In this paper we address the design of the data demodulator for high-efficiency satellite communications suited for telemetry applications. We choose Gaussian minimum shift keying as the modulation format and analyse its relative merits with respect to the more traditional filtered offset quadrature phase-shift keying scheme, when applied to medium-to-high data rate telemetry. Our study encompasses both geostationary orbits and the more challenging low earth orbit scenarios, and investigates the acquisition and tracking performance of the receiver with the aid of computer simulations. In particular, we present design criteria leading to the development of efficient, low-complexity schemes for symbol timing and carrier phase/frequency recovery and we analyse the impact of these synchronization algorithms on the system performance. Next, we discuss the demodulator performance degradation when the transmitted signal is passed through on-board solid-state or travelling-wave-tube power amplifiers. The paper is concluded by a comparative analysis of the above modulation schemes, with special emphasis on their power and spectral efficiency.