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Jackson's method

The following is a recent approach (from 2008) by Leland Jackson [link] based in the frequency domain. Consider vectors a R N and b R M such that

H ( ω ) = B ( ω ) A ( ω )

where H ( ω ) , B ( ω ) , A ( ω ) are the Fourier transforms of h , b and a respectively. For a discrete frequency set one can describe Fourier transform vectors B = W b b and A = W a a (where W b , W a correspond to the discrete Fourier kernels for b , a respectively). Define

H a ( ω k ) = 1 A ( ω k )

In vector notation, let D a = diag ( H a ) = diag ( 1 / A ) . Then

H ( ω ) = B ( ω ) A ( ω ) = H a ( ω ) B ( ω ) H = D a B

Let H d ( ω ) be the desired complex frequency response. Define D d = diag ( H d ) . Then one wants to solve

min E * E = E 2 2

where E = H - H d . From [link] one can write H = H d + E as

H = D a B = D a W b b

Therefore

H d = H - E = D a W b b - E

Solving [link] for b one gets

b = ( D a W b ) H d

Also,

H d = D d I ^ = D d D a A = D a D d A = D a D d W a a

where I ^ is a unit column vector. Therefore

H - E = H d = D a D d W a a

From [link] we get

D a W b b - E = D a D d W a a

or

D a D d W a a + E = D a W b b

which in a least squares sense results in

a = ( D a D d W a ) ( D a W b b )

From [link] one gets

a = ( D a D d W a ) ( D a W b [ ( D a W b ) H d ] )

As a summary, at the i -th iteration one can write [link] and [link] as follows,

b i = ( diag ( 1 / A i - 1 ) W b ) H d a i = ( diag ( 1 / A i - 1 ) diag ( H d ) W a ) ( diag ( 1 / A i - 1 ) W b b i )

Soewito's quasilinearization method

Consider the equation error residual function

e ( ω k ) = B ( ω k ) - D ( ω k ) · A ( ω k ) = n = 0 M b n e - j ω k n - D ( ω k ) · 1 + n = 1 N a n e - j ω k n = b 0 + b 1 e - j ω k + + b M e - j ω k M - D k - D k a 1 e - j ω k - - D k a N e - j ω k N = b 0 + b M e - j ω k M - D k a 1 e - j ω k + a N e - j ω k N - D k

with D k = D ( ω k ) . The last equation indicates that one can represent the equation error in matrix form as follows,

e = F h - D

where

F = 1 e - j ω 0 e - j ω 0 M - D 0 e - j ω 0 - D 0 e - j ω 0 N 1 e - j ω L - 1 e - j ω L - 1 M - D L - 1 e - j ω L - 1 - D L - 1 e - j ω L - 1 N

and

h = b 0 b 1 b M a 1 a N and D = D 0 D L - 1

Consider now the solution error residual function

s ( ω k ) = H ( ω k ) - D ( ω k ) = B ( ω k ) A ( ω k ) - D ( ω k ) = 1 A ( ω k ) B ( ω k ) - D ( ω k ) · A ( ω k ) = W ( ω k ) e ( ω k )

Therefore one can write the solution error in matrix form as follows

s = W ( F h - D )

where W is a diagonal matrix with 1 A ( ω ) in its diagonal. From [link] the least-squared solution error ε s = s * s can be minimized by

h = ( F * W 2 F ) - 1 F * W 2 D

From [link] an iteration Soewito refers to this expression as the Steiglitz-McBride Mode-1 in frequency domain. could be defined as follows

h i + 1 = ( F * W i 2 F ) - 1 F * W i 2 D

by setting the weights W in [link] equal to A k ( ω ) , the Fourier transform of the current solution for a .

