then the output is y(t) = |H(jw)|a(t-groupdelay)cos(w(t-phasedelay))
Now, what about if the input is: x(t) = cos(wt + s(t)), where s(t) is a slowly changing phase shift as a function of time. Could we work out what would the output be? Is there anything like a “shift delay”?
You should expand the cos(wt+s(t)) to cos(s(t)) cos (wt)-sin(s(t) sin(wt)... what you have ere is QPM (quadrature phase modulation) or CPM (continuous phase modulation). The amplitude is 1 and the phase is being shifted... if s(t) is a sinewave it is easy to find the closed form description of its spectrum in terms of Bessel functions. the approximate BW can be estimated by Carson's rule related to amplitude of s(t) and the frequency of s(t).
many manpack radios and many small low earth orbit satellites use CPM with its constant magnitude to keep their power amplifiers efficient and happy with extended battery life.
I think you have to describe the query a little better. I am assuming that x(t) is the input and y(t) the output of a filter. What are the filter properties? In your question you make w, a function w(t-phasedelay). If d s(t)/dt is much less than w, would it be sufficient to consider only w(t)?
Remember that frequency is the derivative of phase, so if s(t) is a slowly advancing ramp, it just means an increase in frequency. Likewise if s(t) is a quadratic function, then the signal becomes a linear FM sweep.
You can also make s(t) a data stream and phase-modulate the signal with it, which is all PSK is.
So, yes, this is done pretty routinely.