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# doppler.ajakes

Construct asymmetrical Doppler spectrum object

## Syntax

```dop = doppler.ajakes(freqminmaxajakes) dop = doppler.ajakes ```

## Description

The `doppler.ajakes` function creates an asymmetrical Jakes (AJakes) Doppler spectrum object. This object is to be used for the `DopplerSpectrum` property of a channel object created with the `rayleighchan` or the `ricianchan` functions.

`dop = doppler.ajakes(freqminmaxajakes)`, where `freqminmaxajakes` is a row vector of two finite real numbers between -1 and 1, creates a Jakes Doppler spectrum that is nonzero only for normalized (by the maximum Doppler shift ${f}_{d}$, in Hz) frequencies ${f}_{norm}$ such that $-1\le {f}_{\mathrm{min},norm}\le {f}_{norm}\le {f}_{\mathrm{max},norm}\le 1$, where ${f}_{\mathrm{min},norm}$ is given by `freqminmaxajakes(1)` and ${f}_{\mathrm{max},norm}$ is given by `freqminmaxajakes(2)`. The maximum Doppler shift ${f}_{d}$ is specified by the `MaxDopplerShift` property of the channel object. Analytically: ${f}_{\mathrm{min},norm}={f}_{\mathrm{min}}/{f}_{d}$ and ${f}_{\mathrm{max},norm}={f}_{\mathrm{max}}/{f}_{d}$, where ${f}_{\mathrm{min}}$ is the minimum Doppler shift (in hertz) and ${f}_{\mathrm{max}}$ is the maximum Doppler shift (in hertz).

When `dop` is used as the `DopplerSpectrum` property of a channel object, space `freqminmaxajakes(1)` and `freqminmaxajakes(2)` by more than 1/50. Assigning a smaller spacing results in `freqminmaxarjakes` being reset to the default value of `[0 1]`.

`dop = doppler.ajakes` creates an asymmetrical Doppler spectrum object with a default ```freqminmaxajakes = [0 1]```. This syntax is equivalent to constructing a Jakes Doppler spectrum that is nonzero only for positive frequencies.

## Properties

The AJakes Doppler spectrum object contains the following properties.

PropertyDescription
`SpectrumType`Fixed value, `'AJakes'`
`FreqMinMaxAJakes`Vector of minimum and maximum normalized Doppler shifts, two real finite numbers between -1 and 1

## Theory and Applications

The Jakes power spectrum is based on the assumption that the angles of arrival at the mobile receiver are uniformly distributed [1]: the spectrum then covers the frequency range from $-{f}_{d}$ to ${f}_{d}$, ${f}_{d}$ being the maximum Doppler shift. When the angles of arrival are not uniformly distributed, then the Jakes power spectrum does not cover the full Doppler bandwidth from $-{f}_{d}$ to ${f}_{d}$. The AJakes Doppler spectrum object covers the case of a power spectrum that is nonzero only for frequencies $f$ such that $-{f}_{d}\le {f}_{\mathrm{min}}\le f\le {f}_{\mathrm{max}}\le {f}_{d}$. It is an asymmetrical spectrum in the general case, but becomes a symmetrical spectrum if ${f}_{\mathrm{min}}=-{f}_{\mathrm{max}}$.

The normalized AJakes Doppler power spectrum is given analytically by:

where ${f}_{\mathrm{min}}$ and ${f}_{\mathrm{max}}$ denote the minimum and maximum frequencies where the spectrum is nonzero. You can determine these values from the probability density function of the angles of arrival.

## Examples

The following MATLAB code first creates a Rayleigh channel object with a maximum Doppler shift of ${f}_{d}=10$ Hz. It then creates an AJakes Doppler object with minimum normalized Doppler shift ${f}_{\mathrm{min},norm}=-0.2$ and maximum normalized Doppler shift ${f}_{\mathrm{max},norm}=0.05$. The Doppler object is then assigned to the `DopplerSpectrum` property of the channel object. The channel then has a Doppler spectrum that is nonzero for frequencies $f$ such that $-{f}_{d}\le {f}_{\mathrm{min}}\le f\le {f}_{\mathrm{max}}\le {f}_{d}$, where ${f}_{\mathrm{min}}={f}_{\mathrm{min},norm}×{f}_{d}=-2$ Hz and ${f}_{\mathrm{max}}={f}_{\mathrm{max},norm}×{f}_{d}=0.5$ Hz.

```chan = rayleighchan(1/1000, 10); dop_ajakes = doppler.ajakes([-0.2 0.05]); chan.DopplerSpectrum = dop_ajakes; chan.DopplerSpectrum```

This code returns:

``` SpectrumType: 'AJakes' FreqMinMaxAJakes: [-0.2000 0.0500] ```

## References

[1] Jakes, W. C., Ed., Microwave Mobile Communications, Wiley, 1974.

[2] Lee, W. C. Y., Mobile Communications Engineering: Theory and Applications, 2nd Ed., McGraw-Hill, 1998.

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