We have analyzed the behavior of a diode laser with low-amplitude optical injection. The noisy injection field has phase fluctuations only, with a linewidth larger than both the free-running slave laser linewidth and the locking range. Simulations show that the laser output spectrum can be much narrower than the injection spectrum and may have a dip at the free-running laser frequency. A model describing these features is based on linear superposition of the stationary responses to monochromatic injection. The mechanism of line narrowing is that the response to injection outside the locking range lies mainly inside.