Non-coherent Massive SIMO Systems in ISI Channels: Constellation Design and Performance Analysis

# Non-coherent Massive SIMO Systems in ISI Channels: Constellation Design and Performance Analysis

Huiqiang Xie, Weiyang Xu, , Wei Xiang, ,
Ke Shao,  Shengbo Xu
This work was supported by the Program for Innovation Team Building at Colleges and Universities in Chongqing, China (Grant No. CXTDX201601006) and the Key Program of Natural Science Foundation of Chongqing under Grant CSTC2017JCYJBX0047.H. Q. Xie, W. Y. Xu and S. B. Xu are with the College of Communication Engineering, Chongqing University, Chongqing, 400044, P. R. China (E-mails: {huiqiangxie, weiyangxu, 20134327}@cqu.edu.cn).Wei Xiang is with the College of Science, Technology & Engineering, James Cook University, Cairns, QLD 4870, Australia (E-mail: wei.xiang@jcu.edu.au).Ke Shao is with the Nanchang Institute of Technology, Nanchang, Jiangxi, 330029, P. R. China (E-mail: shaoke@163.com).
###### Abstract

A massive single-input multiple-output (SIMO) system with a single transmit antenna and a large number of receive antennas in intersymbol interference (ISI) channels is considered. Contrast to existing energy detection (ED)-based non-coherent receiver where conventional pulse amplitude modulation (PAM) is employed, we propose a constellation design which minimizes the symbol-error rate (SER) with the knowledge of channel statistics. To make a comparison, we derive the SERs of the ED-based receiver with both the proposed constellation and PAM, namely and . Specifically, asymptotic behaviors of the SER in regimes of a large number of receive antennas and high signal-to-noise ratio (SNR) are investigated. Analytical results demonstrate that the logarithms of both and decrease approximately linearly with the number of receive antennas, while degrades faster. It is also shown that the proposed design is of less cost, because compared with PAM, less antennas are required to achieve the same error rate.

Energy detection, intersymbol interference (ISI) channel, massive single-input multiple-output (SIMO), constellation design, symbol-error rate (SER).

## I Introduction

Massive multiple-input multiple-output (MIMO) systems, where a large number of antennas are deployed at base stations (BSs) to serve a small number of users sharing the same frequency resources, have recently received a great deal of interest due to its potential gains [1, 2]. For example, massive MIMO is energy efficient as the transmit power scales down with the number of antennas. Meanwhile, channel vectors associated with different users are asymptotically orthogonal, thus both intra- and inter-cell interference can be eliminated with simple detection or precoding algorithms [3, 4, 5]. To reap the benefits that massive MIMO offers, channel state information (CSI) is required at BSs in coherent communications. However, massive antennas make acquiring CSI much more challenging than before. In fact, the computational complexity of channel estimation is so high that estimating all channels in a timely manner becomes infeasible. In addition, the issue of pilot contamination, attributed to reusing pilots among adjacent cells, would make the problem even worse because channel estimates obtained in a given cell will be corrupted by pilots transmitted by users in the other cells[6].

As a promising alternative, non-coherent systems require no knowledge of instantaneous CSI at either the transmitter or receiver[7, 8, 9]. Compared with their coherent counterparts, non-coherent receivers enjoy benefits of low complexity, low power consumption and simple structures at the cost of a sub-optimal performance[10]. Thus, non-coherent receivers are attractive in large-scale antenna systems. Based on the non-overlapping power-space profile without CSI, an optimal decision-feedback differential detector (DFDD) provides significant gains over conventional differential detection [11, 12]. However, the DFDD relies on a particular channel model that cannot be exploited in general. With a large number of antennas, energy detection (ED) finds its application in non-coherent massive single-input multiple-output (SIMO) systems [13]. With a non-negative pulse amplitude modulation (PAM), the transmit symbols can be decoded by averaging the received signal power across all antennas. Since the detection/decoding is performed based on the signal energy, the system should use non-negative signal constellations. For example, non-negative PAM constellations have been documented for two different wireless standards for Millimeterwave (mmWave) short-range communication, ECMA-387 and IEEE802.15.3c [14, 15], respectively. Regarding systems with a large number of antennas, ED was proposed for mmWave communications in [16].

