In information theory and telecommunication engineering, the **signal-to-interference-plus-noise ratio** (**SINR**^{ [1] }) (also known as the **signal-to-noise-plus-interference ratio** (**SNIR**)^{ [2] }) is a quantity used to give theoretical upper bounds on channel capacity (or the rate of information transfer) 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 (from all the other interfering signals) 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.^{ [3] }

- Description
- Mathematical definition
- Propagation model
- SINR model
- Stochastic geometry models
- See also
- References

The complexity and randomness of certain types of wireless networks and signal propagation has motivated the use of stochastic geometry models in order to model the SINR, particularly for cellular or mobile phone networks.^{ [4] }

SINR is commonly used in wireless communication as a way to measure the quality of wireless connections. Typically, the energy of a signal fades with distance, which is referred to as a path loss in wireless networks. Conversely, in wired networks the existence of a wired path between the sender or transmitter and the receiver determines the correct reception of data. In a wireless network one has to take other factors into account (e.g. the background noise, interfering strength of other simultaneous transmission). The concept of SINR attempts to create a representation of this aspect.

The definition of SINR is usually defined for a particular receiver (or user). In particular, for a receiver located at some point *x* in space (usually, on the plane), then its corresponding SINR given by

where *P* is the power of the incoming signal of interest, *I* is the interference power of the other (interfering) signals in the network, and *N* is some noise term, which may be a constant or random. Like other ratios in electronic engineering and related fields, the SINR is often expressed in decibels or dB.

To develop a mathematical model for estimating the SINR, a suitable mathematical model is needed to represent the propagation of the incoming signal and the interfering signals. A common model approach is to assume the propagation model consists of a random component and non-random (or deterministic) component.^{ [5] }^{ [6] }

The deterministic component seeks to capture how a signal decays or attenuates as it travels a medium such as air, which is done by introducing a path-loss or attenuation function. A common choice for the path-loss function is a simple power-law. For example, if a signal travels from point *x* to point *y*, then it decays by a factor given by the path-loss function

- ,

where the path-loss exponent * α>2*, and *|x-y|* denotes the distance between point *y* of the user and the signal source at point *x*. Although this model suffers from a singularity (when *x=y*), its simple nature results in it often being used due to the relatively tractable models it gives.^{ [3] } Exponential functions are sometimes used to model fast decaying signals.^{ [1] }

The random component of the model entails representing multipath fading of the signal, which is caused by signals colliding with and reflecting off various obstacles such as buildings. This is incorporated into the model by introducing a random variable with some probability distribution. The probability distribution is chosen depending on the type of fading model and include Rayleigh, Rician, log-normal shadow (or shadowing), and Nakagami.

The propagation model leads to a model for the SINR.^{ [2] }^{ [6] }^{ [4] } Consider a collection of base stations located at points to in the plane or 3D space. Then for a user located at, say , then the SINR for a signal coming from base station, say, , is given by

- ,

where are fading random variables of some distribution. Under the simple power-law path-loss model becomes

- .

In wireless networks, the factors that contribute to the SINR are often random (or appear random) including the signal propagation and the positioning of network transmitters and receivers. Consequently, in recent years this has motivated research in developing tractable stochastic geometry models in order to estimate the SINR in wireless networks. The related field of continuum percolation theory has also been used to derive bounds on the SINR in wireless networks.^{ [2] }^{ [4] }^{ [7] }

**Rayleigh fading** is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices.

**Rician fading** or **Ricean fading** is a stochastic model for radio propagation anomaly caused by partial cancellation of a radio signal by itself — the signal arrives at the receiver by several different paths, and at least one of the paths is changing. Rician fading occurs when one of the paths, typically a line of sight signal or some strong reflection signals, is much stronger than the others. In Rician fading, the amplitude gain is characterized by a Rician distribution.

**Additive white Gaussian noise** (**AWGN**) is a basic noise model used in information theory to mimic the effect of many random processes that occur in nature. The modifiers denote specific characteristics:

A **sensor array** is a group of sensors, usually deployed in a certain geometry pattern, used for collecting and processing electromagnetic or acoustic signals. The advantage of using a sensor array over using a single sensor lies in the fact that an array adds new dimensions to the observation, helping to estimate more parameters and improve the estimation performance. For example an array of radio antenna elements used for beamforming can increase antenna gain in the direction of the signal while decreasing the gain in other directions, i.e., increasing signal-to-noise ratio (**SNR**) by amplifying the signal coherently. Another example of sensor array application is to estimate the direction of arrival of impinging electromagnetic waves. The related processing method is called **array signal processing**. A third examples includes chemical sensor arrays, which utilize multiple chemical sensors for fingerprint detection in complex mixtures or sensing environments. Application examples of array signal processing include radar/sonar, wireless communications, seismology, machine condition monitoring, astronomical observations fault diagnosis, etc.

**Array processing** is a wide area of research in the field of signal processing that extends from the simplest form of 1 dimensional line arrays to 2 and 3 dimensional array geometries. Array structure can be defined as a set of sensors that are spatially separated, e.g. radio antenna and seismic arrays. The sensors used for a specific problem may vary widely, for example microphones, accelerometers and telescopes. However, many similarities exist, the most fundamental of which may be an assumption of wave propagation. Wave propagation means there is a systemic relationship between the signal received on spatially separated sensors. By creating a physical model of the wave propagation, or in machine learning applications a training data set, the relationships between the signals received on spatially separated sensors can be leveraged for many applications.

In graph theory and network analysis, indicators of **centrality** identify the most important vertices within a graph. Applications include identifying the most influential person(s) in a social network, key infrastructure nodes in the Internet or urban networks, super-spreaders of disease, and brain networks. Centrality concepts were first developed in social network analysis, and many of the terms used to measure centrality reflect their sociological origin. They should not be confused with node influence metrics, which seek to quantify the influence of every node in the network.

