Design of Circular Signal Constellations in the Presence of Phase Noise

Yang Li, Shuzheng Xu, Huazhong Yang Department of Electronic Engineering

Tsinghua University Beijing, China

li-yang05@mails.tsinghua.edu.cn

Abstract—In order to save the limited bandwidth resource, high-order M-ary Amplitude and Phase Shift Keying (APSK) signal constellation as Quadrature Amplitude Modulation (QAM) is widely employed in modern bandwidth-limited communication and broadcasting systems. However, such dense constellation is very sensitive to the disturbance of noise, including not only white Gaussian noise but also phase noise which is caused by many nonideal factors in transmitter, receiver and channel. This paper proposed a design method of circular signal constellations which are extremely robust in the presence of phase noise. As for 64 points, the circular constellation we proposed performs at least 5dB better than the square QAM at the error rate of 310− under the phase noise of 0.1 rad. And compared with the existing patent constellation designed to mitigate phase noise’s effect, the proposed circular constellation not only performs 1dB better in phase noise of 0.1 rad, but also sacrifices no error performance under white Gaussian noise only. And since the circular constellation is with small amplitude fluctuation, it can be further used to mitigate the requirement of power amplifier’s linear range in certain systems. Further, a reliable and easy-to-implement demodulation method is also presented.

Keywords- APSK; circular constellation; phase noise

I. INTRODUCTION Bandwidth is determined by the symbol rate rather than bit

rate [1]. As a result, in the band limited communication system, M-ary APSK modulation where multiple bits are represented by one of M symbols is widely used to get a high data rate. However, this high order modulation scheme suffers deterioration in the system error performance, since the signal constellation becomes denser than the low order ones under the average power constraints in the signal plane. In order to mitigate this negative effect, many papers [2]-[8] worked on the design or optimization of signal constellations in the presence of white Gaussian noise.

However, in addition to the Gaussian noise, Doppler Effect, multi-path fading and nonlinear characteristic in the front end would also introduce phase noise in the demodulated signal constellation points, which can rapidly deteriorate the error performance. This problem is more severe in OFDM systems since more factors such as sampling frequency offset [9] and symbol timing error would further cause this phase noise. Simulation indicates even the classical optimum constellations under white Gaussian noise proposed by [2] are extremely sensitive to phase noise. So in order to guarantee the system error performance, sampling synchronization algorithm, frequency offset evaluation and correction algorithms and other equalization algorithms with adequate

precision are necessary, which makes the receiver more complex and hard to implement. Designing new signal constellations with much more robust characteristics in the presence of phase noise together with Gaussian noise seems to be one way to mitigate the requirement of those algorithms' precision as well as improving the system error performance, but unfortunately not much focus has been given to this topic. To the best of our knowledge, limited works as [10]-[13] ever paid attention to this topic. Constellations proposed by [10]-[12] did provide improved performance in phase noise, but decoders of such modulation scheme are too complex to implement since the constellation points scatter arbitrarily. Patent [13] introduced a specific 64-points constellation which performs relatively well in phase noise and easy to demodulate. But since no design method was mentioned, it cannot be extend to the constellation with other number of points. What is more, such constellation sacrifices its performance in white Gaussian noise to obtain improvement in phase noise, which is not an ideal characteristic of the constellation we desire.

Nowadays, as a novel modulation scheme, circular signal constellations are widely used in the satellite broadcasting systems [14]-[16]. But in the previous work, people’s interest is mainly concentrated on circular constellation’s advantage of smaller amplitude fluctuation compared with square QAM. As a result, the number of circles is implicitly limited and other properties of circular constellations are also overlooked. In this paper, we proposed a circular constellation construction method which is easy to handle and extend to any circle numbers we are interested in. Then based on the symbol error rate expression in the presence of phase noise we deduced, a series of circular constellations are picked out according to a proposed weighted function which we called criterion function. Simulations indicate that these constellations could notably outperform square QAM and the existing patent constellations in the presence of phase noise. What is more, we could use the proposed criterion function to flexibly adjust the characteristics of the constellation according to the specific need. And according to the demodulation method we introduced, this modulation scheme is very suitable for the practical use.

