## 2 Coordinated Group Signal Transformation Basics  
The main idea of the CGST method is to increase the performance of multi-channel systems by joint signal analysis. One way to achieve this is to combine the principles of differential pulse-code modulation (DPCM) with CGST, as shown in Figure 1.  
Following this approach, original signals are first processed through the coordinated extrapolation block to obtain their estimates. Then, the difference signals between the original signals and their estimates are calculated and used for further processing and transmission. At the receiver end, the original signals are restored using the known parameters of the extrapolation block. However, the main challenge is to synthesize the transfer function of the coordinated extrapolation block. One way to solve this problem is by treating it as an optimization problem using methods such as Wiener or Kalman filtering [41; 45; 46]. Nevertheless, this approach may be challenging due to its computational complexity. Another suggested method in [47] uses a group codec scheme for a multi-channel system with semi-type channels (Figure 2).  
Figure 1: Theconcept of differential coordinated group transformation (illustration for a single channel of a multichannel system). Red—source signal, green—the extrapolated one, blue—the compressed (difference) signal.  
Figure 2: The structure of the transceiver channel using coordinated codec with a coding matrix.  
The input vector, \(\overline{U}=[u_{1},\dots,u_{n}]\) (where \(T\) is the transpose operator), excites the system. The signals, \(\widetilde{u}_{i}(t)\) (where \(i\) is the channel number), that are passed through the coding matrix are applied at the second input of the comparison elements. The vector of difference signals, \(\overline{E}=[e_{1},\dots,e_{n}]^{T}\), whose elements are determined by the expression \(e_{i}(t)=u_{i}-\widetilde{u}_{i}(t)\), is the output for the system.  
The coordinated prediction block, known as the coordinating matrix \(\mathbf{K}\), is described as follows:  
\[\mathbf{K}=\left[\begin{array}{cccc}k_{11}&k_{12}&\dots&k_{1n}\\ k_{21}&k_{22}&\dots&k_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ k_{m1}&k_{m2}&\dots&k_{mn}\end{array}\right], \tag{1}\]  
where the \(k_{ii}\) elements are transmission coefficients in the direct branch located on the main diagonal. The \(k_{ij}\) elements (correlation coefficients) are determined by the following equation:  
\[k_{ij}=\frac{1}{\sigma_{i}\sigma_{j}}\cdot\frac{1}{N}\sum_{t=1}^{N}(u_{i}- \overline{u}_{i})(u_{j}-\overline{u}_{j}), \tag{2}\]  
where \(\sigma\) is the root mean square error of the channel signal, and \(N\) is the number of signal's samples when calculating the correlation in a discrete form.  
The problem of coordinated system synthesis can be solved by adjusting the coefficients \(k\) of the main diagonal, considering them to be constant for all of the channels [47]. This adjustment ensures a system's stability while reducing the average signal amplitude at the coder output. The roots of the characteristic equation are determined outside the circumference, which completely encompasses the system hodograph, substantially simplifying the calculation. When defining the roots, the system's hodograph is replaced by a circle circumscribed around this hodograph.