In statistics and probability theory, a **point process** or **point field** is a collection of mathematical points randomly located on a mathematical space such as the real line or Euclidean space. Point processes can be used as mathematical models of phenomena or objects representable as points in some type of space.

**Space-time adaptive processing** (**STAP**) is a signal processing technique most commonly used in radar systems. It involves adaptive array processing algorithms to aid in target detection. Radar signal processing benefits from STAP in areas where interference is a problem. Through careful application of STAP, it is possible to achieve order-of-magnitude sensitivity improvements in target detection.

The **log-distance path loss model** is a radio propagation model that predicts the path loss a signal encounters inside a building or densely populated areas over distance.

In probability theory, the **Schramm–Loewner evolution** with parameter *κ*, also known as **stochastic Loewner evolution** (SLE_{κ}), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics. Given a parameter *κ* and a domain in the complex plane *U*, it gives a family of random curves in *U*, with *κ* controlling how much the curve turns. There are two main variants of SLE, *chordal SLE* which gives a family of random curves from two fixed boundary points, and *radial SLE*, which gives a family of random curves from a fixed boundary point to a fixed interior point. These curves are defined to satisfy conformal invariance and a domain Markov property.

**Precoding** is a generalization of beamforming to support multi-stream transmission in multi-antenna wireless communications. In conventional single-stream beamforming, the same signal is emitted from each of the transmit antennas with appropriate weighting such that the signal power is maximized at the receiver output. When the receiver has multiple antennas, single-stream beamforming cannot simultaneously maximize the signal level at all of the receive antennas. In order to maximize the throughput in multiple receive antenna systems, multi-stream transmission is generally required.

**Cooperative diversity** is a cooperative multiple antenna technique for improving or maximising total network channel capacities for any given set of bandwidths which exploits user diversity by decoding the combined signal of the relayed signal and the direct signal in wireless multihop networks. A conventional single hop system uses direct transmission where a receiver decodes the information only based on the direct signal while regarding the relayed signal as interference, whereas the cooperative diversity considers the other signal as contribution. That is, cooperative diversity decodes the information from the combination of two signals. Hence, it can be seen that cooperative diversity is an antenna diversity that uses distributed antennas belonging to each node in a wireless network. Note that user cooperation is another definition of cooperative diversity. *User cooperation* considers an additional fact that each user relays the other user's signal while cooperative diversity can be also achieved by multi-hop relay networking systems.

In mathematics, **stochastic geometry** is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

In probability theory, a **Laplace functional** refers to one of two possible mathematical functions of functions or, more precisely, functionals that serve as mathematical tools for studying either point processes or concentration of measure properties of metric spaces. One type of Laplace functional, also known as a **characteristic functional** is defined in relation to a point process, which can be interpreted as random counting measures, and has applications in characterizing and deriving results on point processes. Its definition is analogous to a characteristic function for a random variable.

In mathematics, a **determinantal point process** is a stochastic point process, the probability distribution of which is characterized as a determinant of some function. Such processes arise as important tools in random matrix theory, combinatorics, physics, and wireless network modeling.

In probability theory and statistics, **Campbell's theorem** or the **Campbell–Hardy theorem** is either a particular equation or set of results relating to the expectation of a function summed over a point process to an integral involving the mean measure of the point process, which allows for the calculation of expected value and variance of the random sum. One version of the theorem, also known as **Campbell's formula**, entails an integral equation for the aforementioned sum over a general point process, and not necessarily a Poisson point process. There also exist equations involving moment measures and factorial moment measures that are considered versions of Campbell's formula. All these results are employed in probability and statistics with a particular importance in the theory of point processes and queueing theory as well as the related fields stochastic geometry, continuum percolation theory, and spatial statistics.

In mathematics and probability theory, **continuum percolation theory** is a branch of mathematics that extends discrete percolation theory to continuous space. More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs.

In mathematics and telecommunications, **stochastic geometry models of wireless networks** refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.

In probability and statistics, **point process notation** comprises the range of mathematical notation used to symbolically represent random objects known as point processes, which are used in related fields such as stochastic geometry, spatial statistics and continuum percolation theory and frequently serve as mathematical models of random phenomena, representable as points, in time, space or both.

In probability, statistics and related fields, a **Poisson point process** is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is often called simply the **Poisson process**, but it is also called a **Poisson random measure**, **Poisson random point field** or **Poisson point field**. This point process has convenient mathematical properties, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.

- 1 2 M. Haenggi, J. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti. Stochastic geometry and random graphs for the analysis and design of wireless networks.
*IEEE JSAC*, 27(7):1029--1046, September 2009. - 1 2 3 M. Franceschetti and R. Meester.
*Random networks for communication: from statistical physics to information systems*, volume 24. Cambridge University Press, 2007. - 1 2 J. G. Andrews, R. K. Ganti, M. Haenggi, N. Jindal, and S. Weber. A primer on spatial modeling and analysis in wireless networks.
*Communications Magazine, IEEE*, 48(11):156--163, 2010. - 1 2 3 M. Haenggi.
*Stochastic geometry for wireless networks*. Cambridge University Press, 2012. - ↑ F. Baccelli and B. Blaszczyszyn.
*Stochastic Geometry and Wireless Networks, Volume I --- Theory*, volume 3, No 3--4 of*Foundations and Trends in Networking*. NoW Publishers, 2009. - 1 2 F. Baccelli and B. Blaszczyszyn.
*Stochastic Geometry and Wireless Networks, Volume II --- Applications*, volume 4, No 1--2 of*Foundations and Trends in Networking*. NoW Publishers, 2009. - ↑ R. Meester.
*Continuum percolation*, volume 119. Cambridge University Press, 1996.

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.