In the following figure we see an encoder and a decoder of a Delta Modulation System, which is a simple DPCM System with a 1-bit quantizer:
We assume that signal $x(n)$ is known, that $\hat{x}(-1)=0$ and that there is not noise in the telecommunication channel.
I want to define the predicted signal $\tilde{x}(n)$, the difference signal $d(n)$, the quantized difference signal $\hat{d}(n)$, the encoded signal $c(n)$ and the reconstructed signal $\hat{x}(n)$
I have written the following equations: $$d(n) = x(n) - \tilde{x}(n) \; \; \; \; (1) $$ $$\tilde{x}(n) = 0,8 \cdot \hat{x}(n-1) \; \; (2)$$ $$\hat{x}(n) = \tilde{x}(n) + \hat{d}(n) \; \; \;\;(3)$$ In addition, duo to the feedback, the error $e(n)$ for the reconstruction (between the initial signal on the transmitter and the reconstructed signal on the receiver) is equal to the error of the quantizer. So, I also get: $$\hat{d}(n) = d(n) + e(n) \; \; \; \; (4) $$ $$\hat{x}(n) = x(n) + e(n) \; \; \; \; (5) $$
After solving the system of the equations, it seems to me that the number of the uknowns exceeds the number of the linear independent equations. If it so, I need one more equation since not all the equations above are linear independent.
So my question is what am I missing? Thanks in advance!