TABLE OF CONTENTS This section reviews some traditional engineering techniques from dynamical systems, optimization theory, parallel processing, and Markov chain theory that can be leveraged to analyze cognitive radio networks.
Dynamical systems theory is concerned, as the name implies, with analyzing the behavior of dynamical systems, and also designing mechanisms so the systems act in a desirable manner. Typical analysis goals of dynamical systems theory are similar to the ones set out in Section 15.2.3: to determine the expected behavior, convergence, and stability of the system.
Formally, a dynamical system is a system whose change in state is determined by a function of the current state and time. In other words, a dynamical system is any system of the form given by:
which describes the change in the state of a system as a function of the system state, a, and time, t. [5] Implicitly, the system is assumed to be at state a(0) at time t 0.
When Eq. (15.2) is not directly dependent on t (i.e., a
. g( a)), the system is said to be autonomous. For our purposes, it makes sense to treat synchronous systems as autonomous, but for random and asynchronous systems, it is difficult to eliminate the time dependency.
The first goal of a dynamical systems analyst is to solve Eq. (15.2) to yield the evolution function that describes the state of the system as a function...
