# Generalized/Power means in DSP

Generalized/power means may be used to construct moving-average filters with different properties than regular one which is based on arithmetic mean. This observation seems to be trivial - even Wikipedia mentions that. The problem is that power means are usually defined for sets of positive numbers - while in DSP we often deal with negative numbers too. Therefore, many of well-known inequalities and properties of power means may not be applied to sets with negative numbers.

This is of interest to me since I have found that in my partical area of application power means of higher order give better results than arithmetic mean. I have been able to prove some basic properties and I more or less understand where does the improvement come from. However, I would be interested in learning. However, everything I have found seems to focus on positive numbers. So my question is:

Are there any papers on using generalized means for moving average filtering or some other signal processing? Any papers on purely mathematical properties of generalized means for sets with negative values may be relevant too. Thanks!

## 1 Answer

Certain varieties of generalized means have been applied in signal processing, such as the Lehmer mean. This paper discusses several generalized means, mainly from a theoretical point of view.

Note, however, that all applications of generalized means deal with signals that are inherently non-negative. This is also the case in signal processing, where there are many applications with non-negative signals, such as frequency domain denoising, image processing, fuzzy logic, etc. The application of generalized means in image processing is discussed in this thesis.

I think that rather than looking for new definitions of generalized means for bipolar signals, it is probably more useful to define the signals properly such that generalized means can be applied.

A completely different alternative to generalized means might be general non-linear filters, such as Volterra filters, which don't have the restriction that the input signal must be non-negative.

• Thanks! I knew the paper by Wilkin and Beliakov, but the thesis is something new to me. – tms Feb 17 '16 at 12:38