I have attached 2 plots for FFT spectra. One is considered good and one is bad.
The good one is classified on the basis of how closely spaced the frequencies and the bad is based on how multiple frequencies are present.
I am trying to determine a dimensionless objective function that could be maximized or minimized so as to lean towards which is a better plot (Good FFT). One function which is consistent for a particular scale is Max. Power/standard deviation. But this is not a dimensionless quantity.
I would like to determine a dimensionless quantity.
EDIT: Assuming the spectrum is similar to a probability distribution: $\frac{E}{\int_\Omega|f(\omega)|d\omega}$
where E is the max. Amplitude of the PSD.
I am investigating other definitions of spread besides: $$ \omega_{max} - \omega_{min} $$
The objective function proposed by whpowell96 does not seem working generally enough. I identified these 3 properties of a spread:
Given the same shape of spectrum, only by changing number of sample of fft changes $\omega_{max}-\omega_{min}$, and so it gives different peakness for the same spectrum
It is not scale invariant for a triangle spectrum with larger height and same base (you get same objective function while peakness is clearly differernt). See figure below
By scaling up the spectrum (e.g. twice larger in both dimensions), I am expecting to have the same objective function since the peakness is unchanged.
I am investigating other definition of spread instead of $\omega_{max} - \omega_{min}$. Basically I am generalizing whpowell96 's objective function with:
$$ \frac{\|f\|_{L^\infty}\|f\|_{L^0}}{\|f\|_{L^1}} = \frac{\mathrm{max}_{\omega\in\Omega} f(\omega)\cdot|\Omega|}{\int_\Omega|f(\omega)|d\omega}. $$ Basically replacing $\Omega$ with a measure of Statistical dispersion such as (i) IQR (ii) standard deviation, (iii)Mean absolute deviation. It is easy to show that IQR satisfies point 3 but not point 2. I am investigating the other measures.