Documentation

# doppler.gaussian

(To be removed) Construct Gaussian Doppler spectrum object

## Syntax

```dop = doppler.gaussian dop = doppler.gaussian(sigmagaussian) ```

`doppler.gaussian` will be removed in a future release. Use `doppler` `('Gaussian',...)` instead.

## Description

The `doppler.gaussian` function creates a Gaussian Doppler spectrum object that is to be used for the `DopplerSpectrum` property of a channel object (created with either the `rayleighchan` or the `ricianchan` function).

`dop = doppler.gaussian` creates a Gaussian Doppler spectrum object with a default standard deviation (normalized by the maximum Doppler shift ${f}_{d}$, in Hz) ${\sigma }_{G,norm}=1/\sqrt{2}$. The maximum Doppler shift ${f}_{d}$ is specified by the `MaxDopplerShift` property of the channel object. Analytically, ${\sigma }_{G,norm}={\sigma }_{G}/{f}_{d}=1/\sqrt{2}$, where ${\sigma }_{G}$ is the standard deviation of the Gaussian Doppler spectrum.

`dop = doppler.gaussian(sigmagaussian)` creates a Gaussian Doppler spectrum object with a normalized ${f}_{d}$ (by the maximum Doppler shift ${f}_{d}$, in Hz) ${\sigma }_{G,norm}$ of value `sigmagaussian`.

## Properties

The Gaussian Doppler spectrum object contains the following properties.

PropertyDescription
`SpectrumType`Fixed value, `'Gaussian'`
`SigmaGaussian`Normalized standard deviation of the Gaussian Doppler spectrum (a real positive number)

## Theory and Applications

The Gaussian power spectrum is considered to be a good model for multipath components with long delays in UHF communications . It is also proposed as a model for the aeronautical channel . A Gaussian Doppler spectrum is also specified in some cases of the ANSI J-STD-008 reference channel models for PCS applications, for both outdoor (wireless loop) and indoor (residential, office) . The normalized Gaussian Doppler power spectrum is given analytically by:

`${S}_{G}\left(f\right)=\frac{1}{\sqrt{2\pi {\sigma }_{G}^{2}}}\mathrm{exp}\left(-\frac{{f}^{2}}{2{\sigma }_{G}^{2}}\right)$`

An alternate representation is :

`${S}_{G}\left(f\right)=\frac{1}{{f}_{c}}\sqrt{\frac{\mathrm{ln}2}{\pi }}\mathrm{exp}\left(-\left(\mathrm{ln}2\right){\left(\frac{f}{{f}_{c}}\right)}^{2}\right)$`

where ${f}_{c}={\sigma }_{G}\sqrt{2\mathrm{ln}2}$ is the 3 dB cutoff frequency. If you set ${f}_{c}={f}_{d}\sqrt{\mathrm{ln}2}$, where ${f}_{d}$ is the maximum Doppler shift, or equivalently ${\sigma }_{G}={f}_{d}/\sqrt{2}$, the Doppler spread of the Gaussian power spectrum becomes equal to the Doppler spread of the Jakes power spectrum, where Doppler spread is defined as:

`${\sigma }_{D}=\sqrt{\frac{\underset{-\infty }{\overset{\infty }{\int }}{f}^{2}S\left(f\right)df}{\underset{-\infty }{\overset{\infty }{\int }}S\left(f\right)df}}$`

## Examples

The following code creates a Rayleigh channel object with a maximum Doppler shift of ${f}_{d}=10$. It then creates a Gaussian Doppler spectrum object with a normalized standard deviation of ${\sigma }_{G\text{,norm}}=0.5$, and assigns it to the `DopplerSpectrum` property of the channel object.

```chan = rayleighchan(1/1000,10); dop_gaussian = doppler.gaussian(0.5); chan.DopplerSpectrum = dop_gaussian;```

## References

 ANSI J-STD-008, Personal Station-Base Station Compatibility Requirements for 1.8 to 2.0 GHz Code Division Multiple Access (CDMA) Personal Communications Systems, March 1995.

 Bello, P. A., “Aeronautical channel characterizations,” IEEE Trans. Commun., Vol. 21, pp. 548–563, May 1973.

 Cox, D. C., “Delay Doppler characteristics of multipath propagation at 910 MHz in a suburban mobile radio environment,” IEEE Transactions on Antennas and Propagation, Vol. AP-20, No. 5, pp. 625–635, Sept. 1972.

 Pätzold, M., Mobile Fading Channels, Wiley, 2002.