# 2.5 Constrained approximation and mixed criteria  (Page 3/10)

 Page 3 / 10
$\mathcal{L}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{1}{\pi }{\int }_{0}^{\pi }W\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}{\left(A\left(\omega \right)-{A}_{d}\left(\omega \right)\right)}^{2}\phantom{\rule{0.166667em}{0ex}}d\omega +\sum _{i}{\mu }_{i}\phantom{\rule{0.166667em}{0ex}}\left(A,\left({\omega }_{i}\right),-,{A}_{d},\left({\omega }_{i}\right),±,T,\left({\omega }_{i}\right)\right)$

with a corresponding derivative of

$\frac{d\mathcal{L}}{d\phantom{\rule{0.166667em}{0ex}}a\left(n\right)}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{2}{\pi }\int \left(W,\left(\omega \right),\phantom{\rule{0.166667em}{0ex}},\left(,A,\left(\omega \right),-,{A}_{d},\left(\omega \right)\right)\frac{dA}{da}\phantom{\rule{0.166667em}{0ex}}d\omega +{\left(\sum _{i},{\mu }_{i},\frac{dA}{da}|}_{{\omega }_{i}}$

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

$\frac{d\mathcal{L}}{d\phantom{\rule{0.166667em}{0ex}}a\left(n\right)}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{2}{\pi }\sum _{k}{\int }_{{\omega }_{k}}^{{\omega }_{k+1}}\left({W}_{k},\phantom{\rule{0.166667em}{0ex}},\left(\sum _{m=1}^{M}a\left(m\right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega m\right)+K\phantom{\rule{0.166667em}{0ex}}a\left(0\right)-{A}_{d}\left(\omega \right)\right)\right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega n\right)\phantom{\rule{0.166667em}{0ex}}d\omega +{\left(\sum _{i},{\mu }_{i},\frac{dA}{da}|}_{{\omega }_{i}}$

which after rearranging is

$=\sum _{m=1}^{M}\left[\frac{2}{\pi },\sum _{k},{W}_{k},{\int }_{{\omega }_{k}}^{{\omega }_{k+1}},\left(cos\left(\omega m\right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega n\right)\right),\phantom{\rule{0.166667em}{0ex}},d,\omega \right]\phantom{\rule{0.166667em}{0ex}}a\left(m\right)$
$-\frac{2}{\pi }\sum _{k}{W}_{k}{\int }_{{\omega }_{k}}^{{\omega }_{k+1}}{A}_{d}\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega n\right)\phantom{\rule{0.166667em}{0ex}}d\omega +\sum _{i}{\mu }_{i}cos\left({\omega }_{i}n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}0$

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

$\mathbf{R}\phantom{\rule{0.166667em}{0ex}}\mathbf{a}-{\mathbf{a}}_{{\mathbf{d}}_{\mathbf{w}}}+\mathbf{H}\phantom{\rule{0.166667em}{0ex}}\mu \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\mathbf{0}$

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

$r\left(n,m\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{2}{\pi }\sum _{k}{W}_{k}{\int }_{{\omega }_{k}}^{{\omega }_{k+1}}\left(cos\left(\omega m\right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega n\right)\right)\phantom{\rule{0.166667em}{0ex}}d\omega$

except for the first row which should be divided by $2\phantom{\rule{0.166667em}{0ex}}K$ because of the normalizing of the $a\left(0\right)$ 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 $\mathbf{R}$ is a sum of a Toeplitz matrix and a Hankel matrix and this fact might be used to advantage and ${\mathbf{a}}_{{\mathbf{d}}_{\mathbf{w}}}$ is the vector of modified filter parameters with elements

${a}_{{d}_{w}}\left(n\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{2}{\pi }\sum {W}_{k}{\int }_{{\omega }_{k}}^{{\omega }_{k+1}}{A}_{d}\left(\omega \right)\phantom{\rule{0.166667em}{0ex}}cos\left(\omega n\right)\phantom{\rule{0.166667em}{0ex}}d\omega$

and the matrix $\mathbf{H}$ 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

$\left[\begin{array}{cc}\mathbf{R}& \mathbf{H}\\ \mathbf{G}& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathbf{a}\\ \mu \end{array}\right]\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left[\begin{array}{c}{\mathbf{a}}_{{d}_{w}}\\ {\mathbf{A}}_{c}\end{array}\right]$

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

$\begin{array}{ccc}\hfill \mu & =& {\left(\mathbf{G}{\mathbf{R}}^{-1}\mathbf{H}\right)}^{-1}\left({\mathbf{GR}}^{-1}{\mathbf{a}}_{{\mathbf{d}}_{\mathbf{w}}}-{\mathbf{A}}_{\mathbf{c}}\right)\hfill \end{array}$
$\begin{array}{ccc}\hfill \mathbf{a}& =& {\mathbf{R}}^{-1}\left({\mathbf{a}}_{{d}_{w}}-\mathbf{H}\phantom{\rule{0.166667em}{0ex}}\mu \right)\hfill \end{array}$

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 $\mathbf{R}\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{\mathbf{u}}={\mathbf{a}}_{{\mathbf{d}}_{\mathbf{w}}}$ is the optimal unconstrained weighted least squares filter, we can write [link] and [link] in the form

$\begin{array}{ccc}\hfill \mu & \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}& {\left(\mathbf{G}{\mathbf{R}}^{-1}\mathbf{H}\right)}^{-1}\left(\mathbf{G}\phantom{\rule{0.166667em}{0ex}}{\mathbf{a}}_{\mathbf{u}}-{\mathbf{A}}_{\mathbf{c}}\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\left(\mathbf{G}{\mathbf{R}}^{-1}\mathbf{H}\right)}^{-1}\left({\mathbf{A}}_{\mathbf{u}}-{\mathbf{A}}_{\mathbf{c}}\right)\hfill \end{array}$
$\begin{array}{ccc}\hfill \mathbf{a}& \phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}& {\mathbf{a}}_{u}-{\mathbf{R}}^{-1}\mathbf{H}\phantom{\rule{0.166667em}{0ex}}\mu \hfill \end{array}$

## 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 $\mathbf{R}$ 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.

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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
Got questions? Join the online conversation and get instant answers!