## Noise Effect On The Probability of Comparator Decision

In the previous post, it is explained why normal distribution with a standard deviation of $\sigma$ is used to characterize the circuit input-referred noise. In this post, we will calculate the probability of comparator decision (High or Low) with the existence of noise.

We ignore the dc offset of the comparator and only consider the thermal-noise effect. Then the distribution of voltages presented to the comparator input, as shown in Fig.1, can be represented by a normal distribution whose standard deviation is equal to $\sigma$ of the comparator input-referred noise and whose mean value is shifted by the signal Vc. Fig. 1 Input-referred distribution of voltages presented to the comparator. The shaded area represents the probability of the comparator thinking the input voltage is low.

The probability of a decision Low is simply given by the area under the curve to the left of zero, which is indicated by the shaded region in Fig.1. This area is given in terms of the error function by (referring to P1, P2, and P3 in Appendix) $P(\textit{Low})=\frac{1}{2}+\frac{1}{2} erf(\frac{-V_C}{\sqrt{2}~ \sigma})$.

Note that the error function is an odd function (P5 in Appendix), we can further write $P(\textit{Low})=\frac{1}{2}-\frac{1}{2} erf(\frac{V_C}{\sqrt{2}~ \sigma})$.

Finally, based on the above result, the probability can be further calculated in terms of the complementary error function by (P4 in Appendix) $P(\textit{Low})=\frac{1}{2} erfc(\frac{V_C}{\sqrt{2}~ \sigma})$.

Similarly, the probability of a decision High is given by the remaining unshaded area under the curve in Fig.1, which is $P(\textit{High})=\frac{1}{2} erfc(\frac{-V_C}{\sqrt{2}~ \sigma})$.

In the next post, we will use the derived equations to show noise effect on the distribution of ADC output codes.

Appendix

Some equations on standard normal distribution:

1. Probability density function (PDF) $\phi(x)=\frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}}$.

If denote the mean as $\mu$ and the standard deviation as $\sigma$, the PDF of general normal distribution can be expressed as $f(x)= \frac{1}{\sigma} \phi(\frac{x-\mu}{\sigma})$.

2. Cumulative distribution function (CDF) $\Phi(x)=P[X \leq x] = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^x e^{-\frac{t^2}{2}}dt$.

3. Error function $erf(x)=\frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt$.

Hence, the CDF can be further expressed using error function $\Phi(x)=\frac{1}{2}+\frac{1}{2} erf(\frac{x}{\sqrt{2}})$.

4. Complementary error function $erfc(x)=\frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}dt = 1 - erf(x)$.

5. Error function is an odd function. $erf(-x)=- erf(x)$.

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