Venkatarama KrishnanProfessor Emeritus

BSc: Chemistry, (First Class), University of Madras, 1948

Courses Taught GUG Levels
Regular Graduate Courses  Special Graduate Courses  Undergraduate Courses 
Probability & Random Processes Stochastic Modeling in Telecommunications Time Series Analysis & Forecasting Statistical Theory of Communications Discrete Mathematics Linear Systems Adaptive Estimation Algorithms Theory of Communications Control System Synthesis Inertial Navigation & Guidance Optimal Control Theory Theory of Optimization 
Fast Fourier Transforms Convergence of Probability Measures Martingales & Nonlinear Estimation Applications of Time Series Applications of Probability Estimation Theory Wavelets and Applications Fractional Brownian Motion 
Calculus  I Circuit Theory Signals and Systems Introductory Statistics Differential Equations Electromagnetic Field Theory Electronic Circuits Complex Variables Feedback Control Systems Electrical Engineering Laboratory Electromechanical Systems Chemistry 
Book to be Published:
A second edition of the graduate course and reference book, "Probability and Random Processes," will be published by John Wiley and Sons in August 2015. This book will contain additional material with an additional chapter on Probability modeling in Traffic Engineering by Kavitha Chandra
Books Published:
Invited chapter, "Stochastic Processes: Classification," (pp. 15411547) in the reference book, "International Encyclopedia of Statistical Science," edited by Miodrag Lovric and published by Springer in December 2010
A graduate course and reference book, "Probability and Random Processes," published by John Wiley and Sons in July 2006 in the series, Survival Guides in Engineering and Science. Errata sheets for this book: Errata for Probability Book.
A reference book,"Linear Systems PropertiesA Quick Reference," published by CRC Press in February 1998
Fundamental investigations on the martingale approach to nonlinear filtering and smoothing was published in March 1984 as a book by John Wiley and Sons under the title, "Nonlinear Filtering and Smoothing  An Introduction to Martingales, Stochastic Integrals and Estimation"
The book, "Nonlinear Filtering and Smoothing  An Introduction to Martingales, Stochastic Integrals and Estimation," originally published by John Wiley and Sons in March 1984, was republished with a new preface and an errata sheet by Dover Publications in September 2005
Book Temple: Design by Venkatarama Krishnan
Career Overview and Philosophy  Professional Vitae  Errata for Probability Book 
Performance Modeling of Internet Traffic: Modeling of correlated traffic using linear and threshold autoregressive processes, modeling of variable bit rate encoded video streams, estimating buffer occupancy and investigating the impact of correlated arrivals on the delay distribution of buffer occupancy
Medical Imaging: Improved reconstruction accuracy of images in emission and transmission tomography for parallel and fan beams using expectationmaximization algorithm. Investigated 3D Cone Beam Tomography
Hilbert Transforms: Performed fundamental studies on Hilbert transforms and developed Hilbert series and obtained a technique for phase refinement
Martingales and Nonlinear Filtering: Using martingale theory, developed stochastic differential equations satisfied by smoothed estimates in a general model, derived stochastic differential equations in time delay systems, studied properties of conditional distributions in linear models, and formulated a general class of fault detection problems resulting in the following two books:
Time Series Methods: Introduced time series analysis methods for modeling gyro drift angle data. Performed Statistical modeling by means of two factor experiments to determine the effects of parameters on system performance. Using time series analysis developed methods to determine the roughness of runways
Probability Models: Developed causal probability models for reducing the lateral separation of aircraft in transoceanic flight
Estimation: Using Butterworth and Chebyshev filters, successfully completed the problem of improving the estimation accuracy of multiple linear regression coefficients in presence of sinusoids and noise
Biomedical Studies: Investigated the reconstruction of images in Transmission Computed Tomography (TCT) and Emission Computed Tomography (ECT) using statistical techniques. Bayesian maximum aposteriori techniques were implemented by applying the expectation maximization iterative algorithm (CACT Report T9401) using three types of priors like the Gibbs, the cross entropy distance and the hybrid. The potential functions used in the Gibbs prior were the sigmoid and the lncosh functions. The parameter settings were optimized for TCT and ECT. Computer simulations were performed on phantoms in 2D and 3D object spaces. The quantitative accuracy and resolution of the reconstructed images were considerably improved using the sigmoid potential function. The problem of incomplete projections was solved by first using a linear predictive extrapolation scheme and reconstruction achieved from the completed projections
