# Carrier-to-noise ratio

In telecommunications, the carrier-to-noise ratio, often written CNR or C/N, is the signal-to-noise ratio (SNR) of a modulated signal. The term is used to distinguish the CNR of the radio frequency passband signal from the SNR of an analogue base band message signal after demodulation, for example an audio frequency analogue message signal. If this distinction is not necessary, the term SNR is often used instead of CNR, with the same definition.

Digitally modulated signals (e.g. QAM or PSK) are basically made of two CW carriers (the I and Q components, which are out-of-phase carriers) . In fact, the information (bits or symbols) is carried by given combinations of phase and/or amplitude of the I and Q components. It is for this reason that, in the context of digital modulations, digitally modulated signals are usually referred to as carriers. Therefore, the term carrier-to-noise-ratio (CNR), instead of signal-to-noise-ratio (SNR) is preferred to express the signal quality when the signal has been digitally modulated.

High C/N ratios provide good quality of reception, for example low bit error rate (BER) of a digital message signal, or high SNR of an analogue message signal.

## Definition

The carrier-to-noise ratio is defined as the ratio of the received modulated carrier signal power C to the received noise power N after the receive filters:

$\mathrm{CNR} = \frac{C}{N}$.

When both carrier and noise are measured across the same impedance, this ratio can equivalently be given as:

$\mathrm{CNR} = \left( \frac{V_C}{V_N} \right) ^2$,

where $V_C$ and $V_N$ are the root mean square (RMS) voltage levels of the carrier signal and noise respectively.

C/N ratios are often specified in decibels (dB):

$\mathrm{CNR_{dB}} = 10 \log_{10}\left( \frac {C}{N} \right) = C_{dBm} - N_{dBm}$

or in term of voltage:

$\mathrm{CNR_{dB}} = 10 \log_{10}\left( \frac{V_C}{V_N} \right) ^2 = 20 \log_{10}\left( \frac {V_C}{V_N} \right)$

The C/N ratio is measured in a manner similar to the way the signal-to-noise ratio (S/N) is measured, and both specifications give an indication of the quality of a communications channel.

In the famous Shannon–Hartley theorem, the C/N ratio is equivalent to the S/N ratio. The C/N ratio resembles the carrier-to-interference ratio (C/I, CIR), and the carrier-to-noise-and-interference ratio, C/(N+I) or CNIR.