# signal to interference ratio A [fixed channel assignment](Fixed%20channel%20assignment.md) formula ## Basic formula With K interfering neighbours. $\Gamma_{SIR}=\frac{S}{I}=\frac{S}{\sum_{k=1}^KI_k}$ Where S is the desired signal power received from the desired [BS](Base%20station.md), $I_k$ is the received interference power caused by the k-th interfereing co-channel cell [BS](Base%20station.md). ## How to calculate received signal power? Propagation measurements in mobile radio channels have shown the average received power $P_r$ at a distance $d$ can be approximated by $P_r\propto P_Td^{-n}$ where $P_T$ is transmitted power and $n$ is the [path loss](Path%20loss.md) exponent. ## More assumptions If these are violated, the analysis may not be accurate! 1. Mobile station is at the edge of the cell (worst case assumption) 2. All nearest neighbours (first tier) are all active (users using the frequency channels) 3. Ignore all 2nd and higher tier interferers (repeating the cluster structure) 1. Can do this because first tier dominate the interference power 4. All [base stations](Base%20station.md) transmit with the same power P 5. Equal distance $D$ from $K$ interfering cells 6. Ignore the noise to simplify the analysis (see [SINR](Signal%20to%20Interference%20plus%20Noise%20Ratio.md)) ## Relationship between SIR and cluster size N $\frac{S}{I}=\frac{S}{\sum_{k=1}^KI_k}=\frac{PR^{-n}}{\sum_{k=1}^KPD^{-n}_k}=\frac{\left(\sqrt{3N}\right)^n}{K}$ ### Derivation  Summing D K times (assuming all $D_k$ are equal) is $K\times D$ ## Observation: SIR is independent of power! Performance depends on your structure. What can we play around with? $n$ = can't change (channel characteristics) - If n $\uparrow\ \Rightarrow SIR \uparrow$ - This is non-intuitive! - The interference is also impacted by the poorer channel - In a point to point system, as the channel exponent increases the received power will decrease. - High [path loss](Path%20loss.md) exponent could attenuate the interference as well! - High [path loss](Path%20loss.md) exponent is **welcome in situations with interference** $N$ = larger clusters means better performance. - Can't just keep increasing to infinity though - there is a tradeoff. - There is another dimension apart from SIR $K$ = number of interfering cells (assume it to be 6 for simple calculations). - Use cell [sectoring](Sectoring.md) to reduce $K$ - If required SIR can be $\downarrow$ e.g. better coding scheme) the capacity can be $\uparrow$ - This is done by decreasing N to lower the achievable SIR - Use the frequency more aggressively, more users supported by the slice of spectrum These are true regardless of the physical cell size! This equation is nothing to do with area. ## Example For AMPS ([1G](../Telecommunications/1G.md)) the minimum SIR is 18dB. What is the minimum cluster size for a [path loss](Path%20loss.md) exponent of n=4? Solution: $\frac{S}{I}=\frac{\left(\sqrt{3N}\right)^4}{6}\geq10^{1.8}$ $\Rightarrow N \geq 6.4857, N=7$ Don't forget that N can only take values of 1,3,4,7,9,... # References 1. [TELE4652-lecture-02](../../Spaces/University/TELE4652/Lectures/TELE4652-lecture-02-cellular-concepts-and-cellular-network-capacity.pdf)