Resolution Bandwidth (Resolving Sinusoids)

Our ability to resolve two closely spaced sinusoids is determined by the main-lobe-width and sidelobe-level of our window's Fourier transform.

Let $B_w$ denote the main lobe width in Hz, with the main lobe width defined as the width between zero crossings:

For the Rectangular Window (length $M$ ), we have

$$W_R(\omega) = \hbox{asinc}_M(\omega) \mathrel{\stackrel{\Delta}{=}}\frac{ \sin \left( M\omega T/2 \right)}{\sin(\omega T/2)} = \frac{ \sin \left( M\pi f T \right)}{\sin(\pi f T)}$$

Main lobe width is ``two sidelobes wide''

$$\Rightarrow \quad \zbox{B_w = 2\frac{\Omega_M}{2\pi} = 2\frac{f_s}{M}}\quad \hbox{(\mbox{Hz})}$$