1.4 Quantizer design for entropy coded sytems

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Motivated by the cascade of memoryless quantization and entropy coding, the entropy-minimizing scalar memoryless quantizer is derived. Using a compander formulation and tools from the calculus of variations, it is shown that the entropy-minimizing quantizer is the simple uniform quantizer. The penalty associated with memoryless quantization is then analyzed in the asymptotic case of many quantization levels.
• Say that we are designing a system with a memoryless quantizer followed by an entropy coder, and our goal is to minimize theaverage transmission rate for a given σ q 2 (or vice versa). Is it optimal to cascade a σ q 2 -minimizing (Lloyd-Max) quantizer with a rate-minimizing code?In other words, what is the optimal memoryless quantizer if the quantized outputs are to be entropy coded?
• A Compander Formulation: To determine the optimal quantizer,
1. consider a companding system: a memoryless nonlinearity $c\left(x\right)$ followed by uniform quantizer,
2. find $c\left(x\right)$ minimizing entropy H y for a fixed error variance σ q 2 .
• First we must express σ q 2 and H y in terms of $c\left(x\right)$ . [link] suggests that, for large L , the slope ${c}^{\text{'}}\left(x\right):=dc\left(x\right)/dx$ obeys
${c}^{\text{'}}\left(x\right){|}_{x\in {\mathcal{X}}_{k}}=\frac{2{x}_{max}/L}{{\Delta }_{k}},$
so that we may write
${\Delta }_{k}=\frac{2{x}_{max}}{L{c}^{\text{'}}\left(x\right)}{|}_{x\in {\mathcal{X}}_{k}}.$
Assuming large L , the σ q 2 -approximation equation 9 from MSE-Optimal Memoryless Scalar Quantization (lower equation) can be transformed as follows.
$\begin{array}{ccc}\hfill {\sigma }_{q}^{2}& =& \frac{1}{12}\sum _{k=1}^{L}{P}_{k}{\Delta }_{k}^{2}\hfill \\ & =& \frac{{x}_{max}^{2}}{3{L}^{2}}\sum _{k=1}^{L}\frac{{P}_{k}}{{c}^{\text{'}}{\left(x\right)}^{2}}{|}_{x\in {\mathcal{X}}_{k}}\hfill \\ & =& \frac{{x}_{max}^{2}}{3{L}^{2}}\sum _{k=1}^{L}\frac{{p}_{x}\left(x\right)}{{c}^{\text{'}}{\left(x\right)}^{2}}{\Delta }_{k}{|}_{x\in {\mathcal{X}}_{k}}\phantom{\rule{1}{0ex}}\text{since}\phantom{\rule{1}{0ex}}{P}_{k}={p}_{x}\left(x\right){|}_{x\in {\mathcal{X}}_{k}}{\Delta }_{k}\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{large}\phantom{\rule{4.pt}{0ex}}L\hfill \\ & =& \frac{{x}_{max}^{2}}{3{L}^{2}}{\int }_{-{x}_{max}}^{{x}_{max}}\frac{{p}_{x}\left(x\right)}{{c}^{\text{'}}{\left(x\right)}^{2}}dx.\hfill \end{array}$
Similarly,
$\begin{array}{ccc}\hfill {H}_{y}& =& -\sum _{k=1}^{L}{P}_{k}{log}_{2}{P}_{k}\hfill \\ & =& -\sum _{k=1}^{L}{p}_{x}\left(x\right){\Delta }_{k}{log}_{2}\left({p}_{x},\left(x\right),{\Delta }_{k}\right){|}_{x\in {\mathcal{X}}_{k}}\hfill \\ & =& -\sum _{k=1}^{L}{p}_{x}\left(x\right){\Delta }_{k}{log}_{2}{p}_{x}\left(x\right){|}_{x\in {\mathcal{X}}_{k}}-\sum _{k=1}^{L}{p}_{x}\left(x\right){\Delta }_{k}{log}_{2}{\Delta }_{k}{|}_{x\in {\mathcal{X}}_{k}}\hfill \\ & =& \underset{{h}_{x}:\text{differential}\phantom{\rule{4.pt}{0ex}}\text{entropy''}}{\underbrace{-{\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){log}_{2}{p}_{x}\left(x\right)dx}}\phantom{\rule{3.33333pt}{0ex}}-{\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){log}_{2}\underset{{\Delta }_{k}}{\underbrace{\frac{2{x}_{max}}{L{c}^{\text{'}}\left(x\right)}}}dx\hfill \\ & =& {h}_{x}-{log}_{2}\frac{2{x}_{max}}{L}\underset{=1}{\underbrace{{\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right)dx}}\phantom{\rule{0.277778em}{0ex}}+{\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){log}_{2}{c}^{\text{'}}\left(x\right)dx\hfill \\ & =& \text{constant}\phantom{\rule{0.222222em}{0ex}}+\phantom{\rule{0.222222em}{0ex}}{\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){log}_{2}{c}^{\text{'}}\left(x\right)dx\hfill \end{array}$
• Entropy-Minimizing Quantizer: Our goal is to choose $c\left(x\right)$ which minimizes the entropy rate H y subject to fixed error variance σ q 2 . We employ a Lagrange technique again, minimizingthe cost ${\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){log}_{2}{c}^{\text{'}}\left(x\right)dx$ under the constraint that the quantity ${\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){\left({c}^{\text{'}},\left(x\right)\right)}^{-2}dx$ equals a constant C . This yields the unconstrained cost function
$\begin{array}{c}\hfill {J}_{u}\left({c}^{\text{'}},\left(x\right),,,\lambda \right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\int }_{-{x}_{max}}^{{x}_{max}}\underset{\phi \left({c}^{\text{'}}\left(x\right),\lambda \right)}{\underbrace{\left[{p}_{x},\left(x\right),{log}_{2},{c}^{\text{'}},\left(x\right),+,\lambda ,\left({p}_{x},\left(x\right),{\left({c}^{\text{'}},\left(x\right)\right)}^{-2},-,C\right)\right]}}dx,\end{array}$
with scalar λ , and the unconstrained optimization problem becomes
$\underset{{c}^{\text{'}}\left(x\right),\lambda }{min}{J}_{u}\left({c}^{\text{'}}\left(x\right),\lambda \right).$
The following technique is common in variational calculus (see, e.g., Optimal Systems Control by Sage&White). Say ${a}^{\star }\left(x\right)$ minimizes a (scalar) cost $J\left(a,\left(,x,\right)\right)$ . Then for any (well-behaved) variation $\eta \left(x\right)$ from this optimal ${a}^{\star }\left(x\right)$ , we must have
$\frac{\partial }{\partial ϵ}J\left({a}^{\star },\left(x\right),+,ϵ,\eta ,\left(x\right)\right){|}_{ϵ=0}=0$
where ϵ is a scalar. Applying this principle to our optimization problem, we search for ${c}^{\text{'}}\left(x\right)$ such that
$\forall \eta \left(x\right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\frac{\partial }{\partial ϵ}{J}_{u}\left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right),,,\lambda \right){|}_{ϵ=0}=0.$
From [link] we find (using ${log}_{2}a={log}_{2}e·{log}_{e}a$ )
$\begin{array}{ccc}\hfill \frac{\partial {J}_{u}}{\partial ϵ}{|}_{ϵ=0}& =& {\int }_{-{x}_{max}}^{{x}_{max}}\frac{\partial }{\partial ϵ}\phi \left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right),,,\lambda \right){|}_{ϵ=0}dx\hfill \\ & =& {\int }_{-{x}_{max}}^{{x}_{max}}\frac{\partial }{\partial ϵ}\left[{p}_{x},\left(x\right),{log}_{2},\left(e\right),\phantom{\rule{0.166667em}{0ex}},{log}_{e},\left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right)\right),+,\lambda ,\left({p}_{x},\left(x\right),{\left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right)\right)}^{-2},-,C\right)\right]{|}_{ϵ=0}dx\hfill \\ & =& {\int }_{-{x}_{max}}^{{x}_{max}}\left[{log}_{2},\left(e\right),\phantom{\rule{0.166667em}{0ex}},{p}_{x},\left(x\right),{\left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right)\right)}^{-1},\eta ,\left(x\right),-,2,\lambda ,{p}_{x},\left(x\right),{\left({c}^{\text{'}},\left(x\right),+,ϵ,\eta ,\left(x\right)\right)}^{-3},\eta ,\left(x\right)\right]{|}_{ϵ=0}dx\hfill \\ & =& {\int }_{-{x}_{max}}^{{x}_{max}}{p}_{x}\left(x\right){\left({c}^{\text{'}},\left(x\right)\right)}^{-1}\left[{log}_{2},\left(e\right),-,2,\lambda ,{\left({c}^{\text{'}},\left(x\right)\right)}^{-2}\right]\phantom{\rule{0.166667em}{0ex}}\eta \left(x\right)\phantom{\rule{0.166667em}{0ex}}dx\hfill \end{array}$
and to allow for any $\eta \left(x\right)$ we require
${log}_{2}\left(e\right)-2\lambda {\left({c}^{\text{'}},\left(x\right)\right)}^{-2}=0\phantom{\rule{1.em}{0ex}}⇔\phantom{\rule{1.em}{0ex}}{c}^{\text{'}}\left(x\right)=\underset{\text{a}\phantom{\rule{4.pt}{0ex}}\text{constant!}}{\underbrace{\sqrt{\frac{2\lambda }{{log}_{2}e}}}}.$
Applying the boundary conditions,
$\left\{\begin{array}{c}c\left({x}_{max}\right)={x}_{max}\\ c\left(-{x}_{max}\right)=-{x}_{max}\end{array}\right\}\phantom{\rule{1.em}{0ex}}\to \phantom{\rule{1.em}{0ex}}\begin{array}{|c|}\hline c\left(x\right)=x\\ \hline\end{array}.$
Thus, for large- L , the quantizer that minimizes entropy rate H y for a given quantization error variance σ q 2 is the uniform quantizer. Plugging $c\left(x\right)=x$ into [link] , the rightmost integral disappears and we have
${H}_{y}{|}_{\text{uniform}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{h}_{x}-{log}_{2}\underset{\Delta }{\underbrace{\frac{2{x}_{max}}{L}}},$
and using the large- L uniform quantizer error variance approximation equation 6 from Memoryless Scalar Quantization ,
${H}_{y}{|}_{\text{uniform}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{h}_{x}-\frac{1}{2}{log}_{2}\left(12,{\sigma }_{q}^{2}\right).$
It is interesting to compare this result to the information-theoretic minimal average rate for transmission of a continuous-amplitudememoryless source x of differential entropy h x at average distortion σ q 2 (see Jayant&Noll or Berger):
${R}_{min}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{h}_{x}-\frac{1}{2}{log}_{2}\left(2,\pi ,e,\phantom{\rule{0.166667em}{0ex}},{\sigma }_{q}^{2}\right).$
Comparing the previous two equations, we find that (for a continous-amplitude memoryless source) uniform quantizationprior to entropy coding requires
$\frac{1}{2}{log}_{2}\left(\frac{\pi e}{6}\right)\phantom{\rule{3.33333pt}{0ex}}\approx \phantom{\rule{3.33333pt}{0ex}}\begin{array}{|c|}\hline 0.255\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{bits/sample}\\ \hline\end{array}$
more than the theoretically optimum transmission scheme, regardless of the distribution of x . Thus, 0.255 bits/sample (or $\sim 1.5$ dB using the $6.02R$ relationship) is the price paid for memoryless quantization .

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