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Design vector quantizer using Vector Quantizer Design Tool (VQDTool)

Quantizers

`dspquant2`

Double-click on the Vector Quantizer Design block to start VQDTool,
a GUI that allows you to design and implement a vector quantizer.
You can also start VQDTool by typing `vqdtool`

at
the MATLAB^{®} command prompt. Based on your specifications, VQDTool
iteratively calculates the codebook values that minimize the mean
squared error between the training set and the codebook until the
stopping criteria for the design process is satisfied. The block uses
the resulting codebook values to implement your vector quantizer.

For the **Training Set** parameter, enter a *k*-by-*M* matrix
of values you want to use to train the quantizer codebook. The variable
k, where $$k\ge 1$$, is the length of each training vector. It also
represents the dimension of your quantizer. The variable *M*,
where $$M\ge 2$$, is the number of training vectors. This data can
be created using a MATLAB function, such as the default value `randn(10,1000)`

,
or it can be any variable defined in the MATLAB workspace.

You have two choices for the **Source of initial codebook** parameter.
Select `Auto-generate`

to have the block
choose the values of the initial codebook. In this case, the block
picks *N* random training vectors as the initial
codebook, where *N* is the **Number of levels** parameter
and $$N\ge 2$$. When you select `User defined`

,
enter the initial codebook values in the **Initial codebook** field.
The initial codebook matrix must have the same number of rows as the
training set. Each column of the codebook is a codeword, and your
codebook must have at least two codewords.

For the given training set and initial codebook, the block performs
an iterative process, using the Generalized Lloyd Algorithm (GLA),
to design a final codebook. For each iteration of the GLA, the block
first associates each training vector with its nearest codeword by
calculating the distortion. You can specify one of the two possible
methods for calculating distortion using the **Distortion
measure** parameter.

When you select `Squared error`

for
the **Distortion measure** parameter, the block finds
the nearest codeword by calculating the squared error (unweighted).
Consider the codebook $$CB=\left[\begin{array}{cccc}C{W}_{1}& C{W}_{2}& \mathrm{...}& C{W}_{N}\end{array}\right]$$. This codebook has *N* codewords;
each codeword has *k* elements. The *i*-th
codeword is defined as $$C{W}_{i}=\left[\begin{array}{cccc}{a}_{1i}& {a}_{2i}& \mathrm{...}& {a}_{ki}\end{array}\right]$$. The training set has *M* columns
and is defined as $$U=\left[\begin{array}{cccc}{U}_{1}& {U}_{2}& \mathrm{...}& {U}_{M}\end{array}\right]$$, where the *p*-th training vector
is $${U}_{p}={\left[\begin{array}{cccc}{u}_{1p}& {u}_{2p}& \mathrm{...}& {u}_{kp}\end{array}\right]}^{\prime}$$. The squared error (unweighted) is calculated using
the equation

$$D={\displaystyle \sum _{j=1}^{k}{\left({a}_{ji}-{u}_{jp}\right)}^{2}}$$

`Weighted squared error`

for the $$D={\displaystyle \sum _{j=1}^{k}{w}_{jp}{\left({a}_{ji}-{u}_{jp}\right)}^{2}}$$

Once the block has associated all the training vectors with their nearest codeword vectors, the block calculates the mean squared error for the codebook and checks to see if the stopping criteria for the process has been satisfied.

The two possible options for the **Stopping criteria** parameter
are `Relative threshold`

