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Smoothing the denominator spectrum

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

\begin{displaymath}
X(\omega) = \frac{ \overline{F}(\omega) Y(\omega) }
{\left<
\overline{F}(\omega) F(\omega)
\right> }
\end{displaymath} (6)

where the strange notation in the denominator means that the spectrum there should be smoothed a little. Such smoothing fills in the holes in the spectrum where zero-division is a danger, filling not with an arbitrary numerical value $\epsilon$ but with an average of nearby spectral values. Additionally, if the denominator spectrum $\overline{F}(\omega) F(\omega)$ is rough, the smoothing creates a shorter autocorrelation function.

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 $\Delta\omega$ which this corresponds inversely to a time interval over which we smooth. Choosing a numerical value for $\epsilon$ 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 $f(t,x)$ and its 2-D Fourier transform $F(\omega, k_x)$. The data and its autocorrelation are in Figure 1.

The autocorrelation $a(t,x)$ of $f(t,x)$ is the inverse 2-D Fourier Transform of $\overline{F}(\omega, k_x) F(\omega, k_x)$. Autocorrelations $a(x,y)$ satisfy the symmetry relation $a(x,y)=a(-x,-y)$. 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 $X$ and $Y$ to divide, let us divide $F$ by itself. This sounds like $1=F/F$ 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

\begin{displaymath}
1 = F/F \quad \approx \quad \frac{\overline{F}F}{\overline{F}F+\epsilon^2}
\end{displaymath} (7)

and with
\begin{displaymath}
1 = F/F \quad \approx\quad \frac{\overline{F}F}{\left< \overline{F}F \right>}
\end{displaymath} (8)

antoine10
antoine10
Figure 1.
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.
[pdf] [png] [scons]

From Figure 2 we notice that both methods of avoiding zero division give similar results. By playing with the $\epsilon$ and the smoothing width the pictures could be made even more similar. My preference, however, is the smoothing. It is difficult to make physical sense of choosing a numerical value for $\epsilon$. It is much easier to make physical sense of choosing a smoothing window. The smoothing window is in $(\omega,k_x)$ space, but Fourier transformation tells us its effect in $(t,x)$ space.

antoine11
antoine11
Figure 2.
Equation 7 (left) and equation 8 (right). Both ways of dividing by zero give similar results.
[pdf] [png] [scons]


next up previous [pdf]

Next: Imaging Up: UNIVARIATE LEAST SQUARES Previous: Damped solution

2014-12-01