Hilbert Indicators Tell
You When To Trade by John Ehlers
On Lag, Signal Processing, And The Hilbert Transform
Here's one way to control moving average lag, using a little
math and a little-known algorithm called the Hilbert transform to come up with indicators telling you when to trade.
Two characteristics of moving
averages are that they smooth
the input data and they lag the
input data. Their use and application is almost always a
tradeoff between these two
characteristics. The smoothing
function removes the higher-frequency components (that is, the rapid up and down movements)of the input prices, so
moving averages are also referred to as low-pass filters by
engineers. This means moving averages display or allow to
pass through only the low-frequency components (that is, the
slow up and down movements) while removing the high-frequency components. Essentially, what you’ll see instead
of raw prices jumping around is a smoothly moving line
slowly oscillating up and down.
Moving average lag is perhaps the most important characteristic for traders to understand quantitatively. Figure 1 shows how a simple moving average is formed. Data within the observation window is averaged to produce a single point. The observation window (the dotted box) is moved forward in time from bar to bar to form a continuous moving average. If the weighting of the data values within the observation window is uniform, the average value of the data is centered in the horizontal dimension of the window and is also centered in the vertical dimension of the window.