Subband Coding

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Subband Coding

• Jennie Abraham
• 07/23/2009

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Overview

• Previously, different compression schemes were
  looked into
• Vector Quantization Scheme
• Differential Encoding Scheme
• Scalar Quantization Scheme
• - Most efficient when the data exhibit certain
  characteristics

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Overview contd

• Source data characteristics -
• Unfortunately, most source outputs exhibit a
  combination of characteristics.
• ? difficult to select a compression scheme
  exactly suited to the source output.

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Overview - contd

• Decomposing the source output into constituent
  parts using some method.
• Each constituent part is encoded using one or
  more of the methods described previously.
• ? enables the use of these compression schemes
  more effectively.

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Example 14.2.1
Compression Scheme 1
Yn
Yn
Xn
Xn
Compression Scheme 2
Zn
Zn
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Example 14.2.1 Contd

• Xn 10 14 10 12 14 8 14 12 10
  8 10 12
• Yn
• Xn Yn Zn
• Zn

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Introduction to Subband Coding

• The source output can be decomposed into its
  constituent parts using digital filters.
• Each of these constituent parts will be different
  bands of frequencies which make up the source.

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Subband Coding

• A compression approach where digital filters are
  used to separate the source output into different
  bands of frequencies.
• ?Each part then can be encoded separately.

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Filters

• A filter is system that isolates certain
  frequencies.
• Low Pass Filters
• High Pass Filters
• Band Pass Filters

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Filters Contd

• Filter Characteristics
• Magnitude Transfer Function the ratio of the
  magnitude of the input and output of the filter
  as a function of frequency.
• fo Cutoff Frequency.

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Digital Filters

• Sampling and Nyquist rule
• If fo is the highest frequency of the signal then
  the sampling rate gt 2fo per second can accurately
  represent the continuous signal in digital form.
• Extension of Nyquist rule
• For signal with frequency components between
  frequencies f1and f2 then,
• sampling rate 2(f2 f1) per second.
• Violation of Nyquist rule
• Distortion due to aliasing.

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Digital Filtering

• The general form of the input-output
  relationships of the filter is given by
• where,
• Xn input, Ynoutput of the filter,
• Values ai and bi filter coefficients,
• N is called the taps in the filter.
• FIR Filter
• IIR Filter

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Example 14.3.1

• Filter Coefficients ao 1.25, a1 0.5 and the
  input sequence Xn is given by
• then the output Yn is given by

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Example 14.3.2

• Consider a filter with ao 1 and b1 2.
• The input sequence is a 1 followed by 0s.
• Then the output is

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Filters in literature

• Design and analysis of digital filters is
  detailed in Sections 14.5-14.8 of the textbook.
• ?A useful approach is to make use of the
  available literature to select the necessary
  filters rather than design them.

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Filters used in Subband Coding

• Couple of examples of
• Quadrature Mirror Filters (QMF),
• Johnston Filter
• Smith-Barnwell Filters
• Daubechies Filters
• .and so on

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8-tap Johnston Low-Pass Filter
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8-tap Johnston Low-Pass Filter
LP
HP
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Filter Banks

• Subband coding uses filter banks.
• Filter banks are essentially a cascade of stages,
  where each stage consists of a low-pass filter
  and a high-pass filter.

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Subband Coding Algorithm
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Subband Coding Algorithm

• The three major components of this system are -
• the analysis and synthesis filters,
• the bit allocation scheme, and
• the encoding scheme.
• A substantial amount of research has focused on
  each of these components.

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(1) Analysis

• Source output ? analysis filter bank ?
  sub-sampled ?encoded.
• Analysis Filter Bank
• The source output is passed through a bank of
  filters.
• This filter bank covers the range of frequencies
  that make up the source output.
• The passband of each filter specifies each set of
  frequencies that can pass through.

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Subband Coding Algorithm
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(1) Analysis

• Source output ? analysis filter bank ?
  sub-sampled ?encoded.
• Analysis Filter Bank
• Decimation
• The outputs of the filters are subsampled thus
  reducing the number of samples.

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(1) Analysis

• Source output ? analysis filter bank ?
  sub-sampled ?encoded.
• Analysis Filter Bank
• Decimation
• The justification for the subsampling is the
  Nyquist rule and its extension justifies this
  downsampling.

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(1) Analysis

• Source output ? analysis filter bank ?
  sub-sampled ?encoded.
• Analysis Filter Bank
• Decimation
• The amount of decimation depends on the ratio of
  the bandwidth of the filter output to the filter
  input.

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Subband Coding Algorithm
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(1) Analysis

• Source output ? analysis filter bank ?
  sub-sampled ?encoded.
• Analysis Filter Bank
• Decimation
• Encoding
• The decimated output is encoded using one of
  several encoding schemes, including ADPCM, PCM,
  and vector quantization.

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(2) Quantization and Coding

• Selection of the compression scheme
• Allocation of bits between the subbands
• ?allocate the available bits among the subbands
  according to measure of the information content
  in each subband.
• This bit allocation procedure significantly
  impacts quality of the final reconstruction.

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Bit Allocation

• Minimizing the distortion i.e. minimizing the
  reconstruction error drives the bit allocation
  procedure.
• Different subbands?different amount of
  information.
• Bit allocation procedure can have a significant
  impact on the quality of the final reconstruction

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(3) Synthesis

• Quantized and Coded coefficients are used to
  reconstruct a representation of the original
  signal at the decoder.
• Encoded samples from each subband ?decoded
  ?upsampled ? bank of reconstruction filters
  ?outputs combined ? Final reconstructed output

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Application

• The subband coding algorithm has applications in
  -
• Speech Coding
• Audio Coding
• Image Compression

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Application to Image Compression
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Example 14.12.2 Decomposing and Image
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Example 14.12.2 Decomposing and Image
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Example 14.12.2 Decomposing and Image
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Coding the Subbands
Some bands ? VQ
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Example 14.12.3 Coding the Subbands
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Example 14.12.3 Coding the Subbands
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Coding the Subbands
Some bands ? VQ
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Summary

• Subband coding is another approach to decompose
  the source output into components based on
  frequency.
• Each of these components can then be encoded
  using one of the techniques described in the
  previous chapters.

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Summary

• The general subband encoding procedure can be
  summarized as follows
• Select a set of filters for decomposing the
  source.
• Using the filters, obtain the subband signals.
• Decimate the output of the filters.
• Encode the decimated output.
• The decoding procedure is the inverse of the
  encoding procedure.