This paper is arranged as follows. In section II, the expression of symbol error rate in the presence of phase noise is derived. In section III, the circular constellation construction method is proposed. Then a series of circular constellations are picked out in section IV, followed by a set of comparisons with QAM，the classical optimum [2] and the existing patent constellation [13]. In section V, a reliable and easy-to-implement demodulation method is presented. Section VI concludes this paper.

Supported by the NSFC under Grant 90707002.

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II. EXPRESSION OF SYMBOL ERROR RATE

A. Expression derivation

Figure 1. signal constellation plane.

As depicted in Figure 1, the scenario in which transmitted signal TS is wrongly recognized as signal RS in the receiver

is as follows: TS is sent; and with the effect of the phase

noise, TS is rotated θ to turn into 'TS ; then with the effect of

white Gaussian noise, 'TS turns into z which locates in the

detection area of RS denoted as iR . In Figure 1, coordinates X and Y represent the original signal plane, while axis Xr which parallels vector R TS S− and axis Yr are introduced in order to deduce the expression. Assuming the average power of Gaussian noise is 2 0N and denoting this conditional error

probability as ( | )R TP S S , we have

2 '

0( )

2

0

( | ) ( | )2

T

i

z S

N

R T i Tz R

eP S S P z R S dzNπ

−−

∈

= ∈ = ∫∫ (1)

According to the minimum Euclidean distance (MED) detection criterion, iR should be a polygon circumfused by

vertical bisectors of lines connecting RS and its contiguous signal points. But considering the distribution characteristic of white Gaussian noise, z in Figure 1 would mostly locate near '

TS . Hence, in order to make integral (1) realizable, we

enlarge iR to the half plane 'iR as the shaded area, which is

also adopted in reference [2]. Theoretically this expression

gained using the enlarged integral area is an upper bound of the actual symbol error rate, but simulation demonstrates that this bound is very tight, which means the actual error rate curve is very close to the bound curve. Denote all the signal points in Figure 1 in XY-plane and XrYr-plane respectively as follows:

' ' ' ' '

( ) ( )

R R R rR rR

T T T rT rT

T T T rT rTj j

T T rT rT

S x jy x jyS x jy x jy

S x jy x jy

x jy e x jy eθ θ

+ ++ +

+ +

+ +

(2)

Then we derived ( | )R TP S S as follows

2'

0

'

' 2 ' 2

0

T

' 2 ' 2

0 0

T

'

( )2

0

( ) ( )( )2

02

( ) ( )( ) ( )

2 2

02

enlarge

( | )

( | ) ( | )

2

12

12

1(

T

i

r rT r rT

r rR rr

r rT r rT

r rR rr

R T

i T i T

z S

N

z R

x x y yN

r rx x yx

x x y yN N

r rx x yx

P S S

P z R S P z R S

e dzN

e dy dxN

e dx e dyN

QN

π

π

π

−−

∈

− + −+∞ +∞ −

+ =−∞=

− −+∞ +∞− −

+ =−∞=

= ∈ ∈

=

=

=

=

≈

∫∫

∫ ∫

∫ ∫

'T

0

( ))2

r rRrT

x x x+ −

(3)

Where Q function is the Gaussian tail function with a standard approximation as (4)

2

21( ) , 42

x

Q x e xx π

−= > (4)

Conducting some vector operation in Figure 1, we obtain

'T

2 2( ) [ (1 2 )]

2

r rR T RrT

jR T R T

R T

x x S Sx P

S S S S eS S

θ

+ +− = −

− + − =−

i (5)

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Where “ i ” represents dot product and “ ” represents the amplitude of a complex number.

