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On the Performance of Spatial Modulation:Optimal Constellation Breakdown

Mehdi Maleki, Student Member, IEEE, Hamid Reza Bahrami, Member, IEEE,Ardalan Alizadeh, Student Member, IEEE, and Nghi H. Tran, Member, IEEE

Abstract—Spatial modulation (SM) is a new transmission tech-nique for multi-antenna systems in which the transmit antennasare used to modulate the signal. In this paper, the symbol errorrate (SER) performance of SM is investigated. In SM, a signaldomain modulation (i.e. amplitude-phase modulation) and anantenna domain modulation (i.e. space shift keying) are combinedtogether to achieve a certain transmission rate while exploitingthe properties of two independent modulation domains. A keyquestion is the fine balance between the constellation sizes inthe two domains when a constant rate is targeted. For a fixedrate, there are many ways to assign constellation vectors to thespatial and signal domains. In this paper, we investigate optimalconstellation breakdown between space and signal domains. Theanalysis is based on the union bound of the error probabilityof SM with two typical APM schemes, i.e. phase-shift keying(PSK) and square quadrature amplitude modulation (S-QAM).It is shown that, at any transmission rate, there exists an optimalAPM dimension in which the SER is minimized. Furthermore,a trade-off between the number of transmit antennas and thetransmit power is introduced.

Index Terms—MIMO; spatial modulation; space shift keying;amplitude-phase modulation; performance, union bound.

I. INTRODUCTION

SPATIAL modulation (SM) is a recently proposed modu-lation scheme for multi-antenna systems exploiting the

space domain as well as the signal domain to modulatethe information [1]–[4]. In this scheme, an amplitude-phasemodulation (APM) technique is combined with the spaceshift keying (SSK) to send the information over a multi-input multi-output (MIMO) channel. In the SSK, each datasymbol is associated to one specific transmit antenna suchthat in each transmit interval, to transmit a particular sym-bol, only the corresponding transmit antenna is active [5],[6]. Several benefits of SSK and SM transmission and theiradvantages over traditional MIMO transmission schemes havebeen addressed in the literature [2], [5], [7]–[9]. The low trans-mission/reception complexity of these transmission techniquesmakes them a suitable candidate for future wireless systemsespecially when the number of antennas at the transmitter islarge. For example, SSK/SM can be considered as a techniqueapplicable to massive MIMO communication (see e.g. [10] andthe references therein) to support high dimension modulation.

Manuscript received July 5, 2013; revised November 1, 2013. The editorcoordinating the review of this paper and approving it for publication was A.Ghrayeb.

The authors are with the Department of Electrical and Computer Engi-neering, The University of Akron, Akron, Ohio, USA (e-mail: {mm158, hrb,aa148, nghi.tran}@uakron.edu).

Digital Object Identifier 10.1109/TCOMM.2013.121313.130514

Having many antennas at the transmitter, enables modulatinginformation via space which can increase the rate significantlywhile only using one RF chain at the transmitter to avoidcomplexity.

The performance of SM/SSK transmission schemes in dif-ferent scenarios, however, is still under study. In [5], [8], [11]–[15], the error rate performance of SSK/SM communicationsystems is investigated. For this aim, the union bound [16]is used to derive an upper bound on the probability oferror. In [11], the performance of SSK modulation for multi-input single-output (MISO) correlated Nakagami-m fadingchannels is analyzed. The bit error rate (BER) performanceof SSK over multiple-access independent fading channels isinvestigated in [12]. In [8] and [13], assuming that channelstate information (CSI) is not perfectly known at the receiver,the BER performance is studied, and it is shown that SSKor SM communication systems have a reasonable robustnessto channel estimation errors compared to the conventionalmodulation or other MIMO transmission schemes. In [14],an enhanced bound for the BER of SM is proposed, and it isused to derive the bit error probability of SM over generalizedfading channels. In [17], using the enhanced upper boundof [14], an expression for the average bit error probabilityof SM over a Nakagami-m fading channel is derived. Thisis then used to numerically evaluate the performance of SMin a correlated channel, and to find the optimal combinationof APM size and the number of active antennas for a fixedtransmission rate [18].

Recently, some methods to reduce the error probability ofdifferent classes of space modulation transmission schemeshave been proposed. For instance, in [19]–[22], the disper-sion matrix and constellation are designed to improve theperformance of space-time shift keying (STSK) transmission.In [15], [18], [23]–[27] channel state information at thetransmitter (CSIT) is used to adaptively modify the SSK/SMtransmission to improve the error rate.

One important question remained to be answered is theoptimal way to break down the constellation space betweenspace and signal domains. In other words, given a fixedtransmission rate, what is the optimal number of transmitantennas and the APM size to minimize the symbol error rate(SER) of SM. The answer to this question is very importantin practice as it provides a fine balance between the twomodulation domains such that the SER performance targetcan be met with the minimum number of transmit antennas(i.e. complexity). In [14], the BER performance of SMis illustrated for some transmission rates, and the optimal

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2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION

constellation breakdown for those cases is obtained usingsimulation. In this paper, we analytically find the optimalconstellation breakdown which is general, and therefore, isapplicable to any transmission rate and the number of receiveantennas. We also extend the idea to the case of the correlatedMIMO channels, and analytically verify the recent resultsobtained numerically in the literature [18]. To summarize,some of the novel contributions of this paper are as follows:

• We analyze the average SER performance of SM overan uncorrelated Rayleigh fading channel. Based on theresults in [14] for high SNRs, we derive a tight upperbound on the SER performance of SM with phase-shift keying (PSK) and square quadrature amplitudemodulation (S-QAM) schemes.

• Based on the derived upper bound, and by applyingmathematical analysis, we find out the optimal APMsize and the number of transmit antennas to achieve theminimum SER at a fix transmission rate for PSK andS-QAM cases, and show that in both cases, independentof the transmission rate or the number of receiveantennas, considering an APM size of 4 is optimal interm of the error performance.

• By further studying the performance and based onpractical considerations and numerical results, weintroduce a trade-off between the number of transmitantennas and the required transmit power. The introducedtrade-off is a practical means to reduce the number oftransmit antennas by increasing the transmit power andvice versa. Therefore, based on this trade-off, the bestSM design can vary depending on the design criteria;in a low power application, one can achieve the targetperformance and rate by increasing the number oftransmit antennas, while in a system designed with theminimum required number of transmit antennas in mind,these can be achieved by increasing the transmit power.

• Finally, we extend the analysis to the case of correlatedtransmit antennas. To simplify the problem and toprovide some insights on the effect of the spatialcorrelation on the performance of SM, we consider thesame correlation between all the antennas.

The remainder of the manuscript is organized as follows. InSection II, we introduce the system model for SM transmis-sion, while in Section III, we present the SER performanceanalysis assuming two common types of APM, i.e. PSK and S-QAM. In Section IV, we discuss some practical considerationsas well as the trade-off between the optimal number of anten-nas and the transmit power. Section V considers the extensionof the analysis to the case of a correlated MIMO channel. InSection VI, by some simulations, we verify the validity of thetheoretical analysis. Finally, Section VII concludes this paper.

We will use the following notations throughout the paper.(.)T and (.)H represent transpose, and conjugate transposeoperations, respectively. Lower or upper case italic lettersdenote scalar variables. Vectors and matrices are distinguished

Fig. 1. Block diagram of an SM transmission.

by lower case and upper case bold letters, respectively. | . |and ‖ . ‖ denote the absolute value and the Euclidean norm,respectively. �e {.} denotes the real part of a complex number.The expected value or mean of a function of a random variablex is represented by Ex {.}.

