# What are the units of Es/N0?

I want to make sure that my units are all correct. I've been using the formula

$$\frac{P_S}{P_N} = \frac{E_S}{N_0} \frac{R_S}{B}$$

The dimensions that I'm confident of are

• $P_S$, $P_N$ : Watts (Joules/sec)
• $N_0$ : Joules/(sec*Hz)
• $B$ : Hz

The last two are then (I think)

• $R_S$ : symbols/sec
• $E_S$ : Joules/symbol

If those are all correct, and given that

$$\frac{E_S}{N_0} = \frac{P_S}{P_N} \frac{B}{R_S}$$

it seems like the units of $\frac{E_S}{N_0}$ are

$$\frac{\text{Hz}}{\text{symbols/sec}} = \frac{1}{\text{symbol}}$$

I was under the impression that it should be dimensionless. Is this correct?

So, in your calculation, the units of $B/R_S$ are $\text{Hz}/\text{Hz}$, making it unitless.
• That makes sense. So I guess $E_S$ is the energy in a given symbol, and the units of $E_S$ are just Joules (the power times the duration). – Trey Apr 9 '18 at 23:00