A more formal approach to minimizing ε s consists in using a gradient method (these approaches are often referred to as Newton-like methods). First one needs to compute the Jacobian matrix J of s , where the p q -th term of J is given by J p q = s p h q with s as defined in [link] . Note that the p -th element of s is given by

s p = H p - D p = B p A p - D p

For simplicity one can consider these reduced form expressions for the independent components of h ,

s p b q = 1 A p b q n = 0 M b n e - j ω p n = W p e - j ω p q s p a q = B p a q 1 A p = - B p A p 2 a q 1 + n = 1 N a n e - j ω p n = - 1 A p · B p A p · e - j ω p q = - W p H p e - j ω p q

Therefore on can express the Jacobian J as follows,

J = W G

where

G = 1 e - j ω 0 e - j ω 0 M - H 0 e - j ω 0 - H 0 e - j ω 0 N 1 e - j ω L - 1 e - j ω L - 1 M - H L - 1 e - j ω L - 1 - H L - 1 e - j ω L - 1 N

Consider the solution error least-squares problem given by

min h f ( h ) = s T s

where s is the solution error residual vector as defined in [link] and depends on h . It can be shown [link] that the gradient of the squared error f ( h ) (namely f ) is given by

f = J * s

A necessary condition for a vector h to be a local minimizer of f ( h ) is that the gradient f be zero at such vector. With this in mind and combining [link] and [link] in [link] one gets

f = G * W 2 ( F h - D ) = 0

Solving the system [link] gives

h = ( G * W 2 F ) - 1 G * W 2 D

An iteration can be defined as follows Soewito refers to this expression as the Steiglitz-McBride Mode-2 in frequency domain. Compare to the Mode-1 expression and the use of G i instead of F .

h i + 1 = ( G i * W i 2 F ) - 1 G i * W i 2 D

where matrices W and G reflect their dependency on current values of a and b .

Atmadji Soewito [link] expanded the method of quasilinearization of Bellman and Kalaba [link] to the design of IIR filters. To understand his method consider the first order of Taylor's expansion near H i ( z ) , given by

H i + 1 ( z ) = H i ( z ) + [ B i + 1 ( z ) - B i ( z ) ] A i ( z ) - [ A i + 1 ( z ) - A i ( z ) ] B i ( z ) A i 2 ( z ) = H i ( z ) + B i + 1 ( z ) - B i ( z ) A i ( z ) - B i ( z ) [ A i + 1 ( z ) - A i ( z ) ] A i 2 ( z )

Using the last result in the solution error residual function s ( ω ) and applying simplification leads to

s ( ω ) = B i + 1 ( ω ) A i ( ω ) - H i ( ω ) A i + 1 ( ω ) A i ( ω ) + B i ( ω ) A i ( ω ) - D ( ω ) = 1 A i ( ω ) B i + 1 ( ω ) - H i ( ω ) A i + 1 ( ω ) + B i ( ω ) - A i ( ω ) D ( ω )

Equation [link] can be expressed (dropping the use of ω for simplicity) as

s = W B i + 1 - H i ( A i + 1 - 1 ) - H i + B i - D ( A i - 1 ) - D

One can recognize the two terms in brackets as G h i + 1 and F h i respectively. Therefore [link] can be represented in matrix notation as follows,

s = W [ G h i + 1 - ( D + H i - F h i ) ]

with H = [ H 0 , H 1 , , H L - 1 ] T . Therefore one can minimize s T s from [link] with

h i + 1 = ( G i * W i 2 G i ) - 1 G i * W i 2 ( D + H i - F h i )

since all the terms inside the parenthesis in [link] are constant at the ( i + 1 ) -th iteration. In a sense, [link] is similar to [link] , where the desired function is updated from iteration to iteration as in [link] .

It is important to note that any of the three algorithms can be modified to solve a weighted l 2 IIR approximation using a weighting function W ( ω ) by defining

V ( ω ) = W ( ω ) A ( ω )

Taking [link] into account, the following is a summary of the three different updates discussed so far:

SMB Frequency Mode-1: h i + 1 = ( F * V i 2 F ) - 1 F * V i 2 D SMB Frequency Mode-2: h i + 1 = ( G i * V i 2 F ) - 1 G i * V i 2 D Soewito's quasilinearization: h i + 1 = ( G i * V i 2 G i ) - 1 G i * V i 2 ( D + H i - F h i )

Questions & Answers

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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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learn
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Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
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Professor
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Professor
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Alexandre
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Alexandre
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Brian Reply
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Rafiq
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Damian
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LITNING Reply
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LITNING Reply
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LITNING
scanning tunneling microscope
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Rafiq
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Mahi
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Rafiq
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Hafiz
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Bob Reply
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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
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Iterative design of l_p digital filters. OpenStax CNX. Dec 07, 2011 Download for free at http://cnx.org/content/col11383/1.1
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