Directional Modulation Far-field Pattern
Separation Synthesis Approach
Yuan Ding and Vincent F. Fusco
The ECIT Institute Queens University of Belfast
Belfast, BT3 9DT, UK
v.fusco@qub.ac.uk
Abstract—In this paper a far-field power pattern separation approach is proposed for the
synthesis of directional modulation (DM) transmitter arrays. Separation into information
patterns and interference patterns is enabled by far-field pattern null steering. Compared
with other DM synthesis methods, e.g., BER-driven DM optimization and orthogonal
vector injection, the approach developed in this paper facilitates manipulation of artificial
interference spatial distributions. With such capability more interference power can be
projected into those spatial directions most vulnerable to eavesdropping, i.e., the
information side lobes. In such a fashion information leaked through radiation side lobes
can be effectively mitigated in a transmitter power efficient manner. Furthermore, for the
first time, we demonstrate how multi-beam DM transmitters can be synthesized via this
approach.
Keywords- Bit error rate, directional modulation, information patterns, interference patterns,
null steering.
1. INTRODUCTION
Directional modulation (DM) technology, as a promising physical layer security means for
wireless communication in free space, has attracted extensive research in recent years [1-20].
Different to conventional wireless transmitters, which broadcast identical copies of information
into the whole space, DM-enabled transmitters have the capability of scrambling information
formats, i.e., the received constellation patterns in IQ space, along every spatial directions other
than a pre-specified communication direction where no distortion will occur. This spatially
controlled means for information distortion makes interception by potential eavesdroppers sited
away from the desired direction significantly more difficult than otherwise would be possible.
The recent efforts within DM research can be sorted into two categories.
The first is to seek physical transmitter arrangements that enable DM characteristics. In [1, 2],
a DM transmitter structure, termed the parasitic DM structure, consisting of one central-driven
antenna surrounded by a number of reconfigurable parasitic elements in the near-field, was
proposed. Later it was discovered that by imposing baseband signals directly onto the beamforming networks, e.g., variable RF phase shifters and attenuators [3-5], or the antenna radiators
[6] in an actively driven antenna array, signal transmissions with direction-dependent modulation
formats could be achieved. Another two novel types of DM structures, named by the authors as
antenna subset modulation [9] or 4-D antenna arrays [10, 11] and Fourier lens DM arrays [12,
13], have also been described.
The second category is to develop effective and efficient DM synthesis methods. Based on
actively driven antenna arrays, generally speaking, there are four DM synthesis approaches,
including the far-field pattern separation DM synthesis approach that is proposed in this paper.
The orthogonal vector DM synthesis approach [14] that was developed based on the DM vector
representation technique [15]. This shared a similar idea with the artificial noise concept [21] used
in the information theory society is a universal DM synthesis method. It did not attach any
constraints on either DM transmitter array physical arrangements or performance characteristics.
In order to meet various DM system requirements for different applications, the BER-driven DM
synthesis approach [3, 6, 16, 17] and the far-field radiation pattern DM synthesis approach [18,
19] were developed. These two methods, together with the far-field radiation pattern separation
DM synthesis approach presented in this paper, can be regarded as the orthogonal vector method
under additional system constraints, e.g., BER spatial distribution templates, DM array far-field
radiation pattern templates, and interference far-field spatial distributions.
In this paper we introduce a far-field radiation pattern separation DM synthesis approach
wherein DM far-field radiation patterns can be separated into information patterns which describe
information energy projected along each spatial direction, and simultaneously transmitted
interference patterns which represent disturbance on genuine information. By this separation
methodology we can identify the spatial distribution of information transmission and hence focus
interference energy into the most vulnerable directions, i.e., information side lobes, and in doing
so submerge leaked information along unwanted directions. As we pointed out in the last
paragraph this method is linked to the orthogonal vector approach. In fact the separated
interference patterns can be considered as far-field patterns generated by the injected orthogonal
vectors. However, it is more convenient to apply constraints, such as interference spatial
distribution, with the pattern separation approach proposed in this paper. The new approach
presented in this paper is also compatible with multi-beam DM synthesis, an aspect of DM
systems which has to the authors’ knowledge never been discussed previously.
In Section 2 and Section 3 of this paper the proposed far-field pattern separation approaches
are presented for the synthesis of single-beam and multi-beam DM transmitters, respectively.