Inspired by the seminal work in [13], non-coherent massive SIMO systems have attracted a lot of attention from the research community [17, 18, 19, 20, 21, 22, 23]. The symbol-error rate (SER) of the ED-based non-coherent SIMO system is derived in [17], based on which a minimum distance constellation is presented. An asymptotically optimal constellation is proposed in [18], and its performance gap to the optimal design could be made small with large-scale antennas or large-size constellations. In[19], two ED-based receivers are proposed, one of which analyzes the instantaneous channel energy based on Gaussian approximations of the probability density function (PDF), while the other analyzes the average channel energy with chi-square cumulative distribution function (CDF). The authors in [21] propose to optimize the constellation with varying levels of uncertainty on channel statistics. Also, it is proved that the non-coherent massive SIMO system satisfies the same scaling law as its coherent counterpart [23]. However, the aforementioned studies focus on intersymbol interference (ISI) free scenarios[17, 18, 21, 19, 20, 23]. Although an ISI channel can be transformed into multiple flat-fading channels by using OFDM, the inherent high peak-to-average-ratio and sensitivity to the carrier frequency offset present new challenges. Different from the above, the authors in [22] consider the use of the ED-based receiver in multipath environments, where a zero-forcing (ZF) equalizer is employed to remove ISI.

In non-coherent massive SIMO systems, optimizing the constellation design can provide significant performance enhancement over conventional PAM. Accordingly, Manolakos et al. propose an optimal constellation to improve the error performance in flat-fading channels [21]. Although both analytical and numerical results show the potential of constellation optimization in ISI-free communications, whether it could reduce the error rate and what the optimal design would be in multipath scenarios are not clear. Towards this end, this paper focuses on designing a constellation that is able to minimize the SER for the ED-based SIMO system with ISI. The main contributions of our study can be summarized as follows:

• In the presence of multipath channels, the generic SER of the ED-based SIMO receiver, which relates to the constellation and decision thresholds, for the case of a finite number of receive antennas is derived;111, and refer to the error probabilities corresponding to general non-negative constellations, PAM and the proposed constellation, respectively.

• Based upon the derived closed-form expression of , we present a constellation design and decoding scheme with the objective of minimizing the error probability. Then, the SERs of both the proposed design and PAM, namely and , are derived for comparison;

• Asymptotic behaviors of and in regimes of a large number of receive antennas and high signal-to-noise ratios (SNRs) are investigated in detail. It is shown that the logarithm of can be approximated as a linearly decreasing function of the number of antennas, and decreases at a faster rate than the logarithm of when more antennas are equipped. Due to the multipath effect and a finite number of antennas, both and exhibit an irreducible error floor at high SNRs. However, converges to a rate much lower than under the same condition.

The remainder of this paper is organized as follows. The system model is presented in Section II. The derivation of SER, optimal constellation design and threshold setting are detailed in Section III. Section IV presents a thorough SER performance analysis under different scenarios. Numerical simulation results are presented to show the effectiveness of our algorithm in Section V. Finally, Section VI concludes this paper.

: indicates a matrix composed of complex numbers of size . Bold-font variables represent matrices or vectors. For a random variable , means it follows a complex Gaussian distribution with mean and covariance . , , and denote the expectation, variance, covariance and norm of the argument, respectively. denotes the Hermitian transpose. and separately refer to the real and imaginary parts of a complex number. is the least upper bound. Finally, and are taken to indicate the Gaussian error function and complementary Gaussian error function, respectively.

