Directional Modulation Far-field Pattern Separation Synthesis Approach

Yuan Ding and Vincent F. Fusco The ECIT Institute Queens University of Belfast

Belfast, BT3 9DT, UKv.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 beam-

forming 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 far-

field 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),

(1)

where IN denotes the N-by-N identity matrix, and C is a 1-by-N vector with the nth element of

. ‘[∙]−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),

(2)

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 . 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)

(3)

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 single-

beam 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 Gray-

coded 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 single-

beam 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.

7. REFERENCES

[1] Babakhani, A., Rutledge, D.B., Hajimiri, A.: ‘Transmitter architectures based on near-field

direct antenna modulation’, IEEE J. Solid-State Circuits, 2008, 43, pp. 2674–2692

[2] Babakhani, A., Rutledge, D.B., Hajimiri, A.: ‘Near-field direct antenna modulation’, IEEE

Microw. Mag., 2009, 10, pp. 36–46

[3] Daly, M.P., Bernhard, J.T.: ‘Directional modulation technique for phased arrays’, IEEE

Trans. Antennas Propag., 2009, 57, pp. 2633–2640

[4] Daly, M.P., Daly, E.L., Bernhard, J.T.: ‘Demonstration of directional modulation using a

phased array’, IEEE Trans. Antennas Propag., 2010, 58, pp. 1545–1550

[5] Shi, H., Tennant, A.: ‘An experimental two element array configured for directional antenna

modulation’. Proc. 2012 Sixth European Conf. Antennas and Propagation (EUCAP), 2012,

pp. 1624–1626

[6] Daly, M.P., Bernhard, J.T.: ‘Beam steering in pattern reconfigurable arrays using directional

modulation’, IEEE Trans. Antennas Propag., 2010, 58, pp. 2259–2265

[7] Shi, H., Tennant, A.: ‘Enhancing the security of communication via directly modulated

antenna arrays’, IET Microw., Antennas Propag., 2013, 7, pp. 606–611

[8] Hong, T., Song, M., Liu, Y.: ‘Dual-beam directional modulation technique for physical-layer

secure communication’, IEEE Antennas and Wireless Propagation Letters, 2011, 10, pp.

1417–1420

[9] Valliappan, N., Lozano, A., Heath, R.W.: ‘Antenna subset modulation for secure millimeter-

wave wireless communication’, IEEE Trans. Commun., 2013, 61, pp. 3231–3245

[10] Zhu, Q., Yang, S., Yao, R., Nie, Z.: ‘A directional modulation technique for secure

communication based on 4D antenna arrays’, Proc. 2013 Seventh European Conf.

Antennas and Propagation (EUCAP), 2013, pp. 125–127

[11] Zhu, Q., Yang, S., Yao, R., Nie, Z.: ‘Directional modulation based on 4-D antenna arrays’,

IEEE Trans. Antennas Propagat., 2014, 62, pp. 621–628

[12] Zhang, Y., Ding, Y., Fusco, V.: ‘Sidelobe modulation scambling transmitter using Fourier

Rotman lens’, IEEE Trans. Antennas Propagat., 2013, 61, pp. 3900–3904

[13] Ding, Y., Fusco, V.: ‘Sidelobe manipulation using Butler matrix for 60 GHz physical layer

secure wireless communication’. Antennas and Propagation Conference (LAPC),

Loughborough, UK, Nov. 11–12 2013, pp. 61–65

[14] Ding, Y., Fusco, V.: ‘A vector approach for the analysis and synthesis of directional

modulation transmitters’, IEEE Trans. Antennas Propagat., 2014, 62, pp. 361–370

[15] Ding, Y., Fusco, V.: ‘Vector representation of directional modulation transmitters’. Proc.