II. SPATIAL MODULATION

The block diagram of an SM transmitter is depicted inFig. 1. The received signal of such SM transmission over afrequency-flat MIMO channel with N transmit and M receiveantennas can be written as

y =√pHslxn + v, n ∈ {1, 2, ...,N} , l ∈ {1, 2, ...,L}

(1)where H is the M ×N MIMO channel matrix whose entriesare zero mean Gaussian random variables with variance σ2, pis the signal-to-noise ratio (SNR), sl is the APM constellationsymbol, L is the dimension of APM, v is the additive whitezero-mean complex Gaussian noise vector with unitary co-variance matrix, and xn is the n-th SSK constellation symbol,i.e.

xn =

n−th entry↓

[0, . . . , 0, 1, 0, . . . , 0]T (2)

Note that only the n-th entry of vector xn is “1” and the restof the entries are zero. With this type of modulation scheme, atotal of K = NL different symbols can be transmitted, i.e. thesignal space is composed of NL signals. The optimal detectorto minimize the SER for such a transmission is a maximumlikelihood (ML) detector which is given by [3]

(n, l

)= argmax

(n,l)

Re

{(y −

√p

2slhn

)H

slhn

}(3)

The SER of SM is the probability that a pair of (n, l) isdetected as (n′, l′) �= (n, l) at the receiver. The SER of MLdetection given the channel matrix can be expressed as [15]

PM |H =1

K

∑(l′,n′) �=(l,n)

∑(l,n)

Q

(√p

2‖slhn − sl′hn′‖

)(4)

By defining γ = p4σ

2, we can approximate the union boundfor average SER at high SNRs (i.e. γ >> 1) as (see proof inAppendix A)

PM (L) ≈(

2M − 1M

)(8γ)

−MGM (L) (5)

where


MALEKI et al.: ON THE PERFORMANCE OF SPATIAL MODULATION: OPTIMAL CONSTELLATION BREAKDOWN 3

GM (L) = 2M×⎧⎪⎪⎪⎨⎪⎪⎪⎩

K − L

L2

L∑l=1l′=1

(|sl|2 + |sl′ |2

)−M+

1

L

L∑l=1, l′=1

l�=l′

(|sl − sl′ |2

)−M

⎫⎪⎪⎪⎬⎪⎪⎪⎭(6)

The interesting fact about (6) is that the SER of SM isbroken down into two expressions: the first term in the righthand side of (6) is related to SER of space modulation, and thesecond term indicates the error probability of APM scheme.The second term in the right hand side of (6) is a summationover a function of Euclidean distance between each pair ofAPM symbols, which is exactly the union bound for the errorprobability of APM scheme with size of L having one transmitantenna and M receive antennas. However, the first term isinterestingly different and indicates that, in contrast to APM,the absolute values of APM symbols have a direct impacton the SER performance. Note that for conventional APM(L = K), the first term, and for SSK (L = 1), the secondterm in (6) are, respectively, equal to zero. In this paper, ourgoal is to find the optimal balance between these two termsin the right hand side of (6) to achieve the minimum SER.

In the next section, to further analyze the performance weconsider two different APM schemes, i.e. PSK and S-QAM.

III. PERFORMANCE ANALYSIS OF SPATIAL MODULATION

In this section, we first consider the case of the SM withPSK (SM-PSK), and derive the performance expressions forsuch a modulation. The analysis, however, is not valid forthe S-QAM scheme, and therefore, in the second part of thissection, we focus on the SER analysis of SM with S-QAM(SM-SQAM).

A. Phase Shift Keying (PSK)

For the PSK modulation, the l-th symbol can be written assl = exp

(j 2π

L l). So for SM-PSK, we can simplify (6) as

GM (L) = K − L+

L−1∑r=1

(1− cos

(2π

Lr

))−M

(7)

To understand the effect of increasing the APM size, we finda recursive relationship between GM (L) and GM (2L) asfollows

GM (2L) = K − 2L+2L−1∑r=1

(1− cos

( πLr))−M

=

{K − L+

L−1∑r=1

(1− cos

(2π

Lr

))−M}

+

{L∑

r=1

(1− cos

( πL

(2r − 1)))−M

− L

}(8)

The first term in the right side of (8) is the same as (7).Therefore, we can write

GM (2L) = GM (L) +

[L∑

r=1

(1− cos

(πL

(2r − 1)))−M

− L

](9)

Considering L = 1 (i.e. the case of SSK transmission), (9)can be written as

GM (2) = GM (1)− (1− 2−M

)(10)

Since M ≥ 1, GM (2) < GM (1). For L ≥ 2, we can write

L∑r=1

(1− cos(π

L(2r − 1)

))−M=

L/2∑r=1

(1− cos

( π

L(2r − 1)

))−M+

L∑r=L/2+1

(1− cos

(π

L(2r − 1)

))−M

(11)

Note that it is easy to verify that by defining a new variablefor the second term in right side of (11) as r′ = L − r + 1,the second term in (11) is indeed equal to the first term. Sofor L ≥ 2, (9) can be expressed as

GM (2L) = GM (L)

+

⎡⎣2 L/2∑

r=1

(1− cos

( πL

(2r − 1)))−M

− L

⎤⎦ , L ≥ 2

(12)

For the case of L = 2, (12) can be written as

GM (4) = GM (2) (13)

For the case of L ≥ 4, we can writeL/2∑r=1

(1− cos

( π

L(2r − 1)

))−M=

L/4∑r=1

(1− cos

( π

L(2r − 1)

))−M+

L/2∑r=L/4+1

(1− cos

(π

L(2r − 1)

))−M

(14)

To further simplify the second term in the right side of (14),we consider a variable change as r′′ = L/2− r+ 1. So, (12)can be expressed as

GM (2L) = GM (L) + [FM (L)− L] , L ≥ 4 (15)

where

FM (L) = 2

L/4∑r=1

RM

(cos

(πL(2r − 1)

))(16)

where RM (z) = (1− z)−M

+ (1 + z)−M .

Lemma 1: For any 0 < |z| < 1 and M > 0, we haveRM (z) = (1− z)

−M+ (1 + z)

−M> 2.

Proof: Considering the fact that for x > 0, x−M is aconvex function, for x, y > 0, x �= y, we have x−M+y−M

2 >(x+y2

)−M. By substituting x = 1 − z and y = 1 + z in this

equality, the Lemma is proved.

Based on Lemma 1, by considering (16) we can write

FM (L) > 2

L/4∑r=1

2 = L (17)

Due to the fact that FM (L) > L, we have

GM (2L) > GM (L) , L ≥ 4 (18)



TABLE IΦM (L) AS A FUNCTION OF APM SIZE L AND NUMBER OF RECEIVE

ANTENNAS M .

TABLE IIΨM (L) AS A FUNCTION OF APM SIZE L AND NUMBER OF RECEIVE

ANTENNAS M .

Therefore, we can generally write

PM (1) ≥ PM (2) ≈ PM (4) ≤ PM (8) ≤ ... (19)

Interestingly, the smallest error probability is obtained whenbinary or quaternary PSK is used at the transmitter. Therefore,in terms of the error probability L = 2 and L = 4 are theoptimal choices. On the other hand, from complexity point ofview it is more desirable to have less number of antennas.Therefore, from (19) we conclude that for SM-PSK, L = 4 isthe optimal PSK size.

Considering (16), it is easy to show that FM (L) is anincreasing function of M and L, which indicates that thedifference between PM (2L) and PM (L) increases with thenumber of receive antennas or the PSK size.

B. Square Quadrature Amplitude Modulation (S-QAM)

Another common type of APM modulation is S-QAM.Therefore, we consider the performance analysis of SM-SQAM in this sequel. For an S-QAM of size L = a2 (a ≥ 2),we have

sl =1

AL(αl + jβl) , αl, βl ∈ χa, l = 1, 2, ..., L (20)

where χa = {±1,±3, ...,± (a− 1)}, and

A2L =

1

L

L∑l=1

(|αl|2 + |βl|2

)=

2

3(L− 1) (21)

We further simplify the general performance expression in (6)assuming SM-SQAM. For the first term in (6), we have

L∑l=1, l′=1

(|sl|2 + |sl′ |2

)−M

= 16A2ML 4−MΦM (L) (22)

where

ΦM (L) =1

16

L∑l=1, l′=1

(αl

2 + αl′2 + βl

2 + βl′2

4

)−M

=

a/2∑x,y,z,w=1

[I (x, y, z, w)]−M (23)

Fig. 2. Examples of possible rectangles in a 64-QAM constellation.

where I (x, y, z, w) = (2x−1)2+(2y−1)2+(2z−1)2+(2w−1)2

4 . Wehave listed function ΦM (L) for different values of M and Lin the Table I. Note that for sufficiently large values of M (i.e.M ≥ 4), ΦM (L) is almost one. Next, for the second term in(6), we have

L∑l=1, l′=1

l�=l′

|sl − sl′ |−2M= 16A2M

L 4−MΨM (L) (24)

where

ΨM (L) =1

8

∑x,y

κx,y . RL (x, y) .(x2 + y2

)−M(25)

in which

κx,y =

{1, xy = 02, otherwise

(26)

and RL (x, y) is the total number of x× y rectangles one canfind in an L-ary S-QAM constellation diagram. In Table II, wehave listed ΨM (L) for different values of M and L. Also inFig. 2, we have illustrated some examples of rectangles whichcan be considered in an S-QAM constellation diagram. In thisfigure, R1 can be interpreted as a rectangle of size 0× 3 andR2 and R3 are, respectively, 4× 2 and 2× 4 rectangles. It iseasy to confirm that for 0 ≤ x, y ≤ a, we have

Ra2 (x, y) =

{(a− x) (a− y) , 0 ≤ x, y < a, (x, y) �= (0, 0)

0, otherwise(27)

We, next, introduce some interesting properties of functionsΦM (L) and ΨM (L) in Theorem 1.

