# Overlapping power-of-cosine windows with constant weight

I would like to refer you to a relevant question I posed on the Math StackExchange: A generalisation of the Pythagorean trigonometric identity.

As can be seen from the question and the answer, what this solution means is that the $(2n)$th-power-of-cosine window will exhibit a constant weight of $2n\binom{2n}{n}4^{-n}$ with an overlap of $1 - \frac{1}{2n}$, i.e.

• $2$nd-power-of-cosine window exhibits constant weight of $1$ with overlap $50\%$,
• $4$th-power-of-cosine window exhibits constant weight of $\frac{3}{2}$ with overlap $75\%$,
• $6$th-power-of-cosine window exhibits constant weight of $\frac{15}{8}$ with overlap $83.\overline{3}\%$, etc.

I trust it is clear without explanation why constant-weight overlapped window schemes are great for application in signal analysis-synthesis. As such, it seems to me this would be an important result in signal processing and, since I am sort of a newbie to signal processing, I find it unlikely that I have discovered something important - it is rather more likely that I am just "reinventing the wheel".

However, I couldn't find anything relevant online - for example, the most relevant result in a Google search for "overlap power of cosine window constant weight" seems to be a page which, in passing, mentions that:

[w]hen four-times-overlapping Blackman windows are plotted (...) a neatly constant sum appears again(...)

So, where can I find material which discusses issues similar to above in detail and, specifically, is there a paper which provides the above result?

## 1 Answer

your result is interesting but you should take a look at :

F. J. Harris, "On the use of windows for harmonic analysis with the discrete Fourier transform," in Proceedings of the IEEE, vol. 66, no. 1, pp. 51-83, Jan. 1978.

doi: 10.1109/PROC.1978.10837

Abstract: This paper makes available a concise review of data windows and their affect on the detection of harmonic signals in the presence of broad-band noise, and in the presence of nearby strong harmonic interference. We also call attention to a number of common errors in the application of windows when used with the fast Fourier transform. This paper includes a comprehensive catalog of data windows along with their significant performance parameters from which the different windows can be compared. Finally, an example demonstrates the use and value of windows to resolve closely spaced harmonic signals characterized by large differences in amplitude. keywords: {Discrete Fourier transforms;Fourier transforms;Frequency;Harmonic analysis;Oceans;Parameter estimation;Signal processing;Signal resolution;Signal sampling;Smoothing methods}, URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1455106&isnumber=31261

There a number of Figures of Merits (FOMs) that are discussed for windows including $cos^k(),\; k=\{1,2,3,4\}$ but not at the generalized overlaps you show. Looking at the main table in Harris's paper, as $k$ increases, many of the FOMs like the ENBW get worse. Also while the constant energy is preserved for your result, the coherent gain of the DFT decreases. The only FOM that touches on overlap is Overlap Correlation.

If you may permit me being blunt, but $cos^{20}$ as windows go, doesn't seem to offer much advantage in general terms. I think that your result is interesting and someone might publish it, but you might hold off going down to the Porsche dealership.