## Ii System Model

Consider a SIMO network consisting of a single-antenna transmitter and a receiver with antennas [22]. The channel between the transmitter and each receive antenna is modeled as a finite impulse response (FIR) filter with taps[24]. We assume an independent channel realization from one block to another. The received signal at time can be represented by

 y(t)=L−1∑l=0hls(t−l)+n(t) (1)

where , indicates a complex Gaussian noise vector with elements , refers to the channel realization of the th path with , denotes the transmit symbol drawn from a certain non-negative constellation, and is the number of receive antennas. Our study considers the transmit SNR, which is defined as .

We focus on the following encoding and decoding scheme. It is assumed that both the receiver and transmitter possess no knowledge of instantaneous channel and noise, but the channel and noise statistics are available, i.e., means and variances. The non-negative transmit symbol is selected from a constellation set , subject to the average power constraint

 1KK∑i=1pi⩽1 (2)

where denotes the th constellation point and is the constellation size. Based on the ED principle, after the received signal having been filtered, squared and integrated, the average power across all antennas can be written as

 z(t)=∥y(t)∥22M. (3)

In flat-fading channel, is taken as the decision metric for symbol detection [13]. Accordingly, the positive line is partitioned into multiple decoding regions to decide which symbol was transmitted according to the observation of . In fact, can be approximated to one of the Gaussian variables depending on a priori information of transmitted symbols. For example, with a PAM constellation of , the PDF of over an additive white Gaussian noise (AWGN) channel is shown in Fig. 1 (a), where and . Clearly, four distinct Gaussian-like curves can be observed, corresponding to four constellation points. As can be seen from this figure, there is a notable overlap region between and , caused by additive noise and a finite number of antennas. Furthermore, Figs. 1 (a) and (b) indicate that the overlap region enlarges when the number of channel taps increases. This overlap will make it difficult to separate these two decoding regions, and thus decision-making between and is prone to errors. Although deploying more antennas helps reduce this overlap, as shown in Figs. 1 (b) and (c), it incurs an extra cost. Therefore, an optimal constellation is essential to reducing the error probability.

## Iii The Proposed Constellation Design and Threshold Optimization

In this section, a closed-form expression of SER of ED-based receivers with ISI is derived. Accordingly, our constellation design and threshold optimization is proposed to minimize the error probability.

### Iii-a SER of the ED-based Receiver in ISI Channels

We assume that the knowledge of channel and noise statistics is available at the receiver. First, in (3) with a finite number of antennas can be expanded as [22]

 (4)

where the first component contains the desired signal. When , both and converge to zero and the noise component , leaving the only component that affects the SER. However, since can never be infinite, the non-zero , and would adversely affect the error performance. Hence, a closed-form expression of SER, which is the basis for our constellation design, can only be accurately derived when taking into consideration of a finite number of receive antennas. The derivation of the PDF of , which is a non-trivial task, is required to calculate the error rate. Towards this end, we resort to the following lemma.

###### Lemma 1

If the number of receive antennas grows large, then the following approximations are attainable thanks to the Central Limit Theorem (CLT)

 z(t)∼N(μz,σ2z) (5)

with and variance shown as follows

 μz=σ2h0|s(t)|2+L−1∑l=1σ2hl|s(t−l)|2+σ2nσ2z=1M(L−1∑l=0σ2hl|s(t−l)|2+σ2n)2

where indicates a real Gaussian variable.

###### Proof:

The proof can be found in Appendix A. \qed

Specially, the Gaussian approximation is accurate since tends to be large, even when SNR is low. According to Lemma 1, can represented as follows

 z(t)=μz+N(0,σ2z)=σ2h0|s(t)|2+L−1∑l=1σ2hl|s(t−l)|2+σ2n+δ(t) (6)

where is a nondeterministic item and .In multipath channels, a ZF equalizer is employed to remove ISI in before symbol detection [22]. The ZF equalization matrix can be computed by [22]

 wZF=ed(GTG)−1GT (7)

where is an all zero vector with the -th entry being unity, which how to compute can find in [22], and

 G=⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣σ2h0σ2h1⋯σ2hL−10⋯00σ2h0σ2h1⋯σ2hL−1⋯0⋮⋮⋱⋱⋱⋱⋮0⋯0σ2h0σ2h1⋯σ2hL−1⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦.

where , is the length of equalizer. The analytical formula for the decision metric is defined as

 ψ(t)=J−1∑j=0wZFjz(t−j)=|s(t)|2+wσ2n+J−1∑j=0wZFjδ(t−j) (8)

where is the -th elements in . The equalizer will work well when , the third item in (8) will converge to zeros. However, since can never be infinite, the third item in (8) would adversely affect the error performance.

