The fact that optimal transform coding performs as well as
DPCM in the limiting case does not tell us the relative performanceof these methods at practical levels of implementation, e.g., when
transform dimension and predictor length are equal and
$\ll \infty $ .
Below we compare the reconstruction error variances of TC and DPCMwhen the transform dimension
equals the predictor length.
Recalling that
$${G}_{\text{DPCM},N-1}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\sigma}_{x}^{2}}{{\sigma}_{e}^{2}{|}_{min,N-1}}$$
and
$${\sigma}_{e}^{2}{|}_{min,N-1}=\frac{|{\mathbf{R}}_{N}|}{|{\mathbf{R}}_{N-1}|}$$
where
${\mathbf{R}}_{N}$ denotes the
$N\times N$ autocorrelation
matrix of
$x\left(n\right)$ , we find
$$\begin{array}{c}\hfill {G}_{\text{DPCM},N-1}={\sigma}_{x}^{2}\frac{|{\mathbf{R}}_{N-1}|}{|{\mathbf{R}}_{N}|},\phantom{\rule{1.em}{0ex}}{G}_{\text{DPCM},N-2}={\sigma}_{x}^{2}\frac{|{\mathbf{R}}_{N-2}|}{|{\mathbf{R}}_{N-1}|},\phantom{\rule{1.em}{0ex}}{G}_{\text{DPCM},N-3}={\sigma}_{x}^{2}\frac{|{\mathbf{R}}_{N-3}|}{|{\mathbf{R}}_{N-2}|},\phantom{\rule{1.em}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\cdots \phantom{\rule{3.33333pt}{0ex}}\end{array}$$
Recursively applying the equations above, we find
$$\begin{array}{c}\hfill \prod _{k=1}^{N-1}{G}_{\text{DPCM},k}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\left({\sigma}_{x}^{2}\right)}^{N-1}\frac{|{\mathbf{R}}_{1}|}{|{\mathbf{R}}_{N}|}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\left({\sigma}_{x}^{2}\right)}^{N}}{|{\mathbf{R}}_{N}|}\end{array}$$
which means that we can write
$$\begin{array}{c}\hfill |{\mathbf{R}}_{N}|\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\left({\sigma}_{x}^{2}\right)}^{N}{\left(\prod _{k=1}^{N-1},{G}_{\text{DPCM},k}\right)}^{-1}.\end{array}$$
If in the previously derived TC reconstruction error variance expression
$$\begin{array}{ccc}\hfill {\sigma}_{r}^{2}{|}_{\text{TC},N}& =& {\gamma}_{y}{2}^{-2R}{\left(\prod _{\ell =0}^{N-1},{\lambda}_{\ell}\right)}^{1/N}\hfill \end{array}$$
we assume that
${\gamma}_{y}={\gamma}_{x}$ and apply the eigenvalue property
${\prod}_{\ell}{\lambda}_{\ell}=\left|{\mathbf{R}}_{N}\right|$ , the TC gain over PCM becomes
$$\begin{array}{ccc}\hfill {G}_{\text{TC},N}& =& \frac{{\sigma}_{r}^{2}{|}_{\text{PCM}}}{{\sigma}_{r}^{2}{|}_{\text{TC},N}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\gamma}_{x}{\sigma}_{x}^{2}{2}^{-2R}}{{\gamma}_{x}{2}^{-2R}\xb7{\sigma}_{x}^{2}{\left({\prod}_{k=1}^{N-1},{G}_{\text{DPCM},k}\right)}^{-1/N}}\hfill \\ & =& {\left(\prod _{k=1}^{N-1},{G}_{\text{DPCM},k}\right)}^{1/N}\hfill \\ & <& {G}_{\text{DPCM},N}.\hfill \end{array}$$
The strict inequality follows from the fact that
${G}_{\text{DPCM},k}$ is monotonically increasing with
k .
To summarize, DPCM with optimal length-
N prediction performs better
than TC with optimal
$N\phantom{\rule{-0.166667em}{0ex}}\times \phantom{\rule{-0.166667em}{0ex}}N$ transformation and optimal bit
allocation for any finite value of
N .
There is an intuitive explanation for this:the propagation of memory in the DPCM prediction loop makes
the
effective memory of DPCM greater than
N , while in TC the
effective memory is exactly
N .