Download Simulation of Free-Space Communication using the Orbital Angular

Simulation of Free-Space Communication using the
Orbital Angular Momentum of Radio Waves
Áron Demeter, Csaba-Zoltán Kertész
Department of Electronics and Comuters, Transilvania University, Brașov, Romania
[email protected], [email protected]
Abstract—The Orbital Angular Momentum (OAM) property
of radio waves can be efficiently used to combine multiple
free-space radio-communication channels into a single carrier
frequency. This paper presents a study of such a multichannel
system using Simulink model of radio-waves with different OAM
value combined together. The study focuses on proving that OAM
channels remain orthogonal at radio frequencies, and the OAM
modulation of FSK, PSK and QAM modulated signals results in
bit-error rates close to standard communication channel behavior
in noisy environment.
(a)
I. I
In 1902 Poynting discovered, that the circularly polarized
light carries Spin Angular Momentum (SAM), which is associated with photon helicity and depends on the polarization,
therefore it can be present in left- or right-handed circularly
polarized light. The SAM value for one photon is 𝑆 = ±ℏ,
where ℏ is the Planck constant. The other part of angular
momentum was identified by C. Darwin in 1932 as the product
of linear momentum of photon and radial distance, which is
independent of spin [1]. So the total angular momentum 𝐴 is
given by the sum:
𝐴=𝑆 +𝐽
(1)
where 𝐽 represents the Orbital Angular Momentum (OAM)
of the photon, which is associated with the orbital phase
profile of the beam, measured in the direction orthogonal to
the propagation axis [2]. In 1992, Allen et al. discovered that
using cylindrical lenses to generate Laguerre-Gaussian (LG)
modes from Hermite-Gaussian (HG) modes, a helically phased
light beam can be obtained which has got the OAM property.
The OAM value for one photon is defined in (2), where 𝑙 is
an integer number.
𝐽 = ±𝑙ℏ
(2)
In a plane perpendicular to the propagation axis, the phase of
the electric and magnetic vector fields have an 𝑙⋅𝜙 dependence
(𝜙 - angular coordinate), which means that for 𝑙 ≠ 0 the phase
fronts of beams are not planar but helical [3]. For a given 𝑙
value, the beam will contain 𝑙 intertwined helical phase fronts.
A forked diffraction grating can also produce helical beams
with optical vortex. Allen et al. showed that a 𝜓𝑙 beam with
transversal complex radial amplitude 𝐴(𝑟) depending on the
radial (𝑟) and angular (𝜙) coordinates, presented in (3), will
contain 𝑙ℏ OAM state for each photon. [4].
𝜓𝑙 = 𝐴(𝑟) ⋅ 𝑒−𝑖𝑙𝜙
978-1-4799-5183-3/14/$31.00 ' 2014 IEEE
(3)
(b)
(c)
Fig. 1. OAM=3 state. (a) wave structure of 40 waves, where the colors from
blue to red are indicating the increasing amplitude and the continuous red line
interconnects the same phase points of all waves - in this case at the signal
maximum, (b) helical phase structure, (c) phase front, where the colors from
red to blue correspond to the phase between 0 and 2𝜋.
Based on (3), the algorithm for generating twisted waves
can be implemented, resulting the waveforms on Fig.1. Because the LASER beams are composed of a narrow band
of wavelength (monochromatic) and they are collimated, the
light energy is restricted in the direction transverse to the
propagation direction to form a narrow beam, which is ideal
for OAM eigenstates [5].
The first known successful experiment in communications,
with OAM-carrying waves, was done in 2004 by Gibson et
al., the information encoded in OAM states was transmitted in
air over 15m using computer generated holograms on Spatial
Light Modulators (SLM) [6]. In 2012, a 2.56 Tbit/s data
transfer rate was successfully made over 1m using OAM Multiplexing (OAMM). The spectral efficiency can be increased
combining OAMM with polarization multiplexing [7].
In radio domain only a few experiments has been made,
but without much success. In 2011, Bo Thidé and his team
demonstrated that it is possible to simultaneously transmit
more radio channels on the same frequency band over 442m,
by encoding them in different OAM states [2]. However, there
were lots of reports of disagreement about this experiment
846
2
declaring, that it was just an implementation of the Multiple
In Multiple Out (MIMO) technology. In the above experiment,
a modified dish has been used to transmit the helical waves,
and a two-antenna interferometer to receive them. In most of
the articles, the circular antenna arrays are considered as the
best solution for generating OAM-carrying radio beams [8],
which can lead to the same results as in optics [9].
