We introduce a new and quantitative theoretical framework for noise spectral analysis using a threshold detector, which is then applied to a superconducting device: the Josephson bifurcation amplifier. We show that this new framework provides direct access to the environmental noise spectrum with a sensitivity approaching the standard quantum limit of weak continuous measurements. In addition, the accessible frequency range of the spectrum is, in principle, limited only by the ring down time of the resonant mode. This on-chip noise detector is non-dissipative and works with low probing powers, allowing it to be operated at low temperaturesT<15mK. We exploit this technique for measuring the frequency fluctuations of our device and find a low frequency noise with an amplitude and spectrum compatible with a dielectric origin.