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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 .

Extension of the 2-band filter bank into a binary filter tree.

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 1 2 f s
y 1 N 2 1 4 1 2 f s
y 01 N 4 1 8 1 4 f s
y 001 N 8 1 16 1 8 f s
y 0001 N 16 1 32 1 16 f s
y 0000 N 16 0 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 1 4 of the input sampling frequency for that stage; and H 1 is a highpass (bandpass) filter with a pass bandapproximately from 1 4 to 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.

Multi-rate filtering - the result of shifting a filter ahead of a downsampling operation or after an upsamplingoperation.

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):

y ^ n i x n 2 i h i n is even 0 n is odd
Take z-transforms:
Y ^ z n y ^ n z n even n i x n 2 i h i z n
Reverse the order of summation and let m n 2 i : therefore,
Y ^ z i h i even m x m z m z -2 i i h i z -2 i even m x m z m H z 2 1 2 X z X z 1 2 H z 2 X z H z 2 X z 1 2 Y z Y z
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

Y ^ z 1 2 X z X z H z 2 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:
    X ^ z 1 M m 0 M 1 X z 2 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 .)

Binary filter tree, transformed so that all downsampling operations occur at the outputs.

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 i 0 k 2 H 0 z 2 i H 1 z 2 k 1
H k , 0 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 1 2 z -3 z -2 z -1 1
H 001 z 1 2 2 z -7 z -6 z -5 z -4 z -3 z -2 z -1 1
H 0001 z 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 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

Questions & Answers

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Missy Reply
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Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
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Lohitha
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nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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da
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Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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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.
Damian
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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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Brian Reply
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
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LITNING Reply
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LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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The nanotechnology is as new science, to scale nanometric
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Source:  OpenStax, Image coding. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10206/1.3
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