# 13.3 Data quantization in iir filters

 Page 1 / 1
Quantization errors in an IIR filter structure are filtered by portions of the structure before reaching the output. The total quantization noise variance at the output is the sum of the variances of the individual filtered quantizer noises as seen at the filter output.

Finite-precision effects are much more of a concern with IIR filters than with FIR filters, since the effects are more difficult to analyze andminimize, coefficient quantization errors can cause the filters to become unstable, and disastrous things like large-scale limit cycles can occur.

## Roundoff noise analysis in iir filters

Suppose there are several quantization points in an IIR filter structure. By our simplifying assumptions about quantization errorand Parseval's theorem, the quantization noise variance ${}_{\mathrm{y,i}}^{2}$ at the output of the filter from the $i$ th quantizer is

${}_{\mathrm{y,i}}^{2}=\frac{1}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}S{S}_{{n}_{i}}(w)\,d w=\frac{{}_{{n}_{i}}^{2}}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}\,d w={}_{{n}_{i}}^{2}\sum$ h i n 2
where ${}_{{n}_{i}}^{2}$ is the variance of the quantization error at the $i$ th quantizer, $S{S}_{{n}_{i}}(w)$ is the power spectral density of that quantization error, and $H{H}_{i}(w)$ is the transfer function from the $i$ th quantizer to the output point.Thus for $P$ independent quantizers in the structure, the total quantization noise variance is ${}_{y}^{2}=\frac{1}{2\pi }\sum_{i=1}^{P} {}_{{n}_{i}}^{2}\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}\,d w$ Note that in general, each ${H}_{i}(w)$ , and thus the variance at the output due to each quantizer, is different; for example, the system as seen by a quantizer at theinput to the first delay state in the Direct-Form II IIR filter structure to the output, call it ${n}_{4}$ , is with a transfer function ${H}_{4}(z)=\frac{z^{-2}}{1+{a}_{1}z^{(-1)}+{a}_{2}z^{-2}}$ which can be evaluated at $z=e^{iw}$ to obtain the frequency response.

A general approach to find ${H}_{i}(w)$ is to write state equations for the equivalent structure as seen by ${n}_{i}$ , and to determine the transfer function according to $H(z)=CzI-A^{(-1)}B+d$ .

The above figure illustrates the quantization points in a typical implementation of a Direct-Form II IIRsecond-order section. What is the total variance of the output error due to all of thequantizers in the system?

By making the assumption that each ${Q}_{i}$ represents a noise source that is white, independent of the other sources, and additive,

the variance at the output is the sum of the variances atthe output due to each noise source: ${}_{y}^{2}=\sum_{i=1}^{4} {}_{y,i}^{2}$ The variance due to each noise source at the outputcan be determined from $\frac{1}{2\pi }\int_{-\pi }^{\pi } \left|{H}_{i}(w)\right|^{2}{S}_{{n}_{i}}(w)\,d w$ ; note that ${S}_{{n}_{i}}(w)={}_{{n}_{i}}^{2}$ by our assumptions, and ${H}_{i}(w)$ is the transfer function from the noise source to the output .

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
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
Got questions? Join the online conversation and get instant answers!