# Documentation

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# doppler.bigaussian

Construct bi-Gaussian Doppler spectrum object

## Syntax

```dop = doppler.bigaussian(property1,value1,...) dop = doppler.bigaussian ```

## Description

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

`dop = doppler.bigaussian(property1,value1,...)` creates a bi-Gaussian Doppler spectrum object with properties as specified by the property/value pairs. If you do not specify a value for a property, the property is assigned a default value.

`dop = doppler.bigaussian` creates a bi-Gaussian Doppler spectrum object with default properties. The constructed Doppler spectrum object is equivalent to a single Gaussian Doppler spectrum centered at zero frequency. The equivalent command with property/value pairs is:

```dop = doppler.bigaussian('SigmaGaussian1', 1/sqrt(2), ... 'SigmaGaussian2', 1/sqrt(2), ... 'CenterFreqGaussian1', 0, ... 'CenterFreqGaussian2', 0, ... 'GainGaussian1', 0.5, ... 'GainGaussian2', 0.5)```

## Properties

The bi-Gaussian Doppler spectrum object contains the following properties.

PropertyDescription
`SpectrumType`Fixed value, `'BiGaussian'`
`SigmaGaussian1`Normalized standard deviation of first Gaussian function (real positive finite scalar value)
`SigmaGaussian2`Normalized standard deviation of second Gaussian function (real positive finite scalar value)
`CenterFreqGaussian1`Normalized center frequency of first Gaussian function (real scalar value between -1 and 1)
`CenterFreqGaussian2`Normalized center frequency of second Gaussian function (real scalar value between -1 and 1)
`GainGaussian1`Power gain of first Gaussian function (linear scale, real nonnegative finite scalar value)
`GainGaussian2`Power gain of second Gaussian function (linear scale, real nonnegative finite scalar value)

All properties are writable except for the `SpectrumType` property.

The properties `SigmaGaussian1`, `SigmaGaussian2`, `GainGaussian1`, and `GainGaussian2` are normalized by the `MaxDopplerShift` property of the associated channel object.

Analytically, the normalized standard deviations of the first and second Gaussian functions are determined as ${\sigma }_{G1,norm}={\sigma }_{G1}/{f}_{d}$ and ${\sigma }_{G2,norm}={\sigma }_{G2}/{f}_{d}$, respectively, where ${\sigma }_{G1}$ and ${\sigma }_{G2}$ are the standard deviations of the first and second Gaussian functions, and ${f}_{d}$ is the maximum Doppler shift, in hertz. Similarly, the normalized center frequencies of the first and second Gaussian functions are determined as ${f}_{G1,norm}={f}_{G1}/{f}_{d}$ and ${f}_{G2,norm}={f}_{G2}/{f}_{d}$, respectively, where ${f}_{G1}$ and ${f}_{G2}$ are the center frequencies of the first and second Gaussian functions. The properties `GainGaussian1` and `GainGaussian2` correspond to the power gains ${C}_{G1}$ and ${C}_{G2}$, respectively, of the two Gaussian functions.

## Theory and Applications

The bi-Gaussian power spectrum consists of two frequency-shifted Gaussian spectra. The COST207 channel models ([1], [2], [3]) specify two distinct bi-Gaussian Doppler spectra, GAUS1 and GAUS2, to be used in modeling long echos for urban and hilly terrain profiles.

The normalized bi-Gaussian Doppler spectrum is given analytically by:

`${S}_{G}\left(f\right)={A}_{G}\left[\frac{{C}_{G1}}{\sqrt{2\pi {\sigma }_{G1}^{2}}}\mathrm{exp}\left(-\frac{{\left(f-{f}_{G1}\right)}^{2}}{2{\sigma }_{G1}^{2}}\right)+\frac{{C}_{G2}}{\sqrt{2\pi {\sigma }_{G2}^{2}}}\mathrm{exp}\left(-\frac{{\left(f-{f}_{G2}\right)}^{2}}{2{\sigma }_{G2}^{2}}\right)\right]$`

where ${\sigma }_{G1}$ and ${\sigma }_{G2}$ are standard deviations, ${f}_{G1}$ and ${f}_{G2}$ are center frequencies, ${C}_{G1}$ and ${C}_{G2}$ are power gains, and ${A}_{G}=\frac{1}{{C}_{G1}+{C}_{G2}}$ is a normalization coefficient.

If either ${f}_{G1}=0$ or ${f}_{G2}=0$, a frequency-shifted Gaussian Doppler spectrum is obtained.

## Examples

The following MATLAB code first creates a bi-Gaussian Doppler spectrum object with the same parameters as that of a COST 207 GAUS2 Doppler spectrum. It then creates a Rayleigh channel object with a maximum Doppler shift of ${f}_{d}=30$ and assigns the constructed Doppler spectrum object to its `DopplerSpectrum` property.

```dop_bigaussian = doppler.bigaussian('SigmaGaussian1', 0.1, ... 'SigmaGaussian2', 0.15, 'CenterFreqGaussian1', 0.7, ... 'CenterFreqGaussian2', -0.4, 'GainGaussian1', 1, ... 'GainGaussian2', 1/10^1.5) chan = rayleighchan(1e-3, 30); chan.DopplerSpectrum = dop_bigaussian;```

## References

[1] COST 207 WG1, Proposal on channel transfer functions to be used in GSM tests late 1986, COST 207 TD (86) 51 Rev. 3, Sept. 1986.

[2] COST 207, Digital land mobile radio communications, Office for Official Publications of the European Communities, Final report, Luxembourg, 1989.

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