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Amplification is usually necessary when measuring the frequency instability of microwave signals. In this work, we develop a flicker-noise-free frequency measurement system based on a common or shared amplifier. First, we show that correlated flicker phase noise can be cancelled in such a system. We then compare the new system with the conventional system by simultaneously measuring the beat frequency from two cryogenic sapphire oscillators (CSOs) with parts in 1015 fractional frequency instability. We determine for low power, below -80 dBm, the measurements were not limited by correlated noise processes, but by thermal noise of the readout amplifier. In this regime, we show that the new readout system performs as expected and at the same level as the standard system, but with only half the number of amplifiers. We also show that, using a standard readout system, the next generation of CSOs could be flicker phase-noise limited when instability reaches parts in 1016 or better.