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Variance of input or sequence of inputs

**Library:**DSP System Toolbox / Statistics

The Variance block computes the unbiased variance
of each row or column of the input, or along vectors of a specified
dimension of the input. It can also compute the variance of the entire
input. You can specify the dimension using the **Find the
variance value over** parameter. The Variance block
can also track the variance in a sequence of inputs over a period
of time. To track the variance in a sequence of inputs, select the **Running
variance** parameter.

The **Running** mode in the Variance block
will be removed in a future release. To compute the running variance
in Simulink^{®}, use the Moving Variance block
instead.

`In`

— Data inputvector | matrix |

The block accepts real-valued or complex-valued multichannel and multidimensional inputs.

This port is unnamed until you select the **Running
variance** parameter and set the **Reset port** parameter
to any option other than `None`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `fixed point`

**Complex Number Support: **Yes

`Rst`

— Reset portscalar

Specify the reset event that causes the block to reset the running
variance. The sample time of the **Rst** input must
be a positive integer multiple of the input sample time.

To enable this port, select the **Running variance** parameter
and set the **Reset port** parameter to any option
other than `None`

.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `Boolean`

`Port_1`

— Variance along the specified dimensionscalar | vector | matrix |

The data type of the output matches the data type of the input.

When you do not select the **Running variance** parameter,
the block computes the variance in each row or column of the input,
or along vectors of a specified dimension of the input. It can also
compute the variance of the entire input at each individual sample
time. Each element in the output array `y`

is the
variance of the corresponding column, row, or entire input. The output
array `y`

depends on the setting of the **Find
the variance value over** parameter. Consider a three-dimensional
input signal of size *M*-by-*N*-by-*P*.
When you set **Find the variance value over** to:

`Entire input`

— The output at each sample time is a scalar that contains the variance of the*M*-by-*N*-by-*P*input matrix.`Each row`

— The output at each sample time consists of an*M*-by-1-by-*P*array, where each element contains the variance of each vector over the second dimension of the input. For an*M*-by-*N*matrix input, the output at each sample time is an*M*-by-1 column vector.`Each column`

— The output at each sample time consists of a 1-by-*N*-by-*P*array, where each element contains the variance of each vector over the first dimension of the input. For an*M*-by-*N*matrix input, the output at each sample time is a 1-by-*N*row vector.In this mode, the block treats length-

*M*unoriented vector inputs as*M*-by-1 column vectors.`Specified dimension`

— The output at each sample time depends on the value of the**Dimension**parameter. If you set the**Dimension**to`1`

, the output is the same as when you select`Each column`

. If you set the**Dimension**to`2`

, the output is the same as when you select`Each row`

. If you set the**Dimension**to`3`

, the output at each sample time is an*M*-by-*N*matrix containing the variance of each vector over the third dimension of the input.

When you select **Running variance**, the block
tracks the variance of each channel in a time sequence of inputs.
In this mode, you must also specify a value for the **Input
processing** parameter.

`Elements as channels (sample based)`

— The block treats each element of the input as a separate channel. For a three-dimensional input signal of size*M*-by-*N*-by-*P*, the block outputs an*M*-by-*N*-by-*P*array. Each element*y*of the output contains the variance of the element_{ijk}*u*for all inputs since the last reset._{ijk}When a reset event occurs, the running variance

*y*in the current frame is reset to the element_{ijk}*u*._{ijk}`Columns as channels (frame based)`

— The block treats each column of the input as a separate channel. This option does not support input signals with more than two dimensions. For a two-dimensional input signal of size*M*-by-*N*, the block outputs an*M*-by-*N*matrix. Each element*y*of the output contains the variance of the elements in the_{ij}*j*th column of all inputs since the last reset, up to and including the element*u*of the current input._{ij}When a reset event occurs, the running variance for each channel becomes the variance of all the samples in the current input frame, up to and including the current input sample.

**Data Types: **`single`

| `double`

| `int8`

| `int16`

| `int32`

| `uint8`

| `uint16`

| `uint32`

| `fixed point`

`Running variance`

— Option to select running varianceoff (default) | on

When you select the **Running variance** parameter,
the block tracks the variance value of each channel in a time sequence
of inputs.

`Find the variance value over`

— Dimension over which the block computes the variance`Each column`

(default) | `Entire input`

| `Each row`

| `Specified dimension`

`Each column`

— The block outputs the variance over each column.`Each row`

— The block outputs the variance over each row.`Entire input`

— The block outputs the variance over the entire input.`Specified dimension`

— The block outputs the variance over the dimension specified in the**Dimension**parameter.

To enable this parameter, clear the **Running variance** parameter.

`Dimension`

— Custom dimension`1`

(default) | scalarSpecify the dimension (one-based value) of the input signal over which the variance is computed. The value of this parameter must be greater than 0 and less than the number of dimensions in the input signal.

To enable this parameter, set **Find the variance value
over** to `Specified dimension`

.

