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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

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
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
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
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
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
Is there any normative that regulates the use of silver nanoparticles?
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
why?
Adin
what school?
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
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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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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