# 0.1 Amplitude quantization

 Page 1 / 1
Analog-to-digital conversion.

The Sampling Theorem says that if we sample a bandlimitedsignal $s(t)$ fast enough, it can be recovered without error from its samples $s(n{T}_{s})$ , $n\in \{\dots , -1, 0, 1, \dots \}$ . Sampling is only the first phase of acquiring data into acomputer: Computational processing further requires that the samples be quantized : analog values are converted into digital form. In short, we will have performed analog-to-digital (A/D) conversion .

A phenomenon reminiscent of the errors incurred in representing numbers on a computer prevents signal amplitudesfrom being converted with no error into a binary number representation. In analog-to-digital conversion, the signal isassumed to lie within a predefined range. Assuming we can scale the signal without affecting the information itexpresses, we'll define this range to be $\left[-1 , 1\right]$ . Furthermore, the A/D converter assigns amplitude values inthis range to a set of integers. A $B$ -bit converter produces one of the integers $\{0, 1, \dots , 2^{B}-1\}$ for each sampled input. [link] shows how a three-bit A/D converter assigns input values tothe integers.We define a quantization interval to be the range of values assigned to the same integer. Thus, for our examplethree-bit A/D converter, the quantization interval $\Delta$ is $0.25$ ; in general, it is $\frac{2}{2^{B}}$ .

Recalling the plot of average daily highs in this frequency domain problem , why is this plot so jagged? Interpret this effect interms of analog-to-digital conversion.

The plotted temperatures were quantized to the nearest degree. Thus, the high temperature's amplitude wasquantized as a form of A/D conversion.

Because values lying anywhere within a quantization interval are assigned the same value for computer processing, the original amplitude value cannot be recovered without error . Typically, the D/A converter, the device that converts integers to amplitudes, assigns anamplitude equal to the value lying halfway in the quantization interval. The integer 6 would be assigned to the amplitude0.625 in this scheme. The error introduced by converting a signal fromanalog to digital form by sampling and amplitude quantization then back again would be half the quantizationinterval for each amplitude value. Thus, the so-called A/D error equals half the width of a quantization interval: $\frac{1}{2^{B}}$ . As we have fixed the input-amplitude range, the more bitsavailable in the A/D converter, the smaller the quantization error.

To analyze the amplitude quantization error more deeply, we need to compute the signal-to-noise ratio, which equals the ratio of the signal power and the quantizationerror power. Assuming the signal is a sinusoid, the signal power is the square of the rms amplitude: $\mathrm{power}(s)=\left(\frac{1}{\sqrt{2}}\right)^{2}=\frac{1}{2}$ . The illustration details a single quantization interval.

Its width is $\Delta$ and the quantization error is denoted by $\epsilon$ . To find the power in the quantization error, we note that no matter into whichquantization interval the signal's value falls, the error will have the same characteristics. To calculate the rms value, wemust square the error and average it over the interval.
$\mathrm{rms}(\epsilon )=\sqrt{\frac{1}{\Delta }\int_{-\left(\frac{\Delta }{2}\right)}^{\frac{\Delta }{2}} \epsilon ^{2}\,d \epsilon }=\left(\frac{\Delta ^{2}}{12}\right)^{\left(\frac{1}{2}\right)}$
Since the quantization interval width for a $B$ -bit converter equals $\frac{2}{2^{B}}=2^{-(B-1)}$ , we find that the signal-to-noise ratio for theanalog-to-digital conversion process equals
$\mathrm{SNR}=\frac{\frac{1}{2}}{\frac{2^{-(2(B-1))}}{12}}=\frac{3}{2}2^{(2B)}=6B+10\lg 1.5\mathrm{dB}$
Thus, every bit increase in the A/D converter yields a 6 dB increase in the signal-to-noise ratio.The constant term $10\lg 1.5$ equals 1.76.

This derivation assumed the signal's amplitude lay in the range $\left[-1 , 1\right]$ . What would the amplitude quantization signal-to-noiseratio be if it lay in the range $\left[-A , A\right]$ ?

The signal-to-noise ratio does not depend on the signal amplitude. With an A/D range of $\left[-A , A\right]$ , the quantization interval $\Delta =\frac{2A}{2^{B}}$ and the signal's rms value (again assuming it is a sinusoid) is $\frac{A}{\sqrt{2}}$ .

How many bits would be required in the A/D converter to ensure that the maximum amplitude quantization error wasless than 60 db smaller than the signal's peak value?

Solving $2^{-B}=.001$ results in $B=10$ bits.

Music on a CD is stored to 16-bit accuracy. To what signal-to-noise ratio does this correspond?

A 16-bit A/D converter yields a SNR of $6\times 16+10\lg 1.5=97.8$ dB.

Once we have acquired signals with an A/D converter, we canprocess them using digital hardware or software. It can be shown that if the computer processing is linear, the result ofsampling, computer processing, and unsampling is equivalent to some analog linear system. Why go to all the bother if thesame function can be accomplished using analog techniques? Knowing when digital processing excels and when it does not isan important issue.

how can chip be made from sand
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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!