Quantization and Sampling - putting it all together So after I learned this two topic: quantization and sampling, I'm learning the way to look at both of them and try to optimize the split of a given amount of bit B to N and k, where N is the amount of samples and k is the size of the finite set of numbers can represent the values (quantization).
There is one calculation that I'm not totally understand in the next few slides:





Until now it is very clear, including the yellow formula, let's simplify it more a little bit:
$$=\sum _{l=1}^kP_{\psi }\left(\psi _l\right)\cdot \frac{1}{3}\left(2\cdot \frac{1}{2^3}\frac{\left(\psi _H-\psi _L\right)^3}{k^3}\right)=\sum _{l=1}^kP_{\psi }\left(\psi _l\right)\cdot \frac{1}{3}\left(\frac{1}{2^2}\frac{\left(\psi _H-\psi _L\right)^3}{k^3}\right)=\sum _{l=1}^kP_{\psi }\left(\psi _l\right)\cdot \:\frac{1}{12}\frac{\left(\psi _H-\psi _L\right)^3}{k^3}=\frac{1}{12}\sum \:_{l=1}^kP_{\psi \:}\left(\psi \:_l\right)\cdot \:\frac{\left(\psi \:_H-\psi \:_L\right)^3}{k^3}$$
In the next slide, I don't understand where disappeared:$\frac{\left(\psi _H-\psi _L\right)^{ }}{k^{ }}$. Probably it holds that: $\frac{\left(\psi _H-\psi _L\right)}{k}=\left|\left(delta\right)_l\right|$, but I don't see clearly why from the whole discussion:

Does any one see's what this is happening and why it holds the equation ?
Or anything else I'm missing ?
I believe that $|\Delta_l|$ in that equation is a typo and should actually be $|\Lambda_l|$, because from the first equation in $(2)$ we have
$$|\Lambda_l|=\frac{\psi_H-\psi_l}{k}$$