Blaschke products in signal processing

Leave a comment

An all-pass filter has a constant gain over all frequencies. For every pole, the filter has a conjugate-reciprocal copy in a zero. An all-pass filter may have several conjugate-reciprocal pole-zero pairs. The general form of the transfer function for such a filter is
H(z) = A \prod\limits_{k=1}^{n}\left(\frac{z - c_k}{1 - c^*_kz} \right)
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 finite Blashcke product. Finite Blashcke products were around for a while (but not addressed so) even before Blashcke proposed their infinite counterparts in 1915.

Closely associated with the all-pass filters are Blashcke matrices which are of the form \mathbf{B}(z) \mathbf{B}^*(z^{-1}) = \mathbf{I}, such that \mathbf{B}(z) has no poles within the unit circle and \mathbf{B}^*(z^{-1}) 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.

[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.

Bohr (not Niels) revisited

Leave a comment

Danish physicist Niels Bohr is not an unknown scientist to a typical passionate high school physics student. However, until my graduate school studies, I didn’t learn much about the work of his brother Harald Bohr, who was a mathematician. Both did pioneering work in sciences but Niels won the Nobel. Both were passionate footballers but Harald played in Olympics. Their father Christian – a professor of physiology – remarked that Harald was brilliant but Niels was special.

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.