The configuration of a non-coherent massive SIMO receiver is shown in Fig. 2. It is worth noting that after equalization, denoted by , still follows the Gaussian distribution because of the linear ZF equalizer. For ease of derivation, we denote the previous symbols by average power . When the current transmit symbol , the decision metric follows the Gaussian distribution, i.e.

 ψ(t)∼N(μψ(pi),σ2ψ(pi)) (9)

where the mean and variance of is shown as following and in (11) on the top of next page

 μψ(pi)=pi+wσ2n (10)

where is a constant computed by multiplying the equalizer coefficients with an all-one vector [22]. The aforementioned results make the derivation of SER straightforward.

###### Proposition 1

With a finite number of receive antennas, the SER of the ED-based receiver in ISI channels is given below

 Pe=1−1KK∑i=1P(pi)=1−12KK∑i=1(%erf(dL,i√2σψ(pi))+%erf(dR,i√2σψ(pi))) (12)

where and are values of thresholds shown in Fig. 3, denotes the probability of correct decision on . Specifically, and . The shaded area in this figure, which is decided by and , indicates the decision region for .

###### Proof:

The proof can be found in Appendix B. \qed

It is worth noting that the SER in (12) is a generalized result suitable for a variety of non-negative constellations. Given variance and decoding thresholds, one can obtain the corresponding error probability. Intuitively, some interesting remarks are made as follows:

• The SER over flat-fading channels can be obtained by setting in (12). Moreover, the multipath effect of ISI channels could incur performance degradation compared with case of the flat-fading channel;

• Deploying more receive antennas is a straightforward and effective approach to reduce the error probability, because when grows unlimited, we have , , , and finally ;

• If and are fixed and , converges to a steady state independent of . This is equivalent to that an error floor appears when the SNR is larger than a certain value. Therefore, a requirement of high SNRs is not critical in this scenario.

Before further analysis, the influence of a ZF equalizer on SER needs to be clarified. Although it is designed to remove the influence of ISI, the ZF equalizer causes another problem of noise enhancement. This is even worse in our study since the equalization increases the variance of ISI components in (4) at the same time, making the decision between neighboring PDFs more prone to errors. This performance degradation can be minimized by constellation design.

### Iii-B Proposed Constellation Design

It is readily observed from (12) that the error probability decreases with the decrease of , which relates to . This observation, coupled with the relationship between SER and or , clearly demonstrates the potential to improve the error performance via optimizing the constellation. Obviously, minimizing the average symbol error probability equals to maximizing the probability of correct decision, i.e.

 (13) Subject to

where represents the decision region , which is subject to the constraint of transmit power. However, solving (13) is not straightforward. First, the Cauchy-Schwarz inequality is utilized to simplify the problem solving process.

###### Lemma 2

In Euclidean space with standard inner product, the Cauchy-Schwarz inequality states that for all sequences of real numbers and , we have

 (n∑i=1aibi)2⩽(n∑i=1a2i)(n∑i=1b2i) (14)

where the equality holds if and only if for a certain constant .

If we set , the Cauchy-Schwarz inequality can be rewritten as

 1nn∑i=1ai⩽√∑ni=1a2in (15)

where the equality holds if and only if . As a result, the maximum of (13), or equivalently the maximum of is achieved if , which can be expanded as

 erf(dL,1√2σψ(p1))+erf(dR,1√2σψ(p1))=erf(dL,i√2σψ(pi))+erf(dR,i√2σψ(pi))=⋯=erf(dL,K√2σψ(pK))+erf(dR,K√2σψ(pK)). (16)
###### Proposition 2

The average symbol error probability is convex in the space spanned by .