and ```
Maximum
iteration
```

. When you want the design process to stop
when the fractional drop in the squared error is below a certain value,
select `Relative threshold`

. Then, type the
maximum acceptable fractional drop in the **Relative threshold** field.
The fraction drop in the squared error is defined as

$$\frac{\text{erroratpreviousiteration}-\text{erroratcurrentiteration}}{\text{erroratpreviousiteration}}$$

`Maximum iteration`

. Then, enter the
maximum number of iterations you want the block to perform in the `Whichever comes first`

and
enter When a training vector has the same distortion for two different
codeword vectors, the algorithm uses the **Tie-breaking rule** parameter
to determine which codeword vector the training vector is associated
with. When you want the training vector to be associated with the
lower indexed codeword, select `Lower indexed codeword`

.
To associate the training vector with the higher indexed codeword,
select `Higher indexed codeword`

.

With each iteration, the block updates the codeword values in
order to minimize the distortion. The **Codebook update method** parameter
defines the way the block calculates these new codebook values.

If, for the **Distortion measure** parameter,
you choose `Squared error`

, the **Codebook
update method** parameter is set to `Mean`

.

If, for the **Distortion measure** parameter,
you choose `Weighted squared error`

and you
choose `Mean`

for the **Codebook
update method** parameter, the new codeword vector is found
as follows. Suppose there are three training vectors associated with
one codeword vector. The training vectors are

$$T{S}_{1}=\left[\begin{array}{c}1\\ 2\end{array}\right],\text{}T{S}_{3}=\left[\begin{array}{c}10\\ 12\end{array}\right],\text{and}T{S}_{7}=\left[\begin{array}{c}11\\ 12\end{array}\right].$$

The new codeword vector is calculated as $$C{W}_{new}=\left[\begin{array}{c}\frac{1+10+11}{3}\\ \frac{2+12+12}{3}\end{array}\right]$$

where the denominator is the number of training vectors associated
with this codeword. If, for the **Codebook update method** parameter,
you choose `Centroid`

and you specify the
weighting factors $${W}_{1}=\left[\begin{array}{c}0.1\\ 0.2\end{array}\right]$$, $${W}_{3}=\left[\begin{array}{c}1\\ 0.6\end{array}\right]$$, and $${W}_{7}=\left[\begin{array}{c}0.3\\ 0.4\end{array}\right]$$, the new codeword vector is calculated as

$$C{W}_{new}=\left[\begin{array}{c}\frac{\left(0.1\right)\left(1\right)+\left(1\right)\left(10\right)+\left(0.3\right)\left(11\right)}{0.1+1+0.3}\\ \frac{\left(0.2\right)\left(2\right)+\left(0.6\right)\left(12\right)+\left(0.4\right)\left(12\right)}{0.2+0.6+0.4}\end{array}\right]$$

Click **Design and Plot** to design the quantizer
with the parameter values specified on the left side of the GUI. The
performance curve and the entropy of the quantizer are updated and
displayed in the figures on the right side of the GUI.

You must click **Design and Plot** to apply
any changes you make to the parameter values in the VQDTool dialog
box.

The following is an example of how the block calculates the
entropy of the quantizer at each iteration. Suppose you have a codebook
with four codewords and a training set with 200 training vectors.
Also suppose that, at the *i*-th iteration, 40 training
vectors are associated with the first codeword, 60 training vectors
are associated with the second codeword, 20 training vectors are associated
with the third codeword, and 80 training vectors are associated with
the fourth codeword. The probability that a training vector is associated
with the first codeword is $$\frac{40}{200}$$. The probabilities that training vectors are associated
with the second, third, and fourth codewords are $$\frac{60}{200}$$, $$\frac{20}{200}$$, and $$\frac{80}{200}$$, respectively. The GUI uses these probabilities
to calculate the entropy according to the equation

$$H={\displaystyle \sum _{i=1}^{N}-{p}_{i}{\mathrm{log}}_{2}{p}_{i}}$$

where *N* is the number of codewords. Based
on these probabilities, the GUI calculates the entropy of the quantizer
at the *i*-th iteration as

$$\begin{array}{l}H=-\left(\frac{40}{200}{\mathrm{log}}_{2}\frac{40}{200}+\frac{60}{200}{\mathrm{log}}_{2}\frac{60}{200}+\frac{20}{200}{\mathrm{log}}_{2}\frac{20}{200}+\frac{80}{200}{\mathrm{log}}_{2}\frac{80}{200}\right)\\ H=1.8464\end{array}$$

VQDTool can export parameter values that correspond to the figures
displayed in the GUI. Click the **Export Outputs** button,
or press **Ctrl+E**, to export the **Final Codebook**, **Mean
Square Error**, and **Entropy** values to
the workspace, a text file, or a MAT-file.

In the **Model** section of the GUI,
specify the destination of the block that will contain the parameters
of your quantizer. For **Destination**, select ```
Current
model
```

to create a block with your parameters in the
model you most recently selected. Type `gcs`

