# How do I simulate a purely real signal being received on multiple antennae in an array?

I have a signal created at baseband and then transmitted as $$\Re\{s_{bb}*e^{2\pi f_ct}\}$$ which is then received at a 5 element array for further processing. How do I interpret the real signal at each of the array elements without using complex numbers? Simulink needs a purely real signal as an input to the analog filter I am using.

One simple way to think about this is transmitting a pure in-phase carrier: $$s(t) = \cos(2\pi f_c t).$$ The wireless channel will add a random phase, so the received signal is $$r(t) = \cos(2\pi f_c t + \phi)$$ (ignoring changes to the signal amplitude for the moment). Note that, since $\phi$ is random and changes over time, there is no real way for the receiver to determine it.
Now calculate the signal out of the quadrature branch of the receiver: $$r_Q(t) = \text{LPF} \lbrace r(t) \sin(2\pi f_c t + \theta) \rbrace,$$ where $\theta$ is the phase of the receiver oscillator. If you do the math, you'll see that the energy of $r_Q(t)$ is not zero.