I am posting since since we were recently made aware of an error in our journal paper “Deterministic Cramer-Rao Bound for Strictly Non-Circular Sources and Analytical Analysis of the Achievable Gains” (T-SP vol. 64, no. 17). Unfortunately, T-SP does not allow (yet?) to publish errata alongside the original papers, nor to send updates. Therefore, right […]

# Author Archives: admin

## Extended Trigonometric Pythagorean Identities: The Proof

I recently posted on Extended Trigonometric Pythagorean Identities and a “supercharged” version of them, which allows to simplify certain sums of shifted sine functions raised to integer powers. In particular, the claim was that $$\sum_{n=0}^{N-1} \sin^2\left(x+n\frac{\pi}{N}\right) = \frac{N}{2}$$ or more generally for any integer $k$ and $N\geq k+1$: $$\sum_{n=0}^{N-1} \sin^{2k}\left(x+n\frac{\pi}{N}\right) = N \frac{(2k)!}{(k!)^2 2^{2k}} = […]

## Trigonometric Pythagorean Identity, supercharged

You know how they say good things always come back? Well, I recently stumbled over something that reminded me a lot on a post I had made about generalizations of the Trigonometric Pythagorean Identity. In short, the well-known identity $\sin^2(x)+\cos^2(x)=1$ can be generalized to a sum of $N\geq 2$ terms that are uniformly shifted copies […]

## Trigonometric Pythagorean Identity, extended

Here is to another curious and funny identity that has popped up many times and I’ve never quite gotten to find a proof for it: $$ \sum_{n=0}^{N-1} \sin^2\left( x+ \frac{\pi}{N} n \right) = \frac{N}{2}, \; \forall N \in \mathbb{N} \geq 2$$ The Trigonometric Pythagorean Identity What’s the connection to good old Pythagoras you ask? Well, […]

## Widely linear system of equations, revisited

It seems I just cannot stop talking about widely linear equations. Sorry for being to monotonous. My colleague Dominik recently made me aware of the fact that it is possible to solve the widely linear system of equations $$ \ma{A} \cdot \ma{x} + \ma{B} \cdot \ma{x}^* = \ma{c}$$ that I already visited in an earlier […]