<< Chapter < Page Chapter >> Page >

To solve [link] Sanathanan and Koerner defined the same linear system from [link] with the same matrix and vector definitions. However the scalar terms used in the matrix and vectors reflect the presence of the weighting function W ( ω ) in ε s as follows,

λ h = l = 0 L - 1 ω l h W ( ω l ) S h = l = 0 L - 1 ω l h R l W ( ω l ) T h = l = 0 L - 1 ω l h I l W ( ω l ) U h = l = 0 L - 1 ω l h ( R l 2 + I l 2 ) W ( ω l )

Then, given an initial definition of A ( ω ) , at the p -th iteration one sets

W ( ω ) = 1 A p - 1 ( ω k ) 2

and solves [link] using { λ , S , T , U } as defined above until a convergence criterion is reached. Clearly, solving [link] using [link] is equivalent to solving a series of weighted least squares problems where the weighting function consists of the estimated values of A ( ω ) from the previous iteration. This method is similar to a time-domain method proposed by Steiglitz and McBride [link] , presented later in this chapter.

Method of sid-ahmed, chottera and jullien

The methods by Levy and Sanathanan and Koerner did arise from an analog analysis problem formulation, and cannot therefore be used directly to design digital filters. However these two methods present important ideas that can be translated to the context of filter design. In 1978 M. Sid-Ahmed, A. Chottera and G. Jullien followed on these two important works and adapted [link] the matrix and vectors used by Levy to account for the design of IIR digital filters, given samples of a desired frequency response. Consider the frequency response H ( ω ) defined in [link] . In parallel with Levy's development, the corresponding equation error can be written as

ε e = k = 0 L ( R k + j I k ) 1 + c = 1 N a i e - j ω k c - c = 0 M b i e - j ω k c 2

One can follow a similar differentiation step as Levy by setting

ε e a 1 = ε e a 2 = ... = ε e b 0 = ... = 0

with as defined in [link] . Doing so results in a linear system of the form

C x = y

where the vectors x and y are given by

x = b 0 b M a 1 a N y = φ 0 - r 0 φ M - r M - β 1 - β N

The matrix C has a special structure given by

C = Ψ Φ Φ T Υ

where Ψ and Υ are symmetric Toeplitz matrices of order M + 1 and N respectively, and their first row is given by

Ψ 1 m = η m - 1 for m = 1 , ... , M + 1 Υ 1 m = β m - 1 for m = 1 , ... , N

Matrix Φ has order M + 1 × N and has the property that elements on a given diagonal are identical (i.e. Φ i , j = Φ i + 1 , j + 1 ). Its entries are given by

Φ 1 m = φ m + r m for m = 1 , ... , N Φ m 1 = φ m - 2 - r m - 2 for m = 2 , ... , M + 1

The parameters { η , φ , r , β } are given by

η i = k = 0 L cos i ω k for 0 i M β i = k = 0 L | D ( ω k ) | 2 cos i ω k for 0 i N - 1 φ i = k = 0 L R k cos i ω k for 0 i max ( N , M - 1 ) r i = k = 0 L I k sin i ω k for 0 i max ( N , M - 1 )

The rest of the algorithm works the same way as Levy's. For a solution error approach, one must weight each of the parameters mentioned above with the factor from [link] as in the SK method.

There are two important details worth mentioning at this point: on one hand the methods discussed up to this point (Levy, SK and Sid-Ahmed et al.) do not put any limitation on the spacing of the frequency samples; one can sample as fine or as coarse as desired in the frequency domain. On the other hand there is no way to decouple the solution of both numerator and denominator vectors. In other words, from [link] and [link] one can see that the linear systems that solve for vector x solve for all the variables in it. This is more of an issue for the iterative methods (SK&Sid-Ahmed), since at each iteration one solves for all the variables, but for the purposes of updating one needs only to keep the denominator variables (they get used in the weighting function); the numerator variables are never used within an iteration (in contrast to Burrus' Prony-based method presented in [link] ). This approach decouples the numerator and denominator computation into two separate linear systems. One only needs to compute the denominator variables until convergence is reached, and only then it becomes necessary to compute the numerator variables. Therefore most of the iterations solve a smaller linear system than the methods involved up to this point.

Questions & Answers

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
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
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 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