2012 IEEE 17th International Workshop on Computer Aided Modeling and Design of Communication Links and Networks (CAMAD)
Towards the Performance and Energy Efficiency Comparison of Spatial Modulation with Conventional Single-Antenna Transmission over Generalized Fading Channels Konstantinos Ntontin(1) , Marco Di Renzo(2) , Ana Perez-Neira(1) , and Christos Verikoukis(1) (1)
Telecommunications Technological Centre of Catalonia (CTTC) Castelldefels, Spain Email:
[email protected],
[email protected],
[email protected] (2) L2S, UMR 8506 CNRS - SUPELEC - Univ Paris-Sud, Laboratory of Signals and Systems (L2S) French National Center for Scientific Research (CNRS), École Supérieure d’Électricité (SUPÉLEC) University of Paris-Sud XI (UPS), 3 rue Joliot-Curie, 91192 Gif-sur-Yvette (Paris), France Email:
[email protected]
Abstract—In this paper, we present a performance and energy efficiency comparison of the recently devised single RF MultipleInput-Multiple-Output (MIMO) concept of Spatial Modulation (SM) with the conventional single-antenna transmission method with receive Maximum Ratio Combining (MRC), which we denote as single-antenna/MRC, for different types of fading channels. In particular, the fading channels that are examined are: i) Rayleigh, ii) Nakagami-m, and iii) Weibull. Results that are based on Monte Carlo simulations show two main findings: i) the severity of fading has a big impact on the relative signal-tonoise (SNR) difference and energy efficiency gain between these two schemes for a target average bit error probability (ABEP); ii) the more severe the fading is, the more suitable SM tends to be with respect to the single antenna/MRC, according to the energy per bit required to achieve a particular low target ABEP.
I. I NTRODUCTION SM is a recently proposed single RF MIMO technique, which allows information to be conveyed in two ways: i) through a spatial-constellation diagram; and ii) through a signal-constellation diagram [1], [11]. Due to its innovative nature of conveying information through the antenna indices besides the conventional way of using a signal-constellation diagram, it achieves a multiplexing gain with only one active antenna at any time instant, which scales with the logarithm of the number of transmit antennas. This makes SM a suitable transmission method in the Uplink for mobile terminals that are likely to be equipped with multiple antennas, which is a cheap technology, but only a single RF chain due to their size constraints and the associated cost and complexity issues of designing terminals with multiple RF chains [2]. Since the inception of the SM concept few years ago, only few works in literature have tried to give an answer on how its performance is affected by the fading type of the channel. Specifically, for Rayleigh channels, in [1] the authors study analytically the performance of SM and prove that its diversity order is equal to the number of receive
978-1-4673-3125-8/12/$31.00 ©2012 IEEE
antennas for these types of channels. However, they prove that only for one-dimensional constellation diagrams, such as Pulse Amplitude Modulation (PAM), and not for the widely used two-dimensional ones, such as Phase Shift Keying (PSK) and Quadrature Amplitude Modulation (QAM). In addition, the Union Bound formula that they provide for the calculation of the ABEP is quite loose. In [3], the authors solve this issue by providing a tight Union Bound for the calculation of the ABEP and proving that the diversity order of SM is also equal to the number of receive antennas for PSK and QAM constellation diagrams. Furthermore, the results show the interested and unexpected outcome that for some MIMO configurations and target rates a SM system with PSK has a better performance in terms of ABEP than a SM system with QAM, which is the opposite of what would be expected from conventional methods. Moreover, a performance comparison in terms of ABEP between SM and single-antenna/MRC [4] at the receiver is illustrated. The main findings of this comparison is that SM with PSK and SM with QAM always outperform single-antenna/MRC with PSK and QAM, respectively, when the number of receive antenna sis greater than one and the supported rate is at least equal to 3 bits/s/Hz. As far as other types of fading channels are concerned, only the case of Nakagami-m fading has been examined for SM up until now. In particular, in [3] the authors prove that when m 1 the diversity order of SM systems in Nakagami-m fading channels is equal to the number of receive antennas. When m < 1, the diversity order of such a system is equal to the product of m with the number of receive antennas. This differs form the diversity order of single-antenna/MRC systems in Nakagami-m fading channels in which the diversity order is always equal to the product of m with the number of receive antennas. Apart from the fact that the performance of SM in comparison with single-antenna/MRC has only been investigated
