# How to create test vectors for wideband beamforming aplications?

I'm trying to create test vectors for a beamforming algorithm I'm simulating in Matlab. I have a complex signal and I want to create the signal as seen by each antenna element. This will allow me to use my beamforming algorithm to rotate and add the signals to either create gain or nulls for given directions. I understand that the propagation delay for a sine wave between two sensors is $$\Delta_t=\frac{d*sin(\theta)}{c}$$ where $$d$$ is the distance between two antenna elements, $$\theta$$ is the incident angle, and $$c$$ is the propagation speed (typically the speed of light for RF signals). I also understand that the time delay can be represented as a phase offset $$\phi$$.

For narrowband signals, I would add the phase offset $$\phi$$ calculated using the carrier frequency by multiplying my complex signal by $$e^{j \phi}$$. However, for wideband signals this approximation breaks down. The higher and lower frequency components would need to be rotated by some other value. How do I take my complex signal and appropriately rotate it for all frequencies? In other words, how do I create the received signal for each of the antenna elements for a wideband signal?