Can a narrowband beamformer be used for speech separation
If you are dealing with say 8 kHz for a nominal speech bandwidth of of 4kHz i.e. 0 Hz - 4 kHz, then the speech is essentially a wideband signal. Therefore narrowband beamforming won't work very well. Your beam pattern will be okay for the particular frequency of interest - but once you start moving away from that frequency your beam patterns will deteriorate.

What you are looking for is wideband beamforming - which uses actual time delays (or linear phase shifts across frequency in the frequency domain) rather than just phase shifts.

Usually the techniques like LCMV and MVDR are developed for narrowband signals. There are a couple of ways of approaching the wideband problem:


Use a tapped delay line on each channel. If you have $n$ channels and $m$ taps per channel then your correlation matrix is $mn$ x $mn$. So the systems of equations gets very large.
Use a bunch of narrowband beamformers. In this case you would have $m$ beamformers ($m$ frequencies) each one having $n$ channels. Now each beamformer has a $n$ x $n$ correlation matrix, but you have $m$ of them. So it leads to a reduction in complexity from the previous case.
You can form a set of $b$ conventional beams (using time-delay rather than just phase multiplier) and then do the adaptive processing on the beams.


The best reference I can think of is by Van Trees - Optimum Array Processing. Note you sometimes run into slight differences in terminology - some texts will denote MVDR as using the signal correlation matrix, while others use the signal plus noise correlation matrix - Just be careful which one you're looking at. I know Van Trees makes a distinction between the two cases. Most other texts do not and only use one formulation and call it MVDR.