  We study a resonator coupled to a generic detector and calculate the noise
spectra of the two sub-systems. We describe the coupled system by a closed,
linear, set of Langevin equations and derive a general form for the finite
frequency noise of both the resonator and the detector. The resonator spectrum
is the well-known thermal form with an effective damping, frequency shift and
diffusion term. In contrast, the detector noise shows a rather striking
Fano-like resonance, i.e. there is a resonance at the renormalized frequency,
and an anti-resonance at the bare resonator frequency. As examples of this
effect, we calculate the spectrum of a normal state single electron transistor
coupled capacitively to a resonator and of a cavity coupled parametrically to a
resonator.
