The frequency response and the main lobe width for rectangular window are

The rectangular window is the most common windowing technique to design a finite impulse response (FIR) filter.

A rectangular window is defined as

\({w_R}\left( n \right) = \left\{ {\begin{array}{*{20}{c}} {1\;:0 \le n \le N - 1}\\ {0\;\;\;\;:otherwise} \end{array}} \right.\)

Where N = length of the FIR filter

The frequency response of w_R(n) is w_R(ω) which is calculated as:

w_R(ω) = DTFT of w_R(n), i.e.

\({w_R}\left( \omega \right) = \mathop \sum \limits_{n = - \infty }^\infty {w_R}\left( n \right){e^{ - j\omega n}}\)

ω ϵ [-π, π] rad/sec

For the given rectangular sequence, the Fourier transform will be a sinc sequence given by:

\({w_R}\left( \omega \right) \approx \frac{{\sin \left( {\frac{{\omega N}}{2}} \right)}}{{\sin \left( {\frac{\omega }{2}} \right)}}\)

The spectrum is as shown:

Main-lobe width = 2 × (zero-crossing points of sin c function), i.e.

\( = 2 \times \frac{{2\pi }}{N} = \frac{{4\pi }}{N}\)