# Two Interval Upper-Bound \(Q\)-Function Approximations with Applications
## 6 Application in Designing a Quantizer with Equiprobable Cells for Gaussian Sources  
Let \(x\) be a Gaussian random variable with probability density function:  
\[p(x)=\frac{1}{\sqrt{2\pi}}\exp\left\{-\frac{x^{2}}{2}\right\},\quad-\infty<x<+\infty. \tag{20}\]  
To represent this variable with a finite number of bits \(R\), the range of possible amplitude values (\(-\infty\), \(+\infty\)) should be partitioned into \(N=2^{R}\) disjoint cells, whereby one representation value should be assigned to each cell. Due to the symmetry of the Gaussian probability density function, the symmetry should also exist in the quantization, allowing us to observe only a partition of the amplitude range for positive values. Therefore, let \(t_{j}\), \(i=0,1,...,N\) be the thresholds that partition the positive part of the amplitude range and let \(y_{i}\), \(i=1,2,\ldots,N\) be the representation levels, whereby it holds that \(t_{0}<y_{1}<t_{1}<y2<\ldots<t_{N/2-1}<y_{N/2-1}<t_{N/2}\) and \(t_{0}\)=0, \(t_{N/2}=+\infty\). Then, quantization \(F_{Q}\) is the following function:  
\[F_{Q}(x)=y_{i}\,sgn(x)\quad\text{if}\quad t_{i-1}\leq|x|<t_{i}, \tag{21}\]  
where \(sgn()\) is the sign function. In other words, the quantization transforms a variable \(x\) into \(F_{Q}(x)\), producing the error \(x-F_{Q}(x)\). Then, the performance of the quantization can be measured by estimating the mean squared error or distortion \(D\):  
\[D=\frac{1}{M}\sum_{i=1}^{M}(x_{i}-y_{i})^{2} \tag{22}\]  
Figure 5: Absolute relative error of determining ASEP for DE-QPSK modulation: comparison of the applied proposed approximations and the selected ones from the literature. and the signal-to-quantization-noise ratio SQNR:  
\[\text{SQNR}=\frac{\sum_{i=1}^{M}x_{i}^{2}}{\sum_{i=1}^{M}(x_{i}-y_{i})^{2}}, \tag{23}\]  
where \(M\) is a big enough number of variables. Based on the above, it is clear that a symmetric \(N\)-level quantization is defined with thresholds \(t_{i}\), \(i=0,1,\ldots,N/2\) and representation levels \(y_{i}\), \(i=1,2,\ldots,N/2\), which means that the quantization design involves the determination of these levels subject to a certain criterion. In the literature the commonly used criterion is the minimum distortion or a constant signal-to-quantization-noise ratio, while in this paper, as in [24], we designed the quantization subject to the criterion that the probability of a variable with a Gaussian probability density function belonging to a cell is the same for all cells. Actually, we focused on providing a quantization with equiprobable cells. In doing so, we encountered the \(Q\)-function. Namely, the probability that a variable having a Gaussian probability density function belonged to a certain cell was:  
\[P_{i}=\int_{t_{i}-1}^{t_{i}}p(x)dx=Q(t_{i-1})-Q(t_{i})=\frac{1}{N},\quad i=1,2,\ldots,N/2 \tag{24}\]  
thus forming a systems of \(N/2\)-\(1\) integral equations that should be solved per \(t_{i}\), \(i=1,\ldots,N/2-2\), since \(t_{0}=0\) and \(t_{N/2}=+\infty\) implies \(Q(t_{0})=1/2\) and \(Q(t_{N/2})=0\):  
\[Q(t_{i})=\frac{1}{\sqrt{2\pi}}\int_{t_{i}}^{+\infty}\exp\biggl{\{}-\frac{x^{2} }{2}\biggr{\}}dx=\frac{N-2i}{2N},\quad i=1,2,\ldots,N/2-1. \tag{25}\]  
Using a closed-form approximation for the \(Q\)-function can reduce the complexity of the above system of equations, which is what we did in this paper by applying the composite upper-bound approximation given by (10):  
\[Q^{\text{com\_1}}(t_{i})=\frac{N-2i}{2N},\quad i=1,2,\ldots,N/2-1. \tag{26}\]  
After that, we solved the resulting system of nonlinear equations using the bisection method.