Typical examples for each case are also provided. The performance of the synthesized DM
systems, i.e., both single-beam and multi-beam systems, are evaluated via the BER simulations
presented in Section 4, and conclusions are drawn in Section 5.
2. FAR-FIELD PATTERN SEPARATION SINGLE-BEAM DM SYNTHESIS
2.1.
Synthesis procedures
In this section the proposed far-field pattern separation single-beam DM synthesis approach is
presented using an N-by-1 uniformly half wavelength spaced DM transmitter array as shown in
Fig. 1.
1) Select a set of array excitations, P, that generate a far-field radiation pattern whose main
beam points to the desired secure communication direction θ0. Hereafter we term this farfield radiation pattern the ‘information pattern’. As for a conventional phased array we
choose excitations with uniform magnitudes and progressive phases, i.e., −n∙k∙d∙cosθ0, (n =
−(N−1)/2, −(N−1)/2+1, …, (N−1)/2). k is the wave number, and d denotes the antenna
element spacing in an N-element array, λ/2. The array phase centre is chosen as its geometric
centre. In Fig. 2 (a) an example of a normalized information pattern is depicted for N = 5 and
θ0 = 120º. In this paper the ideal isotropic antenna radiation pattern for each array element is
assumed so that proposed scheme can be clearly demonstrated.
2) Find the (N−1) spatial directions of side lobes in the information pattern obtained in the step
1). Then, similar to the step 1), generate (N−1) sets of excitations, Ai = [Ai1 Ai2 … AiN]T, (i =
1, 2, …, N−1), that project far-field radiation patterns whose main beams are directed
towards the (N−1) information side lobe directions respectively. Here ‘[∙]T’ is the vector
transpose operator. In Fig. 2 (a) four radiation power patterns corresponding to each of the
four side lobes in the example information pattern are illustrated. The four side lobe
directions are 0º, 26º, 60º, and 84º. In this example, the power of each main beam associated
with each Ai is, initally, arbitarily set 3 dB larger than that of its corresponding information
side lobe obtained in the step 1). The effect of this power difference on DM system
performance will be investigated in Section 4.
3) Next we alter each of the obtained (N−1) sets of excitations Ai in order to steer far-field
pattern nulls to the desired secure communication direction θ0. We choose the power pattern
projection method stated in [22] to steer the nulls. Here the required optimum excitation
weights Bi satisfying main beam, null direction, and total power transmission requirements,
can be obtained from (1),
Bi   Bi1
Bi 2
BiN    I N  C 1C  Ai
T
(1)
where IN denotes the N-by-N identity matrix, and C is a 1-by-N vector with the nth element of
e  jnkd cos0 . ‘[∙]−1’ in (1) is the Moor-Penrose pseudo inverse operator, returning an N-by-1
vector in this case. By applying this pattern projection in (1), nulls in the (N−1) far-field
patterns obtained in the step 2) are steered to the desired communication direction θ0, while
the main beams are kept unchanged along each of the side lobe directions in the information
pattern. These resultant far-field patterns associated with array excitations Bi are labelled as
the ‘interference patterns’ in this paper. Fig. 2 (b) illustrates the four interference patterns
obtained using (1) from the corresponding patterns in Fig. 2 (a).
4) With the excitations P and Bi, associated with information pattern and interference patterns
respectively, the excitations, Sm, for the mth symbol, Dm, in a data stream transmission in a
single-beam DM system can be synthesized via (2),
N 1
Sm  Dm  P    Rim  Bi 
(2)
i 1
Rim is a random complex number with constraints imposed by the practical hardware
limitations in the DM transmitter, e.g., amplitude and phase shifter increment for the analogue
DM architecture in [3, 5, 16]. Since the (N−1) interference patterns have nulls along the
desired communication direction θ0, the magnitude and phase relations in Dm are well
preserved along direction θ0. Whereas, simultaneously, the received signals along other
spatial directions are randomly and dynamically corrupted by interference, especially along
the information side lobe directions.
In order to summarize the pattern separation single-beam DM synthesis procedure presented
above, a flow chart is provided in Fig . 3.
2.2.
