Nowadays, brain computer interface (BCI) systems, which are based on electroencephalogram (EEG) signals, are becoming more accessible and convenient, and allows to control ”by thoughts”, for example, wheelchair, robotic prosthesis or even a car. Since wireless BCI systems use batteries, efficient energy management is crucial for increased operation time. One part of this system is analog-to-digital converter (ADC), where energy consumption can be significantly reduced. Even further, by using appropriate ADC it is also possible to reduce the amount of information to be transmitted, thus greatly reducing the energy consumption of a transmitter. In order to choose the most appropriate ADC, first a literature review on EEG signals and BCI systems is carried out, to define requirements for selection of ADC. Then, based on these requirements, an in depth analysis of synchronous and asynchronous ADCs is performed, in order to identify their advantages and disadvantaged as well as suitability for BCI applications. Asynchronous Sigma-Delta modulator (ASDM) ADC is selected for further in depth analysis. It shows that for wide dynamic range signals (e.g. EEG), high switching activity of ASDM circuit appears when the input signal amplitude is low, causing increased power consumption of a wireless BCI. To improve efficiency, a new method, called Amplitude Adaptive Asynchronous Sigma Delta modulator (AA-ASDM), is presented and described in detail, including description of signal encoding and fast and real-time reconstruction. In order to verify and assess the proposed method in practice, various simulations, modelings and physical implementations are carried out, including development of one complete wireless BCI system. The experimental research results show that by using AA-ASDM for asynchronous EEG signal acquisition, it is possible to reduce the switching activity by up to 68.85% and thus proportionally the power consumption of a wireless transmitter. Finally, at the end of this work, a summary and conclusion is given. This thesis is the result of the research carried out at the Institute of Electronics and Computer Science within the framework of the National Research Programme „Cyber-physical systems, ontologies and biophotonics for safe&smart city and society” project No. 4.: “Development of technologies for secure and reliable smart-city” and European Social Fund’s (ESF) supported project “R&D Center for Smart Sensors and Networked Embedded Systems”.