Signal to Interference plus Noise Ratio (SINR) in Wireless Communications

Consider a cellular wireless network with transmitter/receiver pairs. Transmit powers are denoted as p_1,ldots,p_n . Transmitter is supposed to transmit to receiver , but due to interference, signal from the other transmitters is also present. In addition, there is (self-) noise power in each receiver. To measure this, we form the Signal to Interference plus Noise Ratio (SINR) at each receiver. This takes the form

$gamma_i := frac{S_i}{I_i+sigma_i}, ;; i=1,ldots,n$

where S_i is a measure of the (desired) signal power received from transmitter , I_i is the total signal power received from all the other receivers, and sigma_i>0 is a measure of the receiver noise.

The SINR is a (in general, complicated) function of the power used at the transmitters.

A Linear Model

We can express the SINRs at the receivers in terms of the powers p_1,ldots,p_n more explicitly by assuming that the received powers S_i , I_i are linear functions of the transmit powers .

The model (also known as the Rayleigh fading model) states that

$S_i = G_{ii} p_i, ;; i=1,ldots,n,$

and

$I_i = sum_{jne i} G_{ij} p_j$

where the coefficients $G_{ij}$ , 1 le i,j le n are known as the path gains from transmitter to receiver .

A posynomial function

The SINR functions

$gamma_i(p) := frac{S_i}{I_i+sigma_i} = frac{G_{ii}p_i}{sigma_i + sum_{jne i} G_{ij} p_j}, ;; i=1,ldots,n,$

are not posynomials, but their inverse

$frac{1}{gamma_i(p)} = frac{sigma_i}{G_{ii}} p_i^{-1} + sum_{jne i} frac{G_{ij}}{G_{ii}} p_jp_i^{-1}, ;; i=1,ldots,n,$

are.