# 3.4 Performance

 Page 1 / 1
Here we analyze the optimal reconstruction error for transform coding. As the number of channels grows to infinity, the performance gain over PCM is shown to depend on the spectral flatness measure. Meanwhile, the performance of transform coding with an infinite number of channels is shown to equal that of DPCM with an infinite-length predictor. However, when the DPCM predictor length is equal to the number of transform coding channels, we show that DPCM always yields better performance.

## Asymptotic performance analysis

• For an $N×N$ transform coder, Equation 1 from "Gain over PCM" presented an expression for the reconstruction error variance ${\sigma }_{r}^{2}{|}_{\text{TC}}$ written in terms of the quantizer input variances $\left\{{\sigma }_{{y}_{k}}^{2}\right\}$ . Noting the N -dependence on ${\sigma }_{r}^{2}{|}_{TC}$ in Equation 1 from "Gain over PCM" and rewriting it as ${\sigma }_{r}^{2}{|}_{\text{TC},N}$ , a reasonable question might be: What is ${\sigma }_{r}^{2}{|}_{\text{TC},N}$ as $N\phantom{\rule{-0.166667em}{0ex}}\to \phantom{\rule{-0.166667em}{0ex}}\infty$ ?
• When using the KLT, we know that ${\sigma }_{{y}_{k}}^{2}={\lambda }_{k}$ where λ k denotes the ${k}^{th}$ eigenvalue of ${\mathbf{R}}_{x}$ . If we plug these ${\sigma }_{{y}_{k}}^{2}$ into Equation 1 from "Gain over PCM" , we get
$\begin{array}{ccc}\hfill {\sigma }_{r}^{2}{|}_{\text{TC},N}& =& {\gamma }_{y}{2}^{-2R}{\left(\prod _{k=0}^{N-1},{\lambda }_{k}\right)}^{1/N}.\hfill \end{array}$
Writing ${\left({\prod }_{k}{\lambda }_{k}\right)}^{1/N}=exp\left(\frac{1}{N}{\sum }_{k}ln{\lambda }_{k}\right)$ and using the Toeplitz Distribution Theorem (see Grenander&Szego)
$\begin{array}{c}\hfill \text{For}\phantom{\rule{4.pt}{0ex}}\text{any}\phantom{\rule{4.pt}{0ex}}f\left(·\right)\text{,}\phantom{\rule{1.em}{0ex}}\underset{N\to \infty }{lim}\frac{1}{N}\sum _{k}f\left({\lambda }_{k}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f\left({S}_{x}\left({e}^{j\omega }\right)\right)d\omega \end{array}$
with $f\left(·\right)=ln\left(·\right)$ , we find that
$\begin{array}{ccc}\hfill \underset{N\to \infty }{lim}{\sigma }_{r}^{2}{|}_{\text{TC},N}& =& {\gamma }_{y}{2}^{-2R}exp\left(\frac{1}{2\pi },{\int }_{-\pi }^{\pi },ln,{S}_{x},\left({e}^{j\omega }\right),d,\omega \right)\hfill \\ & =& {\gamma }_{y}{\sigma }_{x}^{2}\phantom{\rule{0.166667em}{0ex}}{2}^{-2R}\phantom{\rule{0.166667em}{0ex}}{\text{SFM}}_{x}\hfill \end{array}$
where ${\text{SFM}}_{x}$ denotes the spectral flatness measure of $x\left(n\right)$ , redefined below for convenience:
$\begin{array}{ccc}\hfill {\text{SFM}}_{x}& =& \frac{exp\left(\frac{1}{2\pi },{\int }_{-\pi }^{\pi },ln,{S}_{x},\left({e}^{j\omega }\right),d,\omega \right)}{\frac{1}{2\pi }{\int }_{-\pi }^{\pi }{S}_{x}\left({e}^{j\omega }\right)d\omega }.\hfill \end{array}$
Thus, with optimal transform and optimal bit allocation, asymptotic gain over uniformly quantized PCM is
$\begin{array}{c}\hfill \begin{array}{|c|}\hline {G}_{\text{TC},N\to \infty }=\frac{{\sigma }_{r}^{2}{|}_{\text{PCM}}}{{\sigma }_{r}^{2}{|}_{\text{TC},N}\to \infty }=\frac{{\gamma }_{x}{\sigma }_{x}^{2}{2}^{-2R}}{{\gamma }_{y}{\sigma }_{x}^{2}{2}^{-2R}{\text{SFM}}_{x}}=\frac{{\gamma }_{x}}{{\gamma }_{y}}{\text{SFM}}_{x}^{-1}\\ \hline\end{array}.\end{array}$
• Recall that, for the optimal DPCM system,
${G}_{\text{DPCM},N\to \infty }\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\sigma }_{r}^{2}{|}_{\text{PCM}}}{{\sigma }_{r}^{2}{|}_{\text{DPCM},N\to \infty }}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\sigma }_{x}^{2}}{{\sigma }_{e}^{2}{|}_{min}},$
where we assumed that the signal applied to DPCM quantizer is distributed similarly to the signal applied to PCM quantizerand where ${\sigma }_{e}^{2}{|}_{min}$ denotes the prediction error variance resulting from use of the optimal infinite-lengthlinear predictor:
${\sigma }_{e}^{2}{|}_{min}=exp\left(\frac{1}{2\pi },{\int }_{-\pi }^{\pi },ln,{S}_{x},\left({e}^{j\omega }\right),d,\omega \right).$
Making this latter assumption for the transform coder (implying ${\gamma }_{y}={\gamma }_{x}$ ) and plugging in ${\sigma }_{e}^{2}{|}_{min}$ yields the following asymptotic result:
$\begin{array}{|c|}\hline {G}_{\text{TC},N\to \infty }={G}_{\text{DPCM},N\to \infty }={\text{SFM}}_{x}^{-1}\\ \hline\end{array}.$
In other words, transform coding with infinite-dimensional optimal transformation and optimal bit allocation performs equivalently toDPCM with infinite-length optimal linear prediction.

## Finite-dimensional analysis: comparison to dpcm

• 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 $N×N$ 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 ${\prod }_{\ell }{\lambda }_{\ell }=|{\mathbf{R}}_{N}|$ , the TC gain over PCM becomes
$\begin{array}{ccc}\hfill {G}_{\text{TC},N}& =& \frac{{\sigma }_{r}^{2}{|}_{\text{PCM}}}{{\sigma }_{r}^{2}{|}_{\text{TC},N}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{\gamma }_{x}{\sigma }_{x}^{2}{2}^{-2R}}{{\gamma }_{x}{2}^{-2R}·{\sigma }_{x}^{2}{\left({\prod }_{k=1}^{N-1},{G}_{\text{DPCM},k}\right)}^{-1/N}}\hfill \\ & =& {\left(\prod _{k=1}^{N-1},{G}_{\text{DPCM},k}\right)}^{1/N}\hfill \\ & <& {G}_{\text{DPCM},N}.\hfill \end{array}$
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 $N\phantom{\rule{-0.166667em}{0ex}}×\phantom{\rule{-0.166667em}{0ex}}N$ 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 .

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
hi
Loga
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!