# Effect of the Transition Points Mismatch on Quanta Image Sensors
## 2 The Transition Points Mismatch and Its Influence on the BER in a QIS
### Multi-bit QIS  
In an \(n\)-bit multi-bit QIS, the jet signals are quantized with \(2^{n}\) quantization levels, and each level corresponds to one electron as shown in Figure 4. An \(n\)-bit ADC has \(2^{n}-1\) transition points (\(U_{th}(1)\) to \(U_{th}(2^{n}-1)\)) to divide the output voltage range into \(2^{n}\) quantization levels, which are called bins and labeled with the number \(N\) in this paper. The signals corresponding to photoelectron numbers larger than \(2^{n}-1\) are quantized as \(2^{n}-1\). Figure 3: The bit error rate (BER) for a single-bit QIS as a function of the standard deviation of the transition point for different exposure levels with the read noise fixed at \(U_{n}=0.15\)e\({}^{-}\). ADC, analog-to-digital converter. Figure 2: A readout signal of a single-bit quanta image sensor (QIS). Due to the existence of read noise, the signal corresponding to \(k\) photoelectrons can be quantized as other values. The miscounting probability \(BER(k)\) is the area of the probability density distribution corresponding to \(k\) photons spilling out of bin \(N=k\):  
\[BER(k)=P[k]-\int_{U_{th}(k)}^{U_{th}(k+1)}\frac{P[k]}{U_{n}\sqrt{2\pi}}exp\left[ -\frac{[U-k]^{2}}{2U_{n}^{2}}\right]dU. \tag{8}\]  
Note that when \(k=0\) and \(k=2^{n}-1\), \(U_{th}\) (\(0\)) = \(-\infty\) and \(U_{th}\) (\(2^{n}\)) = \(+\infty\), respectively, in (8). In the case of \(k\geq 2^{n}\), the bit errors could be neglected due to the small read noise. Then, the total error rate \(BER_{T}\) is given by:  
\[ \begin{array}{ll}BER_{T}&=\sum\limits_{0}^{2^{n}-1}BER(k)\\ &=\frac{1}{2}\sum\limits_{k=0}^{2^{n}-2}P[k]\cdot erfc\left[\frac{U_{th}(k+1 )-k}{U_{n}\sqrt{2}}\right]+\frac{1}{2}\sum\limits_{k=1}^{2^{n}-1}P[k]\cdot erfc \left[\frac{k-U_{th}(k)}{U_{th}\sqrt{2}}\right].\end{array}  \tag{9}\]  
In a real image sensor, the transition point \(U_{th}(k)\) of each ADC is not totally the same. In a multi-bit QIS, a pre-amplifier would be employed due to its small signal level, so all of the transition points in one ADC have the same deviation from their design values in this paper. We assume that the transition points follow a normal distribution in the spatial domain. The design value of transition point \(U_{th}(k)\) is \(k-0.5\). We label the standard deviation as \(U_{thm}\), which is normalized just as \(U\), \(U_{n}\), etc. Then, the PDF of \(U_{th}(k)\) is given by:  
\[P_{uth}[U_{th}(k)]=\frac{1}{U_{thm}\sqrt{2\pi}}exp\left[-\frac{[U_{th}(k)-k+0. 5]^{2}}{2U_{thm}^{2}}\right]. \tag{10}\]  
Then, the expected value of the total error rate BER is given by:  
\[ \begin{array}{ll}BER=&\frac{1}{2}\sum\limits_{k=0}^{2^{n}-2}P[k]\cdot\int_{ -\infty}^{+\infty}erfc\left[\frac{U_{th}(k+1)-k}{U_{n}\sqrt{2}}\right]\cdot P _{uth}[U_{th}(k+1)]d[U_{th}(k+1)]\\ &+\frac{1}{2}\sum\limits_{k=1}^{2^{n}-1}P[k]\cdot\int_{-\infty}^{+\infty} erfc\left[\frac{k-U_{th}(k)}{U_{n}\sqrt{2}}\right]\cdot P_{uth}[U_{th}(k)]d[U_{th}(k)]. \end{array}  \tag{11}\]  
In fact, Equation (11) can also apply to the case of \(n=1\). Figure 5 shows the relationship between the BER and the standard deviation \(U_{thm}\) for different bit depths of the QIS, and the BER becomes larger with an increasing \(U_{thm}\). The dashed line represents a constant \(5\%\) reference. Because a bit error has a smaller impact on a multi-bit QIS than a single-bit one, a higher reference value is set. As a conclusion, the BER performance is better in the multi-bit QIS with a higher bit depth. As shown in Figure 4, because of the symmetry of the normal distribution, the probability distribution of the signal corresponding to \(k\) photoelectrons spills out into bins \(N>k\) and bins \(N<k\) for \(1\leq k\leq 2^{n}-2\). For \(k=0\) and \(k=2^{n}-1\), their distributions could only spill out into bins on one side, so less bit errors occur. For \(k\geq 2^{n}\), the signals hardly cause bit errors. When the  
Figure 4: The probability distribution of the output signal and the quantization of the multi-bit QIS.