# 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.```

example

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

example

## Examples

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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])```

## Input Arguments

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

Data Types: `double`

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`

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

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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`)

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)`].

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