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Digital filters must be properly scaled to prevent overflow in fixed-point implementations. Scaling by the sum of the absolute value of the impulse response of a filter prevents overflow. However, this is sometimes too conservative in practice, so less stringent rules are often used.

Overflow is clearly a serious problem, since the errors it introduces are very large. As we shall see, it is also responsiblefor large-scale limit cycles, which cannot be tolerated. One way to prevent overflow, or to render it acceptably unlikely, is to scale the input to a filter such that overflow cannot (or is sufficiently unlikely to) occur.

In a fixed-point system, the range of the input signal is limited by the fractional fixed-point number representation to x n 1 . If we scale the input by multiplying it by a value , 0 1 , then x n .

Another option is to incorporate the scaling directly into the filter coefficients.

Fir filter scaling

What value of is required so that the output of an FIR filter cannot overflow( n y n 1 , n x n 1 )? y n k 0 M 1 h k x n k k 0 M 1 h k x n k k M 1 0 h k 1 k M 1 0 h k Alternatively, we can incorporate the scaling directly into the filter, and require that k M 1 0 h k 1 to prevent overflow.

Iir filter scaling

To prevent the output from overflowing in an IIR filter, the condition above still holds:( M ) y n k 0 h k so an initial scaling factor 1 k 0 h k can be used, or the filter itself can be scaled.

However, it is also necessary to prevent the states from overflowing, and to prevent overflow at any point in the signal flow graph where the arithmetic hardware wouldthereby produce errors. To prevent the states from overflowing, we determine the transfer function from the input to all states i , and scale the filter such that i k 0 h i k 1

Although this method of scaling guarantees no overflows, it is often too conservative. Note that a worst-case signal is x n sign h n ; this input may be extremely unlikely. In the relatively common situation in which the input is expected tobe mainly a single-frequency sinusoid of unknown frequency and amplitude less than 1, a scaling condition of w H w 1 is sufficient to guarantee no overflow. This scaling condition is often used. If there are several potential overflowlocations i in the digital filter structure, the scaling conditions are i w H i w 1 where H i w is the frequency response from the input to location i in the filter.

Even this condition may be excessively conservative, for example if the input is more-or-less random, or if occasionaloverflow can be tolerated. In practice, experimentation and simulation are often the bestways to optimize the scaling factors in a given application.

For filters implemented in the cascade form, rather than scaling for the entire filter at the beginning, (whichintroduces lots of quantization of the input) the filter is usually scaled so that each stage is just preventedfrom overflowing. This is best in terms of reducing the quantization noise. The scaling factors are incorporatedeither into the previous or the next stage, whichever is most convenient.

Some heurisitc rules for grouping poles and zeros in a cascade implementation are:

  • Order the poles in terms of decreasing radius. Take the pole pair closest to the unit circle and group it withthe zero pair closest to that pole pair (to minimize the gain in that section). Keep doing this with all remainingpoles and zeros.
  • Order the section with those with highest gain ( H i w ) in the middle, and those with lower gain on the ends.

Leland B. Jackson has an excellent intuitive discussion of finite-precision problems in digitalfilters. The book by Roberts and Mullis is one of the most thorough in terms of detail.

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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