Theorem 1: For functions ΦM (L) and ΨM (L), the follow-ing properties hold:

1) ΦM (L) and ΨM (L) are increasing functions of L anddecreasing functions of M .

2) ΦM (L) is a convergent series for M ≥ 3, and adivergent series for M = 1, 2.

3) ΦM (L) ≈ 1 for M ≥ 4.4) ΨM

(a2) ≈ a (a−1)

4 for M >> 1.5) ΦM (4L)

ΦM (L) is a decreasing function of L and M , and

limL→∞

ΦM (4L)ΦM(L) =

{4, M = 11, M ≥ 2

.

6) ΨM (4L)ΨM(L) > 4.



7) ΨM(L)ΦM (L) is an increasing function of L.

Proof: The proof of Property 1 is straightforward. Theproof for convergence of ΦM (L) for M ≥ 3 and divergence ofΦM (L) for M = 1, 2 can be, respectively, found in AppendixB and C. For Property 3, we need to show that ΦM (L)converges to 1 as M approaches infinity. We know that

a/2∑x,y,z,w=1

[I (x, y, z, w)]−M = 1+

a/2∑x,y,z,w=1

(x,y,z,w) �=(1,1,1,1)

[I (x, y, z, w)]−M

(28)Since I (x, y, z, w) > 1 for any (x, y, z, w) �= (1, 1, 1, 1),it can be concluded that for sufficiently large values of M ,ΦM (L) converges to 1. From Table I, we can notice thatΦ4 (L) ≈ 1, and due to the fact that ΦM (L) is a decreasingfunction of M (Property 1) and the convergence property(Property 2), ΦM (L) ≈ 1 for M ≥ 4. Proof of Property4 is similar to the proof of Property 3. For M ≥ 3, Property 5can be directly concluded from Property 2. Also, the proof ofProperty 5 for the cases of M = 1 and M = 2 is available inAppendix C. The proof of Property 6 is in Appendix D. Thefact that the rate of increase of ΨM (L) is greater than that ofΦM (L) proves Property 7. In addition, based on Property 5for M ≥ 2 and ΨM (4L) > 4ΨM (L) (Property 6), one canconclude that ΨM(L)

ΦM (L) increases with L.Substituting (22) and (24) in (6), we have

GM (L) =16A2M

L 2−M

L2[(K − L)ΦM (L) + LΨM (L)] (29)

To evaluate the performance of SM-SQAM, we first note thatthere is no difference between PSK and S-QAM for L = 1and L = 4. In fact, QAM and PSK constellations in these twocases are similar up to a constant rotation. Therefore, similarto SM-PSK modulation, for SM-SQAM we have

GM (1) ≥ GM (4) (30)

For L ≥ 4 to compare the performance for different QAMsizes, we consider the ratio of the union bounds as

GM (4L)

GM (L)=

1

16

(A2

4L

A2L

)M

WM,L (K) (31)

where

WM,L (K) =(K − 4L)ΦM (4L) + 4LΨM (4L)

(K − L)ΦM (L) + LΨM (L)(32)

Lemma 2: WM,L (K) is a decreasing function of K .Proof: The first derivative of WM,L (K) with respect to

K is

∂WM,L (K)

∂K=

L [ΦM (4L) . (ΨM (L)− ΦM (L)) − 4ΦM (L) . (ΨM (4L) − ϕM (4L))]

[K.ΦM (L) + L (ΨM (L)− ΦM (L))]2

(33)

To show that the nominator of (33) is negative, we need toshow that

4ΨM (4L)

ΦM (4L)− ΨM (L)

ΦM (L)> 3 (34)

TABLE IIIUM (L) AS A FUNCTION OF APM SIZE L AND NUMBER OF RECEIVE

ANTENNAS M .

From Tables I and II, it is easy to verify (34) for L = 4. ForL > 4, using Property 5 and Property 7 in Theorem 1, onecan conclude

4ΨM (4L)

ΦM (4L)− ΨM (L)

ΦM (L)> 3

ΨM (L)

ΦM (L)≥ 3

ΨM (16)

ΦM (16)> 3 (35)

Based on Lemma 2, one can write

WM,L (K) ≥ limK→∞

WM,L (K) =ΦM (4L)

ΦM (L)(36)

So, for any value of M , we have

GM (4L)

GM (L)≥ 1

16

(A2

4L

A2L

)M [ΦM (4L)

ΦM (L)

]Δ= UM (L) (37)

In Table III, we have listed UM (L) for different values of Mand L. For L ≥ 4, we also have

A24L

A2L

=4L− 1

L− 1(38)

Therefore, for L ≥ 4, 4 <A2

4L

A2L

≤ A216

A24= 5. So, from (37) and

(38) one can conclude that for M ≥ 2

GM (4L)

GM (L)≥ 1

16

(A2

4L

A2L

)M [ΦM (4L)

ΦM (L)

]>

1

16

(A2

4L

A2L

)M

>4M

16≥ 1

(39)The second inequality in (39) comes from Property 1 inTheorem 1 (ΦM (4L) > ΦM (L)). Also, based on Property5 for M = 1, we have

G1 (4L)

G1 (L)≥ 1

16

(A2

4L

A2L

)[Φ1 (4L)

Φ1 (L)

]>1

4

(A2

4L

A2L

)> 1 (40)

Therefore, generally we can conclude

PM (1) ≥ PM (4) ≤ PM (8) < ... (41)

Observe from (41) that the smallest error probability happensat L = 4. Therefore, L = 4 is the optimal S-QAM con-stellation size for SM transmission. Interestingly, this resultshows that to make the most benefit of SM in terms of errorprobability, there is no need to use more than K/4 transmitantennas.

By considering Property 5 in Theorem 1 and (38), we canconclude that UM (L) defined in (37) increases by reducingL and increasing M . Based on Property 3 in Theorem 1 and(38), we have

limM→∞

UM (L) = ∞ (42)

and based on Property 5 in Theorem 1 and (38), we have

limL→∞

U1 (L) = 1 limL→∞

UM (L) = 4M−2, M ≥ 2 (43)



The numerical results presented in Table III can verify (42)and (43). (43) indicates that specially when the number ofreceive antennas is large, increasing the QAM size results ina significant increase in the probability of error.

IV. PRACTICAL CONSIDERATIONS

In Section III, we mathematically proved that the unionbound of error probability is minimized at L = 4 for bothSM-PSK and SM-SQAM. However, since in most practicalapplications to avoid complexity, it is desirable to have smallernumber of antennas, we introduce a trade-off between thenumber of antennas and the required transmit power to achievea certain performance. In fact, as will be discussed in SectionVI, our numerical study shows that in some cases it is possibleto use a larger APM dimension (L > 4) with a performanceclose to that of L = 4 case. In such a case, it might be betterto choose the former APM dimension to reduce the numberof transmit antennas while keeping the performance almostthe same. For example, at a fix SNR, if PM (4) and PM (16)are close enough, it might make sense to pick the case ofL = 16 to have a 4-fold reduction in the number of transmitantennas compared to the case of L = 4. Nevertheless, thismeans that to achieve the same probability of error with asmaller number of transmit antennas, it is required to transmitmore power compared to the optimal case of L = 4. Let usassume that to reduce the number of antennas, we are willingto transmit at most Δρdb dB extra power. So, we pick an APMwith the largest L > 4 satisfying the following inequality

PM (L) < PM (4) .10

(MΔρdb

10

)(44)

In this case, we can reduce the number of required antennasat the transmitter by L

4 -fold.To have a better understanding of the behavior of the union

bound for the case of PSK modulation, let us consider theratio of the union bounds of PM (2L) and PM (L). This ratiocan be approximated as

PM (2L)

PM (L)≈ 1 +

2L/2∑r=1

(1− cos

(πL (2r − 1)

))−M − L

K +L−1∑r=1

(1− cos

(2πL r

))−M − L

= 1+FM (L)− L

JM (L) +K − L+ 2−M − 1(45)

where FM (L) is defined in (16) and

JM (L) = 2

L/4∑r=1

RM

(cos

(π

L(2r)

))(46)

A close examination of (45) indicates that by increasing thetransmission rate K , the ratio is close to one for a wider rangeof L which means that for a large K , one might be able touse a larger PSK size with a small performance penalty.

