Subject: HRC tap corrections

1  Introduction

2  The Shape of Data

The middle solid-line curve is a hyperbola that follows the locus of event points. The upper and lower solid-line curves are hyperbolas that are a bounding region for a screening test used to preferentially select x-ray events relative to background events or events that otherwise have poorly determined positions.

Figures 2 and 3 are similar plots for two other subsets from the same observation.

The data for the three subsets are selected based on pulse height, which is a measure of the total charge collected on the position sensing grid. In both of the latter two cases, the locus of points on the positive fine position side follow the same curve as the data displayed in figure 1, while most of those on the negative side are systematically off of the hyperbola that describes the shape of good positions. Moreover, between figures 2 and 3 there are differences in their deviations from the locus of ``good'' points which depend on the size of the signal measured on the grid. One difference between the two to note is that the fine position at which the discontinuity off of the curve occurs becomes more negative as the pulse height increases. This effect can be explained by the fact that the ``ringing'' requires there to be an even number of taps exceeding a threshold and larger signals allow the events to occur farther away from the center of the tap before the number of taps above threshold changes from three to two.

The event data plotted in figures 1, 2, and 3 has already been somewhat processed; better insight into how the data is being corrupted can be obtained by plotting raw tap data. Plotted in figure 4 are raw v-axis tap data selected from the positive side of the central eight coarse positions (CRSV = 26-33, CRSU = 26-33) from the least sensitive amplifier scale (AMP_SF = 3).

Only two of the three tap values are plotted; the abscissa is the central tap (i.e. the coarse position tap) signal, while the ordinate is the negative side tap signal (the one from the side opposite where the event occurred). The solid lines from the origin show the expected linear relationship between the tap values at the center and positive edge of the tap as the signals increase in amplitude. Figure 5 is a similar plot for the negative side of the taps with the ordinate being the tap signal from the positive side of the tap.

The corruption of the tap data for events occurring on the negative side of the tap can be seen as the sinusoidal pattern relative the expected linear relationship with increasing amplitude.

3  Sinusoidal Deviations

Figure 7 shows the AV3 deviations from the expected value for only those events for which AV1/AV2 is in the range 0.98-1.00; these are the events near the negative edge of the tap.

The amplitude of the sinusoid clearly increases with the size of the central tap signal (AV2). What is less obvious from the figure is that the sinusoid does not have a fixed period; the period appears to increase with increasing central tap signal. Also plotted in figure 7 is a sinusoid with amplitude and period that are linear functions of the center tap value:

(   AV2 + b a   )   sin   (   2Pi AV2 - f cAV2 + d   )   .

The period function can be determined from the ``zero crossing'' points of the deviations from the expected value. The amplitude function can then determined from the observed deviations at AV2 values where the sine function has a magnitude of 1. Since there are only two places where the sinusoid goes through zero, there is a multiplicity in period solutions depending on the phase adopted for the zero-crossing points (e.g. 1.0 and 1.5, 1.5 and 2.0, etc.) even after a value for f is selected. For these data, a phase of 1.5 for the zero crossing at AV2 = ~  1640 appears to match the data reasonably well with an imposed constraint of f = 0. Table 1 lists the sinusoid coefficients used in this plot.

Figure 8 shows a similar plot for events with AV1/AV2 in the range 0.295-0.305.

In this figure the sinusoid that is plotted with a dashed line is the same as the one in figure 7; clearly, the zero-crossing points have shifted. The sinusoid plotted with a solid line differs from the one plotted with a dashed line only by the value of f (f = 66). Since the zero-crossing points shift as AV1/AV2 changes, the sinusoid should be modified by making f a function of AV1/AV2, f(AV1/AV2).

Figure 9 is a plot of the lower of the two zero-crossing points as a function of AV1/AV2.

The solid curve is a function of the form:

a+ b   (   1 (AV1/AV2)g - 1   )   ,

with in this case: a = 1640, b = 75, and g = 0.7. The functional form of f(AV1/AV2) is the same this curve but the constants a and b will change. Since we have determined the amplitude and period functions for the sinusoid from the AV1/AV2  ~  1 range while imposing the constraint f = 0, then a is forced to be zero. The value of b is modified by the choice of the phase of this zero crossing point to be 1.5; b = 75/1.5 = 50. Thus we have:

f(AV1/AV2) = 50((AV2/AV1)0.7 - 1)

The u-axis of the detector is affected as well and the sinusoid that matches the distortion can be found in a similar fashion. Figure 10 displays the u-axis data analogous to the v-axis data shown in figure 7. In this figure the data are selected to have AU1/AU2 in the range 0.98-1.00 so that they are from the negative edge of the tap.

The coefficients of the sinusoid for the u-axis are different from those for the v-axis and are given in table 2.

Again, in determining the sinusoid coefficients for these data, a constraint of f = 0 has been imposed. As with the V-axis, the sinusoid shifts in a systematic way as the AU1/AU2 selection range changes. Figure 11 shows how the lower zero-crossing point of the U-axis sinusoidal distortion changes with AU1/AU2.

The solid curve has the same functional form as the curve in figure 9 but with a = 1290, b = 12, and g = 2.0.

4  Sinusoidal Corrections

• AMP_SF = 3
• A₁ > A₃ for the axis - one or both can be affected
• A₁ > constant×A₂ - the constant is different for each axis and could be zero.

Once the affected events are identified the corrupted tap value can be corrected:

A3corrected = A3telemetered -   (   A2+ b a   )   sin   (   2Pi A2 -e((A2/A1)g- 1) cA2 + d   )

Table 3 lists the values of the coefficients for sinusoids for both axes of the HRC-I along with the threshold on A₁/A₂ above which the correction should be applied.

This function form of the correction with the coefficients listed in table 3 was applied to an HRC-I observation of 3C273 (ObsID 461) as a test of this correction scheme. Figures 12 and 13 display the events from the negative side of the v-axis and u-axis taps respectively with the corrections applied to the affected events; in both cases the sinusoidal distortion is greatly diminished.

Figure 12 can be compare directly to a ``before'' picture (from a different data set) in figure 5 and the raw data from the positive side of the v-axis taps in figure 4.

Figure 14 is an image of 3C273 produced using the corrected data and a linear degapping correction on each axis (U: 1.070, V 1.044).

The data have been screened to reduce the background events and events with very poorly defined positions. The central image is more azimuthally symmetric than an image produced using the original data. Figure 15 gives the encircled fraction of events for the source versus radius in pixels.

5  Remaining Work

Since the hardware cause for the ringing distortion is known, there is work in electrical engineering circuit modeling of the response of the HRC electronics. The modeled performance needs to be compared in detail to the laboratory version of the HRC electronics. It is possible that the knowledge gained from an improved understanding of the circuit performance can lead to a better algorithm for the correction, one that originates from first principles rather than being derived empirically.

Dr. Michael Juda
Harvard-Smithsonian Center for Astrophysics
60 Garden Street, Mail Stop 70
Cambridge, MA 02138, USA
Ph.: (617) 495-7062
Fax: (617) 495-7356
E-mail: mjuda@cfa.harvard.edu