In the context of cognitive radio, smart cities and Internet-of-Things, the need for advanced radio spectrum monitoring becomes crucial. However, surveillance of a wide frequency band without using extremely expensive high sampling rate devices is a challenging task. The recent development of compressed sampling approaches offers a promising solution to these problems. In this context, the Modulated Wideband Converter (MWC), a blind sub-Nyquist sampling system, is probably the most realistic approach and was successfully validated in real-world conditions. The MWC can be realized with existing analog components, and there exist calibration methods that are able to integrate the imperfections of the mixers, filters and ADCs, hence allowing its use in the real world. The MWC underlying model is based on signal processing concepts such as filtering, modulation, Fourier series decomposition, oversampling and undersampling, spectrum aliasing, and so on, as well as in-flow data processing. In this paper, we develop an MWC model that is entirely based on linear algebra, matrix theory and block processing. We show that this approach has many interests: straightforward translation of mathematical equations into simple and efficient software programming, suppression of some constraints of the initial model, and providing a basis for the development of an extremely fast system calibration method. With a typical MWC acquisition device, we obtained a speed-up of the calibration computation time by a factor greater than 20 compared with a previous implementation.