A formulation of noise operators and their applications

In previous work, we have discussed the use of operators to solve some signal analysis problems that are at the heart of some signal processing and analysis problems. In this paper, we extend the formalism to include noise by the introduction of a noise operator as an alternative way to think about the various signal in noise problems in radar and sonar. In addition, we show how non-linear combinations of signal with noise can be analyzed as noise operators acting on signals can expressed as combinations of operators with random parameters with a given distribution acting on the signal. Then, we show that the noise reverberation or clutter problem can be thought of as a parameter randomized operator acting on signals in either the radar and sonar domain. Then, we note that this formulation lends itself to finding unexpected consequences of non-linear transformations of random variables. Finally, we note a radar problem first addressed by Cartwright and Littlewood in WWII of identifying a spectrum being produced by a non-linearity in a circuit versus noise can be attacked by these methods.

[1]  B. Frieden,et al.  Another proof of the random variable transformation theorem , 1986 .

[2]  Ali T. Alouani,et al.  Characterization of the PDFs of coordinate transformations in tracking , 2014, IEEE Transactions on Aerospace and Electronic Systems.

[3]  M.J. Hasler,et al.  Electrical circuits with chaotic behavior , 1987, Proceedings of the IEEE.

[4]  Holger Dette,et al.  The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis , 1997 .

[5]  M. L. Cartwright.  On non-linear differential equations of the second order , 1949 .

[6]  Leon Cohen,et al.  The history of noise [on the 100th anniversary of its birth] , 2005, IEEE Signal Processing Magazine.

[7]  K. W. Cattermole.  The Fourier Transform and its Applications , 1965 .

[8]  Eugene Lukacs,et al.  Developments in Characteristic Function Theory , 1976 .

[9]  A common basis for analytical clutter representations , 2005, IEEE International Radar Conference, 2005..

[10]  D. Gillespie,et al.  A Theorem for Physicists in the Theory of Random Variables. Addenda. , 1983 .

[11]  J. Goodman.  Introduction to Fourier optics , 1969 .

[12]  Lord Rayleigh F.R.S..  XII. On the resultant of a large number of vibrations of the same pitch and of arbitrary phase , 1880 .

[13]  N. Metropolis,et al.  The Monte Carlo method. , 1949 .

[14]  F. Wang.  Pioneer Women in Chaos Theory , 2009, 0903.2671.

[15]  John E. Gray,et al.  The post-selection operator current , 2018 .

[16]  A. Papoulis,et al.  Maximum response with input energy constraints and the matched filter principle , 1970 .

[17]  H. V. Trees.  Detection, Estimation, And Modulation Theory , 2001 .

[18]  K. Chung,et al.  Limit Distributions for Sums of Independent Random Variables. , 1955 .

[19]  N. Ushakov.  Selected Topics in Characteristic Functions , 1999 .

[20]  W. Bennett.  Distribution of the sum of randomly phased components , 1948 .

[21]  Leon Cohen.  Transformation of distributions into heavy tailed , 2016, SPIE Defense + Security.

[22]  A. Papoulis,et al.  Random modulation: A review , 1983 .

[23]  Gerhard Kurz,et al.  Recursive Bayesian filtering in circular state spaces , 2015, IEEE Aerospace and Electronic Systems Magazine.

[24]  Erik Ostermann,et al.  Radar Sonar Signal Processing And Gaussian Signals In Noise , 2016 .

[25]  I. N. Sneddon,et al.  Application of integral transforms in the theory of elasticity , 1975 .

[26]  John E. Gray,et al.  The Rayleigh problem is everywhere! , 2010, Defense + Commercial Sensing.

[27]  John E. Gray,et al.  Characteristic functions in radar and sonar , 2002, Proceedings of the Thirty-Fourth Southeastern Symposium on System Theory (Cat. No.02EX540).

[28]  A. Papoulis,et al.  The Fourier Integral and Its Applications , 1963 .

[29]  John E Gray.  An interpretation of Woodward's ambiguity function and its generalization , 2010, 2010 IEEE Radar Conference.

[30]  M. Slack.  The probability distributions of sinusoidal oscillations combined in random phase , 1946 .

[31]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[32]  David Aubin,et al.  Writing the History of Dynamical Systems and Chaos: Longue Durée and Revolution, Disciplines and Cultures , 2002 .

[33]  L. Rayleigh.  XXXI. On the problem of random vibrations, and of random flights in one, two, or three dimensions , 1919 .

[34]  George Papanicolaou,et al.  Frequency Content of Randomly Scattered Signals , 1991, SIAM Rev..

[35]  E. Lukács.  CERTAIN ENTIRE CHARACTERISTIC FUNCTIONS , 2005 .

[36]  John E. Gray,et al.  Does the central limit theorem always apply to phase noise? Some implications for radar problems , 2017, Defense + Security.

[37]  D. Rajan.  Probability, Random Variables, and Stochastic Processes , 2017 .