Summary

• Encoder: $\mathcal{E}:\mathbb{R}\to \{ 1,\dots,N \}$ defined by $$\mathcal{E}(x)=j\iff x\in R_{j}$$i.e. we map each input to one of our $N$ levels
• Decoder: $\mathcal{D}:\{ 1,\dots,N \}\to \{ y_{1},\dots,y_{N} \}$ defined by $$\mathcal{D}(j)=y_{j}$$i.e. We map each of our $N$ levels to a reproduction point

Hence, we can redefine our scalar quantizer $Q$ in terms of the encoder and decoder $$Q(x)=\mathcal{D}(\mathcal{E}(x))$$We also define the rate of $Q$ in terms of the number of levels it admits: $$R(Q)=\log_{2}N\text{ bits/sample}$$