Combining (3) (4) (5), we get the expression of this conditional error probability as

2

2

0

1( | )2

( ) [ (1 2 )]2

A

R T

jR T R T

R T

P S S eA

S S S S eAN S S

θ

π−⎧

≈⎪⎪⎪⎨⎪ − + −⎪ =⎪ −⎩

i (6)

Assuming all the signals in the set share the equal

probability of transmitting, the average symbol error rate is determined as

( ) ( | )

1 ( | )

T R TT R T

R TT R T

Pe P S P S S

P S SN

≠

≠

=

=

∑ ∑

∑∑ (7)

B. Simulation of the expression In order to verify the validity of (6) (7) as the expression of

symbol error rate in the presence of phase noise, we conducted the simulation under three signal constellations which include 16QAM, “optimum” [2], (4, 12)-circle [4] as depicted in Figure 2.

-4 -2 0 2 4-4

-2

0

2

416QAM

-2 0 2-2

-1

0

1

2"optimum"[2]

-2 0 2-3

-2

-1

0

1

2

3(4,12)circle

Figure 2. Three 16points constellations to simulate.

We set the phase noise parameter θ = 2 / 60π ; then we demodulate the signal using the MED detection criterion and make a comparison between this actual results and the theoretical upper bound (6) (7). From simulation results depicted in Figure 3, we notice that the expression (6) (7) is a good approximation of the actual symbol error rate, which means this expression could be reliably employed as the basis of our criterion function proposed in the following section, which is used in the procedure of picking out constellations in the presence of phase noise together with white Gaussian noise.

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Sym

bo

l Err

or

Rat

e

c2c1

a1a2

b1b2

Figure 3. Comparison between (6) (7) and actual symbol error rate.

a1, b1 and c1 (dot line) represent this upper bound of symbol error rate of 16QAM, “optimum”, (4, 12)-circle respectively. a2, b2 and c2 (marked line) represent their actual symbol error rates using MED creterion .

III. CIRCULAR CONSTELLATION CONSTRUCTION METHOD The core principle of constellation optimization in the

presence of white Gaussian noise is maximizing the MED between the signal points under average power constraint. Since we aim at picking out signal constellations which are robust in the presence of phase noise among those which perform well in the white Gaussian noise, the circular constellation construction method we adopt should follow the principle as well. In the previous work, [14] introduced constellations through a brute-force method; [15] mentioned a circular constellation without any construction description. [16] proposed a method to construct circular constellations based on the principle we mentioned, but this method is implicitly constrained in limited circle numbers since it is realized through solving a set of equations. In this paper, we change the principle into another equivalent version, namely, minimizing the average power of constellations under the MED constraint. Based on this new principle, a much easier construction method compared with [16] is proposed. Unlike the previous work where only 4-circles is studied, this paper investigated 4,5,6,7 circles respectively in the 64-points constellation design using the proposed method.

A. Equivalence explanation Denoting the former principle as principle A and the new

one that we proposed as principle B, we give the explanation of their equivalence as follows:

If we constrain the MED of all the constellations to a constant denoted d , then the average power denoted as P varies in different constellations. The optimum solution in principle B is the one with the minimum average power denoted as minP . If we normalize the amplitude of each constellation obtained guided by principle B with the square of its average power P ，then the average power would change into constant 1 and the relative MED would change into

/d P . As a result, the MED of the one with average power

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minP before normalization possess the maximum MED

min/d P afterwards, which is the optimum solution in principle A. Therefore, principle B is equivalent to principle A.

B. Circular constellation construction algorithm Based on principle B, we construct circular constellations

with any number of circles we are interest in as follows. Here we take L circles in 64 points constellations for example.

1) Set the number of points in each circle as 1N , 2N ,…, LN under the constraint of (8).

1 2 64LN N N+ + = (8)

2) Constraining the MED within each circle to constant 1, we could obtain the radius of each circle according to (9).

1 1

2 2

2 sin( / ) 12 sin( / ) 1 2 sin( / ) 1L L

R NR N

R N

ππ

π

=⎧⎪ =⎪⎨⎪⎪ =⎩

(9)

3) If 2 1 1R R− < , then update 2R as 2R +1. Treat the radius orderly from the most inner cirle to the most outter one in the same way as above to guarantee that the MED between contiguous circles also stays 1 at least.

Covering all the combinations of 1N , 2N ,…, LN , all the L-

circle 64-points circular constellations would be constructed. And the one with the minimum average power is the optimum circular constellation with L circles according to principle B, which is supposed to perform the best under white Gaussian noise among all the L-circle 64-points circular constellations.

