# 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.

## Addition and subtraction

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.

#### Questions & Answers

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
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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.
Tarell
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
NANO
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
where are the solutions?
where are the solutions?

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