`Input processing`

— Method to process the input in running mode`Columns as channels (frame based)`

(default) | `Elements as channels (sample based)`

`Columns as channels (frame based)`

— The block treats each column of the input as a separate channel. This option does not support input signals with more than two dimensions. For a two-dimensional input signal of size*M*-by-*N*, the block outputs an*M*-by-*N*matrix. Each element*y*of the output contains the variance of the elements in the_{ij}*j*th column of all inputs since the last reset, up to and including the element*u*of the current input._{ij}When a reset event occurs, the running variance for each channel becomes the variance of all the samples in the current input frame, up to and including the current input sample.

`Elements as channels (sample based)`

— The block treats each element of the input as a separate channel. For a three-dimensional input signal of size*M*-by-*N*-by-*P*, the block outputs an*M*-by-*N*-by-*P*array. Each element*y*of the output contains the variance of the element_{ijk}*u*for all inputs since the last reset._{ijk}When a reset event occurs, the running variance

*y*in the current frame is reset to the element_{ijk}*u*._{ijk}**Variable-Size Inputs**When your inputs are of variable size, and you select the

**Running variance**parameter, then:If you set the

**Input processing**parameter to`Elements as channels (sample based)`

, the state is reset.If you set the

**Input processing**parameter to`Columns as channels (frame based)`

, then:When the input size difference is in the number of channels (number of columns), the state is reset.

When the input size difference is in the length of channels (number of rows), no reset occurs and the running operation is carried out as usual.

To enable this parameter, select the **Running variance** parameter.

`Reset port`

— Reset event`None`

(default) | `Rising edge`

| `Falling edge`

| `Either edge`

| `Non-zero sample`

The block resets the running variance whenever a reset event
is detected at the optional **Rst** port. The reset
sample time must be a positive integer multiple of the input sample
time.

When a reset event occurs while the **Input processing** parameter
is set to `Elements as channels (sample based)`

