Subband Coding

Overview

Previously, different compression schemes were looked into –

(i)Vector Quantization Scheme(ii)Differential Encoding Scheme(iii)Scalar Quantization Scheme

- Most efficient when the data exhibit certain characteristics

Overview – cont’d

Source data characteristics - Unfortunately, most source outputs

exhibit a combination of characteristics.

difficult to select a compression scheme exactly suited to the source output.

Overview - cont’d

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.

Example 14.2.1

Xn

Zn

Yn

Zn

YnCompression Scheme 1

Compression Scheme 2

Xn

Example 14.2.1 – Cont’d

Xn = 10 14 10 12 14 8 14 12 10 8 10 12

Yn =

Xn = Yn + Zn

Zn =

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.

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.

Filters

A filter is system that isolates certain frequencies.

(i) Low Pass Filters(ii) High Pass Filters(iii) Band Pass Filters

Filters – Cont’dFilter 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.

Digital Filters Sampling and Nyquist rule :If fo is the highest frequency of the signal then the

sampling rate > 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.

Digital Filtering

The general form of the input-output relationships of the filter is given by

where, {Xn}= input, {Yn}=output of the filter,Values {ai} and {bi} = filter coefficients, N is called the taps in the filter.

FIR Filter IIR Filter

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

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

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.

Filters used in Subband Coding

Couple of examples of –Quadrature Mirror Filters (QMF),Johnston FilterSmith-Barnwell FiltersDaubechies Filters

….and so on

8-tap Johnston Low-Pass Filter

8-tap Johnston Low-Pass Filter

LP

HP

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.

Subband Coding Algorithm

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.

(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.

Subband Coding Algorithm

(1) Analysis

Source output analysis filter bank sub-sampled encoded.

Analysis Filter BankDecimation The outputs of the filters are subsampled

thus reducing the number of samples.

(1) Analysis

Source output analysis filter bank sub-sampled encoded.

Analysis Filter BankDecimation The justification for the subsampling is the

Nyquist rule and its extension justifies this downsampling.

(1) Analysis

Source output analysis filter bank sub-sampled encoded.

Analysis Filter BankDecimation The amount of decimation depends on the

ratio of the bandwidth of the filter output to the filter input.

Subband Coding Algorithm

(1) Analysis

Source output analysis filter bank sub-sampled encoded.

Analysis Filter BankDecimationEncoding The decimated output is encoded using one

of several encoding schemes, including ADPCM, PCM, and vector quantization.

(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.

Bit Allocation

Minimizing the distortion i.e. minimizing the reconstruction error drives the bit allocation procedure.

Different subbandsdifferent amount of information.

Bit allocation procedure can have a significant impact on the quality of the final reconstruction

(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

Application

The subband coding algorithm has applications in -

Speech CodingAudio CodingImage Compression

Application to Image Compression

LL LH

HL HH

Example 14.12.2 – Decomposing and Image

Example 14.12.2 – Decomposing and Image

Example 14.12.2 – Decomposing and Image

Coding the Subbands

SQ

LL LH

HL HH

DiscardDPCM

Some bands VQ

Example 14.12.3 – Coding the Subbands

Example 14.12.3 – Coding the Subbands

Coding the Subbands

SQ

LL LH

HL HH

DiscardDPCM

Some bands VQ

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.

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.