Developed and constructed a ballistocardiograph to measure cardiac output noninvasively. Reduced false negative responses obtained from scanners used in detection of tumors. Modeled electrocardiograph (EKG) and electroencephalograph (EEG) signals using time series analysis
Inertial Navigation : Investigated problems on error analyses of inertial navigation systems, weather satellite systems and reconnaissance satellite systems. Improved the firecontrol accuracy of navigation to guidance transfer by developing several adaptive adjustment methods. Performed extensive research on strapdown inertial navigation systems and associated alignment problems
William B. Blakemore II Regents Professor Department of Computer Science The University of Texas at Austin, TX
Dr. S. Lakshmivarahan, PhD (Indian Institute of Science)
Dr. Sanguthevar Rajasekharan, PhD (Harvard) 
Dr. Anuradha Annaswamy, PhD (Yale)
Dr. Jayaram K. Udupa, PhD (Indian Institute of Science)
Dr. Kavitha Chandra, DEng (UMass Lowell) 
Was a Senior Systems Analyst (19741976) at the Dynamics Research Corporation (DRC), Wilmington, Massachusetts,
working on modelling and estimation problems associated with submarine navigation and guidance. Continued as a
sconsultant for 10 more years (19771987)
Reports Published at DRC
1. Estimation of Autocovariance Functions, Report M215U, October 1974
2. Minimum Variance Estimators for Multiple Linear Regression Coefficients, Report, M229U, January 1975
3. Estimation of Bias, Ramp, Bow and Amplitudes of Sinusoids in Presence of One Another, Report M239U, March 1975
4. Estimation Errors in Bias, Ramp and Bow in Presence of Correlated Noise, Report, M243U, April 1975
5. Complete Two Factor Experiment for Interchangeability, Report E3596U, August 1975
6. Weighted Representation of Ensemble of Estimated Electrostatic Gyro Bows, Report E3660U, November 1975
7. Confidence Limits and Equivalence of Statistical Tests, Report, M256U, March 1976
8. Confidence Limits on NACF of Ensemble Average of a Stochastic Process, Report M1672U, August 1982
9. Investigation of Navigational Sonar System Velocity Estimation Errors, with M. Froome and G. Koehler, Report E8485U, January 1983
10. Distribution of Spectral Estimators, Report E9334U, June 1984
11. Estimation of Position and Velocity Errors and their Confidence Limits, Report E10242U, July 1985
12. Adaptive Adjustment Algorithms for Latitude, Longitude and Heading Variances of NECM, Report E11731U, July 1986
While at Polytechnic Institute of Brooklyn, designed and coordinated a two year (19681970) program on
Modern Engineering Mathematics to personnel of Bell Laboratories and taught Variational Calculus and
Optimization, Linear Dynamic Systems, and Aerospace Mechanics
Consultant to continuing education programs at Polytechnic Institute of Brooklyn (Now Polytechnic University of New York)
Consultant, Dynamics Research Corporation, Andover
Reviewer of manuscripts for McGrawHill, West Publishing Company, CRC Press, PrenticeHall, John Wiley & Sons
Reviewer of papers for IEEE Transactions
Reviewer of papers for International Encyclopedia of Statistical Science, 2010
Was one of the keynote speakers by video link at the Promotion of the "International Encyclopedia of Statistical Science," held in Banja Luka in the Republic of Srpska, Bosnia on April 20, 2011. Hear the complete speech 6:15 min. For transcript of the speech click here
The editor of the book, "International Encyclopedia of Statistical Science," Miodrag Lovric and
his two collaborators have been nominated for the 2011 Nobel Peace Prize. See
[1]
and
[2]
Listed in Who is Who in America, 2010 Writeup in Lowell Sun6/14/2009
Best Teaching Award from UMass Lowell and ECE Department for the year 2000
Teaching Assistant Award from Princeton University 19561957
Orson Desaix Munn Scholar from Princeton University 19561957
Fulbright Travel Grant, US Department of State 19561957
Sigma Xi (1965)
Eta Kappa Nu (1964)
Was commended for services to the University of Massachusetts Lowell by Governor Jane Swift, acting Governor of Massachusetts, 20012003
Developed and taught a course on Signals and Systems during Spring 2003 in the newly created Engineering Program in Smith College. This course was very well appreciated and commended by the students at the end of the semester
Web Pages Design, Graphic Arts, Photography, Shakespeare, Painting, Carnatic & Western Classical Music, Travel, Hindu Philosophy
Performance Modeling of Internet Traffic, Probabilistic and Statistical Modeling, Tomographic Studies, Signal Processing, Time Series Analysis
In this second edition, additional material like, a historical account of how Isaac Newton solved a binomial probability problem posed by the diarist Samuel Pepys, Spearman and Kendall correlation coefficients, two adaptive estimation techniques Recursive Least Squares and Least Mean Squares, Birth and Death Processes and Renewal Processes have been added. An additional chapter on Probability Modeling of Teletraffic Engineering has also been incorporated. These additional topics make this edition a more complete tome.
