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Uncertainty along the padrow (x and y directions)

The hit position in the padrow direction is fitted with a 3-point Gaussian parametrization [3]. This fit is performed on the projection of the ADC values onto the padrow; the pad with the largest projection (highest ADC), and its two neighbors are used. Neglecting for the moment finite- and finite electron number effects, the uncertainty of measuring the centroid of the Gaussian shows a linear dependence on the uncertainty of the ADC projections [8]:


where and are the pad projections and their uncertainties, respectively, is the centroid position relative to a pad edge, and is the pad pitch.

What is the uncertainty on the pad projection () to use in Equation 6? The noise for the signal on each pixel comes from 3 sources [1]: (1) electronic noise originating before or in the shaper amplifier (SAS), (2) electronic noise in the switched capacitor array (SCA) chip, and (3) digitization noise-- here meaning the ``noise'' associated with rounding a number to integer value. The noise contribution from the second and third terms varies randomly from bucket to bucket, and so is ``white'' noise. However, the noise signal coming through the SAS is filtered with the repsonse function of the amplifier, and so the noise signal on one bucket is correlated with that on the following bucket; see Figure 12.

Figure 12: A random (``white')' noise spectrum is compared to a filtered noise spectrum, offset on the y-axis. Time sampling corresponds to using 1024 time buckets on each side of the TPC. Both spectra are normalized to have unit RMS. Noise coming through the SAS will be correlated in time, due to finite bandwidth; therefore, the error on the sum of consecutive time buckets will not add in quadrature.

If the SAS noise were white () then the error on a pad projection would be


where in ADC counts, and is the number of pixels that contribute to the pad projection.

If the SAS had an extremely long time constant (), then all pixels contributing to a pad projection would be shifted by the same amount by the SAS, so that


Equations 7 and 8 represent extreme cases, as is in reality on the order of the sampling time. Since the SAS contributes heavily to the noise (800 RMS, corresponding to 1.19 ADC counts, as compared to 0.45 and 0.71 ADC counts RMS from SCA and digitization, respectively), it does matter which equation one chooses. Simulations with 512 time samples in P-10 suggest that N9, so that the differs by a factor of 2.5 between the two approximations. Comparison of uncertainties to residual distributions suggests that Equation 8 is the better approximation for our situation, and that is what is used in TPH. See also Section 6.3.

Calculation of the centroid uncertainty for a 3-point Gaussian fit is straightforward. However, the facts that (1) the signal originated through the creation of a finite number of primary electrons (or electron clusters), and (2) from a track that crosses the padrow at a non-zero angle , also affect the spatial resolution in the TPC, quickly dominating it, in fact. The position resolution then takes a form [9]


where represents the centroid uncertainty from Equation 6, and L is the drift distance for the hit. The second and third terms represent the contributions to the uncertainty from finite electron number and finite crossing angle effects, respectively. In Equation 9, and are constants, depending only on detector geometry and TPC gas [9]. These constants must be determined empirically. See Section 6.3.

Note the competing dependencies in the three terms in Equation 9. For example, the first and second terms decrease with increased signal size, while the third increases. This is important, as this third term quickly dominates the resolution. See Figure 13 for calculations of the spatial resolution in the outer sector using P-10 gas in the TPC. It is seen that the so-called ``intrinsic resolution,'' which depends on the signal:noise and is given by Equation 6, rapidly becomes insignificant compared to finite- effects, especially for larger signals. We note that this behaviour, in particular, may change if delta electron production changes in STAR Geant simulations.

Figure 13: Position resolution as a function of crossing angle in the outer sector using P-10 gas, according to our parametrization. Resolution for minimum ionizing particle hits (m.i.p) and hits 4 times as large are shown. Curves are shown for zero noise (dotdash lines), the default settings of the slow simulator (dotted curves), twice the default noise in the simulator (dashed curves), and 4x the noise in the simulator (solid curves). This last case corresponds roughly to the situation considered in the STAR CDR, where, in addition, a slightly different pad geometry was assumed.

For this reason, the position uncertainty cannot be determined at the hitfinding level. Any attempt at this level to produce an ``average error'' whose distribution resembles that of the average residual distribution would be an unsatisfactory comprise that overestimates the uncertainty of some hits while underestimating that of others. TPH enters the value given by Equation 6 in TPHIT.DX and TPHIT.DY. From this information, knowlege of the constants and , the energy loss of the track (in TPHIT.Q), the drift distance (from TPHIT.Z), and a hypothesis of the crossing angle , the position uncertainty can be calculated hit-by-hit. Tracking software must then update these uncertainties DX and DY, as information becomes available about the parent track.

next up previous
Next: Uncertainty in time Up: Position uncertainties Previous: Position uncertainties

Michael A. Lisa
Tue Feb 6 15:49:35 EST 1996