  Directional amplifiers are an important resource in quantum information
processing, as they protect sensitive quantum systems from excess noise. Here,
we propose an implementation of phase-preserving and phase-sensitive
directional amplifiers for microwave signals in an electromechanical setup
comprising two microwave cavities and two mechanical resonators. We show that
both can reach their respective quantum limits on added noise. In the reverse
direction, they emit thermal noise stemming from the mechanical resonators and
we discuss how this noise can be suppressed, a crucial aspect for technological
applications. The isolation bandwidth in both is of the order of the mechanical
linewidth divided by the amplitude gain. We derive the bandwidth and
gain-bandwidth product for both and find that the phase-sensitive amplifier has
an unlimited gain-bandwidth product. Our study represents an important step
toward flexible, on-chip integrated nonreciprocal amplifiers of microwave
signals.
