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L = 1 π 0 π W ( ω ) ( A ( ω ) - A d ( ω ) ) 2 d ω + i μ i A ( ω i ) - A d ( ω i ) ± T ( ω i )

with a corresponding derivative of

d L d a ( n ) = 2 π W ( ω ) ( A ( ω ) - A d ( ω ) d A d a d ω + i μ i d A d a ω i

The integral cannot be carried out analytically for a general weighting function, but if the weight function is constant over each subband, Equation 47 from Least Squared Error Design of FIR Filters can be written

d L d a ( n ) = 2 π k ω k ω k + 1 W k ( m = 1 M a ( m ) cos ( ω m ) + K a ( 0 ) - A d ( ω ) ) cos ( ω n ) d ω + i μ i d A d a ω i

which after rearranging is

= m = 1 M 2 π k W k ω k ω k + 1 ( cos ( ω m ) cos ( ω n ) ) d ω a ( m )
- 2 π k W k ω k ω k + 1 A d ( ω ) cos ( ω n ) d ω + i μ i cos ( ω i n ) = 0

where the integral in the first term can now be done analytically. In matrix notation Equation 49 from Least Squared Error Design of FIR Filters is

R a - a d w + H μ = 0

This is a similar form to that in the multiband paper where the matrix R gives the effects of weighting with elements

r ( n , m ) = 2 π k W k ω k ω k + 1 ( cos ( ω m ) cos ( ω n ) ) d ω

except for the first row which should be divided by 2 K because of the normalizing of the a ( 0 ) term in Equation 49 from FIR Digital Filters and [link] and the first column which should be multiplied by K because of Equation 51 from FIR Digital Filters and [link] . The matrix R is a sum of a Toeplitz matrix and a Hankel matrix and this fact might be used to advantage and a d w is the vector of modified filter parameters with elements

a d w ( n ) = 2 π W k ω k ω k + 1 A d ( ω ) cos ( ω n ) d ω

and the matrix H is the same as used in [link] and defined in [link] . Equations Equation 50 from Least Squared Error Design of FIR Filters and [link] can be written together as a matrix equation

R H G 0 a μ = a d w A c

The solutions to Equation 50 from Least Squared Error Design of FIR Filters and [link] or to [link] are

μ = ( G R - 1 H ) - 1 ( GR - 1 a d w - A c )
a = R - 1 ( a d w - H μ )

which are ideally suited to a language like Matlab and are implemented in the programs at the end of this book.

Since the solution of R a u = a d w is the optimal unconstrained weighted least squares filter, we can write [link] and [link] in the form

μ = ( G R - 1 H ) - 1 ( G a u - A c ) = ( G R - 1 H ) - 1 ( A u - A c )
a = a u - R - 1 H μ

The exchange algorithms

This Lagrange multiplier formulation together with applying the Kuhn-Tucker conditions are used in an iterative multiple exchange algorithm similar tothe Remez exchange algorithm to give the complete design method.

One version of this exchange algorithm applies to the problem posed by Adams with specified pass and stopband edges and with zero error weightingin the transition band. This problem has the structure of a quadratic programming problem and could be solved using general QP methods but themultiple exchange algorithm suggested here is probably faster.

The second version of this exchange algorithm applies to the problem where there is no explicitly specified transition band. This problem is notstrictly a quadratic programming problem and our exchange algorithm has no proof of convergence (the HOS algorithm also has no proof of convergence).However, in practice, this program has proven to be robust and converges for a wide variety of lengths, constraints, weights, andband edges. The performance is completely independent of the normalizing parameter K . Notice that the inversion of the R matrix is done once and does not have to be done each iteration. The details of theprogram are included in the filter design paper and in the Matlab program at the end of this book.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
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Mahi
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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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Digital signal processing and digital filter design (draft). OpenStax CNX. Nov 17, 2012 Download for free at http://cnx.org/content/col10598/1.6
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