Beam width of linear array using MVDR beamformer
I think you're mixing time domain and frequency domain, your signal $x$ is in time and you're multiplying it by a complex exponential, generating a complex signal that is not the Fourier transform of $x$, if you want to delay $x$ in order to simulate the angle of arrival of the signal you have to do $x_{m}(n)=x(n-\tau_{m})$ (in the time domain) or $X_{m}(\omega)=e^{-j \tau_{m}}X(\omega) $ being "m" the microphone number. 

Also you have to include the wave number in the steering vector $k=\dfrac{2\pi}{\lambda}$.