Model fitting by least squares |

Equation (1) gives us one way to divide by zero.
Another way is stated by the equation

Both divisions, equation (1) and equation (6), irritate us by requiring us to specify a parameter, but for the latter, the parameter has a clear meaning. In the latter case we smooth a spectrum with a smoothing window of width, say which this corresponds inversely to a time interval over which we smooth. Choosing a numerical value for has not such a simple interpretation.

We jump from simple mathematical theorizing towards a genuine practical application when I grab some real data, a function of time and space from another textbook. Let us call this data and its 2-D Fourier transform . The data and its autocorrelation are in Figure 1.

The autocorrelation of is the inverse 2-D Fourier Transform of . Autocorrelations satisfy the symmetry relation . Figure 2 shows only the interesting quadrant of the two independent quadrants. We see the autocorrelation of a 2-D function has some resemblance to the function itself but differs in important ways.

Instead of messing with two different functions and to divide, let us divide by itself. This sounds like but we will watch what happens when we do the division carefully avoiding zero division in the ways we usually do.

Figure 2 shows
what happens with

antoine10
2-D data (right) and a quadrant of its autocorrelation (left).
Notice the longest nonzero time lag on the data is about 5.5 sec
which is the latest nonzero signal on the autocorrelation.
Figure 1. |
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antoine11
Equation 7 (left) and
equation 8 (right).
Both ways of dividing by zero give similar results.
Figure 2. |
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Model fitting by least squares |

2014-12-01