To evaluate the performance of the circular antenna arrays
we created a Simulink model. It simulates the propagation of
the helical waves using different types of modulations.
1.5
1
Ay
0.5
−0.5
−1
−1.5
II. T S M
AWGN
AWGN
Channel
HelixFDemodulator
StoringFaFperiod
1
input_signal
In1
Buffer
sim_time
Tx
1
output_signal
Out1
Clock
MATLABFFunction
StoringFaFperiod
2
In2
input_signal
Reshape
Buffer1
sim_time
Clock1
Rx
output_signal
2
Out2
MATLABFFunction1
Fig. 2. The Helix Modulator and the Helix Demodulator connected to the
communication channel
−2
−2
−1.5
−1
−0.5
0
Ax
0.5
1
1.5
2
(a)
3
2
1
Ay
Because the system is very sensible to the phase changes,
we assume, that the transmitter and receiver antenna arrays
are perfectly facing each other in the line-of-sight, like a
fixed wireless system. At lower frequencies and lower OAM
states greater mismatch is allowed. For signal modulation and
demodulation, existing Simulink digital baseband blocks has
been used, containing the complete implementation of the
selected modulation, including the coherent or non-coherent
energy detection at the receiver part. Therefore, the main
parts of the model are the Helix Modulator and the Helix
Demodulator blocks, presented in Fig.2. They contain a Matlab function block which implements the wave-twisting and
-untwisting algorithms. The buffers are storing 𝑁 samples
from the incoming complex signal incorporating at least one
full period. The clocks are required to control the output frame
indexes per simulation step. They are assumed to be perfectly
synchronized, considering that the phase shifting can be done
passively or using phased-array controllers, which are not part
of the simulation.
The Helix Modulator’s Matlab function clones the incoming
signal to the number of waves and shift each of them with 𝑙⋅𝜙,
where 𝑙 is the OAM value and 𝜙 is the phase, uniformly distributed between the antennas. In other words, the transmitter
is a serial to parallel converter and a phase shifter at the same
time. Every shifted signal feeds one antenna from the array.
While the optical beams contains a lots of waves, in radio
domain the number of antennas is limited. To ensure a similar
HelixFModulator
0
0
−1
−2
−3
2
1
40
30
0
20
−1
Ax
10
−2 0
Ns
(b)
Fig. 3. Helix modulator output frame (a) in the case of 4 circles and 40
antennas. (b) is the 3D representation of the same wave for one period and
one antenna circle ( Ns is the samples used in one wave period ).
phase profile to the LG beams, at least 2 ⋅ 𝑙 + 1 antennas are
required [9]. The Helix Demodulator’s Matlab function shifts
the incoming waves to the same phase and makes the sum of
them, acting like a parallel to serial converter and phase shifter
at the same time. In every simulation step, the transmitter
outputs a frame which contains 𝑎𝑛𝑡𝑒𝑛𝑛𝑎𝑛𝑢𝑚𝑏𝑒𝑟×𝑐𝑖𝑟𝑐𝑙𝑒𝑛𝑢𝑚𝑏𝑒𝑟
samples. For example if there are four concentric circles,
each of them with 40 waves (antennas), the outgoing frame
will look like the one on Fig.3. These frames are connected
to an AWGN channel simulating noisy environment. Also a
multipath propagation channel was tested using Rician fading
blocks to simulate reflected signals. Because the OAM works
only in line-of-sight transmission and reflected signals also
change OAM states, multi-path effects were studied more
in terms of interference to the generic OAM transmission
channel. The optical beam width scales with √𝑙 which means,
that the receiver aperture will have a physical limitation on
higher OAM values [6]. In a similar manner the antenna array
dimensions (circle radius) in radio domain will also result in
the same OAM limitation [9].
There are several methods for detecting the OAM states
of radio waves [3], which are not part of our aim, so the
simulation uses well defined states in the Helix Modulator
847
III. T S
A. FSK modulation
Fig. 4. Results of the twisted (center) and untwisted (bottom) waves of an
incoming sine wave (top) in case of matched OAM state (a) and unmatched
OAM state (b)
and Demodulator blocks. When the OAM states are set to the
same value at the transmitter and the receiver, the sum will
give a strong signal, but when they are set different, the result
will be unrecognizable, or close to zero. Even if the OAM
states are set to adjacent, the result will be wrong, so it is easy
to distinguish them. Fig.4 shows what happens with a simple
sinusoidal carrier wave if the OAM states are matched or nonmatched. This proves that our simulation model is consistent
with the theoretical orthogonality of the OAM states.