On the other hand, in the case of S-QAM, we analyzed theratio of union bounds of PM (4L) and PM (L) in Section III,and according to (37), Table III is a good measure to find thebest balance between QAM size and the number of transmitantennas. One interesting fact concluded from (37) is that incontrast to SM-PSK, the performance ratio of SM-SQAM isalmost independent of the transmission rate K . In other words,

the relative increase in the SER from APM size of L to 4Lis almost constant and independent of the target rate.

V. A BRIEF EXTENSION TO CORRELATED CHANNELS

The analysis in previous sections is only valid for theuncorrelated channels. In this section, we extend our analysisto the case of a spatially correlated MIMO channel. Let Tbe the transmit correlation matrix, and consider the channelmodel

H = HT1/2 (47)

where T is a hermitian matrix satisfying trace {T} = N ,and H is a zero mean complex Gaussian random matrixwhose entries have the variance of σ2. Here, for the sakeof tractability, we assume the simple case of equi-correlatedtransmit antennas with equal average channel gains. Althoughthis model might not be realistic, this can give us a good in-sight into the effect of channel correlation on the performanceof SM. Following the same approach as in Appendix A, theexpression for the SER performance of SM over uncorrelatedchannels in (6) can be extended to the case of equi-correlatedtransmit antennas as

GM (L, ρ)

= 2MK − L

L2

L∑L=1, l′=1

(|sl|2 + |sl′ |2 − 2ρRe {slsl′∗})−M

+ 2M1

L

L∑l=1, l′=1

l�=l′

(|sl − sl′ |2)−M

(48)

where ρ is the correlation coefficient between each pair ofantennas, and it is assumed to be real satisfying 0 ≤ ρ < 1.For a small correlation coefficient (i.e. ρ << 1), we have

L∑l=1, l′=1

(|sl|2 + |sl′ |2−2ρRe {slsl′∗}

)−M ≈

L∑l=1l′=1

(|sl|2 + |sl′ |2

)−M+ 2Mρ

L∑l=1l′=1

Re {slsl′∗}(|sl|2 + |sl′ |2

)−M−1

(49)

For any symmetric APM modulation such as the PSK and S-QAM, the second term in the right side of (49) is zero. So,we have

GM (L, ρ) ≈ GM (L, 0) = GM (L) , ρ << 1 (50)

This equation indicates that a small coefficient (e.g. ρ < 0.1)does not affect the performance of SM with symmetric APMmodulation. For SM-PSK, (48) can be written as

GM (L, ρ)

=K − L

L

L−1∑r=0

(1− ρ cos

(2π

Lr

))−M

+

L−1∑r=1

(1− cos

(2π

Lr

))−M

(51)

It is easy to show that by increasing ρ, GM (1, ρ) also

increases. For L ≥ 2, sinceL−1∑r=0

(1− ρ cos

(2πL r

))−M=



L/2−1∑r=0

RM

(ρ cos

(2πL r

)), from Appendix E, it can be con-

cluded that GM (L, ρ) is an increasing function of ρ. Also, inthis case (15) can be extended to

GM (2L, ρ) = GM (L, ρ) + ΔM (L, ρ) (52)

where

ΔM (L, ρ) = Δ′M (L, ρ)− K

2LΔ′′

M (L, ρ) (53)

Δ′ =L∑

r=1

{(1− cos

( π

L(2r − 1)

))−M

−(1− ρ cos

( π

L(2r − 1)

))−M}

(54)

Δ′′ =L−1∑r=0

{(1− ρ cos

(π

L2r

))−M −(1− ρ cos

( π

L(2r + 1)

))−M}

(55)

Note that ΔM (L, 0) = FM (L)− L and Δ′M (L, 0) = 0. For

L = 1 and L = 2, we have

Δ′M (1, ρ) = 2−M − (1 + ρ)

−M< 0

Δ′′M (1, ρ) = (1− ρ)

−M − (1 + ρ)−M

> 0 (56)

Δ′M (2, ρ) = 0

Δ′′M (2, ρ) = (1− ρ)

−M+ (1 + ρ)

−M − 2 > 0 (57)

From (53), (56) and (57), it can be concluded thatΔ′

M (L, ρ) < 0 for L = 1, 2; i.e. the optimal APMsize is not less than four (Lopt ≥ 4).

For L ≥ 4, (56) and (57) can be, respectively, written as

Δ′ = 2

L/4−1∑r=0

{RM

(cos

(π

L(2r + 1)

))− RM

(ρ cos

( π

L(2r + 1)

))}

(58)

Δ′′ =L/4−1∑r=0

{RM

(ρ cos

(π

L(2r + 2)

))+ RM

(ρ cos

( π

L2r

))

− 2 RM

(ρ cos

( π

L(2r + 1)

))}(59)

As shown in Appendix E, by increasing ρ, RM (ρx) in-creases; therefore, Δ′

M (L, ρ) is a decreasing function of ρ.Also, we have

Δ′M (2L, ρ)−Δ′

M (L, ρ) = 2

L/4−1∑r=0

{ΓM (L, r, 1)− ΓM (L, r, ρ)}(60)

where

ΓM (L, r, ρ) = RM

(ρ cos

( π

2L(4r + 3)

))+RM

(ρ cos

( π

2L(4r + 1)

))−RM

(ρ cos

( π

2L(4r + 2)

))(61)

As it is mentioned in Appendix E, both RM (ρx)and ∂RM (ρx)/∂x are increasing functions of ρ.Therefore, by increasing ρ, RM

(ρ cos

(π2L (4r + 3)

))and RM

(ρ cos

(π2L (4r + 1)

)) − RM

(ρ cos

(π2L (4r + 2)

))

also increase. This means that the expression in (59) ispositive and Δ′

M (L, ρ) is an increasing function of L. In thecase of M = 1, numerical results show that Δ′′

1 (L, ρ) > 0and Δ′′

1 (L, ρ) ≈ 0.Lemma 3: For any x1, x2, and any function pairs, f(.) and

g(.), whose second order derivatives exist, we have

∂2f (x)

∂x2>

∂2g (x)

∂x2⇒

f (x1)− 2f(x1 + x2

2

)+ f (x2) > g (x1)− 2g

(x1 + x2

2

)+ g (x2)

(62)

Proof: The proof is straightforward, and omitted forbrevity.

Based on Appendix F and Lemma 3, one can conclude that

Δ′′M+1 (L, ρ) > Δ′′

M (L, ρ) (63)

Also, using (103) in Appendix E, we have

∂Δ′′M (L, ρ)

∂ρ=

M

ρ

{Δ′′

M+1 (L, ρ)−Δ′′M (L, ρ)

}> 0 (64)

This equation indicates that by increasing ρ, Δ′′M (L, ρ) in-

creases. Also, one can write

Δ′′M (L, ρ)−Δ′′

M (2L, ρ)

= 2

L/4−1∑r=0

{RM

(ρ cos

( π

2L(4r + 1)

))+RM

(ρ cos

( π

2L4r + 3

))

− 2 RM

(ρ cos

(πL(2r + 1)

))}(65)

Due to Lemma 3, Δ′′M (L, ρ) − Δ′′

M (2L, ρ) > Δ′′1 (L, ρ) −

Δ′′1 (2L, ρ) > 0. So, the expression in (64) is positive, and

Δ′′M (L, ρ) is a decreasing function of L. Δ′′

M (L, ρ) can alsobe written as

Δ′′ = 2

L/4−1∑r=0

{RM

(ρ cos

( πL2r))

−RM

(ρ cos

( πL(2r + 1)

))}+RM (0)−RM (ρ) (66)

For large values of L (L >> 1), we have

2

L/4−1∑r=0

{RM

(ρ cos

(π

L2r))

−RM

(ρ cos

(π

L(2r + 1)

))}

≈ −L/4∫

r=0

∂RM

(ρ cos

(πL2r

))∂r

dr = RM (ρ)−RM (0) (67)

This means that

limL→∞

Δ′′M (L, ρ) = 0 (68)

One can summarize the properties of Δ′M and Δ′′

M as follows

ρ ↑⇒ Δ′ ↓, Δ′′ ↑⇒ Δ ↓L ↑⇒ Δ′ ↑, Δ′′ ↓⇒ Δ ↑ (69)

Based on (56), (57), GM (4, ρ) < GM (2, ρ) < GM (1, ρ).Due to the properties of Δ′ and Δ′′, and based on the factthat the optimal PSK size for uncorrelated channel (ρ = 0) isfour (Lopt = 4), several conclusion can be made:



• Increasing the correlation between the antennas increasesthe average probability of error.