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in a limited number of fading channels, such as Rayleigh and Nakagami-m, and to the best of the authors’ knowledge, no work in literature has dealt with their energy efficiency comparison. Such a comparison is essential because since its introduction SM has been widely advertised as a low complexity single RF MIMO technique with great potentials for energy efficiency gains. Contribution: Due to the lack of literature work regarding the energy efficiency comparison of these two competitive single RF methods, SM and single-antenna/MRC, in this paper we present such a comparison for a target (uncoded) ABEP by taking into account the corresponding energy consumption model and for different MIMO configurations and correlation values among the transmit antennas. Three types of fading channels are examined: i) Rayleigh, ii) Nakagami-m, and iii) Weibull. The latter type of fading channels has shown to exhibit an exellent fit to experimental fading channel measurements, for both indoor, as well as outdoor environments [5]. Nakagami-m and Weibull are more flexible models than Rayleigh and they can model fading conditions less severe and more severe than Rayleigh. These different conditions will be examined in the energy efficiency comparison. All in all, we believe that this study is beneficial for the designer of a communications system in order to have an idea of under which setups and channel conditions SM is more energy efficient than its competitor single-antenna/MRC. Organization: The rest of the paper is organized as follows. In Section II, the channel model and an overview of the SM concept are given. In Section III, the energy consumption model is presented. In Section IV, we first present Monte Carlo simulations of the performance comparison between SM and single-antenna/MRC for different MIMO setups, correlation values among the transmit antennas, and channel conditions. From these curves, we obtain the energy per bit required to achieve a target ABEP and we show the energy efficiency comparison of these two schemes versus the amount of correlation among the transmit antennas. Finally, Section V concludes this paper. Notation: The following notation is used throughout this paper: i) Es is the average energy per transmitted symbol; ii) n is the Nr -dimensional, where Nr is the number of receive antennas, Additive White Gaussian Noise (AWGN) at the receiver input, with both real and imaginary parts of each of spectral density (PSD) equal the Nr elements having a power √ ∞ to N0 /2; iii) Q(x) = (1/ 2π) x exp(−u2 /2)du denotes the Q-function; iv) E {·} denotes expectation; and v) vectors are denoted in boldface. II. C HANNEL M ODEL AND OVERVIEW OF SM A. Channel Model We consider the Uplink of a single carrier point-to-point MIMO communication system with a mobile terminal having Nt transmit antennas, but only one RF chain, and a Base Station having Nr receive antennas. We assume that the receiver uses a Maximum-Likelihood (ML) detector [6]. Moreover, the receiver is able to estimate the channel impulse
responses perfectly during a pilot-based training estimation period before the data transmission. Consequently, the receiver has a perfect channel state information (CSI). Furthermore, we assume that a feedback channel does not exist between the mobile terminal and the Base Station, which can be due to a fast fading channel for instance. Hence, with a single RF chain and without the opportunity to do antenna selection due to the lack of feedback, the only possible transmission schemes are SM and the single antenna transmission. In addition, due to the limited space in the mobile terminals we assume that there is correlation among the transmit antennas, but not among the receive antennas since in Base Stations space in not such an issue as it is for small mobile devices. We assume that the the correlation among the transmit antennas follows the exponential correlation model with entries rij = r|i−j| , i = 1, 2, ..., Nt , j = 1, 2, ..., Nt , where r is the correlation coefficient between adjacent antennas and 0 r 1. The channel matrix H ∈ CNr ×Nt has identically distributed entries hji , j = 1, 2, ...Nr , i = 1, 2, ...Nt . We denote 2 E |h| = 2σ 2 . Now, let us present the probability density function f|h| for each type of the examined fading channels. 1) Rayleigh: For Rayleigh fading channels: 2 2 x f|h| (x) = 2 e−x /2σ , x 0 (1) σ 2) Nakagami-m: For Nakagami-m fading channels: f|h| (x) =
2mm 2m−1 −mx2 /2σ 2 e , mx Γ (m) (2σ 2 )
x0
(2)
The parameter m determines the severity of fading. When m < 1, the channel conditions are more severe than Rayleigh. More severe than Rayleigh means that some of the multipath components in the radio paths may be blocked in certain directions. When m > 1, the channel conditions are less severe than Rayleigh, which means that in the radio path strong components exist, not only multipath components of similar power. Such components can be for instance the line-of-sight (LOS) component or components that reach the receiver with only a few ground reflections. For m = 1, the Rayleigh case is obtained. 3) Weibull: For Weibull fading channels: b b−1 −xb /2σ2 x e , x0 (3) 2σ 2 When b < 2, the channel conditions are more severe than Rayleigh, less severe when b > 2, and Rayleigh when b = 2. f|h| (x) =
B. Overview of SM Let us give a brief description of the SM principle [1]. Assuming that Nt is a power of two and Q is the size of the signal-constellation diagram, a block of log2 (Nt ) + log2 (Q) information bits at the transmitter is divided into two subblocks of log2 (Nt ) and log2 (Q) bits each. The bits in the first sub-block are used to select the transmit antenna that is activated for transmission, while the other antennas are kept silent. The bits in the second sub-block are used to select the symbol sq (q = 1, 2, ..., Q) that is transmitted from the active
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TX:
DAC
×
Filter
If the channel path loss follows the square-law, Pt can be calculated as [7, Eq. (1)]
Power Amplifier
Mixer Filter
channel
Pt = E b R b ×
LO
Fig. 1.
Transmit RF chain.
antenna. If the symbol sq is transmitted from the th antenna, = 1, 2, ..., Nt , the received signal vector is given by y=
Es h s q + n
(4)
where h ∈ CNR ×1 is the channel vector with elements that correspond to the channel paths from the th transmit antenna to all the receive antennas. At the receiver, the ML detector solves a Nt ×Q hypothesis detection problem and jointly estimates the index of the activated antenna and the symbol sq that is transmitted from it. Hence, the total log2 (Nt ) + log2 (Q) bits are recovered.
