where A is a constant. The above expression is a form of a more general class known as Blashcke (pron. Bloss Kee) products, named after the Austrian mathematician Wilhelm Blaschke. The all-pass filter of order n as above is a

Closely associated with the all-pass filters are Blashcke matrices which are of the form , such that has no poles within the unit circle and is the conjugate transpose. This is similar to the factorization of an all-pass filter [1].

Blashcke products are very useful in digital filter design. A recent book [1] has more information on other applications of Blashcke products.

References:

[1] Mandic D. P. and Goh V. S. L., “Complex Valued Nonlinear Adaptive Filters: Noncircularity, Widely Linear and Neural Models,” Wiley, 2009, p.146-147.

[2] Mashreghi J. and Fricain E., “Blaschke Products and Their Applications,” Springer, 2013.

It is tempting to ask the counter question: if , then does it imply where and are independent RVs? I found the answer in this amazing text:

*Counterexamples in Probability* by Jordan Stoyanov. The book is an adorable compilation of some 300 counterexamples to the probability questions which might be bothering you during a good night-sleep.

So, the answer to my question is no. The counterexample is using the Cauchy distributions. It turns out that the convolution of two Cauchy distributions is always a Cauchy distribution whether and are independent or not.

Coming to think of the convolution of pdfs, our favorite website has a list of convolution of common pdfs.

]]>I learned about Harald’s work in my functional analysis class. By Zorn’s lemma, every Hilbert space admits an orthonormal basis (ONB). However, only separable Hilbert spaces have countable ONB. The ONB for inseparable Hilbert spaces can be constructed by following the principle of transfinite induction with the invocation of the Axiom of Choice/Zorn’s Lemma. One of the examples of ONBs of inseparable Hilbert space is the (Harald) Bohr basis – the almost periodic functions.

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 and and asks for the probability that a sample value of exceeds a sample value of . We are given (notation is borrowed from Hancock’s book),

, and

Since and are statistically independent, we have,

,

where the support of has been changed since for .

Now,

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 , then that of rectified sine wave is . So,

For a deterministic periodic function with period , the spectral density is given by,

where is the Fourier Transform of . 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.

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.

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**Using series expansion:**

For , the power series expansion of the function is given by

and where

Q.E.D.

**Using definite integral:**

For , we have area under the monotonically increasing logarithmic curve lying above the x-axis given by

… (1)

For , we have area under the monotonically increasing logarithmic curve lying below the x-axis given by

Following similar steps as above for , we get,

… (2)

Result follows from (1) and (2).

Q.E.D.

**Using the definition of convexity:**

Let’s assume where and then check the convexity of this function:

Q.E.D.

**Using an information theoretic result:**

This is just a vaporware [4] as of now. However, as shown in [2], this logarithmic inequality can be used to prove Jensen’s Inequality [5] or properties of Kullback-Leibler distance [6]. So is it possible that one of these results can be employed to do the inverse i.e. to prove the above-mentioned logarithmic inequality?

References:

[1] Cover T. M. and Thomas J. A., “Elements of Information Theory,” Wiley-Interscience, 2nd Edition, 2006, Problem 2.13.

[2] Gibbs’ Inequality.

[3] Proof of Gibbs’ Inequality. Please note that this document misspells the inequality as Gib**b’s**.

[4] Vaporware.

[5] Jensen’s Inequality.

[6] Kullback-Leibler Distance.

[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.