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6 The Mk I auto-folder

The basic folding technique is developed from the idea that any function, $y \equiv f(x),~-X \leq x \leq X,~~X > 0$ can be decomposed into two parts, one of which is asymmetric on the region about x=0 and one of which is symmetric on the region about x=0.

In the present case, and for ideal data, once $V_{{\rm sys}}$ has been found and accounted for by subtraction from the measured rotation velocities, then the resulting processed rotation curve should be perfectly asymmetric about $O_{{\rm dyn}}$, its dynamical centre. However, suppose we have $V_{{\rm sys}} \approx V'_{{\rm sys}}$ and $O_{{\rm dyn}} \approx O'_{{\rm dyn}}$ then, after subtracting $V'_{{\rm sys}}$ the curve will be only approximately asymmetric about the assumed dynamical origin, $O'_{{\rm dyn}}$ - and can therefore be considered composed of an exact asymmetric part together with an exact symmetric part. The basic folding technique therefore consists in minimizing the symmetric component with respect to variations in the estimates of $V_{{\rm sys}}$ and $O_{{\rm dyn}}$.

6.1 The Fourier decomposition

The symmetric component, $f_{{\rm sym}}(x)$, of any function $y \equiv f(x)$ defined on the interval $-X \leq x \leq X,~~X > 0$ can be calculated using the finite cosine transform:
$\displaystyle f_{{\rm sym}}(x)$ = $\displaystyle \sum_{m=0}^\infty A(m) \cos (m \pi x /X),$  
A(m) = $\displaystyle {1 \over 2 \,X} \int_{-X}^X f(x) \cos (m \pi x/X) ~{\rm d}x.$ (2)

If the function represents a rotation curve then, in practice, $y \equiv f(x),~-X \leq x \leq X$ is replaced by a numerical function defined at N+1 points on the discretized interval $-X \leq x \leq X$. Suppose that this numerical function is given as (x0,f0),...,(xN,fN). In the ideal case, for which x0=-X and xN=X, then the Fourier component, A(m), can be approximated by a direct numerical integration of (2). However, the more usual case will be that the numerical function is defined over a non-symmetric interval (for example, $-1.83 \leq x \leq 2.47$). In this case, the requirements of the cosine transform makes it necessary to discard and/or interpolate points of the numerical function so that it becomes defined over the largest symmetric interval that can be fitted into the original non-symmetric interval; in the case of the example, this would be $-1.83 \leq x \leq 1.83$.

6.2 How many Fourier modes should be computed?

In very simple situations, for which the numerical function is defined at equal intervals, $\Delta$ say, in the independent variable, there is no point in calculating Fourier modes which have a wave-length less than $2 \Delta $. This leads to the standard rule that the computation of N Fourier modes requires a minimum of 2N + 1 data points, when these are equally spaced. Conversely, a given number of points then informs how many Fourier modes it is worthwhile including.

  \begin{figure}\includegraphics*{fig3.eps}\end{figure} Figure 3: PS solution compared with the Mk I auto-folder solution. The vertical lines indicate the positions of the A, B, C, D and E peaks in the PS solution

However, rotation curve data is very frequently defined over non-equally spaced intervals and, in this case, the basic rule has to be replaced by the rule that there is no point in calculating Fourier modes with a wave-length less that $2 \Delta_{{\rm max}}$, where $\Delta_{{\rm max}}$ is the largest interval separating adjacent points on the rotation curve.

6.3 The minimization procedure

The foregoing describes how to compute the symmetric Fourier modes for given estimates of $V_{{\rm sys}}$ and $O_{{\rm dyn}}$. Suppose M of these are computed, then we form the functional

\begin{displaymath}F(M) = \sum^M_{m=0} \left[ A(m) \right]^2

and minimize this with respect to variations in the estimates of $V_{{\rm sys}}$ and $O_{{\rm dyn}}$. The very noisy nature of rotation curve data requires that a very robust minimization procedure, using no derivatives, should be employed. A routine based on the Simplex algorithm (Nelder & Mead 1965) was found to be very effective in the present case. When applied to the PS sample, this Mk I auto-folder gives the $\ln A$distribution shown in Fig. 3, right. We immediately see that peaks B, C and D of the PS solution (reproduced in Fig. 3 left) are excellently reproduced by the Mk I auto-folder; however, peak E is lost in noise.

It is worth emphasizing that the reproduction, in Fig. 3 right, of the major part of the peak structure of the PS solution confirms that this structure is not an artifact of the PS procedure but is, at the very least, inherent to the sample.

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