# 0.8 Filter banks and transmultiplexers

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## Introduction

In this chapter, we develop the properties of wavelet systems in terms of the underlying filter banks associated with them. This is an expansion andelaboration of the material in Chapter: Filter Banks and the Discrete Wavelet Transform , where many of the conditions and properties developed from a signal expansion point of view in  Chapter: The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients are now derived from the associated filter bank. The Mallat algorithm uses a special structure of filters anddownsamplers/upsamplers to calculate and invert the discrete wavelet transform. Such filter structures have been studied for over three decades in digitalsignal processing in the context of the filter bank and transmultiplexer problems [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link] . Filter bank theory, besides providing efficient computational schemes for wavelet analysis, also gives valuable insights into theconstruction of wavelet bases. Indeed, some of the finer aspects of wavelet theory emanates from filter bank theory.

## The filter bank

A filter bank is a structure that decomposes a signal into a collection of subsignals. Depending on the application, these subsignals helpemphasize specific aspects of the original signal or may be easier to work with than the original signal. We have linear or non-linear filter banks depending on whether or not the subsignals depend linearly on the original signal. Filter banks were originally studied in thecontext of signal compression where the subsignals were used to “represent” the original signal. The subsignals (called subbandsignals) are downsampled so that the data rates are the same in the subbands as in the original signal—though this is not essential. Keypoints to remember are that the subsignals convey salient features of the original signal and are sufficient to reconstruct the original signal.

[link] shows a linear filter bank that is used in signal compression (subband coding). The analysis filters $\left\{{h}_{i}\right\}$ are used to filter the input signal $x\left(n\right)$ . The filtered signals are downsampled to give the subband signals. Reconstruction of the original signal is achieved by upsampling, filtering and adding upthe subband signals as shown in the right-hand part of [link] . The desire for perfect reconstruction (i.e., $y\left(n\right)=x\left(n\right)$ ) imposes a set of bilinear constraints (since all operations in [link] are linear) on the analysis and synthesis filters. This also constrains the downsampling factor, $M$ , to be at most the number of subband signals, say $L$ . Filter bank design involves choosing filters $\left\{{h}_{i}\right\}$ and $\left\{{g}_{i}\right\}$ that satisfy perfect reconstruction and simultaneously give informative and useful subbandsignals. In subband speech coding, for example, a natural choice of desired frequency responses—motivated by the nonuniform sensitivity ofthe human ear to various frequency bands—for the analysis and synthesis filters is shown in [link] .

In summary, the filter bank problem involves the design of the filters ${h}_{i}\left(n\right)$ and ${g}_{i}\left(n\right)$ , with the following goals:

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what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
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Alexandre
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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?
what is a peer
What is meant by 'nano scale'?
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
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