1. Stochastic Processes: Classification, manuscript of an invited chapter published in International Encyclopedia of Statistical Science, Springer in December 2010, pp.15411547. (Posted with permission from Springer)
This is the manuscript of an invited chapter in the book International Encyclopedia of Statistical Science published by Springer in December 2010. It discusses the various types of stochastic processes starting with Stationary and Ergodic Processes, State and TimeDiscretized processes, Gaussian Processes, Markov Processes, Markov Chains, SemiMarkov Processes, Independent Increment Processes, General Random Walk, Birth and Death Processes, Renewal Processes, Martingale Processes and ending with Periodic Processes.
2. Causal Probability Model for Evaluating Future Transoceanic Airlane Separations, with H. J. Rome, IEEE Transactions of Aerospace Electronic Systems, July 1990. (Posted with permission from IEEE)With the advent of automatic dependent surveillance (ADS), a detailed probability model of aircraft crosstrack deviations is required to determine the impact of ADS. A suitable causal probability model is presented where normal navigation, degradation, pilot blunder, and equipment failure are each modeled by Gaussian density functions with mean and standard deviations defined by the physics of the event. The overall model which is a weighted sum of these Gaussian error probabilities is thus amenable to extrapolation. Overlap and encroachment probabilities are derived, and the impact of ADS on this model determined. It is shown that by using a simple form of ADS, separation standards can be reduced and by transmitting a figure of merit (FOM) providing information on failures and degradations, these separation standards can be further reduced. The results suggest an improvement by a factor of two from current separation standards.
3. Stochastic Modeling of an Aircraft Traversing a Runway using Time Series Analysis, with C. Venkatesan, AIAA Journal of Aircraft, February 1981. (Posted with permission from AIAA)Time series is a sequence of observations on a stochastic process made at discrete and equally spaced time intervals. Its future behavior can be predicted by identifying, fitting, and confirming a mathematical model. In this paper, time series is applied to problems concerning runwayinduced vibrations of an aircraft. A mathematical model based on time series is fitted to obtain the impulse response coefficients of an aircraft system for a particular type of operation. Using this model the output, which is the aircraft response can be obtained with lesser computation time with any runway profile input.
4. Initial Alignment of a Gimballess Inertial Navigation System, with K. Grobert, IEEE Transactions of Automatic Control, December 1970. (Posted with permission from IEEE)Initial alignment of gyrostabilized platforms is accomplished by gyrocompassing techniques. These cannot be used in the case of strap down systems (gimballess inertial navigation systems) where the inertial components are "hard mounted." A scheme is proposed whereby initial alignment is accomplished for a strapdown system at standstill, using the gimballess inertial navigation system components only.
5. SemiAnalytic Approach to Root Locus, IEEE Transactions of Automatic Control, January 1966. (Posted with permission from IEEE)The analytical methods of obtaining root locus equations developed in the Soviet Union merit utilization due to the flexibility inherent in the manipulation of algebraic expressions. Arrays are developed whereby these algebraic expressions can be obtained from a knowledge of the open loop transfer function. Breakaway points on the real axis and intersection with the imaginary axis can be obtained directly from these arrays. Finally, arrays are formulated for the case where one of the parameters of the open loop transfer function varies.
In this report a Bayesian reconstruction algorithm of transmission tomographic images is presented. The reconstruction is based on maximum aposteriori (MAP) estimation using the Expectation Maximization (EM) algorithm. The Gibbs prior with sigmoidal potential function is employed. The Gibbs prior improves the image quality as well as the convergence properties of the EM algorithm. The Bayesian reconstruction problem is solved through the OneStepLate ExpectationMaximization algorithm (EMOSL) proposed by Green. Computer simulated phantoms are used in this study. Images reconstructed from this algorithm are compared to those reconstructed from currently available reconstruction algorithms in terms of image quality and convergence rate. The reconstruction algorithm works quite well for lowphotoncount and lowcontrast cases in which the widely used convolution backprojection algorithm performs rather poorly.
