A few good problems from an old book

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In Fall 2000, I was introduced to the signals and systems through a less popular textbook: An Introduction to the Principles of Communication Theory by J. C. Hancock (1961). My fellow undergraduate students used to tremble at its very sight. The book was laconic in explanations and parsimonious in examples. However, it was (and continue to be) a universally used textbook for undergraduate signals and systems course in several universities. I personally believe that this book should be replaced by more recent classic textbooks in this area since it is highly likely that an uninitiated student will further cloud than clear his understanding after the first reading of this book. Indeed, the book’s review that appeared in 1962 in IRE Transactions On Information Theory was an unfavorable one.

That said, the book is very useful as an interesting handbook on communication theory. It packs signals and systems, communication theory, analog electronics, random variables, probability, detection theory and more all in 253 pages – taking up the award of ingenious technical brevity. It also contains some of the most interesting exercises at the end of each chapter. I have revisited them time and again to verify my evolving comprehension of the subject. In one of my more recent regurgitation of this text, I came across two interesting problems, both from Chapter III: Random Signal Theory. The first problem[1] deals with the probability of random variables. It gives probability density functions of two statistically independent random variables X and Y and asks for the probability that a sample value of x(t) exceeds a sample value of y(t). We are given (notation is borrowed from Hancock’s book),

p(x) = 2ae^{-bx}, 0 \leq x \leq \infty, and

p(y) = ae^{-b|y|}, -\infty \leq y \leq \infty

Since X and Y are statistically independent, we have,

p(x, y) = p(x)p(y), 0 \leq x \leq \infty, 0 \leq y \leq \infty,
where the support of Y has been changed since p(x) = 0 for x < 0.

Now, P(X>Y) = P(X-Y>0) = 1 - P(X-Y \leq 0)

\Rightarrow P(X>Y) = 1- {\int\int_{x'-y' \leq 0} p(x,y)dxdy}

= 1- {\int_0^{\infty}\int_{0}^{y} p(x,y)dxdy}

= 1- {\int_0^{\infty}\int_{0}^{y} p(x)p(y)dxdy}

= 1- {\int_0^{\infty}(\int_{0}^{y} 2ae^{-bx}dx) p(y)dy}

= 1- {\int_0^{\infty}( \frac{2a}{-b}e^{-bx}|_{0}^{y}) p(y)dy}

= 1 - \frac{2a}{-b}{\int_0^{\infty}  (e^{-by} - 1) . ae^{-by} dy}

= 1 + \frac{2a^2}{b}{\int_0^{\infty}  (e^{-2by} - e^{-by}) dy}

= 1 + \frac{2a^2}{b} {\frac{1}{-2b} (e^{-2by} - 2e^{-by})|_{0}^{\infty}}

= 1 - \frac{a^2}{b^2} (0 - 1 - (0 - 2))

\Rightarrow P(X>Y) = 1 - \frac{a^2}{b^2}

The second problem[2] deals with finding the spectral density of a function from its time domain representation. Although the equation of the time domain function is not given, it can be deduced from the diagram that the function is a rectified sine wave. If the period of the sine wave is T, then that of rectified sine wave is \frac{T}{2}. So,

f(t) = |\sin(\frac{2\pi}{2T} t)| = |\sin(\frac{\pi t}{T})|

For a deterministic periodic function f(t) with period \frac{T}{2}, the spectral density G(f) is given by,

G(f) = \lim_{T \rightarrow \infty}\frac{|F_T(f)|^2}{T}

where F_T(f) is the Fourier Transform of f(t). Here,

… and I am still working on posting the entire solution.

References:
[1] Hancock J. C., “An introduction to the principles of communication theory,” McGraw-Hill Book Company, 1961, Problem 3-16.
[2] Hancock J. C., “An introduction to the principles of communication theory,” McGraw-Hill Book Company, 1961, Problem 3-27.

A short overview of CFAR algorithms – I

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Recently, in response to a question on CFAR algorithms on a Linkedin group (sign-in required), I posted a list of CFAR schemes proposed and used so far within the radar community. Later, I thought it would be pertinent to discuss some of the CFAR related concepts in this blog. Unfamiliar readers of the basics of detection theory may please consult a few initial chapters of [1]. The hardware implementation of many CFAR techniques is a challenge to radar engineers. A few years ago, I authored a paper [2] on the FPGA implementation of OS-CFAR where the basic idea derives from preserving the traditional running summer architecture of CA-CFAR implementation.

References:
[1] Kay S. M., “Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory,” Prentice Hall, 1998. 

[2] Mishra K. V. and Kuloor R., “Generalized Configurable Architectures for FPGA Implementation of Ordered-Statistic CFAR,” International Radar Symposium India (IRSI), 2005.

An estimation theory problem involving Gamma probability density function

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Most of the available text in estimation theory frequently harps on Gaussian probability density function which, otherwise an excellent mathematical device, often leads the reader not to explore properties of other pdfs. In Spring 2008, during my ECE 652 (Estimation and Filtering Theory) class, I came across this beautiful problem [1] which uses Gamma pdf in a classic estimation theory question. My solution for the problem goes like this ( code for the following courtesy CodeCogs):

[2],

[3].


References:
[1] Mendel J. M., “Lessons in estimation theory for signal processing, communications and control,” Prentice Hall, 1995, Problem 13-7.
[2] Integral of exponential functions.
[3] Gamma distribution.