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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?
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What is STMs full form?
scanning tunneling microscope
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Bob Reply
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The nanotechnology is as new science, to scale nanometric
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Damian Reply
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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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