2044 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 7, JULY 2000 A Statistical and Physical Mechanisms-Based Interference and Noise Model for Array Observations Keith F. McDonald, Member, IEEE, and Rick S. Blum, Member, IEEE Abstract—A statistical noise model is developed from mathe- matical modeling of the physical mechanisms that generate noise in communication receivers employing antenna arrays. Such models have been lacking for cases where the antenna observations may be statistically dependent from antenna to antenna. The model is de- veloped by generalizing an approach for single antenna cases sug- gested by Middleton. The model derived here is applicable to a wide variety of physical situations. The focus is primarily on problems defined by Middleton to be Class A interference. The number of noise sources in a small region of space is assumed to be Poisson distributed, and the emission times are assumed to be uniformly distributed over a long time interval. Finally, an additive Gaussian background component is included to represent the thermal noise that is always present in real receivers. Index Terms—Array processing, Class A noise, impulsive noise, non-Gaussian noise, spatial processing. I. INTRODUCTION O NE OF THE most important contributions to the area of non-Gaussian noise 1 theory was the development of a tractable and accurate model by Middleton in an impressive re- search effort in the 1960’s and 1970’s [1]–[5]. Mathematical expressions were developed for the received and matched fil- tered signal at a single antenna that results from the interfer- ence generated by a collection of emitters in the space around the antenna. By using a statistical formulation, the analysis con- siders the arrangement of the emitters in space and time as well as the propagation environment. The thermal noise that is al- ways present in real receivers is also included in the derivation. To achieve a tractable model from such a complicated analysis, Middleton employed simplifying assumptions where feasible. He was always careful to use physics to justify these assump- tions. Middleton also described and measured real noise from various man-made sources [4], [6] and showed that these mea- surements were consistent with the assumptions he made in his model. Manuscript received May 6, 1999; revised January 3, 2000. This work was supported in part by the Air Force Office of Scientific Research, Air Force Ma- teriel Command, USAF, under Grant F49620-97-1-0461 and by the National Science Foundation under Grant MIP-9703730. The associate editor coordi- nating the review of this paper and approving it for publication was Prof. Dim- itrios Hatzinakos. K. F. McDonald is with the MITRE Corporation, Bedford, MA 01730-1420 USA. R. S. Blum is with the Department of Electrical Engineering and Com- puter Science, Lehigh University, Bethlehem, PA 18015 USA (e-mail: rblum@eecs.lehigh.edu). Publisher Item Identifier S 1053-587X(00)04928-X. 1 Our use of the term noise includes interference whose structure cannot be predicted. One highly desirable property of Middleton’s model is that it is derived from a mathematical analysis of a simplified model of the real physical mechanisms that generate noise in commu- nication receivers. Further, Middleton has shown that his model is consistent with real measurements of noise in communica- tion receivers. Another advantage of Middleton’s model is that it can be expressed in a canonical form, so that noise from many different specific interference scenarios can all be represented by the same model but with a different set of coefficients. In fact, the interference scenarios can result in either Gaussian or non-Gaussian noise. A canonical model for Class A noise, where the duration of a typical interference waveform is much greater than the reciprocal of the bandwidth of the receiver [6], was developed in [3] and [4]. This type of noise generates ig- norable transients in the receiver. The fundamental nature of the Class A model is evidenced by the fact that it was derived sepa- rately in [7] when designing optimal and sub-optimal detectors for linear processes. Another canonical model was developed in [4] for Class B noise, where the duration of a typical interference waveform is much less than the reciprocal of the bandwidth of the receiver [6]. Class B noise produces a significant transient response in the receiver. The most general model includes both Class A and Class B parts. In [3] and [4], the first-order statistics are derived for both the amplitude and envelope of the received voltage at the intermediate frequency (IF) stage of the receiver. Whereas other researchers have also attempted to develop models by considering interference scenarios [8]–[10], none of these models have been as extensively tested [4], [6], as rigor- ously justified using electromagnetic wave theory and physics [1]–[6], or as heavily referenced and employed as Middleton’s model [11]. In fact, due to the popularity and wide applicability of Middleton’s Class A model, there was a previous attempt in [11] to extend his model to multivariate cases (multiple antenna or time observations). In multivariate scenarios, there may be correlation between the different noise observations. However, the multidimensional non-Gaussian noise models proposed in [11] are ad hoc and are not based on a mathematical analysis of interferers located in space. The models in [11] are each just one of several possible two-dimensional (2-D) probability den- sity functions (pdf’s) that have Middleton’s model (a univariate model) as their marginal pdf. It is unknown if the models in [11] have assumed a dependence structure that is consistent with the dependence structure imposed by the real physical mech- anisms that generate noise in communication receivers. Thus, using the models in [11] to analyze performance may result in a completely inappropriate analysis. Similarly, using a model from [11] to design a receiver may result in a scheme that per- forms very poorly in practice, even though the receiver func- 1053–587X/00$10.00 © 2000 IEEE