# Fixed-point number representation

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Specialized DSP hardware typically uses fixed-point number representations for lower cost and complexity and greater speed. Interpretation of two's-complement binary numbers as signed fractions between -1 and 1 allows integer arithmetic to be used for DSP computations, but introduces quantization and overflow errors.

Fixed-point arithmetic is generally used when hardware cost, speed, or complexity is important. Finite-precision quantization issuesusually arise in fixed-point systems, so we concentrate on fixed-point quantization and error analysis in the remainder of this course.For basic signal processing computations such as digital filters and FFTs, the magnitude of the data, the internalstates, and the output can usually be scaled to obtain good performance with a fixed-point implementation.

## Two's-complement integer representation

As far as the hardware is concerned, fixed-point number systems represent data as $B$ -bit integers. The two's-complement number system is usually used: $k=\begin{cases}\text{binary integer representation} & \text{if 0\le k\le 2^{(B-1)}-1}\\ \mathrm{bit-by-bit inverse}(-k)+1 & \text{if -2^{(B-1)}\le k\le 0}\end{cases}$

The most significant bit is known at the sign bit ; it is 0 when the number is non-negative; 1 when the number is negative.

## Fractional fixed-point number representation

For the purposes of signal processing, we often regard the fixed-point numbers as binary fractions between $\left[-1 , 1\right)$ , by implicitly placing a decimal point after the sign bit.

or $x=-{b}_{0}+\sum_{i=1}^{B-1} {b}_{i}2^{-i}$ This interpretation makes it clearer how to implement digital filters in fixed-point, at least when the coefficients have amagnitude less than 1.

## Truncation error

Consider the multiplication of two binary fractions

Note that full-precision multiplication almost doubles thenumber of bits; if we wish to return the product to a $B$ -bit representation, we must truncate the $B-1$ least significant bits. However, this introduces truncation error (also known as quantization error , or roundoff error if the number is rounded to the nearest $B$ -bit fractional value rather than truncated). Note that this occurs after multiplication .

## Overflow error

Consider the addition of two binary fractions;

Note the occurence of wraparound overflow ; this only happens with addition . Obviously, it can be a bad problem.

There are thus two types of fixed-point error: roundoff error, associated with data quantization and multiplication, andoverflow error, associated with data quantization and additions. In fixed-point systems, one must strike a balancebetween these two error sources; by scaling down the data, the occurence of overflow errors is reduced, but the relative sizeof the roundoff error is increased.

Since multiplies require a number of additions, they are especially expensive in terms of hardware(with a complexity proportional to ${B}_{x}{B}_{h}$ , where ${B}_{x}$ is the number of bits in the data, and ${B}_{h}$ is the number of bits in the filter coefficients). Designers try to minimize both ${B}_{x}$ and ${B}_{h}$ , and often choose ${B}_{x}\neq {B}_{h}$ !

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
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
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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
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