,
the running variance for each channel is initialized to the value
in the corresponding channel of the current input. Similarly, when
the **Input processing** parameter is set to ```
Columns
as channels (frame based)
```

, the running variance for
each channel becomes the variance of all the samples in the current
input frame, up to and including the current input sample.

Use this parameter to specify the reset event.

`None`

— Disables the**Rst**port.`Rising edge`

— Triggers a reset operation when the**Rst**input does one of the following:Rises from a negative value to either a positive value or zero.

Rises from zero to a positive value, where the rise is not a continuation of a rise from a negative value to zero.

`Falling edge`

— Triggers a reset operation when the**Rst**input does one of the following:Falls from a positive value to a negative value or zero.

Falls from zero to a negative value, where the fall is not a continuation of a fall from a positive value to zero.

`Either edge`

— Triggers a reset operation when the**Rst**input is a`Rising edge`

or`Falling edge`

.`Non-zero sample`

— Triggers a reset operation at each sample time, when the**Rst**input is not zero.

When running simulations in the Simulink multitasking mode, reset signals have a one-sample latency. Therefore, when the block detects a reset event, there is a one-sample delay at the reset port rate before the block applies the reset. For more information on latency and the Simulink tasking modes, see Excess Algorithmic Delay (Tasking Latency) and Time-Based Scheduling and Code Generation (Simulink Coder).

To enable this parameter, select the **Running variance** parameter.

To use these parameters, the data input must be fixed point.
For all other inputs, the parameters on the **Data Types** tab
are ignored.

`Rounding mode`

— Method of rounding operation`Floor`

(default) | `Ceiling`

| `Convergent`

| `Nearest`

| `Round`

| `Simplest`

| `Zero`

Specify the rounding mode for fixed-point operations as one of the following:

`Floor`

`Ceiling`

`Convergent`

`Nearest`

`Round`

`Simplest`

`Zero`

For more details, see rounding mode.

`Overflow mode`

— Method of overflow action`Wrap`

(default) | `Saturate`

Specify the overflow action for fixed-point operations as one of the following:

`Wrap`

–– The block wraps the result of its fixed-point operations.`Saturate`

–– The block saturates the result of its fixed-point operations.

For more details on overflow actions, see overflow mode for fixed-point operations.

`Input-squared product`

— Data type that stores the input-squared term`Same as input`

(default) | `Binary point scaling`

| `Slope and bias scaling`

The squares of the input elements are stored in the **Input-squared
product** data type. If the input is complex, the squares
of the real and imaginary parts of the input are stored in this data
type. For more details, see Fixed Point.

You can set this parameter to:

`Inherit: Same as input`

— The block specifies this data type to be the same as the input data type.`Binary point scaling`

— The**Input-squared product**data type uses binary point scaling. If you select this option, the block displays the fields to specify the**Word length**and**Fraction length**. The**Signedness**is inherited from the input.`Slope and bias scaling`

— The**Input-squared product**data type uses slope and bias scaling. If you select this option, the block displays the fields to specify the**Word length**and**Slope**. The**Signedness**is inherited from the input and**Bias**is specified to be`0`

.

`Input-sum-squared product`

— Data type that stores the input-sum-squared term`Same as input-squared product`

(default) | `Binary point scaling`

| `Slope and bias scaling`

The squares of the sum of the input elements are stored in the **Input-sum-squared
product** data type. If the input is complex, the squares
of the sum of the real parts and the squares of the sum of the imaginary
parts are stored in this data type. For more details, see Fixed Point.

You can set this parameter to:

`Same as input-squared product`

— The block specifies this data type to be the same as the input squared-product data type.`Binary point scaling`

— The**Input-sum-squared product**data type uses binary point scaling. If you select this option, the block displays the fields to specify the**Word length**and**Fraction length**. The**Signedness**is inherited from the input.`Slope and bias scaling`

— The**Input-sum-squared product**data type uses slope and bias scaling. If you select this option, the block displays the fields to specify the**Word length**and**Slope**. The**Signedness**is inherited from the input and**Bias**is specified to be`0`

.

`Accumulator`

— Accumulator data type`Same as input-squared product`

(default) | `Same as input`

| `Binary point scaling`

| `Slope and bias scaling`

**Accumulator** specifies the data type of
the output of an accumulation operation in the Variance block.
See Fixed Point for
illustrations depicting the use of the accumulator data type in this
block.

You can set this parameter to:

`Same as input-squared product`

— The block specifies the accumulator data type to be the same as the input-squared product data type.`Same as input`

— The block specifies the accumulator data type to be the same as the input data type.`Binary point scaling`

— The**Accumulator**data type uses binary point scaling. If you select this option, the block displays the fields to specify the**Word length**and**Fraction length**. The**Signedness**is inherited from the input.`Slope and bias scaling`

— The**Accumulator**data type uses slope and bias scaling. If you select this option, the block displays the fields to specify the**Word length**and**Slope**. The**Signedness**is inherited from the input and**Bias**is specified to be`0`

.

`Output`

— Output data type`Same as input-squared product`

(default) | `Same as accumulator`

| `Same as input`

| `Binary point scaling`

| `Slope and bias scaling`

**Output** specifies the data type of the output
of the Variance block. See Fixed Point for illustrations depicting the use
of the output data type in this block. You can set it to:

`Same as input-squared product`

— The block specifies the output data type to be the same as the input-squared product data type.`Same as accumulator`

— The block specifies the output data type to be the same as the accumulator data type.`Same as input`

— The block specifies the output data type to be the same as the input data type.`Binary point scaling`

— The**Output**data type uses binary point scaling. If you select this option, the block displays the fields to specify the**Word length**and**Fraction length**. The**Signedness**is inherited from the input.`Slope and bias scaling`

— The**Output**data type uses slope and bias scaling. If you select this option, the block displays the fields to specify the**Word length**and**Slope**. The**Signedness**is inherited from the input and**Bias**is specified to be`0`

.

`Lock data type settings against changes by the fixed-point tools`

— Prevent fixed-point tools from overriding data typesoff (default) | on

Select this parameter to prevent the fixed-point tools from overriding the data types you specify on the block.

The variance of a discrete-time signal is the square of the standard deviation of the signal.

Variance gives a measure of deviation of the signal from its mean value.

For purely real or imaginary input, *u*, of
size *M*-by-*N*, the variance is
given by the following equation:

$$y={\sigma}^{2}=\frac{{\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{N}{\left|{u}_{ij}\right|}^{2}-\frac{{\left|{\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{N}{u}_{ij}}}\right|}^{2}}{M*N}}}}{M*N-1}$$

*u*is the input data element at indices_{ij}*i*,*j*.*M*is the length of the*j*th column.*N*is the number of columns.

For complex inputs, the variance is given by the following equation:

$${\sigma}^{2}={\sigma}_{\mathrm{Re}}{}^{2}+{\sigma}_{\mathrm{Im}}{}^{2}$$

*σ*is the variance of the real part of the complex input._{Re}^{2}*σ*is the variance of the imaginary part of the complex input._{Im}^{2}

When you clear the **Running variance** parameter
in the block and specify a dimension, the block produces results identical
to the MATLAB^{®} `var`

function, when it is called
as `y = var(u,0,D)`

.

`u`

is the data input.`D`

is the dimension.`y`

is the variance along the specified dimension.

The variance along the entire input is identical to calling
the `var`

function as `y = var(u(:))`

.

For a complex input signal, the variance is the sum of the variances of the real and imaginary parts.

$${\sigma}^{2}={\sigma}_{\mathrm{Re}}{}^{2}+{\sigma}_{\mathrm{Im}}{}^{2}$$

Generate C and C++ code using Simulink® Coder™.

Design and simulate fixed-point algorithms using Fixed-Point Designer™.

For purely real or imaginary input, *u* of
size *M*-by-*N*, the variance is
given by the following equation.

$$y={\sigma}^{2}=\frac{{\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{N}{\left|{u}_{ij}\right|}^{2}-\frac{{\left|{\displaystyle \sum _{i=1}^{M}{\displaystyle \sum _{j=1}^{N}{u}_{ij}}}\right|}^{2}}{M*N}}}}{M*N-1}$$

The following diagram shows the data types used within the Variance block when the input is fixed-point.

For complex inputs, the variance is given by the following equation:

$${\sigma}^{2}={\sigma}_{\mathrm{Re}}{}^{2}+{\sigma}_{\mathrm{Im}}{}^{2}$$

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