## 2 Basic Notions and Preliminary Results
### Quantizer Description
#### 2.2.4 Novel Adaptive Dynamic Quantizer  
For a logarithmic quantizer, the quantization level becomes coarser as the magnitude of signal gets larger (away from the original), the uniform quantizer has the disadvantage of fixed error that the error does not decrease when the input is very small, dynamic uniform quantizer is no exception. To overcome these problems, we propose a novel adaptive dynamic quantizer, which combines the logarithmic quantizer and the uniform quantizer by introducing dynamic gain parameters, give the boundary conditions of the novel quantizer. (Please note that this quantizer can be easily extended to a combination of a hysteresis quantizer and a uniform quantizer by replacing \(q_{i}(\cdot)\) with \(q_{h}(\cdot)\)).  
\[kq_{\chi}(u/k)=\left\{\begin{array}{ll}kq_{i}(u_{th})+\left\lfloor\frac{u-ku _{th}}{k\varpi}+K\right\rfloor k\varpi&ku_{th}\leq u\leq kM\\ kq_{i}(\frac{k}{k})&0<u<ku_{th}\\ -kq_{\chi}(\frac{-u}{k})&u<0\end{array}\right. \tag{6}\]  
where \(kq_{\chi}(u/k)\) is the result of quantization of novel adaptive dynamic quantizer, \(k\) is the constant to be specified later, \(k(k>0)\) is adopted as a zooming variable, \(kM\) is quantization range. \(K=\frac{1}{2}\), if \(q_{l}(u_{th})<u_{th}\) and \(K=0\) if \(q_{l}(u_{th})\geq u_{th},u_{th}\) is a positive constant specified by the designer denoting the threshold to switch between the dynamic logarithmic quantizer and dynamic uniform quantizer, the equation \(\frac{1}{2}\varpi=|q_{l}(u_{th})-u_{th}|\), where \(\varpi\) is a parameter for the dynamic uniform quantizer.  
Let us make \(k=1\), \(\varpi=1\). We get Figure 1.  
**Remark 1**.: _The quantization error of the novel adaptive dynamic quantizer is shown below (see the proof in Appendix A)._  
\[|\Delta q_{\chi}(u)|\leq\left\{\begin{array}{ll}k(\delta\frac{|u|}{k}+(1- \delta)d)&|u|<ku_{th}\\ \frac{1}{2}k\varpi&ku_{th}\leq|u|\leq kM\end{array}\right. \tag{7}\]  
_where \(d>0\), \(0<\rho<1\), \(\delta=\frac{1-\rho}{1+\rho}\), the quantization error of \(kq_{\chi}(\frac{|u|}{k})\) is \(\Delta q_{\chi}\), \(\Delta q_{\chi}\) for the novel adaptive dynamic quantizer is always bounded for any \(u\). In addition, low communication cost is still maintained with a suitable choice of large \(u_{th}\). Thus, the newly proposed adaptive dynamic quantizer has the advantages of logarithmic, uniform, dynamic uniform quantizers._  
The quantization error comparisons of magnitude between the newly proposed adaptive dynamic quantizer and the traditional dynamic uniform quantizer as follows:  
Figure 1: Novel adaptive dynamic quantizer.  
According to (7), we divide the input into the following three stages to analyze.  
\[\left\{\begin{array}{ll}\Delta q_{\mu}=\Delta q_{\chi}=\frac{1}{2}k\alpha&|u|>ku_ {th}\\ \\ \Delta q_{\mu}=\Delta q_{\chi}=\frac{1}{2}k\alpha&|u|=ku_{th}\\ \\ \Delta q_{\chi}<\Delta q_{\mu}&|u|<ku_{th}\\ \end{array}\right.\]  
where \(\Delta q_{\mu}\) is the quantization error of the dynamic uniform quantizer mentioned in [25], \(\Delta q_{\chi}=k(\delta\frac{|u|}{k}+(1-\delta)d)\), \(\frac{\Delta\Delta q_{\chi}}{\Delta u}>0\).  
In summary, the error of the dynamic uniform quantizer is \(\Delta q_{\mu}\), the error of the newly proposed adaptive dynamic quantizer is \(\Delta q_{\chi}\). \(\Delta q_{\mu}\geq\Delta q_{\chi}\) if the input is in a finite compact set, it will be more clearly illustrated by the subsequent digital simulation.