Synthesis example
Using the excitations P and Bi associated with example patterns in Fig. 2 (b), the excitations,
Sm, for a single-beam DM array, when four unique QPSK symbols are transmitted, are obtained
and listed in Table 1. Gray-coding is used throughout in this paper, thus the four phase
synchronized QPSK symbols ‘11’, ‘01’, ‘00’, and ‘10’ should lie in the first to the fourth
quadrants in IQ space respectively. In this example, Rim is set to be unity magnitude with random
phase. The corresponding far-field patterns are illustrated in Fig. 4. It can be clearly observed
that only along θ0, 120º, do the magnitudes of the four QPSK symbols overlap each other, and
their phases are 90° spaced, indicating that a standard QPSK constellation, i.e., a central
symmetric square in IQ space, is formed. The constellation patterns detected in all other
locations are scrambled.
3. FAR-FIELD PATTERN SEPARATION MULTI-BEAM DM SYNTHESIS
The approach described in the last section can be adapted for the synthesis of the multi-beam
DM transmitter arrays. The maximum number of secure communication beams which are
utilized for independent data transmissions to multiple intended receivers located along different
spatial directions in a 1-D DM array is (N−1).
3.1.
Synthesis procedures
The proposed synthesis procedures for the multi-beam DM systems are now presented below;
1) Select L sets of array excitations, Ql, that generate far-field radiation patterns with main beam
pointing to each of the pre-specified secure communication directions θl (l = 1, 2, …, L; 2 ≤ L
≤ N−1). For illustration purposes, an example of normalized far-field patterns with main
beams projected towards θ1 = 30º and θ2 = 120º is presented in Fig. 5 (a) for N = 5 and L = 2.
2) Use (1) to steer the null directions in the far-field patterns, generated by each excitation set
Ql, to θv (v = 1, 2, …, L; v ≠ l). In order to achieve this the vector C in (1) is replaced by an
(L−1)-by-N matrix whose (v, n)th entry is e
 jnkd cos v
. Next scale the magnitude of each
resultant excitation set to accommodate the corresponding main beam power to signal to
noise ratio (SNR) requirement along each preferred direction in the DM system. With this
manipulation the far-field patterns in Fig. 5 (a) are altered to those in Fig. 5 (b), which are the
information patterns associated with each selected communication direction θl. Here we
assume that the two information main beams have identical power. The excitation sets
associated with the information patterns are denoted as Pl.
3) Similar to the steps 2) and 3) for the single-beam DM synthesis in Section 2, generate
interference patterns which have pattern nulls along every θl. The excitations associated with
each interference pattern are denoted as Bi. Since there is more than one information pattern
we may project main beams of the interference patterns evenly over the spatial regions not
selected for preferred transmission. For example the three interference patterns that are used
to corrupt the information patterns in Fig. 5 (b) are shown in Fig. 6. Their main interference
beams are selected to have fixed magnitude of −10 dB, and are projected along 60º, 90º, and
150º respectively.
4) With the obtained excitations Pl and Bi, associated with information patterns and interference
patterns respectively, the array excitations, Sm, at the mth symbol time slot, in L parallel
independent data stream transmissions in a multi-beam DM system can be synthesized using
(3)
L
Sm   Dml  Pl    Rim  Bi 
l 1
(3)
i
Here the Dml is the transmitted symbol at the mth time slot along the intended communication
direction θl. It should be pointed out that we use the time slot concept for the multi-beam case
since the symbol rates of independent information transmissions along each selected direction
θl can be different. When Rim is updated randomly, with respect to time, a multi-beam DM
system is synthesized.
3.2.
Synthesis example
Using the example information patterns and interference patterns in Fig. 5 (b) and Fig. 6, the
excitation set for simultaneously transmitting BPSK data along 30º and Gray-coded QPSK data
along 120º are obtained by (3) and are listed in Table 2. The corresponding far-field patterns are
depicted in Fig. 7. It is observed in Fig. 7 that two independent data streams along two different
spatial directions, i.e., 30º and 150º in this example, are formed with no scrambling interference
superimposed, whereas the detected signals along other directions are scrambled randomly. It is
noted that phases are wrapped along direction θ1, 30º, when the BPSK symbol ‘0’ is sent.
