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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
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
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Adin Reply
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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characteristics of micro business
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for teaching engĺish at school how nano technology help us
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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
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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.
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Do you know which machine is used to that process?
s.
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for screen printed electrodes ?
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What is lattice structure?
s. Reply
of graphene you mean?
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or in general
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in general
s.
Graphene has a hexagonal structure
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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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