In this paper, we present an alternative approach to the down-conversion (translation) of the received optical signals collected by the antenna of an all-fiber coherent Doppler lidar (CDL). The proposed method, widely known as image-reject, quadrature detection, or in-phase/quadrature-phase detection, utilizes the advances in fiber optic communications such that the received signal can be optically down-converted into baseband where not only the radial velocity but also the direction of the movement can be inferred. In addition, we show that by performing a cross-spectral analysis, enabled by the presence of two independent signal observations with uncorrelated noise, various noise sources can be suppressed and a more simplified velocity estimation algorithm can be employed in the spectral domain. Other benefits of this architecture include, but are not limited to, a more reliable measurement of radial velocities close to zero and an improved bandwidth. The claims are verified through laboratory implementation of a continuous wave CDL, where measurements both on a hard and diffuse target have been performed and analyzed.