Setting the relative gain

Event distributions of analog signals from different BSD counters are similar in shape, but due to different counter characteristics, such as threshold, light collection efficiency etc., the position and height of the maximum of the distribution varies from counter to counter (see Fig. 1a). However, all BSD counters should have tails of adc spectra (aBSD $\ge$ 200) that are similar both in shape and magnitude. Thus, instead of matching maxima or mean values of adc distributions, it is preferable to match the tails of distributions by matching integrated adc spectra to some chosen reference counter within each layer. This method makes the matching insensitive to distortions of the spectra at low pulse height that come from discriminator thresholds.

The first step in setting relative gains is to choose some point $e_{0}$ above threshold in the adc distribution of the reference channel and calculate the integral from this point to infinity. The second step is to find the appropriate value of e from each other counter so that the corresponding integrals are equal to the reference one

$$\int_{e}^{\infty} \frac{dN}{de} de = I^{ref} = \int_{e_{0}}^{\infty} \frac{dN}{de} de.$$ (1)

The next step is to scale the gain of each counter so that e lines up with $e_0$. In general, the signal is proportional to some gain constant g when pedestal p is subtracted

$$e' = g(e-p).$$ (2)

If the pedestal value is already subtracted, then from Eq. 2 we can get the gain coefficient needed to make scales equal ($e' = e_0$)

$$g = \frac{e_{0}}{e}.$$ (3)

We decided to calculate g for two different integration limits $e_0$ and to take the average value of g. Counter R03 is chosen to be the reference one for all other counters. In Table 1 are given gain constants obtained for two chosen reference values $e_{0_1}$ = 80, and $e_{0_2}$ = 120. Multiplying adc signal by the corresponding g one can set gains in different BSD counters relative to the chosen one. In Fig. 1b one can see adc spectra from several straight counters after their relative gain has been applied.

Table 1: Gain constants set relative to counter R03.

	Counter		$g_{1} = \frac{e_{o_1}}{e_1}$		$g_{2} = \frac{e_{o_2}}{e_2}$		g		$\vert g_1 - g_2 \vert$
	R00		0.8081		0.8219		0.8150		0.0138
	R01		0.6897		0.7018		0.6957		0.0121
	R02		0.6504		0.6704		0.6604		0.0200
	R03		1.0000		1.0000		1.0000		0.0000
	R04		0.5369		0.5333		0.5351		0.0036
	R05		0.9091		0.9375		0.9233		0.0284
	R06		0.9091		0.9231		0.9161		0.0140
	R07		0.9756		0.9917		0.9837		0.0161
	R08		1.0667		1.0909		1.0788		0.0242
	R09		0.7143		0.7229		0.7186		0.0086
	R10		0.7018		0.6977		0.6997		0.0041
	R11		0.8696		0.9023		0.8859		0.0327
	L12		1.3333		1.3483		1.3408		0.0150
	L13		1.1765		1.2000		1.1882		0.0235
	L14		1.1594		1.1538		1.1566		0.0056
	L15		1.2500		1.2903		1.2702		0.0403
	L16		0.8989		0.8824		0.8906		0.0165
	L17		1.0526		1.0435		1.0481		0.0092
	L18		1.0811		1.0909		1.0860		0.0098
	L19		1.3115		1.3333		1.3224		0.0219
	L20		0.9639		0.9756		0.9697		0.0118
	L21		1.1111		1.1215		1.1163		0.0104
	L22		0.8247		0.8163		0.8205		0.0084
	L23		0.9877		0.9524		0.9700		0.0353
	S24		0.9302		0.9023		0.9162		0.0280
	S25		1.0127		1.0526		1.0326		0.0400
	S26		1.0959		1.1321		1.1140		0.0362
	S27		1.1111		1.1429		1.1270		0.0317
	S28		1.1594		1.1881		1.1738		0.0287
	S29		1.1765		1.1765		1.1765		0.0000
	S30		0.7921		0.7643		0.7782		0.0277
	S31		0.7339		0.7547		0.7443		0.0208
	S32		1.2903		1.2903		1.2903		0.0000
	S33		0.8247		0.8571		0.8409		0.0324
	S34		0.8989		0.9231		0.9110		0.0242
	S35		0.8889		0.9023		0.8956		0.0134
	S36		0.8989		0.9160		0.9075		0.0172
	S37		1.0811		1.0811		1.0811		0.0000
	S38		0.8602		0.8759		0.8681		0.0157
	S39		1.0390		1.0345		1.0367		0.0045
	S40		1.0127		1.0256		1.0191		0.0130
	S41		0.8247		0.7843		0.8045		0.0404
	S42		1.0127		0.9836		0.9981		0.0291
	S43		1.2698		1.2903		1.2801		0.0205
	S44		1.0526		1.0619		1.0573		0.0093
	S45		0.8791		0.8571		0.8681		0.0220
	S46		1.0667		1.0811		1.0739		0.0144
	S47		1.1111		1.1215		1.1163		0.0104

Figure 1: Event distribution of adc signal for counters S24-S28 plotted on top of each other before (a) and after (b) gain is matched to the reference channel (R03).