# BER Aided Energy and Spectral Efficiency Estimation in a Heterogeneous Network
## 2 Analysis
### BER-Based SINR Estimation by AWGN Abstraction of Radio Interference
As the noise margin is normalized to the noise effective value \(\sigma\), the probability of in-phase signal overpassing a single decision boundary is:  
\[P(e)=Q\bigg{(}\frac{d-I\cos\vartheta}{\sigma}\bigg{)} \tag{6}\]  
where \(Q(\cdot)\) can be developed by expressing the phase \(\theta\) as the multiple of \(n\) arbitrarily small intervals \(\Delta\theta\), obtained by dividing \(2\pi\) into \(N\) equal parts:  
\[\vartheta=\bigg{(}n-\frac{1}{2}\bigg{)}\cdot\Delta\vartheta;\Delta\vartheta= \frac{2\pi}{N} \tag{7}\]  
Furthermore, we consider that for enough large \(N\), \(Q(\cdot)\) does not vary significantly within any \(\Delta\theta\), so by substituting Equation (7) into Equation (6) we can approximate the latter by:  
\[P(e)\approx\frac{1}{N}\cdot\sum_{n=1}^{N}Q\Bigg{[}\frac{d-I\cos\Bigl{(}2\pi \cdot\frac{n-1/2}{N}\Bigr{)}}{\sigma}\Bigg{]} \tag{8}\]  
where \(I\) and \(\sigma\) are expressed by their related ratios \(S/I\) and \(SNR\), respectively:  
\[I=\sqrt{2}d\cdot 10^{\frac{-(S/I-K)}{20}} \tag{9}\]  
Figure 1:0  
while the factor \(K\) (expressed in dB units) relates the signal power to the relevant modulation scheme (so, e.g., for QPSK it is \(K\) = 0 and for 16 QAM it is \(K\) = 10 log 5 = 6.99). Besides, in front of the sum in Equation (8), weighting factors are to be inserted reflecting the average number of possible transitions over the symbol boundaries, in a particular modulation of interest. For example, for the in-phase component and 16 QAM modulation, there are two boundaries for the inner symbols and a single one for the outer symbols. This implies that \(K\)= 1.5, whereas for 4 QAM and 64 QAM, \(K\) equals 2 and 1.75, respectively. Graphical presentation of Equation (8) for 16 QAM is shown in Figure 5, where adequate trade-off between the non-AWGN radio interference and the AWGN, represented by \(S/I\) and \(SNR\), respectively, can be identified to determine the goodness of abstracting a non-AWGN distortion by equivalent AWGN producing the same BER degradation. Figure 1:1 Symbol error mechanism due to noise and narrow-band radio interference. Specifically, moving to the right along the red horizontal line of constant BER = \(10^{-2}\) from the middle curve for S/I = 30 dB, to the line crossing with the utmost right curve for S/I = 20 dB (i.e., increasing the interference for 10 dB), is tracked by almost equal increase in the SNR value. Moreover, if this quite balanced trade-off between S/I and SNR is evident even for the simple non-AWGN interfering signal of the AWGN modeled above, then we can justifiably expect even more conformance if, instead of the narrowband radio interference, we deal with an almost-AWGN interference (as a sum of a number of mutually independent interferers), which we consider here as the real AWGN. On the other hand, for certain interference (characterized by certain S/I ratio) and at any SNR point of its corresponding BER curve, the equivalent interference--free SNR can be obtained by drawing a vertical line (the turquoise one in Figure 5) down to the crossing with the curve with S/I = \(\infty\) (representing the no-interference case). So, e.g., the (middle) curve in Figure 5, representing the non-AWGN interference (with \(S/I\) = 30 dB) and having \(BER\) = \(10^{-2}\) at S/N = 20 dB, can be AWGN-abstracted by the utmost left AWGN curve (with \(S/I\) = \(\infty\)), if we consider \(BER\) degradation from \(2+10^{-4}\) to \(10^{-2}\), upwards the vertical turquoise line. Consequently, Equations (1)-(3), i.e., the simple SE vs. EE relationship, continue to apply for the AWGN-abstracted non-AWGN radio interference, and SINR can be approximated by the AWGN-equivalent SNR, which can be estimated from easy-to-measure BER.