Chapter 4: Jitter diversity simulation in a simplified
environment
The aim of this simulation part is to implement jitter diversity in an ideal environment so as to
withdraw tendencies for its operation.
We generate a simplified environment and an ideal antenna pattern in order to see how the
jitter process can be optimised with respect to environment and antenna parameters.
4.1. Variables and parameters
Some of the different variables used in this simulation part and the parameters associated
are listed below:

‘x’: realisation domain composed of discrete time steps. The total number of
realisations is NR;

‘θAOA’: angular domain of the multipath rays angles of arrival on the antenna, defined
from 0 to 360°. The total number of angles of arrival is NAOA;

‘θBO’: angular domain of the orientation of the antenna beam, defined from 0 to 360°.
The total number of beam orientations is NBO;
4.2. Environment implementation
In this set of simulations, we chose to use a simple but nevertheless realistic environment,
suitable for our indoor study. The environment response is a function of realisations (i.e.
discrete time) and angles of arrival of the multipath rays on the antenna. We want to perform
“Monte Carlo” simulations, assuming that the user’s location is randomly defined at each time
step. To include this statement in our environment response, each complex component is
composed of a random phase and a random amplitude. So we define a first set of random
values temp1(x, θAOA) uniformly distributed between 0 and 1, which will be used later on to
define the amplitude, and another set of random values temp2(x, θAOA) uniformly distributed
between 0 and 1, which will be used later on to define the phase.
A typical way of modelling indoor channel is to use clustered scatterers [4.1], and our
investigations will concentrate on the rays arriving from a unique cluster. It has been found in
[4.2] that, in an indoor multipath propagation channel using clustered scatterers, which
corresponds to the channel model we will use in the more realistic simulation part, the angle
of arrival distribution within a cluster can be approximated by a Laplacian distribution, defined
in equation (4.1):
PLaplace _ a ( AOA ) 
a  a  AOA
e
2
(4.1)
In (4.1), ‘a’ is a parameter describing the shape of the distribution. It comes directly from the
variance of the distribution, which is equal to 2/a².
Hence we chose to scale the amplitude of the components so that the envelop of the angular
distribution averaged over all realisations follows a Laplacian distribution (see (4.2a)). The
phase of the components will remain random, uniformly distributed between 0 and 2π (see
(4.2b)).
The environment response that we create is defined in equation (4.2).
er ( x,  AOA )   er ( x,  AOA ).e jer ( x , AOA )
Where the amplitude is defined i  1, N R , k  1, N AOA
 er ( xi , AOAk ) 
N R . PLaplace _ a ( AOAk )
NR
 temp1( xn , AOAk )
(4.2)
by:
.temp1( xi ,  AOAk )
(4.2a)
n 1
And the phase by:
 er ( x ,  AOA )  2 .temp 2( x ,  AOA )
(4.2b)
The expression mean environment response will denote the environment response module
averaged over all the realisations.
Figure 4.1 shows an example of environment response amplitude and mean environment
response amplitude over 100 realisations:
Figure 4.1: Mean environment response amplitude
and environment response amplitude representations
The ‘a’ parameter coming from the use of the Laplacian distribution and included in the
amplitude term of our response allows us to control the shape of the environment. To
describe the shape information, we define a parameter BW env corresponding to the halfpower mean environment response width.
Figure 4.2 illustrates the representation of BW env in the concrete case of parameter a set to
0.1:
Figure 4.2: Representation of BWenv
In our simulations, a will go from 10-1 to 10-6 which we find a reasonable range in order to
represent various environments. For large values of a, the shape will be very sharp (i.e. small
width), assuming that the arriving multipath rays are concentrated on one point of the
antenna. On the other hand, if we have very low values of a, the shape will be very flat (i.e.
large width), assuming that the arriving rays on the antenna are much more spread.
Therefore, a large value of the environment width will correspond to a more uniform
repartition of the environment power response information born by the arriving rays. Figures
can be found in Appendix I.i.i that show the environment response amplitude for different
shapes and illustrate that phenomenon.
Table 4.1 summarises the values used for a and the corresponding values of BW env:
a
Table 4.1: Different a values and associated BWenv
10-6 5.10-6 10-4 5.10-4 10-3 5.10-3 10-2 5.10-2
10-1
BW env (°) 89.91 89.91 89.55 87.74 85.94 70.45 53.87 13.87 7.02
4.3. Antenna
Since we are working in ideal conditions, we choose to use an ideal beam pattern: a raised
sine wave. This choice allows us to avoid directivity issues that would rise from the presence
of side and back lobes of a realistic antenna array pattern. The antenna amplitude is a
function of angles of arrival and of beam’s orientation angle, and we choose a real beam
because we are not interested in getting extra diversity gain through the use of the phase in
this simulation. Finally, the exponent α raising the sine wave is a parameter controlling the
antenna beamwidth, but not modifying the maximum received power value since it is always
equal to one in the corresponding direction.
The antenna amplitude is then given by (4.3):
sin ( AOA   BO ) if  AOA   BO  

