# Shot noise variance of a channel for different bits

I've come across a question, where I had to calculate shot noise variance calculations of an optical link.

As per the definition, shot noise in an optical channel is the current that arises from the statistical nature of production and collection of photo electrons. In this question, I had a data given; dark current (Id), that is the current from electron/hole pairs that are thermally generated in the pn junction.

The calculations I had to make is the shot noise variance for bit 1 and bit 0 separately. I could calculate the photo current and using that I can calculate the shot noise variance for bit 1. The issue is with bit 0.

I'm asking this question here, to clarify the fact that if it is correct, if I use dark current as the photo current when calculating the shot noise variance for bit 0 as there is no incident light as per the bit level and the only current that will be available to generate shot noise is the dark current itself.

For reference, following are the equations to calculate variance of shot noise current and variance of dark current;

$$ = 2qI_pB_e$$ $$ = 2qI_dB_e$$

if I use dark current as the photo current when calculating the shot noise variance for bit 0 as there is no incident light as per the bit level and the only current that will be available to generate shot noise is the dark current itself.

If the system in fact sends no light for a 0 bit, then you are correct. The only source of shot noise would be the dark current.

But this is a very rare design. Most transmitters will have very poor performance if they are shut off completely for the 0 bits. So we typically send a low but non-zero power for 0 and a higher power for 1.

This is usually characterized by the extinction ratio of the system, which is the ratio between the power sent for a 1 to the power sent for a 0.

$${\rm ER}=\frac{P_1}{P_0}$$

If you need to calculate the shot noise for a system with finite ER ($$\P_0>0\$$), you will need to use what you know (some combination of 1-level power, ER, and average power) to determine the 0-level power, and then calculate the shot noise from there.

Note: you should also make sure the dark current is much lower than the 1-level current before neglecting it when calculating the shot noise for the 1 level.