4. BER SIMULATION RESULTS
For consistency with the single-beam and dual-beam examples in Sections 2 and 3, we choose
their corresponding information patterns and interference patterns to result in dynamic singlebeam and dual-beam DM systems. Dynamic DM systems, [14, 20], are synthesized by randomly
updating Rim in (2) and (3) for each m. Here the absolute value of Rim is kept constant during each
dynamic DM transmitter synthesis, but we can choose different values. For illustration purposes
we choose 0.5, 1, and 2 for the single-beam DM case, and 1, 2.5 for the dual-beam DM case. The
|Rim| of 0.5 or 2, for single-beam DM, is equivalent to decreasing or increasing the interference
patterns in Fig. 2 (b) by 3 dB. Similarly the |Rim| of 2.5, for dual-beam DM, is equivalent to
increasing the interference patterns in Fig. 6 by 4 dB. The phase of Rim is updated randomly
ranging from 0º to 360º.
In order to assess the secrecy performance of the synthesized dynamic DM systems, both
single-beam and dual-beam, BER simulations are performed through a transmission of a data
stream with 106 random symbols. All simulation results are obtained using MATLAB 2013a
[23]. For the single-beam case Gray-coded QPSK data is used, while for the dual-beam case it is
assumed that two independent random data streams respectively modulated by BPSK and Graycoded QPSK are independently transmitted simultaneously along 30º and 120º. For simplicity
their symbol rates and signal strength along their preferred communication directions are chosen
to be identical. The details of the BER calculation can be found in [20].
In Fig. 8 and Fig. 9 BER spatial distributions for the synthesized dynamic single-beam and
dual-beam DM systems, respectively, are depicted for SNRs, along the desired secure
communication directions, of 12 dB and 22 dB. We choose SNR of 12 dB since a BER of around
3.43×10−5 for QPSK along 120º is predicted [24]. Thus the pattern orthogonality between
information pattern and interference patterns can be validated, i.e., BERs along 120º in the DM
and conventional systems are largely identical, see Fig. 8 (a) and Fig. 9 (c). Simulation at SNR of
22 dB allows the maximum BER side lobes for QPSK transmissions in a conventional system in
Fig. 8 (b) and Fig. 9 (d) to reach below 10−3, making BER side lobe comparison noticeable on a
logarithmic scale. In both cases the conventional system refers to transmitters that radiate only
through the information patterns with no interference patterns present, i.e., ‘|Rim| = 0’ for singlebeam DM and ‘only one information beam and |Rim| = 0’ for dual-beam DM. It should be noted
that for the dual-beam DM case, signals, Dm1, radiated through one information pattern act as
interference to signals, Dm2, projected through the other information pattern. However, this
‘interference’ on its own is not sufficient to submerge information leaked through side lobes in
the other information pattern, see dotted curves in Fig. 9 (b) and (d) respectively. It can be
concluded that DM functionality enabled by the energy radiated through the interference patterns
can reduce the decodable spatial region and suppress BER side lobes effectively, especially
under high SNR scenarios. The more interference energy injected, the more enhanced secrecy
performance the DM systems can achieve.
5. CONCLUSION
In this paper it has been shown how DM transmitters either single-beam or multi-beam can be
synthesized by summing information patterns and interference patterns that are orthogonal along
the selected communication directions. This approach allows DM transmitters to lower the
possibility of information recovery by eavesdroppers located away from the desired
communication direction through enhanced scrambling in the most vulnerable directions, i.e.,
along the information side lobes. The extensions to current state of the art DM technology
elaborated in this paper should permit more versatile and secure radio systems to be created
which have enhanced security properties applied at the physical layer.
6. ACKNOWLEDGMENT
This work was sponsored by the Queen’s University of Belfast High Frequency Research
Scholarship.
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Q
Eavesdropper
Q
Legitimate
Receiver
I
I
Constellation Point in IQ Space
λ: Wavelength
Array Element
λ/2
λ/2
λ/2
θ0
Information Pattern
……………
ele. 1
Figure 1.
ele. 2
Phase
ele. 3
Centre
Interference Patterns
ele. N-1
ele. N
N-by-1 uniformly half wavelength spaced single-beam DM transmitter array. The
information pattern and interference patterns are illustrated. The interference distorts signal
constellations along all spatial directions other than the prescribed direction θ0.
3
0
Magnitude (dB)
−10
−20
−30
−40
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
Information Pattern
Far-field Patterns Generated by Ai, with Main
Beams Pointing to the Each Sidelobe Direction
(a)
3
0
Magnitude (dB)
−10
−20
−30
−40
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
Information Pattern
Interference Patterns Generated by Bi, with Main
Beams Pointing to the Each Information Sidelobe
(b)
Figure 2.
(a). The information pattern obtained in the step 1) and four far-field patterns
generated by Ai obtained in the step 2); (b). The information pattern obtained in the step 1) and
four interference patterns generated by Bi obtained in the step 3).
