# 4.2 The binary filter tree

 Page 1 / 1
This module introduces the binary filter tree.

Recall that for image compression (see The 2-band Filter Bank ), the purpose of the 2-band filter bank in the Haar transform is tocompress most of the signal energy into the low-frequency band.

We may achieve greater compression if the low brand is further split into two. This may be repeated a number of times to givethe binary filter tree, shown with 4 levels in .

In 1-D, this is analogous to the way the 2-D Haar transform was extended to the multi-level Haar transform .

For an $N$ -sample input vector $x$ , the sizes and bandwidths of the signals of the 4-level filter tree are:

Signal No. of samples Approximate pass band
$x$ $N$ $0\to \frac{1}{2}{f}_{s}$
${y}_{1}$ $\frac{N}{2}$ $\frac{1}{4}\to \frac{1}{2}{f}_{s}$
${y}_{01}$ $\frac{N}{4}$ $\frac{1}{8}\to \frac{1}{4}{f}_{s}$
${y}_{001}$ $\frac{N}{8}$ $\frac{1}{16}\to \frac{1}{8}{f}_{s}$
${y}_{0001}$ $\frac{N}{16}$ $\frac{1}{32}\to \frac{1}{16}{f}_{s}$
${y}_{0000}$ $\frac{N}{16}$ $0\to \frac{1}{32}{f}_{s}$

Because of the downsampling (decimation) by 2 at each level, the total number of output samples = $N$ , regardless of the number of levels in the tree.

The ${H}_{0}$ filter is normally designed to be a lowpass filter with a passband from 0 to approximately $\frac{1}{4}$ of the input sampling frequency for that stage; and ${H}_{1}$ is a highpass (bandpass) filter with a pass bandapproximately from $\frac{1}{4}$ to $\frac{1}{2}$ of the input sampling frequency.

When formed into a 4-level tree, the filter outputs have the approximate pass bands given in . The final output ${y}_{0000}$ is a lowpass signal, while the other outputs are all bandpass signals, each covering a band of approximately oneoctave.

An inverse tree, mirroring , may be constructed using filters ${G}_{0}$ and ${G}_{1}$ instead of ${H}_{0}$ and ${H}_{1}$ , as shown for just one level in part (b) of this figure . If the PR conditions of this previous equation and this previous equation are satisfied, then the output of each level will be identical to the input of the equivalent level in , and the final output will be a perfect reconstruction of the input signal.

## Multi-rate filtering theorem

To calculate the impulse and frequency responses for a multistage network with downsampling at each stage, as in , we must first derive an important theorem for multirate filters.

The downsample-filter-upsample operation of (a) is equivalent to either the filter-downsample-upsample operation of (b) or the downsample-upsample-filter operation of (c), if the filter is changed from $H(z)$ to $H(z^{2})$ .

From (a):

Take z-transforms:
Reverse the order of summation and let $m=n-2i$ : therefore,
where $Y(z)=H(z^{2})X(z)$

This describes the operations of (b). Hence the first result is proved.

The result from line 3 in

$\stackrel{^}{Y}(z)=\frac{1}{2}(X(z)+X(-z))H(z^{2})=\stackrel{^}{X}(z)H(z^{2})$
shows that the filter $H(z^{2})$ may be placed after the down/up-sampler as in (c), which proves the second result.

## General results for m:1 subsampling

It can be shown that:

• $H(z)$ becomes $H(z^{M})$ if shifted ahead of an M:1 downsampler or following an M:1 upsampler.
• M:1 down/up-sampling of a signal $X(z)$ produces:
$\stackrel{^}{X}(z)=\frac{1}{M}\sum_{m=0}^{M-1} X(ze^{\frac{i\times 2\pi m}{M}})$

## Transformation of the filter tree

Using the result of , can be redrawn as in with all downsamplers moved to the outputs. (Note requires much more computation than .)

We can now calculate the transfer function to each output (before the downsamplers) as:

${H}_{01}(z)={H}_{0}(z){H}_{1}(z^{2})$
${H}_{001}(z)={H}_{0}(z){H}_{0}(z^{2}){H}_{1}(z^{4})$
${H}_{0001}(z)={H}_{0}(z){H}_{0}(z^{2}){H}_{0}(z^{4}){H}_{1}(z^{8})$
${H}_{0000}(z)={H}_{0}(z){H}_{0}(z^{2}){H}_{0}(z^{4}){H}_{0}(z^{8})$
In general the transfer functions to the two outputs at level $k$ of the tree are given by:
${H}_{k,1}=\prod_{i=0}^{k-2} {H}_{0}(z^{2^{i}}){H}_{1}(z^{2^{(k-1)}})$
${H}_{k,0}=\prod_{i=0}^{k-1} {H}_{0}(z^{2^{i}})$
For the Haar filters of this equation and this equation from our discussion of the 2-band filter bank, the transfer functionsto the outputs of the 4-level tree become:
${H}_{01}(z)=\frac{1}{2}(z^{-3}+z^{-2}-z^{-1}+1)$
${H}_{001}(z)=\frac{1}{2\sqrt{2}}(z^{-7}+z^{-6}+z^{-5}+z^{-4}-z^{-3}+z^{-2}+z^{-1}+1)$
${H}_{0001}(z)=\frac{1}{4}(z^{-15}+z^{-14}+z^{-13}+z^{-12}+z^{-11}+z^{-10}+z^{-9}+z^{-8}-z^{-7}+z^{-6}+z^{-5}+z^{-4}+z^{-3}+z^{-2}+z^{-1}+1)$
${H}_{0000}(z)=\frac{1}{4}(z^{-15}+z^{-14}+z^{-13}+z^{-12}+z^{-11}+z^{-10}+z^{-9}+z^{-8}+z^{-7}+z^{-6}+z^{-5}+z^{-4}+z^{-3}+z^{-2}+z^{-1}+1)$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!

#### Get Jobilize Job Search Mobile App in your pocket Now! By Madison Christian By OpenStax By OpenStax By Ryan Lowe By By Dravida Mahadeo-J... By Anh Dao By Olivia D'Ambrogio By Hoy Wen By Rhodes