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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

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
what is the stm
Brian Reply
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?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
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What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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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
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
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nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
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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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