# doppler.rjakes

Construct restricted Jakes Doppler spectrum object

## Syntax

`dop = doppler.rjakesdop = doppler.rjakes(freqminmaxrjakes)`

## Description

The `doppler.rjakes` function creates a restricted Jakes (RJakes) Doppler spectrum object that is used for the `DopplerSpectrum` property of a channel object (created with either the `rayleighchan` or the `ricianchan` function).

`dop = doppler.rjakes` creates a Doppler spectrum object equivalent to the Jakes Doppler spectrum. The maximum Doppler shift of the RJakes Doppler spectrum object is specified by the `MaxDopplerShift` property of the channel object.

`dop = doppler.rjakes(freqminmaxrjakes)`, where `freqminmaxrjakes` is a row vector of two finite real numbers between 0 and 1, creates a Jakes Doppler spectrum. This spectrum is nonzero only for normalized frequencies (by the maximum Doppler shift, ${f}_{d}$, in Hertz), ${f}_{norm}$, such that $0\le {f}_{\mathrm{min},norm}\le |{f}_{norm}|\le {f}_{\mathrm{max},norm}\le 1$, where ${f}_{\mathrm{min},norm}$ is given by `freqminmaxrjakes(1)` and ${f}_{\mathrm{max},norm}$ is given by `freqminmaxrjakes(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, `freqminmaxrjakes(1)` and `freqminmaxrjakes(2)` should be spaced by more than 1/50. Assigning a smaller spacing results in `freqminmaxrjakes` being reset to the default value of `[0 1]`.

## Properties

The RJakes Doppler spectrum object contains the following properties.

PropertyDescription
`SpectrumType`Fixed value, `'RJakes'`
`FreqMinMaxRJakes`Vector of minimum and maximum normalized Doppler shifts (two real finite numbers between 0 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], where the spectrum 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, the Jakes power spectrum does not cover the full Doppler bandwidth from $-{f}_{d}$ to ${f}_{d}$. This exception also applies to the case where the antenna pattern is directional. This type of spectrum is known as restricted Jakes [3]. The RJakes Doppler spectrum object covers only the case of a symmetrical power spectrum, which is nonzero only for frequencies f such that $0\le {f}_{\mathrm{min}}\le |f|\le {f}_{\mathrm{max}}\le {f}_{d}$.

The normalized RJakes Doppler power spectrum is given analytically by:

where

${A}_{r}=\frac{1}{\frac{2}{\pi }\left[{\mathrm{sin}}^{-1}\left(\frac{{f}_{\mathrm{max}}}{{f}_{d}}\right)-{\mathrm{sin}}^{-1}\left(\frac{{f}_{\mathrm{min}}}{{f}_{d}}\right)\right]}$

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

## Examples

The following code first creates a Rayleigh channel object with a maximum Doppler shift of ${f}_{d}=10$. It then creates an RJakes Doppler object with minimum normalized Doppler shift ${f}_{\mathrm{min},norm}=0.14$ and maximum normalized Doppler shift ${f}_{\mathrm{max},norm}=0.9$.

The Doppler object is assigned to the `DopplerSpectrum` property of the channel object. The channel then has a Doppler spectrum that is nonzero for frequencies f such that $0\le {f}_{\mathrm{min}}\le |f|\le {f}_{\mathrm{max}}\le {f}_{d}$, where and .

```chan = rayleighchan(1/1000, 10); dop_rjakes = doppler.rjakes([0.14 0.9]); chan.DopplerSpectrum = dop_rjakes; chan.DopplerSpectrum```

The output is:

``` SpectrumType: 'RJakes' FreqMinMaxRJakes: [0.1400 0.9000] ```

## 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.