# Regarding SINR (Signal to Interference Plus Noise Ratio) in wireless communication

I am working on a research paper related to wireless communication, wherein I am facing some doubt while writing expression of SINR (which is a ratio of Signal variance in numerator to Interference variance + Noise variance in denominator).

Let us consider a wireless communication scenario where received signal over random wireless channel is given by

$$y=h_1s_1+ h_2s_2+h_1w$$ ---(1)

where,

$$h_1, h_2$$ are wireless channels and are random in nature, so we had modelled as $$h_1 \sim \mathcal{C}\mathcal{N}(0,\sigma^2_{h_1}), h_2 \sim \mathcal{C}\mathcal{N}(0,\sigma^2_{h_2})$$,

$$s_1$$ is message signal, $$s_2$$ is interfering signal and are modelled as $$s_1 \sim \mathcal{N}(0,P_1), s_2 \sim \mathcal{N}(0,P_2)$$ and $$w$$ is Gaussian noise modelled as $$w\sim \mathcal{N}(0,N_w)$$.

My query is that which is the correct way to write the SINR expression eq. (2) or eq. (3)

$$\text{SINR} = \frac{|h_1|^2 P_1}{|h_2|^2P_2+\sigma^2_{h_1}N_w}$$ ---(2)

$$\text{SINR} = \frac{|h_1|^2 P_1}{|h_2|^2P_2+|h_1|^2N_w}$$ ---(3)

Please note that in equation (2), I had written $$\sigma^2_{h_1}$$ in denominator instead of $$|h_1|^2$$ as $$h_1$$ is associated with noise. So is it correct to write in this way ?

Any help in this regard will be highly appreciated.

• Hi Pranu, welcome to DSP.SE! Am I following the logic correctly that you are assuming only SNR from the transmitter (it appears your transmit signal and noise both pass through the same channel, and the s2 is therefore noise free)? Typically the noise $w$ would be dominated by the receiver (thermal noise floor, noise figure) in which case s1 and s2 as they are both received (by whatever channel model is used for each of them) would then have the noise added and then pass through a common channel as the matched filter in the receiver. Commented Apr 7 at 13:06
• Thank u so much sir for ur reply....yes sir you are following it correctly Commented Apr 8 at 11:28

1. The received signal should be $$y=h_1s_1+h_2s_2+w$$ So, SINR is to be represented as $$SINR=\frac{|h_1|^2P_1}{|h_2|^2P_2+N_w}$$