Choosing Window Length to Resolve Sinusoids

A conservative requirement for resolving 2 sinusoids (in noisy conditions) with a spacing of $\Delta f$ Hz is to choose a window length $M$ long enough so that their main lobes are clearly discernible. For example, we may require that their main lobes meet at the first zero crossings.

To obtain the separation shown above, we must have $B_w \le \Delta f$ , where $B_w$ is the main lobe width in Hz, and $\Delta f$ is the sinusoidal frequency separation in Hz.

For the rectangular window, $B_w$ can be expressed as

$$B_w = 2 \frac{f_s}{M}$$

Hence we need:

$$\begin{eqnarray*} B_w&=& 2 \frac{f_s}{M} \le \Delta f \\ &\Rightarrow& M \ge 2 \frac{f_s}{\Delta f} \end{eqnarray*}$$

or

$$\zbox{M \ge 2 \frac{f_s}{\vert f_2-f_1\vert}}$$

• A length $M$ rectangular window satisfying this inequality is said to resolve the sinusoidal frequencies $f_1$ and $f_2$
• This is equivalent to our previous observation since
  $$M \ge 2 \frac{f_s}{\Delta f}\quad\Leftrightarrow\quad \Delta f \ge 2\frac{f_s}{M} \quad\Leftrightarrow\quad \Delta \omega \ge 2\Omega_M$$

• In summary, to resolve sinusoidal frequencies $f_1$ and $f_2$ under a rectangular window, it is sufficient for the window length $M$ to span at least $2$ periods of the difference frequency $f_2-f_1$ , where $2$ is the width of the main lobe, measured in sidelobe-widths.
• By the Fourier scaling theorem, $K$ periods must suffice for a main lobe of width $K\Omega_M$ .