Bernoulli
Binary
2-FSK
Bernoulli Binary
Generator2
M-FSK
Modulator
Baseband1
The Bit-Error Rate (BER) curves of the baseband modulation are very close to the theoretical non-coherent BER curve.
Fig.5 shows the scheme of the simulation model and the BER
of the 2-FSK signal for 𝑙 = 3, 𝑙 = 6 and 𝑙 = 9. The simulations
were made with the minimum number of waves for every
OAM state.
For the antenna array, the 2-FSK passband modulation
uses the same configuration as the baseband model, but the
baseband signal is upconverted to 1GHz carrier frequency. The
BER curves and the simulation model are shown on Fig.6 in
a similar manner to the baseband simulation.
In the OAM-based communication the focus is on the
orthogonality of the OAM-states which makes possible to
transmit more information on the same carrier frequency. The
simulation model on Fig.7 is demonstrating, that different
communication channels can be used on the same frequency
band without any interference. This model is using a single
Helix Demodulator whose OAM parameter is the one of the
parameters set in the Helix Modulators. In this case, the
demodulator in the figure was set to decode the OAM =
5 modulated channel. Every channel is containing different
information generated by the Bernoully Binary blocks with
different probability of zero. The system is using the minimum
Bernoulli
Binary
Bernoulli Binary
Generator1
Helix Modulator
0
Tx ErrordRate
Calculation
Rx
0
101
BER
Wrong
Total
Data
AWGN
Channel3
Modulator5
Tx
AWGN
Rx
Error Rate
Calculation5
Display5
Helix Modulator2
0
Error1Rate
Calculation
Error Rate
Calculation6
Rate Transition2
0
101
BER
AWGN
Wrong Channel2
Total
AWGN
Display6
ZOH
2-FSK
Unbuffer
FSK
Data
M-FSK
Demodulator
Baseband
Buffer1
FSK
Helix Demodulator
Demodulator1
0
10
−1
10
Helix Demodulator2
0
10
Theoretical=2−FSK
l===0
l===3
l===6
l===9
Theoretical92−FSK
l9=90
l9=93
l9=96
l9=99
−1
10
−2
BER
BER
10
−2
10
−3
10
−3
10
−4
10
−5
10
0
−4
2
4
6
8
10
10
12
Eb/N0 (dB)
Fig. 5.
Baseband FSK simulation model and the BER diagram of the
baseband 2-FSK signals
0
2
4
6
Eb/N0 (dB)
8
10
12
Fig. 6. Passband FSK simulation model and the BER diagram of the passband
2-FSK signals
848
Error Rate
Calculation4
905347
BER
Wrong
Total
Bernoulli
Binary
Bernoulli Binary
Generator4
Bernoulli
Binary
Bernoulli Binary
Generator5
Bernoulli
Binary
Bernoulli Binary
Generator6
Bernoulli
Binary
Bernoulli Binary
Generator7
Data
FSK
Modulator6
Data
9
BER
Wrong
Total
Helix Modulator OAM = 3
ErrorORateTx
Calculation
Rx
54
191
Error Rate
Calculation1
Display4
9
ErrorORateTx
Calculation
Rx
191
Display1
FSK
ZOH
Modulator1
Data
FSK
FSK
AWGN
Channel4
Add1
Modulator2
Data
AWGN
Helix Modulator OAM = 5
Helix Demodulator3
Demodulator2
43
191
Display3
FSK
Display2
Modulator4
Rate Transition3
ErrorORateTx
Calculation
Rx
904257
BER
Wrong
Total
Helix Modulator OAM = 7
Data
BER
Wrong
Total
Helix Modulator OAM = 8
905149
52
191
Error Rate
Calculation2
ErrorORateTx
Calculation
Rx
Error Rate
Calculation3
Fig. 7.
Passband FSK simulation model for a 4 combined communication channels with single channel demodulation
ZOH
Bernoulli
Binary
Bernoulli Binary
Generator8
Data
FSK
Modulator9
FSK
Helix Modulator1 OAM = 3
Data
Demodulator5
Helix Demodulator1 OAM = 3
Rate Transition6
ZOH
Bernoulli
Binary
Bernoulli Binary
Generator9
Bernoulli
Binary
Bernoulli Binary
Generator10
Data
FSK
Modulator3
Data
FSK
FSK
Modulator7
AWGN
Helix Modulator1 OAM = 5
Add2
Data
Demodulator4
Helix Demodulator1 OAM = 5
Rate Transition5
ZOH
AWGN
Channel5
FSK
Helix Modulator1 OAM = 7
Data
Demodulator3
Helix Demodulator1 OAM = 7
Rate Transition4
ZOH
Bernoulli
Binary
Bernoulli Binary
Generator11
Data
FSK
Modulator8
FSK
Helix Modulator1 OAM = 8
Fig. 8.
