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Exact analysis of quantization errors is difficult because quantization is highly nonlinear. Approximating quantization errors as independent, additive white Gaussian noise processes makes analysis tractable and generally leads to fairly accurate results. Dithering can be used to make these approximations more accurate.

Fundamental assumptions in finite-precision error analysis

Quantization is a highly nonlinear process and is very difficult to analyze precisely. Approximations and assumptions are madeto make analysis tractable.

Assumption #1

The roundoff or truncation errors at any point in a system at each time are random , stationary , and statistically independent (white and independent of all other quantizers in a system).

That is, the error autocorrelation function is r e k e n e n + k q 2 k . Intuitively, and confirmed experimentally in some (but notall!) cases, one expects the quantization error to have a uniform distribution over the interval 2 2 for rounding, or 0 for truncation.

In this case, rounding has zero mean and variance Q x n x n 0 Q 2 e n 2 B 2 12 and truncation has the statistics Q x n x n 2 Q 2 B 2 12

Please note that the independence assumption may be very bad (for example, when quantizing a sinusoid with an integerperiod N ). There is another quantizing scheme called dithering , in which the values are randomly assigned to nearby quantizationlevels. This can be (and often is) implemented by adding a small (one- or two-bit) random input to the signal before atruncation or rounding quantizer.

This is used extensively in practice. Altough the overallerror is somewhat higher, it is spread evenly over all frequencies, rather than being concentrated in spectrallines. This is very important when quantizing sinusoidal or other periodic signals, for example.

Assumption #2

Pretend that the quantization error is really additive Gaussian noise with the same mean and variance as the uniform quantizer. That is, model

As

This model is a linear system, which our standard theory can handle easily. We model the noise asGaussian because it remains Gaussian after passing through filters, so analysis in a system context is tractable.

Summary of useful statistical facts

  • Correlation function

    r x k x n x n + k
  • Power spectral density

    S x w DTFT r x n
  • Note r x 0 x 2 1 2 w S x w
  • r xy k x * n y n k
  • Cross-spectral density

    S xy w DTFT r xy n
  • For y h x : S yx w H w S x w S yy w H w 2 S x w
  • Note that the output noise level after filtering a noise sequence is y 2 r yy 0 1 w H w 2 S x w so postfiltering quantization noise alters the noise power spectrum and may change its variance!
  • For x 1 , x 2 statistically independent r x 1 + x 2 k r x 1 k r x 2 k S x 1 + x 2 w S x 1 w S x 2 w
  • For independent random variables x 1 + x 2 2 x 1 2 x 2 2

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
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Do somebody tell me a best nano engineering book for beginners?
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there is no specific books for beginners but there is book called principle of nanotechnology
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what is fullerene does it is used to make bukky balls
Devang Reply
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.
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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
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
s.
Graphene has a hexagonal structure
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Source:  OpenStax, Digital filter structures and quantization error analysis. OpenStax CNX. Jan 02, 2005 Download for free at http://cnx.org/content/col10259/1.1
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