Generate information far-field pattern
with main beam pointing to  .
Excitations are denoted as P.
Generate far-field patterns with main beams
pointing to the information pattern side lobes.
Steer the far-field pattern nulls along  .
The resultant patterns are denoted as
interference patterns with excitations of Bi.
Synthesize DM excitations by combining
information and interference weighted P and Bi.
Figure 3.
Flow chart of the synthesis procedures for the far-field pattern separation single-
beam DM synthesis approach.
3
Magnitude (dB)
0
−10
−20
−30
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
For Symbol ‘11’
For Symbol ‘01’
For Symbol ‘10’
For Symbol ‘00’
(a)
180º
135º
90º
Phase
45º
0º
−45º
−90º
−135º
−180º
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
For Symbol ‘11’
For Symbol ‘01’
For Symbol ‘10’
For Symbol ‘00’
(b)
Figure 4.
Far-field (a) magnitude and (b) phase patterns for the example single-beam DM
array. Array excitations for each symbol are listed in the Table 1.
3
0
Magnitude (dB)
−10
−20
−30
−40
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
Pattern with Main Beam Pointing to θ1
Pattern with Main Beam Pointing to θ2
(a)
3
0
Magnitude (dB)
−10
−20
−30
−40
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
Information Pattern with Main Beam Pointing to θ1
Information Pattern with Main Beam Pointing to θ2
(b)
Figure 5.
(a). An example of the initial far-field patterns generated by Ql; (b). The
orthogonal information patterns generated by Pl with main beams pointing to each θl, i.e., θ1 =
30º and θ2 = 120º.
0
Magnitude (dB)
−10
−20
−30
−40
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
Figure 6.
Interference patterns for the example dual-beam DM system with secure
communication directions of 30º and 120º. The main beams of the interference patterns are
projected into 60º, 90º, and 150º respectively.
3
Magnitude (dB)
0
−10
−20
−30
0º
30º
60º
90º
120º
150º
180º
150º
180º
Spatial Direction θ
3
Magnitude (dB)
0
−10
−20
−30
0º
30º
60º
90º
120º
Spatial Direction θ
(a)
180º
135º
90º
Phase
45º
0º
−45º
−90º
−135º
−180º
0º
30º
60º
90º
120º
150º
180º
150º
180º
Spatial Direction θ
180º
135º
90º
Phase
45º
0º
−45º
−90º
−135º
−180º
0º
30º
60º
90º
120º
Spatial Direction θ
(b)
Figure 7.
BPSK ‘0’, QPSK ‘11’
BPSK ‘0’, QPSK ‘01’
BPSK ‘0’, QPSK ‘00’
BPSK ‘0’, QPSK ‘10’
BPSK ‘1’, QPSK ‘11’
BPSK ‘1’, QPSK ‘01’
BPSK ‘1’, QPSK ‘00’
BPSK ‘1’, QPSK ‘10’
The far-field (a) magnitude and (b) phase patterns of the example synthesized
dual-beam DM array for simultaneously BPSK and Gray-coded QPSK data transmissions. Array
excitations for each symbol combination are listed in the Table 2.
100
10−1
SNR = 12 dB
BER
10−2
|Rim | = 2
−3
|Rim | = 1
10
|Rim | = 0.5
−4
|Rim | = 0
10
3.43×10−5
10−5
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
(a)
100
SNR = 22 dB
10−1
BER
10−2
10−3
|Rim | = 2
|Rim | = 1
10−4
10−5
|Rim | = 0.5
|Rim | = 0
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
(b)
Figure 8.
Simulated BER spatial distributions for example dynamic single-beam DM
systems and conventional QPSK system (|Rim| = 0) under SNRs of (a) 12 dB and (b) 22 dB.
Gray-coded QPSK data streams with 106 random symbols are used for simulation.
