Can someone help with clarifying the difference between two approaches to frequency domain "deconvolution:
For the frequency domain problem:
We want to find a filter $F(\omega)$ which will convert some observed signal $B(\omega)$
into a desired signal $D(\omega)$, which is known
i.e. $D(\omega) = B(\omega)F(\omega)$
The intuitive / direct approach is to just calculate $F(\omega)=D(\omega)/B(\omega)$
Application of the least-squares criterion leads to $F(\omega)=R_{DB}(\omega)/R_{BB}(\omega)$ i.e. $D(\omega)B^*(\omega)/B(\omega)B^*(\omega)$
Q1: If the direct division can be done successfully (i.e. no zeros in the denominator), the two approaches seem to always give identical answers in practice. Is that because the direct division is actually handled in code (e.g. Python) using the conjugates.
Q2: Does the distinction between the two approaches only arise when the division requires the addition of some stabilizing noise in the denominator?