# 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 to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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!