<< Chapter < Page Chapter >> Page >

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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
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
Devang Reply
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
Abhijith Reply
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?
s. Reply
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 ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





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
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Iterative design of l_p digital filters' conversation and receive update notifications?

Ask