# 16.1 Scaling

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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 $\left|x(n)\right|\le 1$ . If we scale the input by multiplying it by a value  , $0< < 1$ , then $\left|x(n)\right|\le$ .

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( $\forall n\colon \left|y(n)\right|\le 1$ , $\forall n\colon \left|x(n)\right|\le 1$ )? $\left|y(n)\right|=\left|\sum_{k=0}^{M-1} h(k)x(n-k)\right|\le \sum_{k=0}^{M-1} \left|h(k)\right|\left|\right|\left|x(n-k)\right|\le \sum_{k=0}^{M-1} \left|h(k)\right|\times 1$  $< \sum_{k=0}^{M-1} \left|h(k)\right|$ Alternatively, we can incorporate the scaling directly into the filter, and require that $\sum_{k=0}^{M-1} \left|h(k)\right|< 1$ to prevent overflow.

## Iir filter scaling

To prevent the output from overflowing in an IIR filter, the condition above still holds:( $M$ ) $\left|y(n)\right|< \sum$ 0 h k so an initial scaling factor $< \frac{1}{\sum }$ 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 $\forall i\colon \sum$ 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 $x(n)=\mathrm{sign}(h(-n))$ ; 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 $\forall w\colon \left|H(w)\right|\le 1$ 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 $\forall i, w\colon \left|{H}_{i}(w)\right|\le 1$ 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 ( $(\left|{H}_{i}(w)\right|)$ ) 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.

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
ya I also want to know the raman spectra
Bhagvanji
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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