Stochastic Models

Sometimes it is convenient to use stochastic representations of the electromagnetic field in place of deterministic ones to describe average observations in complex environments, and in this case the number of degrees of freedom depends on the parameters of the stochastic process used to represent the field. These should be chosen so that the model is consistent with the physics, and can predict average observations.

While representations in a deterministic setting consider the field radiated by an arbitrary environment, in a stochastic setting we consider the field radiated by a random environment. In this case, the analog of the number of degrees of freedom is the amount of stochastic diversity of the received waveform. A larger diversity corresponds to more unpredictable waveforms that require, on average, a larger number of coefficients to be represented to a given accuracy.

Both deterministic degrees of freedom and stochastic diversity have applications in communications. The number of degrees of freedom provides an upper bound over all possible environments on the number of channels that can be used to multiplex different streams of information over different dimensions of the signals’ space. In the stochastic setting, the amount of diversity provides a limit on the reliability that can be achieved by performing transmissions over multiple realizations of the channel. If the received signal is modeled as a random process in time, frequency, and space, then redundant transmissions over multiple frequency bands, multiple time slots, or multiple antennas can improve the probability that at least one of these transmissions is received successfully. In short, the number of degrees of freedom is used to measure the rate gain that can be achieved by performing multiple parallel transmissions over the channel, and the stochastic diversity is used to measure the reliability gain that can be achieved by performing repeated transmissions over multiple realizations of the channel.

In a stochastic setting, the analog of the Hilbert–Schmidt representation leading to the number of degrees of freedom is the Karhunen–Loève representation leading to the stochastic diversity.