Construct restricted Jakes Doppler spectrum object

`dop = doppler.rjakes`

dop = doppler.rjakes(freqminmaxrjakes)

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 $$0\le {f}_{\mathrm{min},norm}\le \left|{f}_{norm}\right|\le {f}_{\mathrm{max},norm}\le 1$$, 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]`

.

The RJakes Doppler spectrum object contains the following properties.

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

`SpectrumType` | Fixed value, `'RJakes'` |

`FreqMinMaxRJakes` | Vector of minimum and maximum normalized Doppler shifts (two real finite numbers between 0 and 1) |

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 $$0\le {f}_{\mathrm{min}}\le \left|f\right|\le {f}_{\mathrm{max}}\le {f}_{d}$$.

The normalized RJakes Doppler power spectrum is given analytically by:

$$S(f)=\frac{{A}_{r}}{\pi {f}_{d}\sqrt{1-{(f/{f}_{d})}^{2}}},\text{}0\le {f}_{\mathrm{min}}\le \left|f\right|\le {f}_{\mathrm{max}}\le {f}_{d}$$

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.

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 $$0\le {f}_{\mathrm{min}}\le \left|f\right|\le {f}_{\mathrm{max}}\le {f}_{d}$$, where $${f}_{\mathrm{min}}={f}_{\mathrm{min},norm}\times {f}_{d}=1.4\text{Hz}$$ and $${f}_{\mathrm{max}}={f}_{\mathrm{max},norm}\times {f}_{d}=9\text{Hz}$$.

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]

[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.

`doppler`

| `doppler.ajakes`

| `doppler.bell`

| `doppler.bigaussian`

| `doppler.flat`

| `doppler.gaussian`

| `doppler.jakes`

| `doppler.rounded`

| `rayleighchan`

| `ricianchan`

| `stdchan`

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