# Fixed point arithmetic  (Page 3/4)

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In the 2's complement fractional representation, an $N$ bit binary word can represent $2^{N}$ equally space numbers from $\frac{-2^{(N-1)}}{2^{(N-1)}}=1$ to $\frac{2^{-(N-1)}}{2^{(N-1)}}=1-2^{(N-1)}$ .

For example, we interpret an 8-bit binary word ${b}_{7}{b}_{6}{b}_{5}{b}_{4}{b}_{3}{b}_{2}{b}_{1}{b}_{0}$ as a fractional number $x=\frac{-({b}_{7}2^{7})+{b}_{6}2^{6}++{b}_{1}\times 2+{b}_{0}}{2^{7}}=(-({b}_{7})+\sum_{i=0}^{6} 2^{(i-7)}{b}_{i})\in \left[-1 , 1-2^{-7}\right]$

This representation is also referred as Q-format . We can think of having an implied binary digit right after the MSB. If we have an $N$ -bit binary word with MSB as the sign bit, we have $N-1$ bits to represent the fraction. We say the number has Q-( $N-1$ ) format. For example, in the example, $x$ is a Q-7 number. In C6211, it is easiest to handle Q-15 numbers represented by each 16bit binary word, because the multiplication of two Q-15 numbers results in a Q-30 number that can still be stored ina 32-bit wide register of C6211. The programmer needs to keep track of the implied binary point when manipulatingQ-format numbers.

(Q format): What are the decimal fractional numbers corresponding to the Q-7 format binary numbers; $01001101$ , $11100100$ , $01111001$ , and $10001011$ ?

Intentionally left blank.

## Two's complement arithmetic

The convenience of 2's compliment format comes from the ability to represent negative numbers and computesubtraction using the same algorithm as a binary addition. The C62x processor has instructions to add, subtract andmultiply numbers in the 2's compliment format. Because, in most digital signal processing algorithms, Q-15 format ismost easy to implement on C62x processors, we only focus on the arithmetic operations on Q-15 numbers in the following.

The addition of two binary numbers is computed in the same way as we compute the sum of two decimal numbers.Using the relation $0+0=0$ , $0+1=1+0=1$ and $1+1=10$ , we can easily compute the sum of two binary numbers. The C62x instruction ADD performs this binary addition on different operands.

However, care must be taken when adding binary numbers. Because each Q-15 number can represent numbers in therange $\left[-1 , 1-2^{15}\right]$ , if the result of summing two Q-15 numbers is not in this range, we cannot represent the result in theQ-15 format. When this happens, we say an overflow has occurred. Unless carefully handled, the overflow makes the result incorrect.Therefore, it is really important to prevent overflows from occurring when implementing DSP algorithms. One wayof avoiding overflow is to scale all the numbers down by a constant factor, effectively making all the numbers verysmall, so that any summation would give results in the $\left[-1 , 1\right)$ range. This scaling is necessary and it is important to figure out how muchscaling is necessary to avoid overflow. Because scaling results in loss of effective number of digits, increasingquantization errors, we usually need to find the minimum amount of scaling to prevent overflow.

Another way of handling the overflow (and underflow) is saturation . If the result is out of the range that can be properly represented in the given datasize, the value is saturated, meaning that the value closest to the true result is taken in the rangerepresentable. Such instructions as SADD , SSUB perform the operations followed by saturation.

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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
where are the solutions?
where are the solutions?