Stochastic Differential Equation Theory Applied to Wireless Channels
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Ever faster data transmission in wireless communication is desired to satisfy emerging markets for various media services, such as voice, picture and video calls, multimedia messaging, music and video downloads, and even television. With the explosive increase in the use of mobile devices such as cellular phones, PDAs, GPS, and laptop computers, power consumption has become a prime consideration in the design of mobile communication systems. In order to reliably maintain a high rate of transmission and low power consumption, it is imperative that the receiver obtains as much knowledge as possible about the current state of the channel. A more accurate model of wireless communication
channels will indisputably help in obtaining more knowledge about the transient channel state, providing a more accurate and efficient reproduction of the transmitted signal, and decreased power consumption by the receiver. With careful choice and consideration of the channel model, systemic optimization based on the selected channel model will improve the system performance of the transmitter and receiver through better encoding and decoding, as well as through better control of transmitted signal's power level. This thesis focuses
on understanding the physical and statistical characteristics of wireless channels, and investigates how to represent wireless channels using simple mathematical models. This thesis initially studied a simple time-varying stationary channel, i.e.a multipath fiat fading channel without terminal motion, which is typically used for indoor wireless communication. With an introduction of stochastic differential equations, we derived a first-order AR stochastic process to represent this stationary channel. For a general multipath fiat fading channel with terminal motion, the traditional Clarke's model was then extended by incorporating the effects of fluctuations in the component phases and analyzed statistically. The resulting theoretical power spectrum was shown to fit practical measured spectra, in contrast to the traditional theoretical fiat fading channel spectra (Jakes' spectrum in ) .
Finally, we developed a state-space model that represents a wireless channel using these modified spectral characteristics. This was achieved by developing a relationship between the state-space model and the theory of a rational transfer function. A novel method for designing a rational transfer function for linear systems was then proposed. In this method, the rational transfer function is represented via the Observable Canonical Form (OCF) to obtain the state-space model, which can be used to represent and simulate a fiat fading wireless channel. The presented state-space approach is simple and provides rapid computation. The present AR and state-space models provide valuable contributions that can be integrated with other algorithms for better system optimization of wireless communication networks.