  We studied noise properties of microwave signals transmitted through the
cryogenic resonator. The experiments were performed with the 11.342 GHz
sapphire loaded cavity resonator cooled to 6.2 K. Based on the measured
transmission coefficient of the cryogenic resonator we computed its noise
suppression function. This was done via Monte-Carlo simulations some details of
which are discussed in this Letter. Next, we measured technical fluctuations of
a signal incident on the cryogenic resonator. Having processed these data with
the previously computed noise filtering "template" we inferred noise spectra of
the transmitted signal. We found that spectral densities of both phase and
amplitude fluctuations of the transmitted signal were close to the thermal
noise limit of -180 dB/Hz at Fourier frequencies F $\ge$ 10 kHz. Such thermal
noise limited microwaves allow more precise tests of special relativity and
could be useful at some stages of quantum signal processing.
