I am trying to create a state space system for modeling the problem of binary phase shift keying (0/1) when the transmitter is moving at a constant velocity towards the receiver. The state space system takes the form:

dot{q(t)} = Aq(t) + bx(t)

y(t) = c^T * q(t) +d * x(t)

I was thinking that for my states I can use (1) distance between the transmitter and the receiver and (2) velocity of the transmitter (that is moving at a constant velocity towards the transmitter).

Updating q seems pretty straightforwards. But I don’t know how to go about updating y.
I created my equations which are as follows:
My equations for x(t) and y(t)

But as you can see, y(t) is expressed in terms of sine. I don’t know how to go about creating a state space system out of this model. Any help would be appreciated.