# Presence of bandwidth in capacity formula

In the first paper, achievable rate is

w_n is bandwidth. p_n is transmit power. h is channel gain.

And in second paper, achievable rate is

P is transmit power. H is channel gain.

What I curious is the difference between two same formula, the presence of bandwidth.

In first formula, there is bandwidth, however in second formula, there is no bandwidth in there(in the fraction part, denominator). What's the difference?

• $w_i$ seems to be the bandwidth term in the second formula, so what's the question about? Commented Jun 7, 2022 at 11:13
• the question is that there is no bandwidth in second formula(in the fraction part, denominator) Commented Jun 7, 2022 at 11:15
• Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking.
– Community Bot
Commented Jun 7, 2022 at 11:37
• @MarcusMüller sorry for my misclarifing my problem Commented Jun 8, 2022 at 2:45

$$\N_0\$$ is the noise density, something that gives power per bandwidth, and since power is energy per time, and time is the inverse qunatity to bandiwdth, $$\N_0\$$ is an energy.
It makes no sense to divide a power $$\P_i\lvert H_i\rvert^2\$$ by an energy, and add it to 1, because you can't add a dimensionless number 1 to something that is a bandwidth.