# 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 ?

• Where does the lecture come from? Nov 15 '20 at 16:38
• A university course on signal processing. Why ? Nov 15 '20 at 16:41
• I am interested in the topic, I was interested in knowing whether the teacher provided papers, etc. and if the slides are online (the snapshots are not easy to use) Nov 15 '20 at 16:44
• Actually those are from a new course that is proceeding this semester, and as much as I checked there aren't any provided open materials, but if it help's, probably I'm going to ask another questions with slides from it, so you are welcomed to discuss them in that manner :) Nov 15 '20 at 16:59

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
$$|\Lambda_l|=\frac{\psi_H-\psi_l}{k}$$
• With the notation $|\dot|$ standing for the length of the interval, if I understand well (I did not find it in the slides) Nov 15 '20 at 16:08