# BER Aided Energy and Spectral Efficiency Estimation in a Heterogeneous Network
## 2 Analysis
### BER-Based SINR Estimation by AWGN Abstraction of Radio Interference  
Degraded SINR usually implies constellation symbol errors, and, thereby, SINR is often tested, which requires complex equipment to measure the noise and inter-symbol interference (ISI) [20]. Instead, estimating BER can be an alternative, i.e., an easy-to-measure performance trade-off \"currency\", rather than SINR (where, by \"easiness\", we consider the possibility to estimate BER in-service, simply by counting the retransmissions at the physical/MAC layer [21] with a count that determines the Block Error ratio (BLER). Then, an appropriate model can be applied to estimate BER from BLER [20]. This could be useful in practice encompassing the various phases of product-related research, development, manufacturing, and, finally, exploitation of a product in the LTE and 5G New Radio Environment. Note the classical expression for BER determined by the Signal-to-Noise Ratio (SNR), for the M-QAM signal transmission over the AWGN channel [22]:  
\[BER=\frac{4}{\log_{2}M}\cdot Q\Bigg{(}\sqrt{\frac{3\cdot SNR}{M-1}}\Bigg{)} \tag{4}\]  
where \(Q(\cdot)\) stands for the Gaussian tail function, represented by the \"waterfall\"--steep curves in Figure 1, which visualize the threshold effect that is immanent to digital radio receivers. Specifically, in very small cells, it is quite justifiable to presume strong received signals (i.e., high SNR), as well as that the base-band inter-symbol interference (ISI) due to channel time dispersion, is, usually, to the large extent, eliminated by a (long-enough) standard cyclic prefix (CP) [20], though some (mostly equipment-related) non-AWGN impairments can still remain. Furthermore, it is also fairly justifiable to consider radio interference a dominant impairment, which itself (being a sum of enough many mutually independent RF interfering signals and according to the Central Limit Theorem) can be reliably considered a Gaussian random variable. Figure 1: “Waterfall”—steep \(BER\) vs \(SNR\) curves (for Nyquist BW). In this case, \(SINR\) practically reduces to \(SNR\), so Equation (4) implies that:  
\[SINR\approx SNR=\frac{M-1}{3}\bigg{[}Q^{-1}\bigg{(}\frac{BER\cdot\log_{2}M}{4} \bigg{)}\bigg{]}^{2} \tag{5}\]  
where \(Q^{-1}(\cdot)\) denotes the inverse function of the Gaussian tail. Applying link abstraction, any distortion, be it additive Gaussian or not, can be considered equivalent to that much additive Gaussian noise that would result with equal BER degradation, i.e., shift the \(BER(SNR)\) curves to the right for the adequate SNR degradation. This is shown in Figure 2, as the non-AWGN degradation of the 16 QAM modulation symbols (visible on the part of the constellation on the left) is modeled by the carrier-to-noise (C/N) degradation for the same BER value. As an illustrative example, let us derive a simplified model of radio interference by superposing a narrow-band interfering signal (such as, e.g., dominantly from an adjacent channel) to the information-carrying one, thus reducing the effective noise margin, as shown in Figure 3.  
As reducing noise margin implies increased probability of a symbol error, let us consider the probability of the in-phase signal overpassing a single decision boundary. In Figure 4, it can be seen that the effective (now reduced) noise margin, equal to \(d\)-\(I\) cos\(\theta\), where \(I\) denotes the interfering signal amplitude, with the phase \(\theta\) taking any value, uniformly distributed between 0 and \(2\pi\). Figure 3: Superposing a narrow-band interfering signal on the information-carrying one. Figure 2: AWGN abstraction of non-AWGN impairments.