Digital beamforming and phase shift stage insertion I have read some literatures about the different topologies of beamforming and it came to my mind a question : where is applied the phase shift for each channel depending on the type of topology/technology used?

For analog beamforming, it is obvious as there is only one signal, the modulated carrier signal which is splitted to the different channels and phase shifting occurs afterward. The phase shifting stage comprises a gain/attenuation stage and a phase shifter stage. If the modulated signal is built by a digital stage, there is one DAC used for all channels prior to mixing and phase shifting.

In digital beamforming, the topology is a bit different: one DAC is used for each channel. This enables more possibilities, among them the possibility to phase shift digitally each channel when defining the modulation. However two types of digital beamforming exist:


one with LO mixing if the DAC is not fast enough to produce a RF signal, 
another using RF-DAC where no LO mixer is required.


As in the first situation LO signal is generally the same for all channels, no phase shift is applied on LO-IF signal. Is the phase shift on IF modulation signal is enough or is it needed to phase shift the signal after mixing (to obtain an equivalent behavior as in analog beamforming?).

Best regards
In a first analysis, it does not matter where the phase shifting is inserted, as long as the mixing stage is linear.

Indeed, shifting a temporal signal is equivalent to multiplying it with a complex number integration the phase shift $\theta$ :


Unshifted signal : $s(t)$
$\theta$ phase shifted signal : $s'(t) = s(t).e^{j\theta}$
$f_0$ carrier mixed unshifted signal : $s_c(t) = s(t).c(t) = s(t).e^{2j\pi f_0.t}$
$f_0$ carrier mixed $\theta$ shifted signal : $s'_c(t) =s(t).e^{2j\pi f_0.t}.e^{j\theta}$ = $s(t).c'(t)$ 


where $c'(t) = e^{j(2\pi f_0.t+\theta)}$ is the shifted carrier.

In the end phase shifting the signal to be modulated is equivalent to phase shifting after carrier modulation.