In the communication systems that use a linear modulation scheme for transmission, the fractionally-spaced (FS) samples of the received signal constitute a wide-sense cyclostationary time series. Hence, the standard Fourier transform techniques cannot be used to study the spectral characteristics of the received FS samples or to derive the transfer function (TF) of the corresponding digital minimum mean-square error (MMSE) receiver. In this paper, an analytical expression for the TF of the FS MMSE equalizer is derived, which includes the effects of the continuous-time to discrete-time (C/D) converter used at the receiver front end. Using this TF, the sources of instability of the FS least-mean-square (LMS) algorithm and the effects of the equalizer length and sampling phase on convergence of the LMS algorithm are explained. For stabilization of the FS LMS algorithm, conditions on the front-end C/D converter are provided, such that, when satisfied the LMS algorithm becomes more stable and the learning characteristics of the modified receiver are better than the leaky-FS LMS algorithm. Theoretical results are corroborated by simulations.