quantiz

Produce quantization index and quantized output value

Syntax

index = quantiz(sig,partition)

[index,quants] = quantiz(sig,partition,codebook)

[index,quants,distor] = quantiz(sig,partition,codebook)

Description

index = quantiz(sig,partition) returns the quantization levels of input signal sig by using the scalar quantization partition specified in input partition.

[index,quants] = quantiz(sig,partition,codebook) specifies codebook, which prescribes a value for each partition in the scalar quantization. codebook is a vector whose length must exceed the length of partition by one. The function also returns quants, which contains the scalar quantization of sig and depends on the quantization levels and prescribed values in the codebook.

[index,quants,distor] = quantiz(sig,partition,codebook) returns an estimate of the mean square distortion of the quantization data.

Examples

Produce Quantization Index and Quantized Output Value

Show 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.

Generate sample data and specify partition and codebook vectors. Specify the partition vector by defining the distinct endpoints of the different intervals as the element values of the vector. Specify the codebook vector with an element value for each interval defined in the partition vector. The codebook vector must be one element longer than the partition vector.

samp = [-2.4, -1, 0, 0.2, 0.8, 1.2, 2,3, 3.5, 5];
partition = [0, 1, 3];
codebook = [-1, 0.5, 2, 3]; 

Quantize the data samples. Display the input sample data, the quantization index, and the corresponding quantized output value of the input data.

[index,quantized] = quantiz(samp,partition,codebook);
[samp; index; quantized]'

ans = 10×3

   -2.4000         0   -1.0000
   -1.0000         0   -1.0000
         0         0   -1.0000
    0.2000    1.0000    0.5000
    0.8000    1.0000    0.5000
    1.2000    2.0000    2.0000
    2.0000    2.0000    2.0000
    3.0000    2.0000    2.0000
    3.5000    3.0000    3.0000
    5.0000    3.0000    3.0000

Quantize Sampled Sine Wave

To illustrate the nature of scalar quantization, this example shows how to quantize a sine wave. Plot the original and quantized signals to contrast the x symbols 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.

Generate a sine wave sampled at times defined by t. Specify the partition input by defining the distinct endpoints of the different intervals as the element values of the vector. Specify the codebook input with an element value for each interval defined in the partition vector. The codebook vector must be one element longer than the partition vector.

t = [0:.1:2*pi];
sig = sin(t);
partition = [-1:.2:1];
codebook = [-1.2:.2:1];

Perform quantization on the sampled sine wave.

[index,quants] = quantiz(sig,partition,codebook);

Plot the quantized sine wave and the sampled sine wave.

plot(t,sig,'x',t,quants,'.')
title('Quantization of Sine Wave')
xlabel('Time')
ylabel('Amplitude')
legend('Original sampled sine wave','Quantized sine wave');
axis([-.2 7 -1.2 1.2])

Optimize Quantization Parameters

Testing and selecting parameters for large signal sets with a fine quantization scheme can be tedious. One way to produce partition and codebook parameters easily is to optimize them according to a set of training data. The training data should be typical of the kinds of signals to be quantized.

This example uses the lloyds function to optimize the partition and codebook according to the Lloyd algorithm. The code optimizes the partition and codebook for one period of a sinusoidal signal, starting from a rough initial guess. Then the example runs the quantiz function twice to generate quantized data by using the initial partition and codebook input values and by using the optimized partitionOpt and codebookOpt input values. The example also compares the distortion for the initial and the optimized quantization.

Define variables for a sine wave signal and initial quantization parameters. Optimize the partition and codebook by using the lloyds function.

t = 0:.1:2*pi;
sig = sin(t);
partition = -1:.2:1;
codebook = -1.2:.2:1;
[partitionOpt,codebookOpt] = lloyds(sig,codebook);

Generate quantized signals by using the initial and the optimized partition and codebook vectors. The quantiz function automatically computes the mean square distortion and returns it as the third output argument. Compare mean square distortions for quantization with the initial and optimized input arguments to see how less distortion occurs when using the optimized quantized values.

[index,quants,distor] = quantiz(sig,partition,codebook);
[indexOpt,quantOpt,distorOpt] = ...
    quantiz(sig,partitionOpt,codebookOpt);
[distor, distorOpt]

ans = 1×2

    0.0148    0.0022

Plot the sampled sine wave, the quantized sine wave, and the optimized quantized sine wave.

plot(t,sig,'x',t,quants,'.',t,quantOpt,'s')
title('Quantization of Sine Wave')
xlabel('Time')
ylabel('Amplitude')
legend('Original sampled sine wave', ...
    'Quantized sine wave', ...
    'Optimized quantized sine wave');
axis([-.2 7 -1.2 1.2])

Input Arguments

sig — Input signal

vector

Input signal, specified as a vector. This input specifies the sampled signal for this function to perform quantization.

Data Types: double

partition — Distinct endpoints of different ranges

row vector

Distinct endpoints of different ranges, specified as a row vector. This input defines several contiguous, nonoverlapping ranges of values within the set of real numbers. The values present in this input must be strictly in ascending order. The length of this vector must be one less than the number of partition intervals.

Example: [0, 1, 3] partitions the input row vector into the four sets {X: X ≤ 0}, {X: 0 < X ≤ 1}, {X: 1 < X ≤ 3}, and {X: 3 < X}.

Data Types: double

codebook — Quantization value for each partition

row vector

Quantization value for each partition, specified as a row vector. This input prescribes a common value for each partition in the scalar quantization. The length of this vector must equal the number of partition intervals, that is, the length of this vector must exceed the length of the partition input by one.

Data Types: double

Output Arguments

index — Quantization index

nonnegative row vector

Quantization index of the input signal, returned as a nonnegative row vector. This output determines on which partition interval, each input value is mapped. Each element in index is one of the N integers in the range [0, N–1].

If the partition input has length N, index is a vector whose Kth entry is:

• 0 if sig(K) ≤ partition(1)

• M if partition(M) < sig(K) ≤ partition(M+1)

• N if partition(N) ≤ sig(K)

quants — Output of quantizer

row vector

Output of the quantizer, which contains the quantization values of the input signal, returned as a row vector. The size of quants matches that of input argument sig. When codebook is not specified as an input argument, you can define the codebook values as a vector whose length must exceed the length of the partition by one.

quants is calculated based on the codebook and index inputs and is given by quants(i) = codebook(index(i) + 1), where i is an integer in the range [1, length(sig)].

distor — Mean square distortion

positive scalar

Mean square distortion of the quantized signal, returned as a positive scalar. You can reduce this distortion by choosing appropriate partition and codebook values. For more information on optimizing partition and codebook values, see the lloyds function.

Version History

Introduced before R2006a