• Since Δ is an increasing function of L, the optimal PSKsize satisfies

ΔM (Lopt/2, ρ) < 0, ΔM (Lopt, ρ) ≥ 0 (70)

Therefore, to find the optimal PSK size, we only needto increase L, and look for the smallest L that makesΔM (L, ρ) non-negative.

• Since Δ is a decreasing function of L, by increasing ρ,the optimal PSK size increases (or remains unchanged),i.e.

ρ > ρ′ ⇒ Lopt(K,L,M, ρ) ≥ Lopt(K,L,M, ρ′) (71)

This means that to have an optimal transmission schemein highly correlated channel, as expected we have to relymore on signal (APM) domain rather than the spatialdomain.

• Since K is an important factor in reducing Δ, we cangenerally write

K > K ′ ⇒ Lopt(K,L,M, ρ) ≥ Lopt(K′, L,M, ρ)

(72)In fact, for not very large correlation coefficients,Δ′

M (L, ρ) is relatively much larger than Δ′′M (L, ρ). For

relatively small K , optimum L is smaller than that of thelarger K .

• Based on (68), for very large values of L, changing Kor ρ does not change the optimal PSK size.

For SM-SQAM in correlated channels, ΨM (L) in (29) doesnot change, while instead of ΦM (L), we have ΨM (L, ρ) as

GM (L, ρ) =16A2M

L 2−M

L2

[(K − L)ΦM (L, ρ) + LΨM (L)

](73)

ΦM (L, ρ) is defined as

L∑l=1, l′=1

(|sl|2 + |sl′ |2−2ρRe {slsl′∗})−M

= 16A2ML 4−M ΦM (L, ρ)

(74)

where

ΦM (L, ρ) =1

4

4∑i=1

a/2∑x,y,z,w=1

[Ii (x, y, z, w, ρ)]−M (75)

Note that

Ii (x, y, z, w, ρ) = I (x, y, z, w) + ρ ci(x, y, z, w), i = 1, 2, 3, 4(76)

where c1(x, y, z, w) = 2xy + 2zw − y − x − z − w + 1,c2(x, y, z, w) = −c1(x, y, z, w), c3(x, y, z, w) =|2xy − 2zw − x− y + z + w|, c4(x, y, z, w) =−c3(x, y, z, w). It is easy to show that

Ii (x, y, z, w, ρ) ≥ (1− ρ) I (x, y, z, w) , i = 1, 2, 3, 4 (77)

Also, one can write

[I1 (x, y, z, w, ρ)]−M + [I2 (x, y, z, w, ρ)]−M

= [I (x, y, z, w)]−MRM

(ρ c1(x, y, z, w)

I (x, y, z, w)

)≥ 2[I (x, y, z, w)]−M

(78)

and

[I3 (x, y, z, w, ρ)]−M + [I4 (z, w, x, y, ρ)]−M

= [I (x, y, z, w)]−MRM

(ρ c3(x, y, z, w)

I (x, y, z, w)

)≥ 2[I (x, y, z, w)]−M

(79)

Based on (75)-(79), we have

ΦM (L) ≤ ΦM (L, ρ) ≤ (1− ρ)−MΦM (L) (80)

From (78) and (79), and based on Appendix E, it can beconcluded that ΦM (L, ρ) is an increasing function of ρ.

Using the aforementioned properties, we can simply showthat if we replace ΦM (L, ρ) instead of ΦM (L), Theorem 1will still be valid.

Lemma 4: Lets define WM,L (K, ρ) as the extension of (32)for correlated channel

WM,L (K, ρ) =(K − 4L)ΦM (4L, ρ) + 4LΨM (4L)

(K − L)ΦM (L, ρ) + LΨM (L)(81)

For L ≥ 4 and M ≥ 2, WM,L (K, ρ) is a decreasingfunction of K .

Proof: For M ≥ 4, based on Property 3 of Theorem 1and for M ≥ 2 and L >> 1, based on Property 5 of Theorem1, we have ΦM (4L, ρ) ≈ ΦM (L, ρ). Therefore, for cases ofM ≥ 4 or M ≥ 2, L >> 1, one can write

4ΨM (4L)

ΦM (4L, ρ)− ΨM (L)

ΦM (L, ρ)≈ 4ΨM (4L)−ΨM (L) > 15ΨM (L)

(82)The last equality is due to Property 6 of Theorem 1. Also,based on Properties 1 and 4 of Theorem 1, we have ΨM (L) >0.5. Therefore, from (82), we can write

4ΨM (4L)

ΦM (4L, ρ)− ΨM (L)

ΦM (L, ρ)> 3, (M ≥ 4) or (M ≥ 2, L >> 1)

(83)Also, by numerical trial, we can simply verify (83) for M =2, 3 and small values of L.Based on Lemma 4, for M ≥ 2 one can write

WM,L (K, ρ) ≥ limK→∞

WM,L (K, ρ) =ΦM (4L, ρ)

ΦM (L, ρ)> 1

(84)From (84), for SM-SQAM with more than two receive anten-nas (i.e. M ≥ 2), the following conclusions can be made:

• Increasing the correlation between the antennas increasesthe average probability of error.

• The optimal QAM size (Lopt) does not change withcorrelation coefficient (ρ), i.e. Lopt = 4.

• The SM constellation size (K) does not affect Lopt.

Note that for the case of single receive antenna, the introducedunion bound in this paper is not accurate, and therefore, wecannot use it for the analysis.

The results of this section is interesting, as in contrast to thecase of uncorrelated channels, we observe different behaviors



10 15 20 25 30 35 40

10−6

10−4

10−2

100

102

SNR (dB)

Prob

abili

ty o

f E

rror

N = 1, L = 64 (Simulation)N = 1, L = 64 (Theory)N = 2, L = 32 (Simulation)N = 2, L = 32 (Theory)N = 4, L = 16 (Simulation)N = 4, L = 16 (Theory)N = 8, L = 8 (Simulation)N = 8, L = 8 (Theory)N = 16, L = 4 (Simulation)N = 16, L = 4 (Theory)N = 32, L = 2 (Simulation)N = 32, L = 2 (Theory)N = 64, L = 1 (Simulation)N = 64, L = 1 (Theory)

Fig. 3. SER performance of SM-PSK modulation for different values of Land N (K = 64, M = 2).

10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

SNR (dB)

Prob

abili

ty o

f E

rror

N = 1, L = 64 (Simulation)

N = 1, L = 64 (Theory)


N = 4, L = 16 (Theory)


N = 16, L = 4 (Theory)


N = 64, L = 1 (Theory)

Fig. 4. SER performance of SM-SQAM for different values of L and N(K = 64, M = 2).

for SM-PSK and SM-SQAM. From the analysis in this section,we can conclude that the optimal QAM size for SM-SQAMis Lopt = 4, while for SM-PSK, the optimal PSK size canbe larger than 4 (i.e. Lopt > 4). Since for L = 4, SM-PSKand SM-SQAM have practically the same performance, forthe case that the optimal PSK size for SM-PSK is larger than4 (i.e. Lopt > 4), we can conclude that not only optimal SM-PSK performs better than SM-SQAM, it also requires lessnumber of transmit antennas. Therefore, SM-PSK is a betteroption than SM-SQAM in the correlated channels.

VI. SIMULATION RESULTS

In this section, we provide some numerical results to furtherinvestigate the error performance of SM, and compare themwith the results obtained by mathematical analysis in SectionIII. Throughout the simulations, we assume a Gaussian MIMOchannel with zero mean unit variance links between each pairof the transmit and receive antennas.

