  Current and future interferometeric gravitational-wave detectors are limited
predominantly by shot noise at high frequencies. Shot noise is reduced by
introducing arm cavities and signal recycling, however, there exists a tradeoff
between the peak sensitivity and bandwidth. This comes from the accumulated
phase of signal sidebands when propagating inside the arm cavities. One idea is
to cancel such a phase by introducing an unstable optomechanical filter. The
original design proposed in [Phys.~Rev.~Lett.~{\bf 115},~211104 (2015)]
requires an additional optomechanical filter coupled externally to the main
interferometer. Here we consider a simplified design that converts the
signal-recycling cavity itself into the unstable filter by using one mirror as
a high-frequency mechanical oscillator and introducing an additional pump
laser. However, the enhancement in bandwidth of this new design is less than
the original design given the same set of optical parameters. The peak
sensitivity improvement factor depends on the arm length, the signal-recycling
cavity length, and the final detector bandwidth. For a 4~km interferometer, if
the final detector bandwidth is around 2~kHz, with a 20~m signal-recycling
cavity, the shot noise can be reduced by 10 decibels, in addition to the
improvement introduced by squeezed light injection. We also find that the
thermal noise of the mechanical oscillator is enhanced at low frequencies
relative to the vacuum noise, while having a flat spectrum at high frequencies.