###### Proof:

The proof can be found in [25]. \qed

According to Proposition 2, is convex with respect to in region , . Thus, can be maximized if

 erf(dL,i√2σψ(pi))=erf(dR,i√2σψ(pi)). (17)

Submitting (17) into (16), the following result is obtained

 erf(dR,1√2σψ(p1))=erf(dL,2√2σψ(p2))=erf(dR,2√2σψ(p2))=⋯=erf(dL,K√2σψ(pK)). (18)

This equation indicates that to minimize the overall SER, the number of errors with respect to each constellation point should be the same. This observation is quite different from the case of PAM constellations. As can be observed from (18) and Fig. 3, two important results can be obtained, i.e.

 dR,iσψ(pi) =dL,i+1σψ(pi+1)=pi+1−piσψ(pi+1)+σψ(pi)=√2T, (19) dR,i =dL,i,i=1,2,…,K

where is defined for the ease of analysis.

Since is a monotonically increasing function of its argument, maximizing is equivalent to maximizing the probability of correct decision. Thus, the optimization problem in (13) can be transformed into

 maximize{P,△1,…,△K} (20) Subject to

The first constraint in (20) can be rewritten as

 pi+1−pi=√2T(σψ(pi+1)+σψ(pi)). (21)

Given a known , can be calculated by (11). Thus (21) is transformed into

 (pi+1−pi−√2Tσψ(pi))2=2T2σ2ψ(pi+1). (22)

Afterwards, with a fixed and an initial value of , the problem can be converted to the following quadratic equation

 A(T)p2i+1+B(T,pi)pi+1+C(T,pi)=0 (23)

where

 A(T) =w2σ4h0M−12T2 (24) B(T,pi) =piT2+√2σψ(pi)T+2w2σ2h0σ2nM+2w2σ2h0MKL−1∑l=1σ2hl C(T,pi) =C1−C2(T,pi) C1 C2(T,pi) =(pi√2T+σψ(pi))2.

If we set the initial value of 222In general, is initialized to be zero., can be calculated iteratively. For example, (24) shows that both and relate to , thus the constraint condition with can be solved as follows

 pi+1=√2T(√Mσψ(pi)+wσ2n+wKL−1∑l=1σ2hl)+pi√M−√2Twσ2h0 (25)

Notice that the range of , which are the bounds of bisection, , guarantee the solution is real positive and are related with the number of antennas. Because the large number of antennas at the receiver, the range is enough large to find out optimize constellation solution. The result can be found in Appendix C. With the same approach, one can obtain . Then, the optimal problem (20) can be represent as

 maximize{P,△1,…,△K} (26) Subject to
###### Proposition 3

is an increasing function w.r.t. .

###### Proof:

The proof can be found in Appendix C. \qed

If satisfies the power constraint, an optimal constellation is obtained. If not, needs to be adjusted accordingly. If the average power is less than power constraint, needs to be increased. If not, will be decreased. At last, a simple method of bisection can be employed to calculate the optimal . In conclusion, the solution to (26) can be summarized in a step-by-step manner in Algorithm 1, where and indicate the lower and upper bounds of bisection range, respectively.

Fig. 4 (a) compares the proposed constellation and a non-negative PAM at various SNRs. It is shown that the distance between and of the proposed design is larger than that of PAM at low SNRs. However, the distances between other neighboring of our design are smaller compared to the case of PAM. This is reasonable because the error decision between and plays the most important role in computing the SER. Moreover, our design converges to PAM gradually with the increase of SNR. Similar results is obtained if we analyze the behavior of when the number of channel taps varies, as shown in Fig. 4 (b). Thus, the following remarks are made:

• In case of any change of parameters that would result in a larger , the distance between and will enlarge to make the decision between and less prone to errors;

• As a result, the distances between other neighboring would decrease accordingly to keep the same constraint of transmit power.