In Figs. 3 and 4, the theoretical as well as the experimental

0 1 2 3 4 5 6 7 810

−4

10−3

10−2

10−1

100

PSK Size ( log2(L) )

Prob

abili

ty o

f E

rror

K = 8 (Simulation)K = 8 (Theory)K = 16 (Simulation)K = 16 (Theory)K = 32 (Simulation)K = 32 (Theory)K = 64 (Simulation)K = 64 (Theory)K = 128 (Simulation)K = 128 (Theory)K = 256 (Simulation)K = 256 (Theory)

Fig. 5. SER performance of SM-PSK versus PSK modulation size fordifferent values of K (SNR = 40dB, M = 1).

SER performance versus SNR are, respectively, illustrated forSM-PSK and SM-SQAM assuming a symbol size of 64 andtwo receive antennas (M = 2). For comparison, differentcombinations of APM sizes (L) and number of transmitantennas (N ) are considered. As shown, the analytical resultsagree with the simulation results to a good extent. The reasonthat the theoretical results predict a worse performance is dueto the use of the union bound in the analysis. As illustratedin Fig. 3, in terms of the SER, SM with QPSK (L = 4) isthe optimal choice; however, the SER of the SM with 8-PSK(L = 8) is very close to that of the optimal case. In fact,at the SNR of 30dB the performance of 8-PSK SM is only1dB lower than that of QPSK SM. Therefore, if in a realscenario, a 1dB extra transmit power is affordable, one canreduce the number of required transmit antennas by half whilekeeping the transmission rate K constant. By comparing Figs3 and 4, we can observe that for the case of N = 1, L = 64(only APM modulation), as expected, QAM performs betterthan PSK. However, for the cases of N = 64, L = 1 andN = 16, L = 4, the performance curves in the two figuresare identical. This actually makes sense because in the case ofN = 64, L = 1, no APM scheme is used, while in the caseof N = 16, L = 4, QAM and PSK are practically the sameconstellation (with only a π/4 rotation).

In Figs. 5-8, the SER performance of SM is displayedagainst the APM size for different transmission rates. Figs.5 and 6 show the theoretical as well as the experimentalperformance of SM-PSK, for SNR = 40dB, M = 1, and SNR= 20dB, M = 3, respectively. Note that at small PSK sizes,the SER performance worsens by increasing the transmissionrate or equivalently the number of transmit antennas. Notealso that in all cases at sufficiently large PSK sizes (L), theperformance of all the SM schemes converges regardless oftransmission rate K . This means that at sufficiently large PSKsizes, the APM dimension is the indicating factor in terms ofthe SER performance regardless of the number of transmitantennas (N ). Comparing Figs. 5 and 6 also reveals that, ata fix transmission rate, the performance difference of optimalPSK size with other cases increases by increasing the number



0 1 2 3 4 5 6 7 8 910

−5

10−4

10−3

10−2

10−1

100

101

PSK Size ( log2(L) )

Prob

abili

ty o

f E

rror

K = 16 (Simulation)K = 16 (Theory)K = 32 (Simulation)K = 32 (Theory)K = 64 (Simulation)K = 64 (Theory)K = 128 (Simulation)K = 128 (Theory)K = 256 (Simulation)K = 256 (Theory)K = 512 (Simulation)K = 512 (Theory)

Fig. 6. SER performance of SM-PSK versus PSK modulation size fordifferent values of K (SNR = 20dB, M = 3).

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−4

10−3

10−2

10−1

100

QAM Size ( log4(L) )

Prob

abili

ty o

f E

rror

K = 16 (Simulation)K = 16 (Theory)K = 32 (Simulation)K = 32 (Theory)K = 64 (Simulation)K = 64 (Theory)K = 128 (Simulation)K = 128 (Theory)K = 256 (Simulation)K = 256 (Theory)K = 512 (Simulation)K = 512 (Theory)K = 1024 (Simulation)K = 1024 (Theory)

Fig. 7. SER performance of SM-SQAM versus QAM size for differentvalues of K (SNR = 40dB, M = 1).

of receive antennas. For instance, as it is shown in Fig. 5 andFig. 6, the difference between the optimal case L = 4 andthe case of L = 8 is higher when we increase the number ofreceive antennas from M = 1 to M = 3. In Fig. 7 and Fig.8, the performance of SM-SQAM is illustrated versus QAMsize for SNR=40dB and 20dB, and M = 1 and 3, respectively.Many of the conclusions made for the case of PSK modulationare still valid in this case. However, as observed from theexperimental curves, by increasing the number of receiveantennas, the performance difference between at differentQAM sizes (L) for the case of M = 1 is small. This is,however, different from the theoretical results obtained fromthe union bound. In fact, the union bound analysis predicts thatthere should be a difference between PM (4L) and PM (L)(see equations (40) and (41)) which is not clearly the casefrom experimental results. Nevertheless, for larger number ofantennas, as it is shown in Fig. 8, the behavior of experimentaland theoretical curves is the same.

In Fig. 9 and Fig. 10, the performance of SM-PSK and SM-

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

−6

10−5

10−4

10−3

10−2

10−1

100

101

102

QAM Size ( log4 (L) )

Prob

abili

ty o

f E

rror

K = 16 (Simulation)K = 16 (Theory)K = 32 (Simulation)K = 32 (Theory)K = 64 (Simulation)K = 64 (Theory)K = 128 (Simulation)K = 128 (Theory)K = 256 (Simulation)K = 256 (Theory)K = 512 (Simulation)K = 512 (Theory)K = 1024 (Simulation)K = 1024 (Theory)

Fig. 8. SER performance of SM-SQAM versus QAM size for differentvalues of K (SNR = 20dB, M = 3).

0 1 2 3 4 5 6 7 8

10−4

10−3

10−2

10−1

100

PSK Size ( log2 (L) )

Prob

abili

ty o

f E

rror

ρ=0 (Simmulation)ρ=0 (Theory)ρ=0.2 (Simmulation)ρ=0.2 (Theory)ρ=0.4 (Simmulation)ρ=0.4 (Theory)ρ=0.6 (Simmulation)ρ=0.6 (Theory)ρ=0.8 (Simmulation)ρ=0.8 (Theory)

Fig. 9. SER performance of SM-PSK versus PSK size for different valuesof ρ (SNR = 30dB, M = 2).

SQAM are, respectively, illustrated for different correlationcoefficients (ρ = 0.2, ρ = 0.4, ρ = 0.6, and ρ = 0.8). Aswe expected from the analysis in Section V, as the correlationincreases the optimal PSK size gradually moves to the rightside, while the optimal QAM size does not change. Also fromthese figures, it is observable that for correlated channels, SM-PSK performs better than SM-SQAM.

Next, we investigate the practical consideration and thetrade-off discussed in Section IV. We assume that Δρdb =1.5 dB is the maximum amount of extra power that can betransmitted compared to the optimal case of L = 4. It isassumed that the best APM size is the one that results ina smaller number of transmit antennas while keeping thetransmit power within the allowable range. The best APMsize is obtained using (45). In Fig. 11 and Fig. 12, the bestPSK and S-QAM sizes as a function of the transmission rate(K), and the number of receive antennas (M ) are obtainedusing both theory and simulations. The simulation results areshown with bars, while the results obtained from theoretical



0 0.5 1 1.5 2 2.5 3 3.5 410

−4

10−3

10−2

10−1

QAM Size ( log4 (L) )

Prob

abili

ty o

f E

rror

ρ=0 (Simmulation)ρ=0 (Theory)ρ=0.2 (Simmulation)ρ=0.2 (Theory)ρ=0.4 (Simmulation)ρ=0.4 (Theory)ρ=0.6 (Simmulation)ρ=0.6 (Theory)ρ=0.8 (Simmulation)ρ=0.8 (Theory)

Fig. 10. SER performance of SM-SQAM versus QAM size for differentvalues of ρ (SNR = 30dB, M = 2).

