When enhancing a speech signal using a single microphone system, various approaches based on an autoregressive speech model are referenced in the literature. Using a Kalman filter, they operate in two steps: (1) the noise variances and the autoregressive parameters are estimated, (2) the speech signal is retrieved using standard Kalman filtering. However the existing methods are usually iterative and a voice activity detector (VAD) is often required to find the silent frames for the estimation of the variance of the white noise. To avoid these drawbacks, we propose to consider Kalman filter-based speech enhancement as a realisation issue, i.e. as the estimation of the system matrices in the state space representation using the estimation of the correlation function of the observations. For this purpose, we first present various solutions, based on works initially developed in the field of identification by Van Overschee et al. and Verhaegen. Their non-iterative extensions to coloured noise are also addressed and used with car noise. In the second part of the paper we propose an alternative approach based on Mehra et al. and Belanger's approaches dealing with the estimation of the steady Kalman gain and previously derived in the framework of identification. This approach still avoids the use of a VAD.