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# doppler.gaussian

Construct Gaussian Doppler spectrum object

## Syntax

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

## 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 [3]. It is also proposed as a model for the aeronautical channel [2]. 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) [1]. 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 [4]:

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

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

[1] 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.

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

[3] 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.

[4] Pätzold, M., Mobile Fading Channels, Wiley, 2002.