I have shown so far how do develop a low-pass filter in order to eliminate lag in smoothed time-series.  It works very well in trending markets, but has slight lag in cyclic markets.  I now wish to create a filter that has zero lag in cyclic markets.  This will take the form of a high-pass filter because it will remove slow changing trend components and only display smoothed cyclic components with near zero lag.

 The reason I’m designing these filters is so that I can smooth my neural network input data whilst avioding lag which comes as a consequence of smoothing.  This would improve the learning capabilities of my neural networks, and forecasts would eventually become more accurate when I decide to use Genetic Algorithms to optimise a population of network structures.