M = 1 (Simulation) M = 2 (Simulation) M = 3 (Simulation) M = 4 (Simulation) M = 1 (Theory) M = 2 (Theory) M = 3 (Theory) M = 4 (Theory)

Symbol Size

Numberof

ReceiveAntennas

K = 211K = 210

K = 23K = 24K = 26K = 27K = 28

K = 216K = 215

K = 214K = 213

K = 212

K = 29

K = 25

L = 29

L = 28

L = 27

L = 26

L = 25

L = 24

L = 23

L = 22

L = 21

L = 20

BestPSKSize

Fig. 11. Optimal PSK modulation size for different transmission rates Kand number of receive antennas M .

analysis are illustrated using curves. We have shown the resultsfor 4 different number of receive antenna cases, i.e. M = 1to M = 4. Note that for most cases the theoretical andexperimental results match perfectly. Also, for the case of S-QAM, as the number of receive antennas increases, the ratioof PM (4L)

PM (L) increases significantly. Therefore, for large numberof receive antennas L = 4 is the best option. Note also thatfor smaller values of Δρdb (not shown here), the results ofsimulation and theory deviate more especially for the casesof having one or two receive antennas. The reason is that ourtheoretical results in this paper are based on the ratio of upperbounds on the SERs and not the exact SERs. Especially forsmall number of antennas, the bound and the exact values ofSERs for different values of APM size do not always followthe same pattern. For example, in Fig. 7, we can see that incontrast to the bound, the exact SERs do not always increasewith L.

VII. CONCLUSION

In this paper, we investigated the SER performance of SMto answer an important question, i.e. what is the best way tobreak down the symbol transmission in the signal and spatialdomains. In other words, we studied the fine balance betweenthe modulation in the spatial domain and the traditional signal

M = 1 (Simulation) M = 2 (Simulation) M = 3 (Simulation) M = 4 (Simulation) M = 1 (Theory) M = 2 (Theory) M = 3 (Theory) M = 4 (Theory)

L = 45

L = 44

L = 42

K = 27K = 26

K = 29K = 28

K = 23K = 24K = 25

K = 211K = 210

K = 212K = 213K = 214K = 215K = 216

L = 41

L = 40

L = 43

L = 47

L = 46

L = 48

Numberof

ReceiveAntennas

Symbol Size

BestS−QAM

Size

Fig. 12. Optimal S-QAM size for different transmission rates K and numberof receive antennas M .

domain to obtain the best SER performance. We first derivedthe performance equations for the SM-PSK as well as theSM-SQAM in terms of the APM size and the number oftransmit antennas used for SM transmission. Based on theseexpressions, for each modulation we investigated the optimalAPM size or equivalently the optimal number of transmitantennas to minimize the SER at a fix transmission rate.Interestingly, the analytical and numerical results indicate thatan APM size of 4 provides the best performance; however,since in some cases reducing the number of antennas resultsin a slight increase in the probability of error, we introduceda trade-off between the number of transmit antennas and theextra transmit power required to achieve a target SER. Thistrade-off provides a guideline to design optimal SM systemsin practice as it strikes a balance between complexity (i.e. re-quired number of transmit antennas) and the performance (i.e.required transmit power). Finally, we extend the performanceanalysis of SM with uncorrelated antennas to the case ofequi-correlated transmit antennas. The mathematical analysisfor both the correlated and uncorrelated transmit antennasscenarios can be verified by the Monte Carlo simulation forall the cases, except for the case of SM-SQAM scenario withsingle antenna receiver (M = 1). Using a tighter union boundin the analysis can provide a more accurate analytical resultin this special case.

APPENDIX APROBABILITY OF ERROR

The union bound for SER in (4) can be expressed as

PM |H ≤ 1

K

N∑n=1, n′=1

n�=n′

L∑l=1l′=1

Q

(√p

2‖slhn − sl′hn′‖2

)

+1

K

N∑n=1

L∑l=1, l′=1l�=l′

Q

(√p

2|sl − sl′ |2‖hn‖2

)(85)

The two terms in the right side of (85) correspond to spaceand signal domains, respectively. Taking expectation from bothsides of (85), the union bound for average SER can be writtenas

PM ≤ K − L

L2

L∑l=1, l′=1

Ω1 (l, l′) +

1

L

L∑l=1, l′=1

l�=l′

Ω2 (l, l′) (86)



where

Ω1 (l, l′) = E

{Q

(√p

2‖slhn − sl′hn′‖2

)}(87)

Ω2 (l, l′) = E

{Q

(√p

2|sl − sl′ |2‖hn‖2

)}(88)

By defining γ = p4σ

2, and following the pdf-based approachin [28, Ch. 9.2.2], for i = 1, 2, we have

Ωi

(l, l′)=

(1− μl,l′

i

2

)M M−1∑m=0

(M − 1 +m

m

)(1 + μl,l′

i

2

)m

(89)

where

μl,l′i =

√γl,l′i

1 + γl,l′i

, i = 1, 2 (90)

γl,l′1 = γ

(|sl|2 + |sl′ |2

), γl,l′

2 = γ(|sl − sl′ |2

)(91)

For high SNRs (γ >> 1), 1+μi

2 ≈ 1, and 1−μi

2 ≈ 14γi

. So, wehave [16, Ch. 14.4.1]

Ωi (l, l′) ≈

(2M − 1

M

)(4γl,l′

i

)−M

, i = 1, 2 (92)

Substituting (92) in (86) results in

PM ≤(

2M − 1M

)(4γ)−M×

⎧⎪⎪⎪⎨⎪⎪⎪⎩

K − L

L2

L∑l=1, l′=1

(γl,l′1

/γ)−M

+1

L

L∑l=1, l′=1

l�=l′

(γl,l′2

/γ)−M

⎫⎪⎪⎪⎬⎪⎪⎪⎭

(93)

APPENDIX BPROOF OF PROPERTY 2 IN THEOREM 1

To prove the convergence of ΦM (L), it is enough to showthat

∑x,y,z,w=1

[I (x, y, z, w)]−M is a convergent series. To

show that, we have

a/2∑x,y,z,w=1

[I (x, y, z, w)]−M = 1

+

a/2∑x,y,z,w=2

[I (x, y, z, w)]−M + 4

a/2∑x,y,z=2

[I (x, y, z, 1)]−M

+ 6

a/2∑x,y=2

[I (x, y, 1, 1)]−M + 4

a/2∑x=2

[I (x, 1, 1, 1)]−M

(94)

Assuming x > 1, we have x (x− 1) > (x− 1)2; so, one can

write

∑x,y,z,w=2

[I (x, y, z, w)]−M ≤∑

x,y,z,w=1

(x2 + y2 + z2 +w2

)−M

(95)Since for positive values of x, y, z and w, we have(x2 + y2 + z2 + w2

)M> x2M−3kykzkwk (where 1 < k <

5/3 is a rational number), one can write

∑x,y,z,w=2

[I (x, y, z, w)]−M <∑

x,y,z,w=1

x−(2M−3k)y−kz−kw−k

=

(∑y=1

y−k

)(∑z=1

z−k

)(∑w=1

w−k

)(∑x=1

x−(2M−3k)

)

(96)

Since∞∑n=1

1nr converges if r > 1, and diverges if r ≤ 1,

and due to the fact that k > 1, the first three series inthe right side of (96) converge, and the last one convergesif and only if M ≥ 3. Therefore, we can conclude thatfor M ≥ 3,

∑x,y,z,w=2

[I (x, y, z, w)]−M converges. Similarly,∑

x,y,z=2[I (x, y, z, 1)]

−M and∑

x,y=2[I (x, y, 1, 1)]

−M are con-

vergent series for M ≥ 2, and∑x=2

[I (x, 1, 1, 1)]−M is a

convergent series for M ≥ 1. Therefore, we can concludethat generally for M ≥ 3, ΦM (L) converges as L → ∞.