In this way, the proposed design is capable of adaptively optimizing according to the channel and noise statistics.

### Iii-C Threshold Optimization

Based on (19), the decision metric can be decoded according to the maximum likelihood or other rules[23]. Given the right and left distances of and shown in Fig. 3, the decision threshold for can be obtained as

 dL,i=dR,i=√2Tmaxσψ(pi). (27)

The optimized decision boundaries between two neighboring constellation points is then denoted as

 treLi =pi+wσ2n−dL,i (28) treRi =pi+wσ2n+dR,i.

With the optimized threshold, a transmit symbol can be decoded as follows

 ^s(t)=⎧⎪ ⎪⎨⎪ ⎪⎩√p1, ψ(t)⊆(−∞,treR1],√pi, ψ(t)⊆(treLi,treRi], i=2,3,…,K−1√pK, ψ(t)⊆(treLK,+∞). (29)

In the light of (18), the probability of correct decision is consisted of the same value

 P(pi)=2erf(Tmax),i=1,2,…,K. (30)

Then, according to the maximized and decision thresholds, the error probability in (12) can be approximated as

 Pe_opt≈1−1K((K−1)erf(Tmax)+1). (31)

Therefore, finding the minimum is equivalent to maximizing , because is a monotonically increasing function of its argument.

### Iii-D Relation with the Rate Function Scheme

Among the existing publications, the one that is most related to our proposed scheme is the one presented in [13], which is based on the rate function. This scheme was shown to be the first constellation design for non-coherent massive SIMO based on ED. However, it only explores the scenario of flat-fading channel. Motivated by this, we will next compare this scheme with the proposed design. First, we briefly review the key idea of the rate function scheme, which is represented by the following lemma.

###### Lemma 3

For any and zero mean i.i.d. random variables , we have[13]

 P(∑Mi=1uiM⩾d)⩽e−M⋅I(d) (32)

where is the rate function.

Based on lemma 3 and moment generating function of , the upper bound of SER can be obtained, i.e.

 PU≜1KK∑i=1⎛⎜⎝e−Md2R,i2k(pi)+e−Md2L,i2k(pi)⎞⎟⎠ (33)

where , and with denoting the signal received by the th antenna at time instant , and being the channel between the transmitter and the th antenna in the scenario of flat-fading channel.

To have a deep understanding of the difference between these two schemes, we consider the application of the proposed constellation design in flat-fading channel. Hence, the received signal and decision metric can be rewritten as

 r(t)=hs(t)+n(t) (34) ψflat(t)=∥r(t)∥22M (35)

where and . Based on the CLT shown in Appendix A, also follows the Gaussian distribution. The relationship between and the variance of , which is denoted by , is

 (36)

As mentioned earlier, as long as the threshold and variance are known, a closed-form expression of SER can be obtained. According to (12), the SER of a non-coherent SIMO system in flat-fading channels can be written as

 (37)
###### Lemma 4

The complementary error function approaches its limit when as follows[26]

 erfc(x)≈e−x2√πx,ifx→+∞.

Using Lemma 4, (37) is able to be approximated as

 Pe,flat≈1KK∑i=1√k(pi)2πM⎛⎜ ⎜ ⎜⎝e−Md2R,i2k(pi)dR,i+e−Md2L,i2k(pi)dL,i⎞⎟ ⎟ ⎟⎠. (38)

The following inequalities are easily obtained when is large

 k(pi)2πMd2R,i<1,k(pi)2πMd2L,i<1. (39)

As a consequence, the upper bound of in (38) is

 Pe,flat<1KK∑i=1⎛⎜⎝e−Md2R,i2k(pi)+e−Md2L,i2k(pi)⎞⎟⎠=PU. (40)

From (40), an interesting conclusion can be obtained. Through the scale of (38), the same upper bound of SER is observed as in [13]. Therefore, the effectiveness of our proposed scheme in flat-fading channel is validated.

