In a system theoretic setting, identification from time domain data can be viewed as interpolating derivatives of a rational function. Typically, rational interpolation of derivatives requires computing the singular value decomposition (SVD) of large Loewner matrices constructed directly from the data. As a result, significant computational overhead is introduced through the SVD. The main result of the present thesis is simple—we construct interpolants efficiently without forming large Loewner matrices. A previously known recursive procedure is revisited with new insights, then further developed in a state-space setting. The key is to construct an interpolant recursively from ground up, by using the minimum amount of data. The resulting recursive interpolant is minimal and given in a state space form with rich structure. An important special case is the interpolation of impulse response measurements. This case is addressed separately and an efficient implementation requiring only matrix-vector multiplications is put forward. Furthermore, we extend the method to data corrupted by noise, where an additional model order reduction step is used to identify a low order model from the data. The newly developed recursive procedure is then tested on two examples involving actual noisy time-domain responses of a beaded elastic string and a cantilever beam.