I can imagine the proliferation of Alcubierre drives in the distant future, which would distort space so it takes the shape of a Mexican hat in 4D map. I am wondering, what would the gravitation wave output from such a drive looks like? How many chirps would occur between turning he warp drive on and off? Can the warp drive be passive? If so, would there be any gravitational waves, however feeble?
Alcubierre's solution to the Einstein field equations (Alcubierre 1994) gives us a metric - an expression describing how spacetime curves - of the form $$ds^2=-dt^2+(dx - v_s(t)f(r_s)dt)^2+dy^2+dz^2$$ where our drive is moving in the $x$ direction at a speed $v_s(t)$. $f(r_s)$ is a function encoding the size of the bubble itself. When we talk about an Alcubierre drive, we are usually talking about this solution to the field equations, rather than a device that has more freedom to behave in different ways.
Gravitational waves are generated by any object with an accelerating mass quadrupole moment. What that means is that we require certain kinds of acceleration to produce gravitational waves. Sure, any object with mass or energy will bend spacetime, but to actually produce gravitational waves we need acceleration. The upshot of all this is that an Alcubierre drive, while in operation, will not produce gravitational waves if $v_s(t)$ is constant. (If a constant speed drive is what you mean by "passive", then this is the key bit for you!)
In the case where $v_s(t)$ is not constant, I'm not sure what the gravitational wave signature would look like, because I don't think anyone's investigated it. The problem is that the Alcubierre metric describes a universe with just a moving drive and nothing else outside it - we assume that $f(r_s)$ goes to $0$ outside the bubble, and so we have flat Minkowski space. In other words, the metric behind the drive assumes that there are no gravitational waves. It's a toy model, not necessarily an accurate representation of reality. When we talk about an Alcubierre drive, we think about the metric - it's not like we have a physical object and calculate the properties of spacetime (i.e. the metric and its perturbations) based on that.
Regarding chirps: The gravitational wave chirp of two inspiraling masses occurs because the gravitational wave output itself is changing. In the final milliseconds before coalescence, the two bodies rapidly get closer and closer, drastically increasing their orbital frequency. As gravitational waves are emitted and twice the orbital frequency, we would therefore expect the frequency of the waves to change in time - producing a chirp.
We would, then, only detect a chirp if there was a change in the acceleration; presumably, this would not be significant in an accelerating ship if that acceleration was roughly constant, and so we would detect no chirp. The only way to produce a sequence of chirps would be if the ship was repeatedly accelerating and then decelerating, which is strange behavior for an object trying to reach high speeds.
Your best answer would be to look Dr Alcubierre's original paper or summaries of it to get a description of how 'space' is curved before and behind a ship using this drive (it's not a gate). The drive 'bends' local space around the ship severely so as per General Relativity there should be gravitational wave effects (I think).
If you insist on a 'gate' then I don't know a possible answer. Dr Alcubierre's paper relates strictly to a hypothetical object moving at FTL speeds when surrounded by an effect the object generates itself as it moves, not a generator that imparts 'motion' onto other objects while staying fixed in place itself. Your 'gate' would have to travel with the ship to be a true Alcubierre drive. For a true gate you are better off with wormholes.