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This module looks at Bi-Orthogonal PR-FIR filterbanks and shows how they are similar to orthogonal designs yet provide linear-phase filters.

Bi-orthogonal filter banks

Due to the minimum-phase spectral factorization, orthogonal PR-FIR filterbanks will not have linear-phase analysis and synthesis filters. Non-linear phase may be undesirable forcertain applications. "Bi-orthogonal" designs are closely related to orthogonal designs, yet give linear-phase filters.The analysis-filter design rules for the bi-orthogonal case are

  • F z : zero-phase real-coefficient halfband such that F z n N 1 N 1 f n z n , where N is even.
  • z N 1 F z H 0 z H 1 z
It is straightforward to verify that these design choices satisfy the FIR perfect reconstruction condition H z c z l with c 1 and l N 1 :
H z H 0 z H 1 z H 0 z H 1 z z N 1 F z 1 N 1 z N 1 F z z N 1 F z F z z N 1
Furthermore, note that z N 1 F z is causal with real coefficients, so that both H 0 z and H 1 z can be made causal with real coefficients. (This was another PR-FIR requirement.) The choice c 1 implies that the synthesis filters should obey G 0 z 2 H 1 z G 1 z -2 H 0 z From the design choices above, we can see that bi-orthogonal analysis filter design reduces to the factorization of acausal halfband filter z N 1 F z into H 0 z and H 1 z that have both real coefficients and linear-phase. Earlier we saw thatlinear-phase corresponds to root symmetry across the unit circle in the complex plane, and that real-coefficientscorrespond to complex-conjugate root symmetry. Simultaneous satisfaction of these two properties can be accomplished by quadruples of roots. However, there are special cases in which a root pair, or even a single root, cansimultaneously satisfy these properties. Examples are illustrated in :

The design procedure for the analysis filters of a bi-orthogonal perfect-reconstruction FIR filterbank issummarized below:

  • Design a zero-phase real-coefficient filter F z n N 1 N 1 f n z n where N is a positive even integer (via, e.g. , window designs, LS, or equiripple).
  • Compute the roots of F z and partition into a set of root groups G 0 G 1 G 2 that have both complex-conjugate and unit-circle symmetries. Thus a root group may have one ofthe following forms: G i a i a i 1 a i 1 a i a i a i 1 G i a i a i a i a i G i a i 1 a i a i a i ± 1 G i a i Choose
    Note that H ^ 0 z and H ^ 1 z will be real-coefficient linear-phase regardless of which groups are allocated to whichfilter. Their frequency selectivity, however, will be strongly influenced by group allocation. Thus, you manyneed to experiment with different allocations to find the best highpass/lowpass combination. Note also thatthe length of H 0 z may differ from the length of H 0 z .
    a subset of root groups and construct H ^ 0 z from those roots. Then construct H ^ 1 z from the roots in the remaining root groups. Finally,construct H ^ 1 z from H ^ 1 z by reversing the signs of odd-indexed coefficients.
  • H ^ 0 z and H ^ 1 z are the desired analysis filters up to a scaling. To take care of the scaling, first create H ~ 0 z a H ^ 0 z and H ~ 1 z b H ^ 1 z where a and b are selected so that n h ~ 0 n 1 n h ~ 1 n . Then create H 0 z c H ~ 0 z and H 1 z c H ~ 1 z where c is selected so that the property z N 1 F z H 0 z H 1 z is satisfied at DC ( i.e. , z 0 1 ). In other words, find c so that n h 0 n m h 1 n 1 m 1 .

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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