In this paper, we made an investigation of 64 points circular constellations with L circles, where L equals 4,5,6,7 respectively. The optimum solutions according to principle B are summarized in TABLE I. The simulation of their error performance under white Gaussian noise only is depicted in Figure 4. And the simulation of their error performance in the presence of Gaussian noise together with phase noise is depicted in phase 5.

TABLE I. OPTIMUM CIRCULAR CONSTELLATIONS

L ( 1N , 2N ,…,

LN )

( 1R , 2R ,…, LR )

after normalization

Average Power before

normalization 4 (7,13,19,25) (0.3511, 0.6558, 0.9605,

1.2652) 10.7717

5 (1,6,13,19,25) (0, 0.3109, 0.6496, 0.9605, 1.2713)

10.3459

6 (1,6,12,15,15, 15)

(0, 0.2821, 0.5643, 0.8464, 1.1286, 1.4107)

12.5625

7 (1,6,11,11,11, 12,12)

(0, 0.2461, 0.4921, 0.7382, 0.9843, 1.2303, 1.4764)

16.5156

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Sym

bo

l Err

or

Rat

e

b1c1

d1e1

a1

Figure 4. Symbol error rate under white Gaussian noise only.

a1 represents 64QAM; b1,c1,d1,e1 represent L=4,5,6,7 in TABLE I.

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10-4

10-3

10-2

SNR /dB

Sym

bo

l Err

or

Rat

e

a2b2

d2

e2

c2

Figure 5. Symbol error rate in the presence of white Guassian noise together with phase noise (θ =0.1).

a2 represents 64QAM; b2,c2,d2,e2 represent L=4,5,6,7 in TABLE I.

From the simulation results depicted in Figure 4 and Figure 5, we notice that: 64QAM and the optimum circular constellations with L=4,5 circles perform well under white Gaussian noise only, however they deteriorate very fast when phase noise is taken into account; on the other hand, the optimum circular constellations with L=6,7 circles appear extremely robust in the presence of phase noise, but they are worse than QAM when there is only white Gaussian noise. Therefore, none of these optimum circular constellations seems to be the ideal constellation we desire. We solved this problem in the following section.

IV. CONSTELLATION SELECTION

A. Criterion Function The ideal constellations we want are those which could not

only perform as well as QAM when there is white Gaussian noise only, but also be notably robust in the presence of phase noise. Based on this aim, we made an investigation of selecting

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circular constellations among those which are constructed in section III as follows.

Denoting the average symbol error rate under certain SNR and phase noise θ as ( , )Pe SNRθ which can be calculated using (6) (7), we could logically infer that: if we want the constellations which perform well under white Gaussian noise only, select those with smaller ( 0, )Pe SNRθ = ; on the other hand, if we are interested in the constellations which are robust in the presence of phase noise *θ , select those with smaller

*( , )Pe SNRθ θ= . Therefore, if we want to guarantee both advantages, a weighted function is used as the constellation selecting criterion. This function which we call criterion function is defined as (10).

*(0, ) ( , )G PF W Pe SNR W Pe SNRθ= × + × (10)

GW and PW represent the weight of ( 0, )Pe SNRθ =

and *( , )Pe SNRθ θ= respectively, implying the relative importance of each error performance characteristic. For example, if we care more about the error performance when there is no phase noise, we could increase GW in selecting the

constellations. Otherwise, we could decrease GW to sacrifice to some degree the performance under white Gaussian noise for further improvement of the performance in the presence of phase noise.

As a heuristic method, the parameters used in calculating Pe in (10) were chosen more empirically rather than theoretically. Interested readers could easily extend this method to other specific application situations. Here, we set phase noise parameter *θ to 0.1 which appears to be a relative severe case for QAM indicated in Figure 5. And the SNR parameter is determined much more freely. Since simulation indicates that the error rate curves of constellations with the same points rarely intersect unless they are very close, we could take a point to represent the whole curve when investigating the curves’ relative positions. In this paper, we set SNR=26dB. Accordingly, GW is set to 500 and PW is set to 1.