Helix Demodulator1 OAM = 8
Data
Demodulator6
Rate Transition7
Tx ErrorHRate
Calculation
Rx
Error Rate
Calculation10
Tx ErrorHRate
Calculation
Rx
Error Rate
Calculation7
Tx ErrorHRate
Calculation
Rx
Error Rate
Calculation8
Tx ErrorHRate
Calculation
Rx
Error Rate
Calculation9
0
0
101
BER
Wrong
Total
Display11
0
0
101
BER
Wrong
Total
Display8
0
0
101
BER
Wrong
Total
Display9
0
0
101
BER
Wrong
Total
Display10
Passband FSK simulation model for a 4-channel simultaneous communication
number of antennas for the maximum OAM state, in this case
19 antennas. The BER calculator shows zero for the selected
channel, but the other channels BER values are high. With
19 antennas the system is capable to distinguish 9 channels;
therefore every introduced antenna is increasing the total
capacity. Another simulation model is shown on Fig.8, where
are four Helix Demodulators, one for each channel. Every
antenna array is receiving all of the vortex channels, but each
decodes only one of them. In this case, the BER results are
zero in every calculator block.
The OAM state orthogonality was also tested on single
channel transmission with multi-path effect simulated with
Rician fading blocks. These enable simulating various reflection path between the transmitter and the receiver. The
ratio between the power in the line-of-sight component and
the power in the diffuse component was set to 𝑘 = 6, and
also considering some moving objects near the system, the
maximum diffuse Doppler shift was set to 40Hz. Theoretically
signals reflected loose their OAM state, so only the line-ofsight channel will be picked up by the receiver, however in
practice the reflected signals also present a large incoming
interference which increases BER levels. This increase is on
the other hand less than the effect of multi-path on the plain
FSK signals. The model and result of multi-path analysis is
presented on Fig.11.
B. PSK modulation
The PSK passband simulation model and the BER diagram
for the passband simulation is shown in Fig.9.
It can be seen, that the results are close to the FSK
modulation’s curves, but compared to the theoretical 2-PSK
curve, they are worse at higher 𝐸𝑏 /𝑁0 ratio. This shows the
sensibility of the system to the phase shifting operation. It is
expected that the imprecision introduced by the the floating
point calculations within all the phase shifting operations to
influence also the PSK demodulation. This also implies that
on physical implementation a greater care is needed on the
phase shifting lines leading to the antennas.
The figures are terminating at the 𝐸𝑏 /𝑁0 point where no
more errors are detected at 100 kbits. Because of the one
period process at each Helix block, the Error Rate Calculation
block is set to delay the detection by two bits. As the number
of period processing increase, the delay also increase, but this
does not affect the BER result.
849
Bernoulli
Binary
Data
Bernoulli1Binary
Generator1
Modulator5
Tx
Rx
Error1Rate
Calculation
0
261
Data
BernoulliABinary
Generator1
Helix1Modulator2
0
Error1Rate
Calculation6
Rate1Transition2
Bernoulli
Binary
PSK
BER
AWGN
Wrong
Total Channel2
Modulator5
Tx
AWGN
QAM
Rx
0
ErrorARate
Calculation
ErrorARate
Calculation6
Display6
ZOH
HelixAModulator2
0
165
BER
Wrong AWGN
Total Channel2
AWGN
Display6
ZOH
Data
PSK
Demodulator1
Data
RateATransition2
Helix1Demodulator2
−1
QAM
Demodulator1
HelixADemodulator2
0
10
10
TheoreticalS2−PSK
lS=S0
lS=S3
lS=S6
lS=S9
−2
10
Theoretical=8−QAM
l===0
l===3
l===6
l===9
−1
10
−2
BER
BER
10
−3
10
−3
10
−4
10
−4
10
−5
10
0
−5
1
2
3
4
5
Eb/N0 (dB)
6
7
8
10
9
0
2
4
6
Eb/N0 (dB)
8
10
12
Fig. 9. Passband PSK simulation model and the BER diagram of the passband
2-PSK signals
Fig. 10. Passband QAM simulation model and the BER diagram of the
passband 8-QAM signals
The Rate Transition block is not necessary, but useful when
the difference between the carrier frequency and bit rate is
high. To get closer to the theoretical 2-PSK curve, the number
of samples and the amplitude of the waves can be increased.
