quantiz

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

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])

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])