2014 8th European Conf. Antennas and Propagation (EUCAP), Hague, Netherlands, 2014,

pp. 332–336

[16] Ding, Y., Fusco, V.: ‘BER driven synthesis for directional modulation secured wireless

communication’, International Journal of Microwave and Wireless Technologies, 2014, 6,

(02), pp. 139–149

[17] Ding, Y., Fusco, V.: ‘Directional modulation transmitter synthesis using particle swarm

optimization’. Antennas and Propagation Conference (LAPC), Loughborough, UK, 2013,

pp. 500–503

[18] Ding, Y., Fusco, V.: ‘Directional modulation transmitter radiation pattern considerations’,

IET Microw., Antennas Propag., 2013, 7, (15), pp. 1201–1206

[19] Ding, Y., Fusco, V.: ‘Constraining directional modulation transmitter radiation patterns’,

IET Microw., Antennas Propag., accepted for publication.

[20] Ding, Y., Fusco, V.: ‘Establishing metrics for assessing the performance of directional

modulation systems’, IEEE Trans. Antennas Propagat., 2014, 62, (5), pp. 2745–2755

[21] Goel, S., Negi, R.: ‘Guaranteeing secrecy using artificial noise’, IEEE Trans. Wireless

Commun., 2008, 7, pp. 2180–2189

[22] Tung Sang Ng: ‘Generalized Array Pattern Synthesis Using the Projection Matrix’, IEE

Proceedings Microw., Antennas Propag., 1985, 132, (1), pp. 44–46

[23] MATLAB Release: The MathWorks, Inc., Natick, MA, USA, 2013

[24] Shafik, R.A., Rahman, S., Razibul Islam, A.H.M.: ‘On the extended relationships among

EVM, BER and SNR as performance metrics’. Int. Conf. Electrical and Computer

Engineering, 2006 (ICECE’06), 2006, pp. 408–411

[25]

Figure 1. 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.

(a)

(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.

(a)

(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.

(a)

(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.

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.

(a)

(b)

Figure 7. 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.

(a)

(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.

(a)

(b)

(c)

(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.

Sm1 (×10−1) Sm2 (×10−1) Sm3 (×10−1) Sm4 (×10−1) Sm5 (×10−1)m=1

Symbol ‘11’−1.912−j1.198

−1.750+j2.013

+0.560+j1.361

+1.871−j0.231

−2.355−j0.891

m=2Symbol ‘01’

2.629−j1.830

−1.090−j0.469

+0.082+j1.614

+0.767+j1.277

+2.778−j1.770

m=3Symbol ‘00’

+0.610+j1.554

+1.467−j0.707

−3.131−j1.920

−0.380+j2.786

−0.163+j1.751

m=4Symbol ‘10’

−0.193+j2.023

−0.312+j3.129

+1.898−j2.190

−1.942−j0.218

−1.633+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.

Sm1 (×10−1) Sm2 (×10−1) Sm3 (×10−1) Sm4 (×10−1) Sm5 (×10−1)m=1,

BPSK ‘0’, QPSK ‘11’−2.749−j0.693

−0.888+j0.996

−1.236+j2.235

+3.931−j1.210

−3.352+j0.676

m=2, BPSK ‘0’, QPSK ‘01’

−0.414−j3.524

+2.631−j1.898

−4.294+j1.489

+3.249+j0.564

+0.729−j1.439

m=3, BPSK ‘0’, QPSK ‘00’

+0.392−j1.200

+2.315−j1.787

−4.466−j2.274

+0.440+j1.263

−0.837+j4.122

m=4, BPSK ‘0’, QPSK ‘10’

−1.301−j1.296

+2.456+j2.408

−1.465−j1.643

−1.402−j2.320

−2.507+j2.867

m=5, BPSK ‘1’, QPSK ‘11’

−0.903+j0.493

−1.697+j0.026

+4.958+j1.806

−0.745−j1.019

−0.165−j4.806

m=6, BPSK ‘1’, QPSK ‘01’

+1.135−j0.628

−2.225−j0.854

+0.639+j2.718

−1.458+j2.541

+3.180−j2.957

m=7, BPSK ‘1’, QPSK ‘00’

+2.877+j1.524

−0.917−j1.128

+0.727−j2.964

−3.104+j0.556

+3.237+j0.396

m=8, BPSK ‘1’, QPSK ‘10’

+1.438+j2.283

−1.193+j1.324

+4.223−j2.320

−3.976−j2.630

−0.331−j0.315

a. Sm = [Sm1 Sm2 Sm3 Sm4 Sm5]T