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# doppler.rjakes

Construct restricted Jakes Doppler spectrum object

## Syntax

dop = doppler.rjakes
dop = 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
SpectrumTypeFixed value, 'RJakes'
FreqMinMaxRJakesVector 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]
```