a ( AOA , BO )  
otherwise

0
(4.3)
The antenna amplitude is set to zero for certain angles in order to suppress the back lobe
occurring otherwise.
Figure 4.3 shows the antenna power pattern for different beam’s orientation angles:
Figure 4.3: Antenna power pattern
The antenna half-power beamwidth is called BW ant.
In the previous section, we defined several environments using a range of values of BW env.
Similarly, we will test several antenna patterns on each environment and compare the
results.
The values we choose for the exponent are summarised in the table below with the
corresponding values of BW ant:
α
BW ant
(°)
3
Table 4.2: Different α values and associated BWant
7
9
15
17
19
21
23
33
39
47
53.87 35.49 31.53 24.32 22.88 21.8 20.72 19.64 16.39 14.95 13.87
4.4. Width ratio
In order to synthesise both widths previously defined, BW env and BW ant, we create a global
ratio RW, bearing information coming from these two parameters. The ratio is defined in (4.4):
RW 
BWant
BWenv
(4.4)
Figure 4.4 represents the behaviour of the ratio depending on the values of BW env and BW ant
described in Tables 4.1 and 4.2:
Figure 4.4: Width ratio
4.5. Transfer function
The transfer function h depends on realisations and on beam’s orientation angle and
represents the power collected at the antenna output. For each realisation, the antenna
beam describes the entire angular domain around the base station, i.e. all beam’s
orientations are performed.
According to [4.3], the discrete transfer function is given by (4.5):
h( x, BO ) 
N AOA
 er ( x,
n 1
AOAn
).a ( AOAn , BO )
(4.5)
Figure 4.5 shows a representation of the module of h for particular values of BW ant and
BW env:
Figure 4.5: Transfer function module
We can observe the numerous fades encountered. Depending on the environment width and
on the beamwidth, the fades are more or less pronounced as shown in Appendix I.i.iii.
4.6. Jitter process
In our Monte Carlo simulation, we are in a memory less situation, that is to say that since our
spatial moves are random, they are not related one to each other. Therefore, our jitter
algorithm will neither take into account any update nor the previous user’s position. We will
thus not rely on gradient information to determine the direction in which the beam will be
oriented.
At this step, we need introduce new parameters:

‘θref’ is the angular reference domain. So at the kth iteration, θrefk is the angle
corresponding to the maximum value of the environment response module,
k  1, N R ;

‘θjitt’ is the jitter range, say the angle of which the beam is turned on both sides of the
reference angle. The jitter angle is equal to 3° as it is recommended in [4.4];

‘θFB’ is the angle corresponding to the direction of the fixed beam, that is to say a
beam staying in the direction of the first maximum of the environment response
during the whole simulation, so  FB   ref 1 ;

‘θpath’ is the angular path domain, say the angles chosen by the jitter process at each
iteration;