Searching all the 5-circle constellations constructed in section III, the one with the minimum F value was picked out and depicted in Figure 6. Its structure information is summarized in TABLE II. Then we compared the performance of this proposed circular constellation with QAM, the classical optimum [2], and the existing patent constellation [13] respectively.

TABLE II. CONSTELLATIONS STRUCTURE INFORMATION

L ( 1N , 2N ,…, LN ) ( 1R , 2R ,…, LR )

after normalization

5 (4,10,15,17,18) (0.2034,0.491,0.7787,1.0663,1.354)

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

Figure 6. the 5-cirlcle constellation with minimum F value.

B. Compare with QAM In order to investigate the performance of selected

constellation thoroughly, we compared QAM and the selected (4,10,15,17,18) circular constellation under no phase noise and severe phase noise( θ =0.1 rad) respectively. The simulation results are depicted in Figure 7.

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sym

bo

l Err

or

Rat

e

a1b1

a2

b2

Figure 7. Symbol error rate of QAM and (4,10,15,17,18).

a1, b1 represent 64QAM and (4,10,15,17,18) respectively under whiteGaussain noise only; a2, b2 represent their symbol error rates in the presence of Gaussian noise together with phase noise (θ =0.1).

From figure 7 we notice that such selected constellation not only performs as well as QAM when there is only white Gaussian noise without any loss, but also significantly outperform QAM when phase noise is taken into consideration (more than 5dB at the error rate of 310− ). In other words, (4,10,15,17,18) circular constellation can sufficiently satisfy our requirement of ideal constellations.

C. Compare with the classical optimum Employing the optimization method in designing the

optimum constellations under white Gaussian noise proposed in [2], we obtain a 64 points local optimum constellation after

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we set the initial state to 64-QAM, the iteration step length to 0.0005, and the iteration time to 20000. The comparison between such classical optimum constellation and the selected (4,10,15,17,18) circular constellation is depicted in Figure 8.

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SNR /dB

Sym

bo

l Err

or

Rat

e

b1

a1

a2b2

Figure 8. Symbol error rate of the classical optimum and (4,10,15,17,18).

a1, b1 represent the classical optimum and (4,10,15,17,18) respectively under white Gaussain noise only; a2, b2 represent their symbol error rates in the presence of Gaussian noise together with phase noise (θ =0.1).

From figure 8 we notice that even the classical optimum constellation under white Gaussian noise is very sensitive to phase noise. Though it performs a little better (about 0.5dB) than (4,10,15,17,18) when there is no phase noise, it cannot compare with (4,10,15,17,18) when the phase noise amounts to 0.1 rad.

D. Compare with the existing patent. The patent constellation [13] proposed to mitigate the phase

noise’s effect is depicted in Figure 9. The comparison between this constellation with (4,10,15,17,18) is depicted in Figure 10. The performance of QAM is also depicted to make the comparison more clearly.

-5 0 5-8

-6

-4

-2

0

2

4

6

8

Figure 9. the patent constelltion [13].

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Sym

bo

l Err

or

Rat

e

b1c1

a1

a2

c2

b2

Figure 10. Symbol error rate of the QAM, (4,10,15,17,18) and patent[13]. a1, b1,c1 represent QAM, (4,10,15,17,18) and patent[13] respectively under white Gaussain noise only; a2, b2,c2 represent their symbol error rates in the presence of Gaussian noise together with phase noise (θ =0.1).

From Figure10 we notice that the proposed (4,10,15,17,18) constellation performs 1dB better than the patent constellation under phase noise of θ =0.1. What is more, (4,10,15,17,18) does not sacrifice any performance under white Gaussian noise compared with QAM, while the patent constellation does. And with its demodulation method proposed in [13], the actual symbol error rate of the patent constellation would still increase somewhat compared with its theoretical error rate to make its performance even worse.