APPENDIX CPROOF OF PROPERTY 4 IN THEOREM 1

Using the definition of ΦM (L) in (24), one can write

φM = limL→∞

ΦM (L) = lima→∞ΦM

(a2

)= lim

a→∞

a/2∑x,y,z,w=1

[I (x, y, z, w)]−M

= lima→∞a4

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0[I (ax, ay, az, aw)]−Mdxdydzdw

= lima→∞a4−2M

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0

dxdydzdw[(x− 1

2a

)2+

(y − 1

2a

)2+

(z − 1

2a

)2+

(w − 1

2a

)2]M

=

∫ 1/20

∫ 1/20

∫ 1/20

∫ 1/20

dxdydzdw

(x2+y2+z2+w2)M

lima→∞ a2M−4

(97)

By considering x = r1 cos θ1, y = r1 sin θ1, z = r2 cos θ2and w = r2 sin θ2, we have

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0

∫ 1/2

0

dxdydzdw

(x2 + y2 + z2 + w2)M

= 4

∫ π/4

0

∫ π/4

0

∫ b2

0

∫ b1

0

r1r2dr1dr2dθ1dθ2

(r21 + r22)M

(98)

where bi = 12cosθi

. By replacing r1 = r cosλ and r1 = r sin λ,we have

∫ b2

0

∫ b1

0

r1r2dr1dr2

(r21 + r22)M

=

∫ tan−1(

b2b1

)

0

(∫ b1cos λ

0

dr

r2M−3

)dλ

+

∫ tan−1(

b1b2

)

0

(∫ b2cosλ

0

dr

r2M−3

)dλ

(99)

It is then easy to show that for M ≥ 2, the integral∫ x

0r3−2Mdr does not exist. For M = 1 and M = 2, φM

in (97) goes to infinity which means that Φ1 (L) and Φ2 (L)are divergent series. Also, one can conclude that

limL→∞

Φ1 (4L)

Φ1 (L)=

(2a)2

a2= 4, lim

L→∞Φ2 (4L)

Φ2 (L)= 1 (100)



APPENDIX DPROOF OF PROPERTY 5 IN THEOREM 1

In this appendix, we provide the proof of Property 5. Basedon the definition of ΨM (L) in (26), we can write

ΨM (4L)

ΨM (L)=

∑x,y

κx,y .R4L (x, y)(x2 + y2

)−M

∑x,y

κx,y .RL (x, y) (x2 + y2)−M

>

a−1∑x=0,y=0

κx,y .R4a2 (x, y)(x2 + y2

)−M

a−1∑x=0,y=0

κx,y .Ra2 (x, y) (x2 + y2)−M

(101)

We also have

R4a2 (x, y) = (2a− x) (2a− y)

= 4 (a− x) (a− y) + 2y (a− x) + 2x (a− y) + xy

= 4Ra2 (x, y) + 2y (a− x) + 2x (a− y) + xy

⇒ R4a2 (x, y) > 4Ra2 (x, y) , 0 ≤ x, y ≤ a− 1(102)

Therefore, from (101) and (102), one can conclude thatΨM (4L)ΨM(L) > 4.

APPENDIX EPROOF OF PROPERTIES OF DM (x, ρ)

In this appendix, we prove some of the properties offunction DM (x, ρ) = RM (ρx). By taking the first and secondderivatives, it is easy to show that DM (x, ρ) is an increasingand convex function of M . Also, we have

∂DM

∂ρ=

M

ρ{DM+1(x, ρ)−DM (x, ρ)} (103)

Based on the fact that DM+1(x, ρ) > DM (x, ρ), it can beinduced that DM (x, ρ) is an increasing function of both xand ρ. By taking the derivative of both sides of (103) withrespect to ρ, we have

∂2DM

∂x∂ρ=

M

xρEM (x, ρ) (104)

where

EM (x, ρ) = M {DM+2(x, ρ)− 2DM+1(x, ρ) +DM (x, ρ)}+ {DM+2(x, ρ) −DM+1(x, ρ)} (105)

Based on the convexity of DM (x, ρ) with respect to M ,∂2DM

∂x∂ρ > 0.

APPENDIX FPROOF OF PROPERTY OF DM (x, ρ)

In this appendix, we prove that

∂2DM+1 (cos (x), ρ)

∂x2>

∂2DM (cos (x), ρ)

∂x2(106)

The second derivative of DM (cos (x), ρ) with respect to xcan be expressed as

∂2DM (cos (x), ρ)

∂x2= MDM (x, ρ) (107)

where

DM (x, ρ) =[ρ2

(1 +Msin2 (x)

)− 1]RM+2 (ρ cos (x))

+RM+1 (ρ cos (x)) (108)

For M > 0, it is sufficient to show that DM+1 (x, ρ) >DM (x, ρ). We have

DM+1 (x, ρ)− DM (x, ρ) = α3RM+3 (ρ cos(x))

− α2RM+2 (ρ cos(x))−DM+1 (ρ cos(x))

≥ α3RM+3 (ρ cos(x))− α2RM+2 (ρ cos(x))

+ α1RM+1 (ρ cos(x)) (109)

where α1 = −ρ2sin2 (x)− 1, α2 = ρ2(1 +Msin2 (x)

)− 2,and α3 = ρ2

(1 +Msin2 (x)

)+ ρ2sin2 (x)− 1. Note that

α1 + α3 = α2. We, also, have

α3RM+3 (ρ cos(x)) + α1RM+1 (ρ cos(x))

=α3 + α1

2{RM+3 (ρ cos(x)) +RM+1 (ρ cos(x))}

+α3 − α1

2{RM+3 (ρ cos(x))−RM+1 (ρ cos(x))} (110)

Since α3 > α1, RM+3 > RM+1, and RM is a convex functionof M , we have

α3RM+3 (ρ cos(x)) + α1RM+1 (ρ cos(x))

> (α3 + α1)

{RM+3 (ρ cos(x)) +RM+1 (ρ cos(x))

2

}> (α3 + α1)RM+2 (ρ cos(x)) = α2RM+2 (ρ cos(x))

(111)

From (109) and (111), we can verify the validity of (106).

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Mehdi Maleki received the B.Sc. and M.Sc. degreesin electrical engineering from Amirkabir Universityof Technology (Tehran Polytechnic), Tehran, Iran, in2006 and 2009, respectively. He is currently workingtoward the Ph.D. degree with The University ofAkron, OH, USA. He is also a Research Assistantin wireless communications laboratory (WCL) of theDepartment of Electrical and Computer Engineering,The University of Akron. He regularly serves asreviewer for IEEE transactions/journals and majorconferences. His research interests include digi-

tal communications, digital signal processing, multi-user communications,multiple-input multiple-output wireless systems, cooperative communicationsand cognitive radio networks.

Hamid Reza Bahrami is currently an Assistant Pro-fessor at the Department of Electrical and ComputerEngineering at The University of Akron, OH, USA.His main area of research includes wireless com-munication, information theory, and applications ofsignal processing in communication. He received hisPh.D. degree in electrical engineering from McGillUniversity, Montreal, Canada in 2008. From 2007 to2009 and prior to joining the University of Akron,he was a scientist at Wavesat Inc. Montreal, Canada.Dr. Bahrami has B.Sc. and M.Sc. degrees both in

Electrical Engineering from Sharif University of Technology and University ofTehran, Iran, respectively. He has served as the Editor for TRANSACTIONS ONEMERGING TELECOMMUNICATIONS TECHNOLOGIES, and as the TechnicalProgram Committee member of numerous IEEE conferences including IEEEGLOBECOM and International Conference on Communications (ICC). He iscurrently a member of the IEEE, the IEEE Communications Society, and theIEEE Vehicular Technology Society.

Ardalan Alizadeh is a Ph.D. student at the Depart-ment of Electrical and Computer Engineering at theUniversity of Akron, Ohio. He received his M.Sc.degree from Shahid Beheshti University, Tehran,Iran in 2011 and his B.Sc. degree in ElectricalEngineering from Amirkabir University of Tech-nology, Tehran, Iran in 2008. His current area ofresearch includes cognitive radio networks, MIMOinterference alignment networks, spatial modulationand software defined radio (SDR) implementationby USRP.

Nghi H. Tran received the B.Eng. degree fromHanoi University of Technology, Vietnam, in 2002,and the M.Sc. degree (with Graduate Thesis Award)and the Ph.D. degrees from the University ofSaskatchewan, Canada, in 2004 and 2008, respec-tively, all in Electrical and Computer Engineering.

From May 2008 to July 2010, he was at McGillUniversity, Canada, as a Postdoctoral Scholar underthe prestigious Natural Sciences and EngineeringResearch Council of Canada (NSERC) PostdoctoralFellowship. From August 2010 to July 2011, Dr.

Tran was at McGill University as a Research Associate. He also worked as aConsultant in the satellite industry. In particular, his project with AdvantechSatellite Networks Inc. involved the design and implementation of MODEMalgorithms for the next generation of DVB-RCS satellite systems. SinceAugust 2011, Dr. Tran has been an Assistant Professor in the Department ofElectrical and Computer Engineering, The University of Akron, OH, USA. Dr.Tran’s research interests span the areas of signal processing, communicationand information theories for wireless systems and networks.


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