# Carrier-to-Noise Density Ratio (C/N0) Explained

## What is Carrier-to-Noise Density Ratio (C/N0)?

Carrier-to-noise density ratio (C/N0) is the ratio of carrier power to noise power density and denotes the strength of the power of carrier wave relative to the noise.

C/N0 is often used in calculating link budget, and it describes the clarity of signal.

Thus, the receiver should keep the value of C/N0 to the available level by reducing the loss of the receiver system.

## Derivation of Carrier-to-Noise Density Ratio (C/N0)

Let’s assume the free space (imaginary space where no matter exists), the power of carrier wave $$C$$ is

$C = P_r = P_tG_tG_r\left(\frac{\lambda}{4\pi d}\right)^2$

where $$P_t$$ is the transmission power, $$G_t$$ is the gain of transmission antenna, $$G_r$$ is the gain of the reception antenna, $$P_r$$ is the reception power, $$\lambda$$ is wave length, and $$d$$ is the distance between antennas.

Please refer to the following article for this derivation.

Derivation of FSPL from Friss tranmission equation

Also, noise power $$N$$ is

$N = kT_sB$

where $$k$$ is Boltzman constant, $$T_s$$ is the system noise temperature, $$B$$ is the bandwidth.

Thus, the carrier-to-noise ratio can be written as

$\frac{C}{N} = \frac{P_tG_tG_r}{kT_sB}\left(\frac{\lambda}{4\pi d}\right)^2$

where the noise power $$N$$ is, using noise spectral density $$N_0$$,

$N = N_0B$

C/N0 can be written as

$\frac{C}{N_0} = \frac{P_tG_tG_r}{kT_s}\left(\frac{\lambda}{4\pi d}\right)^2$

This is also converted to

$\frac{C}{N_0} = \frac{\mathrm{EIRP}}{L_{fsp}}\frac{G_r}{T_s}\frac{1}{k}$

Converting this equation to dB,

$\mathrm{C/N_0} = \mathrm{EIRP}-L_{fsp}+G/T+228.6$

EIRP derivation

Eb/N0 Explained