## 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,

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