# Effect of the Transition Points Mismatch on Quanta Image Sensors
## 2 The Transition Points Mismatch and Its Influence on the BER in a QIS
### Single-bit QIS  
The readout signal containing shot noise and read noise in a single-bit QIS is shown in Figure 2. After processing by a 1-bit ADC, the signals that are lower than the transition point \(U_{th}\) are quantized as \"0\", or \"1\" otherwise. Because of the existence of read noise, bit-flip errors may occur. In [18], it is suggested that the read noise should be less than 0.30 e\(-\) r.m.s. for photoelectron counting in conventional image sensors. In a QIS, it has been proven that the read noise \(U_{n}\) should be less than 0.15e\(-\) r.m.s to ensure a low BER, which is lower than 0.001 [12]. In fact, the read noise \(U_{n}\) may be different among pixels in the array. In [6], 15-16% variations in read noise were observed for several kinds of specialized jets. However, in the modeling, a uniform read noise is assumed for simplicity. Based on the assumption of such a low read noise, the number of photoelectrons \(k\) could only be miscounted as an adjacent value \(k\)\(-\)1 or \(k+1\), and the probability of being miscounted as \(k\)-2 or \(k+2\), and further values could be neglected. For example, if the transition point \(U_{th}\) is 0.5e\(-\) and the photoelectrons number is 1e\({}^{-}\) and 2e\({}^{-}\), the probabilities of miscounting \(k\) as 0 are 4.3 \(\times\) 10\({}^{-4}\) and 7.9 \(\times\) 10\({}^{-24}\), respectively. So, in a single-bit QIS, there are only two kinds of bit-flip errors: miscounting \(k\) = 1 as \(k\) = 0 and miscounting \(k\) = 0 as \(k\) = 1. The probability of miscounting \(k\) = 1 as \(k\) = 0, \(BER\) (1), is given by the area A shown in Figure 2 and is equal to:  
\[BER\ (1)=\int_{-\infty}^{U_{th}}P[U,1]dU. \tag{3}\]  
The probability of miscounting \(k\) = 0 as \(k\) = 1, \(BER\) (0), is given by the area B and is equal to:  
\[BER\ (0)=\int_{U_{th}}^{+\infty}P[U,0]dU. \tag{4}\]  
The total error rate, \(BER_{T}\), can be obtained by adding (3) and (4) together:  
\[BER_{T}=\frac{1}{2}He^{-H}erfc\left(\frac{1-U_{th}}{\sqrt{2}U_{n}}\right)+ \frac{1}{2}e^{-H}erfc\left(\frac{U_{th}}{\sqrt{2}U_{n}}\right). \tag{5}\]  
The transition points of 1-bit ADCs that are used in the same QIS are slightly different in the spatial domain. If \(U_{th}\) follows a normal distribution in the spatial domain and the average value is 0.5e\({}^{-}\), its PDF \(P_{uth}\left[U_{th}\right]\) is given by:  
\[P_{uth}[U_{th}]=\frac{1}{U_{thn}\sqrt{2\pi}}exp\left[-\frac{(U_{th}-0.5)^{2}}{ 2U_{thn}^{2}}\right], \tag{6}\]  
where \(U_{thn}\) is the standard deviation of the transition point \(U_{th}\). Then, the expected value of the total error rate BER is given by:  
\[BER=\int_{-\infty}^{+\infty}BER_{T}.P_{uth}[U_{th}]dU_{th}. \tag{7}\]The relationship between the BER and \(U_{thm}\) expressed in (7) is plotted in Figure 3, and the normalized read noise \(U_{n}\) is fixed at 0.15e\({}^{-}\). The BER is higher with lower exposure, and it has a maximum value under sparse exposure (\(H\leq 0.1\)). It can be explained as follows. The lower the exposure is, the larger the probabilities of \(k=1\) and \(k=0\) are, and these are the only two cases that can cause bit errors. From Figure 3, \(U_{thm}\) should be lower than 0.15e\({}^{-}\) to ensure that the BER is lower than 1% for all exposure levels. The standard deviation of the transition point in the voltage domain should be lower than 0.15e\({}^{-}\)\(\times\)\(CG\), so a high conversion gain for a jet is essential for column or cluster ADC design. In [19], the proposed pixel achieves a high conversion gain of 220 \(\mu\)V/e\({}^{-}\), and in [5] and [6], tapered-pump-gate jets show a 345 \(\mu\)V/e\({}^{-}\) conversion gain on average. Anyway, the standard deviation of the transition point of 1-bit ADC should be as low as dozens of microvolts.