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Construct asymmetrical Doppler spectrum object

`dop = doppler.ajakes(freqminmaxajakes)`

`dop = doppler.ajakes`

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.

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

`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]`

.

creates
an asymmetrical Doppler spectrum object with a default `dop = doppler.ajakes`

```
freqminmaxajakes
= [0 1]
```

. This syntax is equivalent to constructing a Jakes
Doppler spectrum that is nonzero only for positive frequencies.

The AJakes Doppler spectrum object contains the following properties.

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

`SpectrumType` | Fixed value, `'AJakes'` |

`FreqMinMaxAJakes` | Vector of minimum and maximum normalized Doppler shifts, two real finite numbers between -1 and 1 |

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:

$$\begin{array}{c}S(f)=\frac{{A}_{a}}{\pi {f}_{d}\sqrt{1-{(f/{f}_{d})}^{2}}},\text{}-{f}_{d}\le {f}_{\mathrm{min}}\le f\le {f}_{\mathrm{max}}\le {f}_{d}\\ {A}_{a}=\frac{1}{\frac{1}{\pi}\left[{\mathrm{sin}}^{-1}\left(\frac{{f}_{\mathrm{max}}}{{f}_{d}}\right)-{\mathrm{sin}}^{-1}\left(\frac{{f}_{\mathrm{min}}}{{f}_{d}}\right)\right]}\end{array}$$

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.

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 $${f}_{\mathrm{min}}={f}_{\mathrm{min},norm}\times {f}_{d}=-2$$ Hz and $${f}_{\mathrm{max}}={f}_{\mathrm{max},norm}\times {f}_{d}=0.5$$ 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]

[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.bell`

| `doppler.bigaussian`

| `doppler.flat`

| `doppler.gaussian`

| `doppler.jakes`

| `doppler.rjakes`

| `doppler.rounded`

| `rayleighchan`

| `ricianchan`

| `stdchan`

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