On the other hand, if under certain circumstances where the robustness in phase noise is a much more important characteristic for us even at the cost of some sacrifice of performance under white Gaussian noise, we still have another better choice among the circular constellations constructed in section III. What we should do is just to modify the parameters in the criterion function used in the constellation selection. To the extreme, we set GW to 0 to obtain a new optimum 5-circle constellation summarized in TABLE III. The performance simulation is depicted in Figure 11, From which we notice that the new-selected circular (7,11,14,16,16) constellation further improves the performance in phase noise ( 2dB better than the patent [13]). The tradeoff is 0.6dB performance loss under white Gaussian noise compared with QAM.

In other words, we could use the proposed criterion function to flexibly adjust the characteristics of the constellation according to the specific need.

TABLE III. CONSTELLATIONS STRUCTURE INFORMATION

L ( 1N , 2N ,…, LN ) ( 1R , 2R ,…, LR )

after normalization

5 (7,11,14,16,16) (0.3073, 0.5740, 0.8406, 1.1073, 1.3740)

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10-3

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SNR /dB

Sym

bo

l Err

or

Rat

e

a1

a2

b1

c2b2

c1

Figure 11. Symbol error rate of the QAM, (7,11,14,16,16) and patent[13]. a1, b1,c1 represent QAM, (7,11,14,16,16) and patent[13] respectively under white Gaussain noise only; a2, b2,c2 represent their symbol error rates in the presence of Gaussian noise together with phase noise (θ =0.1).

V. DEMODULATION OF CIRCULAR CONSTELLATION Like the square QAM which could be easily demodulated

because the inphase and quadrature component can be set apart, the circular constellation demodulation is easy to realize because the amplitude and phase of signal points act independently in the demodulation procedure. We conduct the demodulation as follows.

1) Determine which circle the received signal point belongs to according to its amplitude.

2) Then determine which constellation point in this circle it should be according to its phase

Therefore, the detection area of signal RS becomes a sector ring depicted as the shaded area in Figure 12, which was also mentioned in [10]. Here, r1 takes the arithmetic average of R1 and R2; r2 takes the arithmetic average of R2 and R3; 1je θ and

2je θ bisect the arcs connecting RS and its contiguous constellation points in circle R2. The error performance simulation of (4,10,15,17,18) obtained in section IV using this demodulation scheme is depicted in Figure 13.

Figure 12. circular constellation demodulation scheme.

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Sym

bo

l Err

or

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e

a2a1

b1b2

Figure 13. acutal symbol error rate of (4,10,15,17,18).

a1, a2 (marked line) represent the theoretical and actual error rate of (4,10,15,17,18) under white Gaussain noise only; b1, b2 (marked line) represent its theoretical and actual error rate in the presence of Gaussian noise together with phase noise (θ =0.1).

From Figure 13 we notice that such demodulation scheme is reliable since the actual error curve is very close to the theoretical one. Because directly calculating the phase of received signal might take extra effort, we suggest using the cosine or sine of phase instead in the second step of demodulation procedure. Since mainly some comparators are needed, the complexity of the demodulation method is low, which means the circular constellation is also an attractive modulation scheme of practical use.

VI. CONCLUSION Focusing on the phase noise which could seriously deteriorate the error performance of high-order APSK constellations, this paper made an investigation in the circular constellations which are nowadays widely used in the satellite broadcasting systems. A simple circular constellation construction method was proposed, based on which 4,5,6,7-circles 64-points constellations are constructed. Then with the criterion function one specific 5-circle circular constellation was picked out. This proposed constellation not only performs as well as QAM when there is only white Gaussian noise but also significantly outperforms QAM when phase noise is also taken into consideration (more than 5dB at the error rate of 310− ), which are both better than the existing patent constellation [13] designed to mitigate phase noise’s effect. Further, we could use the proposed criterion function to flexibly adjust the characteristics of the constellation according to the specific need. What is more, a reliable and easy-to-complement demodulation method of circular constellation was also introduced, which means the circular constellation is also an attractive modulation scheme of practical use.

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[8] Israel Morejon., Jwalant Dholabkia., Yueping Zeng., “Symmetric spherical QAM constellation.” U.S. Patent, No. 6996189 (Feb.7, 2006).

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