in the MATLAB Command
Window to display the name of your current model. Select ```
New
model
```

to create a block in a new model file.

From the **Block type** list, select `Encoder`

to
design a Vector Quantizer Encoder block. Select `Decoder`

to
design a Vector Quantizer Decoder block. Select `Both`

to
design a Vector Quantizer Encoder block and a Vector Quantizer Decoder
block.

In the **Encoder block name** field, enter
a name for the Vector Quantizer Encoder block. In the **Decoder
block name** field, enter a name for the Vector Quantizer
Decoder block. When you have a Vector Quantizer Encoder and/or Decoder
block in your destination model with the same name, select the **Overwrite
target block** check box to replace the block's parameters
with the current parameters. When you do not select this check box,
a new Vector Quantizer Encoder and/or Decoder block is created in
your destination model.

Click **Generate Model**. VQDTool uses the
parameters that correspond to the current plots to set the parameters
of the Vector Quantizer Encoder and/or Decoder blocks.

**Training Set**Enter the samples of the signal you would like to quantize. This data set can be a MATLAB function or a variable defined in the MATLAB workspace. The typical length of this data vector is

`1e5`

.**Source of initial codebook**Select

`Auto-generate`

to have the block choose the initial codebook values. Choose`User defined`

to enter your own initial codebook values.**Number of levels**Enter the number of codeword vectors,

*N*, in your codebook matrix, where*N*≥ 2.**Initial codebook**Enter your initial codebook values. From the

**Source of initial codebook**list, select`User defined`

in order to activate this parameter. The codebook must have the same number of rows as the training set. You must provide at least two codeword vectors.**Distortion measure**When you select

`Squared error`

, the block finds the nearest codeword by calculating the squared error (unweighted). When you select`Weighted squared error`

, the block finds the nearest codeword by calculating the weighted squared error.**Weighting factor**Enter a vector or matrix. The block uses these values to compute the weighted squared error. When the weighting factor is a vector, its length must be equal to the number of rows in the training set. This weighting factor is used for each training vector. When the weighting factor is a matrix, it must be the same size as the training set matrix. The individual weighting factors cannot be negative. The weighting factor vector or matrix cannot contain all zeros.

**Stopping criteria**Choose

`Relative threshold`

to enter the maximum acceptable fractional drop in the squared quantization error. Choose`Maximum iteration`

to specify the number of iterations at which to stop. Choose`Whichever comes first`

and the block stops the iteration process as soon as the relative threshold or maximum iteration value is attained.**Relative threshold**This parameter is available when you choose

`Relative threshold`

or`Whichever comes first`

for the**Stopping criteria**parameter. Enter the value that is the maximum acceptable fractional drop in the squared quantization error.**Maximum iteration**This parameter is available when you choose

`Maximum iteration`

or`Whichever comes first`

for the**Stopping criteria**parameter. Enter the maximum number of iterations you want the block to perform.**Tie-breaking rules**When a training vector has the same distortion for two different codeword vectors, select

`Lower indexed codeword`

to associate the training vector with the lower indexed codeword. Select`Higher indexed codeword`

to associate the training vector with the lower indexed codeword.**Codebook update method**When you choose

`Mean`

, the new codeword vector is calculated by taking the average of all the training vector values that were associated with the original codeword vector. When you choose`Centroid`

, the block calculates the new codeword vector by taking the weighted average of all the training vector values that were associated with the original codeword vector Note that if, for the**Distortion measure**parameter, you choose`Squared error`

, the**Codebook update method**parameter is set to`Mean`

.**Destination**Choose

`Current model`

to create a Vector Quantizer block in the model you most recently selected. Type`gcs`

in the MATLAB Command Window to display the name of your current model. Choose`New model`

to create a block in a new model file.**Block type**Select

`Encoder`

to design a Vector Quantizer Encoder block. Select`Decoder`

to design a Vector Quantizer Decoder block. Select`Both`

to design a Vector Quantizer Encoder block and a Vector Quantizer Decoder block.**Encoder block name**Enter a name for the Vector Quantizer Encoder block.

**Decoder block name**Enter a name for the Vector Quantizer Decoder block.

**Overwrite target block**When you do not select this check box and a Vector Quantizer Encoder and/or Decoder block with the same block name exists in the destination model, a new Vector Quantizer Encoder and/or Decoder block is created in the destination model. When you select this check box and a Vector Quantizer Encoder and/or Decoder block with the same block name exists in the destination model, the parameters of these blocks are overwritten by new parameters.

**Generate Model**Click this button and VQDTool uses the parameters that correspond to the current plots to set the parameters of the Vector Quantizer Encoder and/or Decoder blocks.

**Design and Plot**Click this button to design a quantizer using the parameters on the left side of the GUI and to update the performance curve and entropy plots on the right side of the GUI.

You must click

**Design and Plot**to apply any changes you make to the parameter values in the VQDTool GUI.**Export Outputs**Click this button, or press

**Ctrl+E**, to export the**Final Codebook**,**Mean Squared Error**, and**Entropy**values to the workspace, a text file, or a MAT-file.

Double-precision floating point

[1] Gersho, A. and R. Gray. *Vector Quantization
and Signal Compression*. Boston: Kluwer Academic Publishers,
1992.

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