### Abstract

When trying to analyze a complex communication system, scientists often apply concepts from stochastic modeling and analysis to obtain a description of the system, frequently assuming that this will supplement our knowledge and improve our understanding. The philosophy is to obtain a result that would occur on the average, when this system is working under normal conditions. However, we must consider the fact that the introduction of probability in communication-system analysis often involves invoking certain assumptions and additional information about the system, which may not be valid. Hence, under these circumstances one may obtain a result that may not be commensurate with the conceived communication system. The objective of this paper is to highlight the basic assumptions that are invariably associated with the signal analysis in a system using stochastic analysis, and the introduction of probabilistic methods. Surprisingly, in many cases, analysis using stochastic methods may provide results equivalent to those obtained using deterministic methods. Examples are presented to illustrate our approach, and to explain the basic assumptions and formulate the mathematical framework associated with a stochastic analysis. We also demonstrate the equivalence between a random and a deterministic process, and under what conditions they approach the Cramer-Rao bound. Analysis using stochastic models to describe a system may be superior to a deterministic description. However, such a characterization comes with a large cost: namely, one must have more definitive knowledge about the system, knowledge that is often unavailable. For convenience, with the application of a random model, the concepts of stationarity and ergodicity are used to simplify the mathematical analysis of measured data. It is shown in this paper that the introduction of ergodicity in probability is similar to a deterministic analysis of a single waveform, and, in both cases, characterizes the entire underlying mathematical agenda. An example is presented to illustrate the salient features of an ergodic process as opposed to a deterministic process. It is seen that for practical problems, it might be easier and more relevant to introduce a deterministic model and to then carry out a stochastic analysis. However, this may not be practical, since the underlying ensemble is not available nor are its probability density functions. Moreover a deterministic solution may present the best solution for a given data set, whereas the stochastic approach yields an "average" solution for all the waveforms in the ensemble. Hence, the stochastic solution may not be the desired solution for the given data set. However, when accurate statistics are available, a better solution may be obtained using probabilistic methods.

Original language | English (US) |
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Pages (from-to) | 40-50 |

Number of pages | 11 |

Journal | IEEE Antennas and Propagation Magazine |

Volume | 44 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2002 |

### Keywords

- Adaptive arrays
- Adaptive signal processing
- Adaptive systems
- Communication systems
- Deterministic algorithms
- Deterministic analysis
- Ergodic process
- Probability
- Random process
- Random variables
- Stochastic process

### ASJC Scopus subject areas

- Condensed Matter Physics
- Electrical and Electronic Engineering

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## Cite this

*IEEE Antennas and Propagation Magazine*,

*44*(4), 40-50. https://doi.org/10.1109/MAP.2002.1043146