100
10−1
BPSK receiver detection
SNR = 12 dB
−2
10
BER
Two information beams, |Rim | = 2.5
10−3
Two information beams, |Rim | = 1
10−4
Two information beams, |Rim | = 0
10−5
Only one BPSK information beam, |Rim | = 0
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
(a)
100
BPSK receiver
detection
10−1
BER
0º
30º
60º
90º
120º
Spatial Direction θ
(b)
150º
180º
Two information beams, |Rim | = 2.5
10−5
Two information beams, |Rim | = 1
10−4
Two information beams, |Rim | = 0
10−3
Only one BPSK information beam, |Rim | = 0
SNR = 22 dB
10−2
100
SNR = 12 dB
10−1
QPSK receiver detection
10−2
BER
Two information beams, |Rim | = 2.5
10−3
Two information beams, |Rim | = 1
10−4
Two information beams, |Rim | = 0
3.43×10−5
Only one QPSK information beam, |Rim | = 0
10−5
0º
30º
60º
90º
120º
150º
180º
Spatial Direction θ
(c)
100
10−1
BER
10
0º
30º
60º
90º
120º
Spatial Direction θ
150º
180º
Two information beams, |Rim | = 2.5
QPSK receiver detection
−5
Two information beams, |Rim | = 1
SNR = 22 dB
10−4
Two information beams, |Rim | = 0
10−3
Only one QPSK information beam, |Rim | = 0
10−2
(d)
Figure 9.
Simulated BER spatial distributions for the dynamic dual-beam DM and
conventional systems under SNRs of (a), (c) 12 dB and (b), (d) 22 dB. BPSK and Gray-coded
QPSK data streams with 106 random symbols along 30º and 120º respectively are used for
simulation.
Table 1.
Synthesized single-beam DM array excitations Sm a for four unique QPSK symbol
transmissions shown in Fig. 4.
m=1
Symbol ‘11’
m=2
Symbol ‘01’
m=3
Symbol ‘00’
m=4
Symbol ‘10’
Sm1 (×10−1)
−1.912
−j1.198
2.629
−j1.830
+0.610
+j1.554
−0.193
+j2.023
Sm2 (×10−1)
−1.750
+j2.013
−1.090
−j0.469
+1.467
−j0.707
−0.312
+j3.129
Sm3 (×10−1)
+0.560
+j1.361
+0.082
+j1.614
−3.131
−j1.920
+1.898
−j2.190
Sm4 (×10−1)
Sm5 (×10−1)
+1.871
−2.355
−j0.231
−j0.891
+0.767
+2.778
+j1.277
−j1.770
−0.380
−0.163
+j2.786
+j1.751
−1.942
−1.633
−j0.218
+j1.228
a. Sm = [Sm1 Sm2 Sm3 Sm4 Sm5]T
Table 2.
Synthesized dual-beam DM array excitations Sm
a
for simultaneously BPSK and
Gray-coded QPSK transmissions along directions of 30º and 120º respectively, with far-field
patterns shown in Fig. 7.
m=1,
BPSK ‘0’, QPSK ‘11’
m=2,
BPSK ‘0’, QPSK ‘01’
m=3,
BPSK ‘0’, QPSK ‘00’
m=4,
BPSK ‘0’, QPSK ‘10’
m=5,
BPSK ‘1’, QPSK ‘11’
m=6,
BPSK ‘1’, QPSK ‘01’
m=7,
BPSK ‘1’, QPSK ‘00’
m=8,
BPSK ‘1’, QPSK ‘10’
Sm1 (×10−1)
−2.749
−j0.693
−0.414
−j3.524
+0.392
−j1.200
−1.301
−j1.296
−0.903
+j0.493
+1.135
−j0.628
+2.877
+j1.524
+1.438
+j2.283
Sm2 (×10−1)
−0.888
+j0.996
+2.631
−j1.898
+2.315
−j1.787
+2.456
+j2.408
−1.697
+j0.026
−2.225
−j0.854
−0.917
−j1.128
−1.193
+j1.324
Sm3 (×10−1) Sm4 (×10−1) Sm5 (×10−1)
−1.236
+3.931
−3.352
+j2.235
−j1.210
+j0.676
−4.294
+3.249
+0.729
+j1.489
+j0.564
−j1.439
−4.466
+0.440
−0.837
−j2.274
+j1.263
+j4.122
−1.465
−1.402
−2.507
−j1.643
−j2.320
+j2.867
+4.958
−0.745
−0.165
+j1.806
−j1.019
−j4.806
+0.639
−1.458
+3.180
+j2.718
+j2.541
−j2.957
+0.727
−3.104
+3.237
−j2.964
+j0.556
+j0.396
+4.223
−3.976
−0.331
−j2.320
−j2.630
−j0.315
a. Sm = [Sm1 Sm2 Sm3 Sm4 Sm5]T