# Carrier-to-noise ratio

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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 analog base band message signal after demodulation, for example an audio frequency analog message signal. If this distinction is not necessary, the term SNR is often used instead of CNR, with the same definition.

## Contents

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 analog 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 receiver 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:

$\mathrm {CNR_{dB}} =10\log _{10}\left({\frac {V_{C}}{V_{N}}}\right)^{2}=20\log _{10}\left({\frac {V_{C}}{V_{N}}}\right)$ ## Measurements and estimation

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.

C/N estimators are needed to optimize the receiver performance.  Typically, it is easier to measure the total power than the ratio of signal power to noise power (or noise power spectral density), and that is why CNR estimation techniques are timely and important.

## Carrier-to-noise density ratio

In satellite communications, carrier-to-noise-density ratio (C/N0) is the ratio of the carrier power C to the noise power density N0, expressed in dB-Hz. When considering only the receiver as a source of noise, it is called carrier-to-receiver-noise-density ratio.

It determines whether a receiver can lock on to the carrier and if the information encoded in the signal can be retrieved, given the amount of noise present in the received signal. The carrier-to-receiver noise density ratio is usually expressed in dBHz.

The noise power density, N0=kT, is the receiver noise power per hertz, which can be written in terms of the Boltzmann constant k (in joules per kelvin) and the noise temperature T (in kelvins).

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In signal processing, a matched filter is obtained by correlating a known delayed signal, or template, with an unknown signal to detect the presence of the template in the unknown signal. This is equivalent to convolving the unknown signal with a conjugated time-reversed version of the template. The matched filter is the optimal linear filter for maximizing the signal-to-noise ratio (SNR) in the presence of additive stochastic noise. In digital communication or data transmission, Eb/N0 is a normalized signal-to-noise ratio (SNR) measure, also known as the "SNR per bit". It is especially useful when comparing the bit error rate (BER) performance of different digital modulation schemes without taking bandwidth into account.

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The signal-to-interference ratio, also known as the carrier-to-interference ratio, is the quotient between the average received modulated carrier power S or C and the average received co-channel interference power I, i.e. cross-talk, from other transmitters than the useful signal.

In information theory and telecommunication engineering, the signal-to-interference-plus-noise ratio (SINR) is a quantity used to give theoretical upper bounds on channel capacity in wireless communication systems such as networks. Analogous to the signal-to-noise ratio (SNR) used often in wired communications systems, the SINR is defined as the power of a certain signal of interest divided by the sum of the interference power and the power of some background noise. If the power of noise term is zero, then the SINR reduces to the signal-to-interference ratio (SIR). Conversely, zero interference reduces the SINR to the SNR, which is used less often when developing mathematical models of wireless networks such as cellular networks.

1. Islam, A. K. M. Najmul; Lohan, E. S.; Renfors, M. (Mar 2008). "Moment based CNR estimators for BOC/BPSK modulated signal for Galileo/GPS". 2008 5th Workshop on Positioning, Navigation and Communication. pp. 129–136. doi:10.1109/WPNC.2008.4510366. ISBN   978-1-4244-1798-8. This article incorporates  public domain material from the General Services Administration document: "Federal Standard 1037C".(in support of MIL-STD-188)