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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

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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.
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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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