# Bit Streaming Processing Algorithms for Intelligent Hardware Converters
## 3 Results
### Bit-Stream Algorithms
As a result of bit-stream multiplication, the value \(\mathrm{N_{1}}\mathrm{\Theta_{N_{2}}}\) is formed. The result calculation starts when the bit corresponding to the code value \(\mathrm{N_{1}}\) appears, i.e., the output code \(\mathrm{N_{out}}\) is incremented when \(\mathrm{N_{1}}\mathrm{\Theta_{N_{2}}}=1\) (blocks 3, 4). If after the pulse frequency sweep of the code \(\mathrm{N_{1}}\) the bit is not formed, the output code \(\mathrm{N_{out}}\) does not change. To form the compensatory actions, the denominator code \(\mathrm{N_{3}}\) is converted into a PWM signal \(\mathrm{\Theta_{N_{3}}}\). The code \(\mathrm{N_{out}}\) is converted into a bit stream, and the result of multiplying in the compensation branch of the algorithm is the value of \(\mathrm{N_{out}}\mathrm{\Theta_{N_{3}}}\). When \(\mathrm{N_{out}}\mathrm{\Theta_{N_{3}}}=1\) (blocks 7, 8), it is necessary to form a compensating bit. When the compensating bit appears, the compensation mechanism starts working, and the code \(\mathrm{N_{out}}\) is decreased by one. If \(\mathrm{N_{out}}\mathrm{\Theta_{N_{3}}}=0\), then the bit in the compensatory stream is not formed, and the output code \(\mathrm{N_{out}}\) does not change. Operations 1-3 and 5-7 can be executed in parallel. At the initial stage of the algorithm operation, the number of compensating actions is small, but as the output code \(\mathrm{N_{out}}\) increases, the stream in the compensating loop becomes more intense. This continues until the process reaches equilibrium, characterized by the equality of the intensities of the streams causing the increments and decrements. If equilibrium is reached, the code \(\mathrm{N_{out}}\) is the result of calculations. If the streams change randomly during the computation, the equilibrium is disturbed and automatically compensated, and the result is restored. Any changes in the input signal lead to the transition of the computational process to a new equilibrium state. The period \(\mathrm{T}\) of the algorithm operation is determined by the selected bit width of codes and is \(\mathrm{T}=2^{\mathrm{n}}\) t, where t\({}_{\mathrm{i}}\) is the time interval determined frequency of the process quantization. The description of the computational process is based on the time interval t\({}_{\mathrm{i}}\), chosen as the unit time, and then in relative units \(\mathrm{T}=2^{\mathrm{n}}\). The number of time samples t\({}_{\mathrm{i}}\) per period \(\mathrm{T}\) is denoted as \(\mathrm{X_{t}}\). The input and output codes are scaled by the maximum value in the used bit grid. A scaling factor equal to one is taken, so that the maximum values of \(\mathrm{N_{in}}\) and \(\mathrm{N_{out}}\) coincide with the maximum value of the period \(\mathrm{T}\) in relative time units. Figure 3: Algorithm of the multiplication-division operation. Thus, in the equilibrium state, the output code \(\mathrm{N_{out}}\) is determined by the following dependence:  
\[\mathrm{N_{out}}=\mathrm{N_{1}}\frac{\Theta_{\mathrm{N_{2}}}}{\Theta_{\mathrm{N_{ 3}}}}\quad\mathrm{or}\quad\mathrm{N_{out}}=\frac{\mathrm{N_{1}}\mathrm{N_{2}}} {\mathrm{N_{3}}}\]  
The time to reach the equilibrium state is determined by the number \(\mathrm{N_{t}}\) periods, T.  
If the process quantization frequency is high, we can treat the sequence of output codes as a continuous function. Therefore, the dynamics of the transient process can be determined using the following equation:  
\[\mathrm{N_{out}}=\int_{0}^{\mathrm{N_{t}}}\bigl{(}\mathrm{N_{1}}\Theta_{ \mathrm{N_{2}}}-\mathrm{N_{out}}\Theta_{\mathrm{N_{3}}}\bigr{)}\mathrm{dt}.\]  
The equation describes the process of transition to the tracking mode as the process of accumulating the difference of incrementing and decrementing streams during each period T. If the values \(\mathrm{N_{1}}\), \(\Theta_{\mathrm{N_{2}}}\), \(\Theta_{\mathrm{N_{3}}}\) do not change during the transition process, the equation for the variable t can be solved as follows.