When simulating the multiple channels of OAM states
combined with a similar model as presented for the FSK
modulation, the results were nearly the same meaning that
the OAM multiplexing has no further effect on the signals.
Multi-path effect on PSK signals is also similar to the FSK
case (Fig.12), but with BER values higher comparatively to
theoretical values than were with FSK due to the sensibility
to phase changing of the PSK modulation.
phase+amplitude modulation is used. To get closer to the
theoretical 8-QAM curve, also the amplitude and the number
of samples can be increased.
C. QAM modulation
In radio frequency free-space communications, the FSK
and PSK modulations are not so frequently used as QAM
modulation. A QAM modulated channel has much higher bit
rate capability than the PSK modulated channel, and the error
probability is much lower than at the FSK. The passband
QAM simulation model together with the BER diagram of
the passband model is shown in Fig.10.
As at the other simulations, the simplest constellation,
8-QAM was used. If the number of waves (antennas) increase,
then the amplitude of the waves in the Helix Demodulator
and the QAM Demodulator will also increase. The simulation
needs the processing quantity to be one period of carrier
frequency in order to change the phase correctly even if
IV. C
The simulation presented in this paper prove that OAMbased radio channels remain orthogonal with acceptable BER
on well-known modulation types. The OAM multiplexing
of radio waves does not bring anything basically new, as
long as it can be implemented with the traditional beam
forming techniques using MIMO antenna matrixes or phased
arrays. The simulated BER curves shows that there are no big
differences between the different OAM states, so theoretically
an infinite number of channels can be used without any
BER degradations. Compared to the modified parabolic dish,
a circular antenna array can generate more than one OAM
state without any mechanical modifications, given that at least
2 ⋅ 𝑙𝑚𝑎𝑥 + 1 antennas are used.
The results shows that the OAMM is compatible with
the digital multiplexing techniques even if they are phasechanging modulations. However the phase shifting circuits has
to be implemented with high precision to avoid introducing
error into the phase fronts needed by the latter. Also the phasechanging modulations are much more susceptible to multipath effects, which need to be addressed in the design of the
communication system. Compared to the optical domain, only
one circle of waves is enough to generate the OAM states.
850
Bernoulli
Binary
Data
Bernoulli
Binary
FSK
Reshape
BernoulliBBinary
Generator1
Modulator5
Tx
Rx
0
412
ErrorBRate
Calculation6
AWGN
Channel2
AWGN
Rician
Fading
Tx
MultipathBRician
FadingBChannel1
Rx
HelixBModulator2
0
ErrorBRate
Calculation
0
3606
ErrorBRate
Calculation6
Display
AWGN
Channel2
Rician
Fading
AWGN
MultipathBRician
FadingBChannel1
Display
ZOH
Data
RateBTransition2
FSK
Data
Demodulator1
RateBTransition2
HelixBDemodulator2
0
−1
10
Demodulator1
HelixBDemodulator2
0
Theoretical=2−PSK
l===0
l===3
l===6
l===9
−1
10
−2
−2
10
BER
10
−3
−3
10
10
−4
−4
10
10
−5
−5
0
PSK
10
Theoretical=2−FSK
l===0
l===3
l===6
l===9
10
BER
Modulator5
0
ErrorBRate
Calculation
PSK
Reshape
BernoulliBBinary
Generator1
HelixBModulator2
ZOH
10
Data
2
4
6
8
10
Eb/N0 (dB)
12
14
16
10
18
0
2
4
6
Eb/N0 (dB)
8
10
12
Fig. 11. Multi-path FSK simulation model and the BER diagram of the
passband 2-FSK signals affected by multi-path effect
Fig. 12. Multi-path PSK simulation model and the BER diagram of the
passband 2-PSK signals affected by multi-path effect
As with the optical domain, the line of sight it is also
important in the radio domain, as the OAM of the reflected
waves can be affected making impossible the correct decoding
in the receiver. This can mean that the channel security is
increased (eavesdropping out of the line-of-sight is impossible)
on the other hand practical use is limited to fixed wireless communications. There is a possibility to use intelligent antennas
for beam-forming purpose, however this needs further studies.
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R
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