# 0.12 Noise-shaping

 Page 1 / 1

## Noise-shaping

Noise shaping is a procedure which allows to put some color on an additive white noise. While this adds to the overall noise,one can shape the resulting noise such that its prevalent “color” lies in the high frequencies and at the same time reduce thepresence of noise in the low frequencies.

In the context of oversampling, noise shaping achieves an improvement if it is done so that it reduces noise in the band $\left[-{f}_{0}/2,{f}_{0}/2\right]$ and pushes it to the “colors” with frequencies in $\left[{f}_{0}/2,{f}_{e}/2\right]$ and $\left[-{f}_{e}/2,-{f}_{0}/2\right]$ .

The most simple version of noise shaping results in two noise terms, the one produced by the system, one added with a delay on purpose.Set ${\epsilon }_{0}=0$ , then:

$\begin{array}{ccc}\hfill {x}_{k}& \to & \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-{\epsilon }_{k-1}\to \left({x}_{k}-{\epsilon }_{k-1}\right)\to \hfill \\ & \to & \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}Quant.\to \left\{\begin{array}{cc}{y}_{k}\hfill & \text{output}\hfill \\ {\epsilon }_{k}={y}_{k}-\left({x}_{k}-{\epsilon }_{k-1}\right)\hfill & \text{feedback}\hfill \end{array}\right)\hfill \end{array}$

This assumes that we know or can know the error. Such is the case with quantization, where we can compute ${\epsilon }_{k}$ simply as the difference between output and input of the quantization (see [link] ). This is not feasible in the example of a wireless channel with inference.

To study the effect of noise shaping we compute the Fourier transform of the overall error ${n}_{k}={y}_{k}-{x}_{k}$ . We find

${n}_{k}={y}_{k}-{x}_{k}={\epsilon }_{k}-{\epsilon }_{k-1}$

or

$n\left(t\right)=y\left(t\right)-x\left(t\right)=e\left(t\right)-e\left(t-{\tau }_{e}\right)$

with Fourier transform (use that ${f}_{e}=1/{\tau }_{e}$ )

$\begin{array}{ccc}\hfill N\left(f\right)& =& E\left(f\right)-E\left(f\right){e}^{-j2\pi {\tau }_{e}f}=E\left(f\right)\left(1-{e}^{-j2\pi f/{f}_{e}}\right)\hfill \\ & =& E\left(f\right)2sin\left(\pi f/{f}_{e}\right){e}^{-j\pi f/{f}_{e}}\hfill \end{array}$

and power spectrum, (use that $2{sin}^{2}\left(x\right)=1-cos\left(2x\right)$ )

${|N\left(f\right)|}^{2}={|E\left(f\right)|}^{2}2\left(1,-,cos,\left(\frac{2\pi f}{{f}_{e}}\right)\right)$

Note that the spectrum is no longer flat; in other words, the noise ${n}_{k}$ is colored. We note that the colored spectrum is small for small $f$ , and it is still spread over a period of length ${f}_{e}$ . Thus, noise shaping results in a reduction of noise inthe small frequencies (see [link] ).

Now we continue as with simple oversampling: since the signal has been sampled at ${f}_{e}=\beta {f}_{0}$ , with ${f}_{0}\ge 2B$ and $\beta >1$ we may low-pass filter after noise shaping at cutoff frequency ${f}_{0}/2$ . Thus, we take advantage of the fact that most of the power of the“shaped” or “colored” noise is in the `high'-frequency bands $\left[{f}_{0}/2,{f}_{e}/2\right]$ and $\left[-{f}_{e}/2,-{f}_{0}/2\right]$ with very little noise at frequencies around 0.

To assess the gain we compute the power of the noise after this low-pass filter. Note that the low-pass filter sets $N\left(f\right)=0$ for ${f}_{e}/2>|f|>{f}_{0}/2$ : this removes much of the noise as demonstrated in the following computation; it also guarantees that power won't be changedwhen downsampling after filtering to ${f}_{0}$ . Using again that $E\left(f\right)={\Delta }^{2}/12$ we get:

$\begin{array}{ccc}\hfill {P}_{\mathrm{shaping}}& =& \frac{1}{{f}_{e}}{\int }_{-{f}_{0}/2}^{{f}_{0}/2}{|N\left(f\right)|}^{2}df=\frac{1}{{f}_{e}}2{\int }_{0}^{{f}_{0}/2}{|E\left(f\right)|}^{2}2\left(1,-,cos,\left(\frac{2\pi f}{{f}_{e}}\right)\right)\hfill \\ & =& \frac{4{\Delta }^{2}}{12{f}_{e}}\left(f,-,\frac{{f}_{e}}{2\pi },sin,\left(\frac{2\pi f}{{f}_{e}}\right)\right){|}_{0}^{{f}_{0}/2}=\frac{{\Delta }^{2}}{12}4\left(\frac{1}{2\beta },-,\frac{1}{2\pi },sin,\left(\frac{\pi }{\beta }\right)\right)\hfill \end{array}$

Using the approximation $sin\left(x\right)\simeq x-{x}^{3}/6+{x}^{5}/5!-+...$ we find

$\frac{{P}_{\mathrm{shaping}}}{{P}_{\mathrm{noise}}}=4\left(\frac{1}{2\beta },-,\frac{1}{2\pi },sin,\left(\frac{\pi }{\beta }\right)\right)\simeq 4\left(\frac{1}{2\beta },-,\frac{1}{2\pi },\left(\frac{\pi }{\beta },-,\frac{{\pi }^{3}}{6·{\beta }^{3}}\right)\right)=\frac{{\pi }^{2}}{3}{\beta }^{-3}$

Noise reduction under oversampling with noise-shaping

In conclusion, the SNR improves (as compared to no oversampling) under noise shaping by the inverse of the factor [link] , thus roughly by $\frac{3}{{\pi }^{2}}{\beta }^{3}$ . In decibel, this corresponds to approximatively $10{log}_{10}\left(3\right)-20{log}_{10}\left(\pi \right)+30{log}_{10}\left(\beta \right)$ , thus, roughly 3 times more than with oversampling alone.

Numerical Examples:

With an OSR of $\beta =128$ the SNR improves

• under oversampling alone by $10{log}_{10}\left(128\right)\simeq 21$ dB
• under oversampling coupled with basic noise-shaping as above by approximatively $10{log}_{10}\left(3\right)-20{log}_{10}\left(\pi \right)+30{log}_{10}\left(128\right)=58.0445$ dB, or roughly 3 times more.

When doubling the OSR the SNR improves

• under oversampling alone by $10{log}_{10}\left(2\right)\simeq 3.01$ dB
• under oversampling with noise-shaping by approximatively $30{log}_{10}\left(2\right)=9.03$ dB, or approximatively 3 times more.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
Got questions? Join the online conversation and get instant answers!