‘θBPP’ is the angular domain of the best possible value, say at each step the angle
corresponding to the best possible value of the module of h.
We will now describe what we will call the jitter with respect to the reference direction
algorithm or JRDA.
The reference direction is the direction corresponding to the maximum value of the
environment response module. Therefore, we will apply the jitter algorithm around this
reference direction for each realisation.
Figure 4.6 sketches the JRDA process: at each iteration k, k  1, N R , h( xk ,  refk ) is
compared to h( xk ,  refk   jitt ) and h( xk ,  refk   jitt ) . The beam’s orientation chosen θpathk
will correspond to the orientation of the maximum value. Thus, h( x,  path ) is the whole of the
values of h module collected by the jitter process during the simulation. h( x, path )
refer to the collected power in dB.
dB
will
Figure 4.6: JRDA process
In order to compare the JRDA to the best possible value we could obtain at each step, we
define the best possible process (BPP). This algorithm consists in doing a full scan of the
environment at each iteration, and in taking the maximum value of the collected power
module. Thus, h( x, BPP ) is the whole of the values of h module collected by the best
possible process during the simulation.
We also define h( x, FB ) as the whole of the values of h module collected by the fixed beam
during the simulation. We will use these values as a reference in the calculation of the
diversity gain, as explained in the next section. Indeed, these values are collected without
any algorithm, so no diversity is achieved in this process.
4.7. Results
We run the simulation for a number of realisations NR equal to 10000.
4.7.1. Angular path
On of our concerns is to study the path taken by the angle of the JRDA, and see how it
behaves depending on the environment and on the antenna widths.
Figure 4.7 shows a particular situation for the first 500 realisations:
Figure 4.7: Path angle
The jitter process is clearly illustrated. Though the angular spread is not very large, Figure
4.8 shows that it strongly depends on the environment width. Indeed, the standard deviation
calculated for different beamwidths and environment widths is dramatically increasing under
the influence of the environment width whereas the beamwidth changes have a poor effect
on it.
Figure 4.8: 3D standard deviation representation
Figure 4.9 emphasises what was mentioned before, say that the environment changes have
a strong effect on the spreading of the jittering angle path in the JRDA case. The standard
deviation is averaged over the beamwidths values, and we can observe the tendency of the
standard deviation variations vs. the environment width.
Figure 4.9: 2D averaged standard deviation representation
4.7.2. Diversity gain
We define the normalised power levels of the JRDA (eq. (4.6a)), of the BPP (eq. (4.6b)) and
of the fixed beam (eq. (4.6c)):
h( x, path )
Where h( x,  path )
dB
dB _ norm
 h( x, path )
dB
 h( x, path )
dB
(4.6a)
h( x, BPP ) dB _ norm  h( x, BPP ) dB  h( x, BPP ) dB
(4.6b)
h( x, FB ) dB _ norm  h( x, FB ) dB  h( x, FB ) dB
(4.6c)
, h( x,  BPP ) dB and h( x,  FB ) dB are respectively the mean power levels in
dB of the JRDA, the BPP and the fixed beam.
As done in [4.4], we define the diversity gain of the JRDA called DGJRDA _1% as the difference
between the cumulative density values of h( x, path )
dB _ norm
and h( x,  FB ) dB _ norm at the 1%
level of probability as illustrated by Figure 4.10:
Figure 4.10: Diversity gain representation
In the same way, we define the total power gain of the JRDA called TPGJRDA _1% as the
difference between the cumulative density values of h( x, path )
dB
and h( x, FB ) dB at the 1%
level of probability as illustrated by Figure 4.11:
Figure 4.11: Total power gain representation
Finally, we define the median power gain of the JRDA called MPGJRDA _ 50% as the difference
between the cumulative density values of h( x, path )
dB _ norm
and h( x,  FB ) dB _ norm at the 50%
level of probability.
We obtain the following results for the diversity gain displayed in Figures 4.12, 4.13 and 4.14:
Figure 4.12: 3-dimensional diversity gain
The main tendency observed is that the diversity gain increases when the environment width
increases and the antenna beamwidth decreases. The more accentuated tendency between
both width parameters happens with the change in the antenna beamwidth, which can be
explained by the fact that the narrower the beamwidth, the higher the collected power.
Figure 4.13 represents the diversity gain averaged over the environment widths, which is to
say that we look at the values of the diversity gain averaged on the environments shapes for
different beamwidths. The tendency mentioned above is clear, and the artefact we can
distinguish does not seem to be important, because it was appearing at different places
depending on the simulations. Therefore, it might come from the environment randomness.
Figure 4.13: Averaged diversity gain
Figure 4.14 confirms the tendency because highest gain values occur for small values of the
ratio RW, that is to say when the antenna beamwidth is the smallest.
Figure 4.14: 2-dimensional diversity gain
Finally, we can observe on Figure 4.15 the contribution of the diversity gain to the total power
gain. Whatever the values of the diversity gain are, they are always preponderant compared
to the median power gain. Figure 4.15 is a particular case since the diversity gain values are
larger than the total power gain values. This phenomenon is only occurring for a narrow
environment width as other figures demonstrate it in Appendix I.i.iv. However, we presented
this particular case in order to be coherent with the CDF representations of Figures 4.10 and
4.11.
Figure 4.15: Gains evolutions
4.8. Conclusion
In this simulation, we studied the effects of different simple environments and different ideal
antenna patterns on an adapted jitter process operation. The main tendency is that the jitter
process seems to take advantage of a narrow antenna beamwidth coupled with a uniform
repartition of the environment response power around the antenna. The diversity gain ranges
from 1.6 to 7.1 dB, all environments and beamwidths taken into account. Finally, we
observed that in our adapted jitter algorithm, the angular path variations are strongly
dependent on the environment definition.
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