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# Band-Limited White Noise

Introduce white noise into continuous system

Sources

## Description

### Simulation of White Noise

The Band-Limited White Noise block generates normally distributed random numbers that are suitable for use in continuous or hybrid systems.

Theoretically, continuous white noise has a correlation time of 0, a flat power spectral density (PSD), and a total energy of infinity. In practice, physical systems are never disturbed by white noise, although white noise is a useful theoretical approximation when the noise disturbance has a correlation time that is very small relative to the natural bandwidth of the system.

In Simulink® software, you can simulate the effect of white noise by using a random sequence with a correlation time much smaller than the shortest time constant of the system. The Band-Limited White Noise block produces such a sequence. The correlation time of the noise is the sample rate of the block. For accurate simulations, use a correlation time much smaller than the fastest dynamics of the system. You can get good results by specifying

$tc\approx \frac{1}{100}\frac{2\pi }{{f}_{max}},$

where fmax is the bandwidth of the system in rad/sec.

### Comparison with the Random Number Block

The primary difference between this block and the Random Number block is that the Band-Limited White Noise block produces output at a specific sample rate. This rate is related to the correlation time of the noise.

### Usage with the Averaging Power Spectral Density Block

The Band-Limited White Noise block specifies a two-sided spectrum, where the units are Hz. The Averaging Power Spectral Density block specifies a one-sided spectrum, where the units are the square of the magnitude per unit radial frequency: Mag^2/(rad/sec). When you feed the output of a Band-Limited White Noise block into an Averaging Power Spectral Density block, the average PSD value is π times smaller than the Noise power of the Band-Limited White Noise block. This difference is the result of converting the units of one block to the units of the other: 1/(1/2)(2π) = 1/π, where:

• 1/2 is the factor for converting from a two-sided to one-sided spectrum

• 2π is the factor for converting from Hz to rad/sec

## Algorithm

To produce the correct intensity of this noise, the covariance of the noise is scaled to reflect the implicit conversion from a continuous PSD to a discrete noise covariance. The appropriate scale factor is 1/tc, where tc is the correlation time of the noise. This scaling ensures that the response of a continuous system to the approximate white noise has the same covariance as the system would have to true white noise. Because of this scaling, the covariance of the signal from the Band-Limited White Noise block is not the same as the Noise power (intensity) parameter. This parameter is actually the height of the PSD of the white noise. This block approximates the covariance of white noise as the Noise power divided by tc.

## Data Type Support

The Band-Limited White Noise block outputs real values of type double.

## Parameters and Dialog Box

Noise power

Specify the height of the PSD of the white noise. The default value is 0.1.

Sample time

Specify the correlation time of the noise. The default value is 0.1. For more information, see Specify Sample Time in the Simulink documentation.

Seed

Specify the starting seed for the random number generator. The default value is 23341.

Interpret vector parameters as 1-D

Select to output a 1-D array when the block parameters are vectors. Otherwise, output a 2-D array one of whose dimensions is 1. See Determining the Output Dimensions of Source Blocks in the Simulink documentation.

## Examples

The following Simulink examples show how to use the Band-Limited White Noise block:

## Characteristics

 Sample Time Specified in the Sample time parameter Scalar Expansion Yes, of the Noise power and Seed parameters and output Dimensionalized Yes Multidimensionalized No Zero-Crossing Detection No