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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 -1 0 1 . 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 three-bit A/D converter assigns voltage in the range -1 1 to one of eight integers between 0 and 7. For example, allinputs having values lying between 0.5 and 0.75 are assigned the integer value six and, upon conversion back to an analogvalue, they all become 0.625. The width of a single quantization interval Δ equals 2 2 B . The bottom panel shows a signal going through theanalog-to-digital converter, where B is the number of bits used in the A/D conversion process (3 inthe case depicted here). First it is sampled, then amplitude-quantized to three bits. Note how the sampledsignal waveform becomes distorted after amplitude quantization. For example the two signal values between 0.5and 0.75 become 0.625. This distortion is irreversible; it can be reduced (but not eliminated) by using more bits inthe A/D converter.

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 1 1 . 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 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 Δ is 0.25 ; in general, it is 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: 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: power s 1 2 2 1 2 . The illustration details a single quantization interval.

A single quantization interval is shown, along with atypical signal's value before amplitude quantization s n T s and after Q s n T s . ε denotes the error thus incurred.
Its width is Δ and the quantization error is denoted by ε . 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.
rms ε 1 Δ ε Δ 2 Δ 2 ε 2 Δ 2 12 1 2
Since the quantization interval width for a B -bit converter equals 2 2 B 2 B 1 , we find that the signal-to-noise ratio for theanalog-to-digital conversion process equals
SNR 1 2 2 2 B 1 12 3 2 2 2 B 6 B 10 10 logbase --> 1.5 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 10 logbase --> 1.5 equals 1.76.

This derivation assumed the signal's amplitude lay in the range -1 1 . What would the amplitude quantization signal-to-noiseratio be if it lay in the range A A ?

The signal-to-noise ratio does not depend on the signal amplitude. With an A/D range of A A , the quantization interval Δ 2 A 2 B and the signal's rms value (again assuming it is a sinusoid) is A 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 16 10 10 logbase --> 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.

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
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Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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Smarajit Reply
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Source:  OpenStax, Analog-to-digital conversion. OpenStax CNX. Sep 20, 2008 Download for free at http://cnx.org/content/col10578/1.2
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