In conclusion, this paper provides a general framework for the constellation design in ED-based non-coherent massive SIMO systems. The results are applicable in both flat-fading and multipath-fading channels.

## Iv Performance Analysis and Discussion

In this section, we discuss the influence of key parameters on the error probability, including the number of receive antennas , SNR and constellation size. The non-negative PAM and our proposed constellation design are included for comparative performance study.

In Section III-A, we have derived the SER of the ED-based receiver with the proposed constellation design. For comparative purposes, the SER expression for a non-negative PAM constellation is given as follows

 Pe_pam=12KK∑i=1(erfc(dR_pam,i√2σψ(pi_pam)))+12KK∑i=1(% erfc(dL_pam,i√2σψ(pi_pam))) (41)

where , . The constellation of a non-negative PAM scheme is denoted by . The result in (41) can be obtained straightforwardly by using the same approaches as in Appendix A, with the PAM constellation and decision thresholds from [22].

### Iv-a SER Approximation

By applying Lemma 4 to (31) and (41), it can be shown that SERs of the proposed constellation design and PAM are expressed as

 (42)

When the constellation size of the non-negative PAM is small, e.g., , is dominated by the component of in (42) [19]. For example, simulation results indicate that the error decision between and accounts for up to 99.1% of the overall errors when and at . Moreover, this phenomenon is also observed in Fig. 1. Therefore, the SER of PAM reduces to

 Pe_pam≈1K√12πσψ(p1_pam)e−d2R_pam,12σ2ψ(p1_pam)dR_pam,1+1K√12πσψ(p2_pam)e−d2L_pam,22σ2ψ(p2_pam)dL_pam,2. (43)

It follows from (41) that and . Simulation results indicate that the error decision on accounts for up to 77.64% of the overall errors when and at . For convenience of analysis, the first term in (43) is removed333This approximation in fact decreases the SER of the ED receiver employing non-negative PAM, and thus it represents the worse-case scenario for our constellation design in terms of performance comparison.. Therefore, we can obtain the following logarithmic SER

 logPe_pam≈−d2L_pam,22σ2ψ(p2_pam)loge+log√12πσψ(p2_pam)KdL_pam,2. (44)

Similarly, the logarithmic operation is applied to the SER of the proposed design, i.e.

 logPe_opt≈−T2maxloge+logK−1√πKTmax. (45)

Based upon the results in (44) and (45), we now can compare the error performances between the proposed constellation design and PAM in the event of a larger number of receive antennas and high SNRs.

### Iv-B Influence of a Finite Number of Receive Antennas

#### Iv-B1 SER as a Function of a Finite Number of Antennas

In order to study the influence of a finite number of receive antennas on SER, we fix the SNR and constellation size. Therefore, the variance in (11) can be expressed as a function of

 σψ(pi,M)=w√1Mζ(pi) (46)

where . Since the SNR is constant in this scenario and is a propagation-related parameter, only relates to . It follows from (19) that

 Tmax=doptw√2ζ(popt)√M. (47)

Through substituting (47) into (45), the logarithm of is expressed as a function of

 logPe_opt=−d2opt2w2ζ(popt)Mloge−12logM+log√2πζ(popt)(K−1)wKdopt (48)

where the third term is a constant irrelevant of . In addition, if grows large, the second component in (48) plays a much less significant role compared with the first one. As a result, approximates to a linear decreasing function of . This confirms that deploying more antennas is an effective way to reduce decoding errors.

#### Iv-B2 Non-negative PAM versus the Proposed Optimal Constellation

Applying the same approach in (46)-(48), the error performance of non-negative PAM can be represented as

 logPe_pam=−d2L_pam,22w2ζ(p2_pam)Mloge−12logM+log√ζ(p2_pam)2πwKdL_pam,2 (49)

which can also approximate to a linear decreasing function of when , the same as in (48). It is readily observed that the key to performance comparison between and lies in their slopes with respect to . After removing constants in common, it is equivalent to comparing and .

First of all, it proves that