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### Choosing Window Length to Resolve Sinusoids

A conservative requirement for resolving 2 sinusoids (in noisy conditions) with a spacing of Hz is to choose a window length 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 , where is the main lobe width in Hz, and is the sinusoidal frequency separation in Hz.

For the rectangular window, can be expressed as

Hence we need:

or

• A length rectangular window satisfying this inequality is said to resolve the sinusoidal frequencies and
• This is equivalent to our previous observation since

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

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