Above, I have drawn a diagram showing Doppler Effect (here we are using space-time but in a non-relativistic sense. Time and distance are the same for A and B).
Edit: I am adding a relativistic space-time diagram below this with lines of simultaneity drawn. I am also editing the description to be more relavent to the updated diagram.
The diagram shows the frame of a stationary observer B. A travels with velocity c/2. A emits flashes of light every second (according to the time of observer B - flashes of light are shown as dashed lines and are emitted where the lines of simultaneity meet the worldline of A and hence they are emitted every second in the time of B)
We see that in the frame of B, we begin to see the light one second after it is emitted. light is continuous and after the lag shown, light from a time delta t' (according to the time in frame B) is observed across time delta t (which is 2 x delta t').
This seems to indicate that what we OBSERVE in frame B seems to depend on only the slope of A. Is this indeed what we will see in B?