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Quantizing a Signal

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Section Overview Representing Partitions Representing Codebooks Scalar Quantization Example 1 Scalar Quantization Example 2 Determining Which Interval Each Input Is In

Section Overview

Scalar quantization is a process that maps all inputs within a specified range to a common value. It maps inputs in a different range of values to a different common value. In effect, scalar quantization digitizes an analog signal. Two parameters determine a quantization: a partition and a codebook.

This section describes how to represent these parameters. It also shows, via examples, how to use the partition and codebook with the quantiz function.

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Representing Partitions

A quantization partition defines several contiguous, nonoverlapping ranges of values within the set of real numbers. To specify a partition in the MATLAB environment, list the distinct endpoints of the different ranges in a vector.

For example, if the partition separates the real number line into the four sets

• {x: x ≤ 0}

• {x: 0< x ≤ 1}

• {x: 1 < x ≤ 3}

• {x: 3 < x}

then you can represent the partition as the three-element vector

partition = [0,1,3];

The length of the partition vector is one less than the number of partition intervals.

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Representing Codebooks

A codebook tells the quantizer which common value to assign to inputs that fall into each range of the partition. Represent a codebook as a vector whose length is the same as the number of partition intervals. For example, the vector

codebook = [-1, 0.5, 2, 3];

is one possible codebook for the partition [0,1,3].

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Scalar Quantization Example 1

The code below shows how the quantiz function uses partition and codebook to map a real vector, samp, to a new vector, quantized, whose entries are either -1, 0.5, 2, or 3.

partition = [0,1,3];
codebook = [-1, 0.5, 2, 3];
samp = [-2.4, -1, -.2, 0, .2, 1, 1.2, 1.9, 2, 2.9, 3, 3.5, 5];
[index,quantized] = quantiz(samp,partition,codebook);
quantized

The output is below.

quantized =

  Columns 1 through 6 

   -1.0000   -1.0000   -1.0000   -1.0000    0.5000    0.5000

  Columns 7 through 12 

    2.0000    2.0000    2.0000    2.0000    2.0000    3.0000

  Column 13 

    3.0000

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Scalar Quantization Example 2

This example illustrates the nature of scalar quantization more clearly. After quantizing a sampled sine wave, it plots the original and quantized signals. The plot contrasts the x's that make up the sine curve with the dots that make up the quantized signal. The vertical coordinate of each dot is a value in the vector codebook.

t = [0:.1:2*pi]; % Times at which to sample the sine function
sig = sin(t); % Original signal, a sine wave
partition = [-1:.2:1]; % Length 11, to represent 12 intervals
codebook = [-1.2:.2:1]; % Length 12, one entry for each interval
[index,quants] = quantiz(sig,partition,codebook); % Quantize.
plot(t,sig,'x',t,quants,'.')
legend('Original signal','Quantized signal');
axis([-.2 7 -1.2 1.2])

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Determining Which Interval Each Input Is In

The quantiz function also returns a vector that tells which interval each input is in. For example, the output below says that the input entries lie within the intervals labeled 0, 6, and 5, respectively. Here, the 0th interval consists of real numbers less than or equal to 3; the 6th interval consists of real numbers greater than 8 but less than or equal to 9; and the 5th interval consists of real numbers greater than 7 but less than or equal to 8.

partition = [3,4,5,6,7,8,9];
index = quantiz([2 9 8],partition)

The output is

index =

     0
     6
     5

If you continue this example by defining a codebook vector such as

codebook = [3,3,4,5,6,7,8,9];

then the equation below relates the vector index to the quantized signal quants.

quants = codebook(index+1);

This formula for quants is exactly what the quantiz function uses if you instead phrase the example more concisely as below.

partition = [3,4,5,6,7,8,9];
codebook = [3,3,4,5,6,7,8,9];
[index,quants] = quantiz([2 9 8],partition,codebook);

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Source Coding

Optimizing Quantization Parameters

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