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Construct Gaussian Doppler spectrum object

`dop = doppler.gaussian`

dop = doppler.gaussian(sigmagaussian)

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`

.

The Gaussian Doppler spectrum object contains the following properties.

Property | Description |
---|---|

`SpectrumType` | Fixed value, `'Gaussian'` |

`SigmaGaussian` | Normalized standard deviation of the Gaussian Doppler spectrum (a real positive number) |

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:

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

An alternate representation is [4]:

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

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:

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

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;

[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.

`doppler`

| `doppler.ajakes`

| `doppler.bell`

| `doppler.bigaussian`

| `doppler.flat`

| `doppler.jakes`

| `doppler.rjakes`

| `doppler.rounded`

| `rayleighchan`

| `ricianchan`

| `stdchan`

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