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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 binary integer representation 0 k 2 B 1 1 bit-by-bit inverse k 1 2 B 1 k 0

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 -1 1 , by implicitly placing a decimal point after the sign bit.

or x b 0 i B 1 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 B h !

Questions & Answers

How we are making nano material?
LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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
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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
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
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how to know photocatalytic properties of tio2 nanoparticles...what to do now
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it is a goid question and i want to know the answer as well
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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Devang Reply
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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.
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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
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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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