Thanks Neeraj, My question is that many literature consider N as a constant value rather than an R.V. Why it is customary to represent (N) as a constant value rather than a random variable ? especially when we are interested in the statistical behaviour of the SINR ?

Hi,

as Neeraj Varshney told you, this constant number is the variance of the Gaussian noise which represents the noise power. It is a R.N. not a constant.

Thanks Mohamoud, Check these leading publications, they all treat (N=\sigma^2) as a constant when working with SINR.

T. D. Novlan and J. G. Andrews, "Analytical Evaluation of Uplink Fractional Frequency Reuse," Communications, IEEE Transactions on, vol. 61, pp. 2098-2108, 2013.

J. G. Andrews, F. Baccelli, and R. K. Ganti, "A tractable approach to coverage and rate in cellular networks," Communications, IEEE Transactions on, vol. 59, pp. 3122-3134, 2011.

M. Haenggi, J. G. Andrews, F. Baccelli, O. Dousse, and M. Franceschetti, "Stochastic geometry and random graphs for the analysis and design of wireless networks," IEEE Journal on Selected Areas in Communications., vol. 27, pp. 1029-1046, 2009.

and much more....

The fact that the noise power is included as a deterministic value instead of a random value is a consequence of the time scale that the communication system is considered. The expressions in the papers you mention are reliabilities (or outage probabilities) at the packet level. Since packets typically contain at least a few 100 symbols and channel coding is employed, averaging occurs and the effect of the noise is reduced to its power.

In contrast, the random effects of fading (or the node positions) occur at a much different (slower) time scale, so their distributions need to be taken account.

Many thanks Dr. M. Haenggi for the informative answer, I think the justification you mentioned is quite reasonable. This question came to my mind after reading the paper from N. Miyoshi et al [1] (In the context of stochastic geometry) where they dealt with the the noise as a random variable rather than with its variance value over the packet period.

However, is there any analytical limit for the time window at which we can confidently consider the average noise power instead its random behaviour ?

[1] N. Miyoshi and T. Shirai, "A cellular network model with Ginibre configurated base stations," Research Rep. on Math. and Comp. Sciences (Tokyo inst. of tech.), 2012.

I am not aware of any theoretical results regarding this type of convergence, but there may very well be some, perhaps in the form of concentration inequalities. For example, Chebyshev's inequality shows that the deviation of a sum of n i.i.d. random variables from the mean decreases as 1/n. Depending on your desired level of confidence, you can pick an n. In practice, I'd say for n>100, the actual distribution of the noise hardly matters. As a result, most researchers studying packet-level outage performance only consider the noise power as a deterministic quantity.

For your information: Shannon himself did something similar in his paper "Communication in the presence of noise". There he assumed that the symbol length was sufficiently long to describe the noise in terms of a deterministic amount of energy instead of a rapidly varying value. You may find it interesting to study this passage. Maybe he even gives a conditon when this assumption holds, I don't remember.

Your question brings back fond memories when I was a young communication engineer.

Usually, we consider noise in a communication channel to be represented by two properties: 1) its noise power spectral density function, and 2) the distribution function of its voltage (or current).

A Gaussian, white noise source has a flat power spectral density function and its amplitudes (voltage or current) have a Gaussian distribution. Of course, no such source exists but it is good for theoretical calculations.

The formula for 'signal in noise and interference' is a very crude approximate to the real situation is a communication system but it is widely used to make performance estimates or characterize a communications channel. In this formula,

SINR = P / (I + N)

SINR (Signal-to-interference-plus-noise ratio) is the effective signal power (P) divided by the sum of the interference power (I) and the noise power (N). Both powers I and N must be measured in the effective noise bandwidth of the communication channel (Shannon Banwidth).

None of these values has anything to do with the random distribution function of the noise or the signals. P, I, and N are determined by integrating the product of the noise bandwith frequency response and the power spectral density functions of the random quanities.

On another topic, the spectral density, distribution function, and stationarity of the inference all play a large roll in determing the performance of the communication system. A simple "I" is a gross oversimplification of the true situation.

The SINR does not play a serious role in the design and practical implementation of a communication system.

Good luck

In any proper analysis of wireless network performance, the randomness of the interference power I is carefully modeled and incorporated (there are many papers that do this). As a result, the SINR random variable is the key quantity in analyzing and designing a wireless network. 

For mathematical modeling, Gaussian noise process is assumed as wide sense stationary for calculation of variance or power.Thus for modeling, many times we have considered white Gaussian noise with constant power considering the process as stationary.