Parts of this report have appeared as Chapter 23, Probabilistic Methods in Transmission Tomography in the book Probability and Random Processes published by John Wiley & Sons Inc. in July, 2006
Indian music was developed and practiced nearly three thousand years ago. It consisted of musical styles and schools evolved by blending Vedic chants and folk music. The South Indian Classical music practiced in Tamil Nadu, Andhra Pradesh, Kerala and Karnataka is called Carnatic Music. It starts from the diatonic scale of seven note, octaverepeating scale comprising of the swaras Sa, Ri, Ga, Ma, Pa, Da, Ni along with five semi tones. It was systematized in the fifteenth century and mathematically formalized in the seventeenth century by Venkatamakhin (1650) from Karnataka in his landmark work 'Chaturdandi Prakasika' and continues to be refined. This essay discusses the mathematical aspects of the Melakartha Ragas, which contain all the seven notes and the octave in its scale.
2. On Hinduism , November 2009Hinduism is generally regarded as the world's oldest organized way of life called 'Sanatana Dharma' with over 900 million people as adherents. This short essay discusses the basic tenets and practice of Hinduism.
3. Cactus Flight 1549, February 2009On January 15, 2009, a US airways flight 1549 with code sign Cactus 1549 took off from LaGuardia airport bound for Charlotte, North Carolina. Two minutes into takeoff at an altitude of 2, 700 feet the plane was hit by a flock of geese, and all power was lost. This writeup discusses the events that followed, and how the plane made a water landing on the Hudson river in Manhattan six minutes after takeoff. All passengers were rescued. Comparisons are also made to 3 other similar water landings with no loss of lives during the years 1956  2009.
4. Generating Random Variates for some distributions, February 2007This note is written for PTC Express, March 2007 about using Mathcad to generate random variates for Gaussian, Rayleigh, exponential and beta distributions. Each step of the program in Mathcad is shown with corresponding figures for generated random variates for these distributions.
5. Sundara Kandam, December 2006The origin of the ballad of Sundara Kandam is unknown. It was given by word of mouth by his grandmother Sundari Ammal alias Lakshmi Ammal familiarly called Etchi Patti. It describes in very colloquial Tamil a particular episode in Ramayana, where Hanuman sets forth to find Sita after Ravana had abducted her. The whole ballad is in the words of Hanuman as he describes to Rama the obstacles he encountered on his way to Lanka, how he found Sita in the Ashoka forest there, and what transpired after that.
6. Bhaja Govindam, August 2006Adi Sankaracharya is one of the greatest of Hindu philosophers if not the greatest, who lived around 780 AD. He is unique in the history of the world as he combined in himself the attributes of a philosopher, a devotee, a mystic, a poet and a religious reformer. This writeup discusses the controversies surrounding his birth, and the translation of his work, "Bhaja Govindam". The verses are given in Sanskrit with their corresponding exact translations in English.
7. Exploring Basic Probability, September 2001This lecture was presented to CACT on September 28, 2001. It discusses some basic ideas of Probability like independence, conditional probability, Bayes' theorem, aspects of combinatorics, hypergeometric distribution, upper and lower bounds on Gaussian tails and concludes with some continuous distributions. Some of the results are new. It is also available at the CACT webpage
My Family
Picture of my family: Standing from left: Ravi, Gayathri, Hemalekha, John Praveen. Sitting from left: Manya, Krishnan, Kamala, Arya, July 2010
Picture by Kishan Photography
(Left) My mother Dharmambal Venkataraman (19111998) with my father B. Venkataraman (19011972) and (right) with his sister's brotherinlaw Sir C. V. Raman (Physics Nobel Prize 1930 for Raman Effect) (19881970), August 1965
Pictures with my cousin (aunt's son) Prof. S. Chandrasekhar (Physics Nobel Prize 1983 for Chandrasekhar Limit)(19101995) and his wife Lalitha Chandrasekhar (19102013) at their then residence in Williams Bay, Wisconsin, USA (Summer 1963). NASA's flagship XRay telescope launched on July 23, 1999 is named "Chandra Observatory" honoring Prof. Chandrasekhar
Pictures by Venkatarama Krishnan
