Definition (Average Code Rate)

Let $\mathcal{C}$ be a $D$-ary $n$-th order VLC $$f:\mathcal{X}^{n}\to\{0,1,\cdots,D-1\}^*$$for a discrete source $\{X_{i}\}^{\infty}_{i=1}$ with alphabet $\mathcal{X}$ and joint pmfs $\{p_{X^n}\}$ and let $\mathscr{l}(c_{x^n})$ denote the length of codeword $c_{x_{n}}:=f(x^{n})$ associated with source $n$-tuple $x^{n}\in\mathcal{X}^{n}$. Then the average codeword length for $\mathcal{C}$ is given by $$\overline{\mathscr{l}}:=E[\mathscr{l}(c_{X^n})]=\sum\limits_{x^{n}\in\mathcal{X}^{n}}p_{X^{n}}(x^{n})\mathscr{l}(c_{x^{n}})$$and its average code rate is given by $$\overline R:= \frac{\overline{\mathscr{l}}}{n}= \frac{1}{n}\sum\limits_{x^{n}\in\mathcal{X}^{n}}p_{X^n}(x^{n})\mathscr{l}(c_{x^{n}})$$