the magnitude response coincides with the minimum value. Therefore, the peak side-lobe ratio is given by

$\displaystyle \rho = \frac{\left|\left.H(e^{j\omega})\right|_{\omega =0}\right|}{\left|\left.H(e^{j\omega})\right|_{\omega = \frac{3\pi}{M}}\right|}.$

$\displaystyle = M\sin\left(\frac{3\pi}{2M}\right).$

Now, for large values of $M$, the argument of the sin function in the above equation becomes close to zero. Using the Taylor approximation of the $\sin(\theta)$ function near $\theta=0$, we have

$\displaystyle \rho \approx \frac{3\pi}{2}=4.71 = 13.46\,\text{dB}$ for large values of $M$.

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 ${M}$ be for this approximation to be valid? Judging from Figure 2, the approximation is valid for ${M>8}$.

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

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### 3 Responses to Peak Side-lobe Ratio of the Frequency Response of the Rectangular Window

1. Karthikeyan A R says:

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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2. very good

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