While the Nyquist rate serves as a lower bound to sample a general bandlimited signal with no information loss, the sub-Nyquist rate may also be sufficient for sampling and recovering signals under certain circumstances. Previous works on sub-Nyquist sampling achieved dimensionality reduction mainly by transforming the signal in certain ways. However, the underlying structure of the sub-Nyquist sampled signal has not yet been fully exploited. In this paper, we study the fundamental limit and the method for recovering data from the sub-Nyquist sample sequence of a linearly modulated baseband signal. In this context, the signal is not eligible for dimension reduction, which makes the information loss in sub-Nyquist sampling inevitable and turns the recovery into an under-determined linear problem. The performance limits and data recovery algorithms of two different sub-Nyquist sampling schemes are studied. First, the minimum normalized Euclidean distances for the two sampling schemes are calculated which indicate the performance upper bounds of each sampling scheme. Then, with the constraint of a finite alphabet set of the transmitted symbols, a modified time-variant Viterbi algorithm is presented for efficient data recovery from the sub-Nyquist samples. The simulated bit error rates (BERs) with different sub-Nyquist sampling schemes are compared with both their theoretical limits and their Nyquist sampling counterparts, which validates the excellent performance of the proposed data recovery algorithm.