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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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