Pamp = (1 + α) Pt
(5)
where α = ξ/η − 1 with η being the drain efficiency of the power amplifier and ξ being the peak-to-average-power ratio (PAPR) that is dependent on the modulation type and the size of the constellation diagram. For single carrier systems and assuming a set of complex and equally likely symbols sq , q = 1, 2, ..., Q, where Q is the modulation order, ξ is defined as the ratio between the maximum squared amplitude of sq and the average of all the squared amplitudes [8]. Assuming that the constellation diagram has a normalized power that is equal to unity, then ξ = max |sq |
2
(6)
From (6), we can deduce that ξ is equal to one for Phase Shift keying (PSK) constellation diagrams, regardless of the modulation order, but for Quadrature Amplitude Modulation (QAM) diagrams it depends on the modulation order since QAM is a multi-amplitude modulation scheme in contrast to PSK. As a result, its PAPR is greater than one for a modulation order higher than four.
(7)
where Eb is the required energy per bit at the receiver to achieve a target ABEP, Rb is the bit rate, d is the distance between the transmitter and the receiver, Gt and Gr are the transmit and receive antenna gains, respectively, λ is the carrier wavelength, Ml is the link margin, and Nf is the noise figure of the receiver. The circuit power consumption of the single RF transmit chain is calculated as [7, Eq. (3)] Pcircuit = PDAC + Pmix + Pf ilt + Psyn
(8)
where PDAC , Pmix , Pf ilt , Psyn denote the power consumption of the digital-to-analog converter, the mixer, the filter, and the frequency synthesizer, respectively. The total energy consumption per bit Etot at the transmit RF chain can be obtained as [7, Eq. (4)] Etot = (Pamp + Pcircuit ) /Rb
III. E NERGY C ONSUMPTION M ODEL By considering the Uplink, we are interested in the energy consumption at the transmit side only, not the receive side, since battery-powered mobile terminals are much more in need of energy savings than the Base Stations. To model the energy consumed at the transmit side, we need to know the electronic elements that the transmission chain is consisted of. The chain is depicted in Fig. 1 and it is based on [7]. According to [7], the total power consumption at a transceiver consists of two components: i) the power that is consumed at the power amplifier, denoted as Pamp , and ii) the power consumed at all the other circuit blocks, denoted as Pcircuit . A common approximate model used assumes a linear relationship between Pamp and the transmit power Pt [7, Eq. (2)]:
2
(4πd) Ml N f Gt Gr λ 2
(9)
By using (5), (7), and (8), we get from (9) Etot = ξEb ×
2
(4πd) Pcircuit Ml N f + ηGt Gr λ2 Rb
(10)
Let us now consider two different single RF schemes. Since only the transmission method changes, both schemes have the same value for the Pcircuit , d, Gt , Gr , η, λ, Ml , and Nf parameters. Assuming the same Rb and target ABEP for both schemes, (10) reveals something very interesting: The scheme that requires a lower Eb to satisfy the target ABEP does not necessarily have a lower Etot because the ξ parameter that corresponds to the PAPR is also involved in (10). IV. N UMERICAL R ESULTS The aim of this section is to provide an insight into the corresponding trends in the performance, based on Monte Carlo simulations, and energy efficiency comparison for a target ABEP of SM with single-antenna/MRC for different MIMO configurations, correlation values among the transmit antennas, and fading channels. We assume that a rate of 4 bits/s/Hz can be supported and the target ABEP for the energy efficiency comparison is 10−5 . Regarding the energy consumption, we use the parameters of Table I, which have been obtained from [7]. As far as the generation of correlated Nakagami-m random variables is concerned. we used the method of [9]. For correlated Weibull variables, we used the method of [10]. Finally, we set σ 2 = 0.5. A. Rayleigh For Rayleigh channels, we see in Fig. 2a) how both methods compare in terms of performance. In particular, we see that for both the Nr = 2 and the Nr = 3 cases, SM outperforms single-antenna/MRC in the uncorrelated case. For the highcorrelation case, we see a significant loss of performance for the SM case, whereas the single- antenna/MRC method
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TABLE I S YSTEM PARAMETERS
4 bits/s/Hz
a)
0
10
Single−antenna/MRC, 16−QAM SM, Nt=2, 8−QAM, r=0
−1
10
η = 0.35 Symbol Rate = 15 ksymbols/sec N0 = −174 dBm/Hz Ml = 40 dB Nf = 10 dB Psyn = 50 mW
SM, Nt=4, QPSK, r=0 SM, N =2, 8−QAM, r=0.8 t
−2
ABEP
fc = 2.5 GHz d = 750 m Gt Gr = 5 dBi Pmix = 30.3 mW Target ABEP = 10−5 Pf ilt = 2.5 mW
SM, Nt=4, QPSK, r=0.8
10
N =2 r
−3
10
Nr=3 −4
10
−5
10
0
5
10
15 20 E /N [dB] s
100
30
35
Nt=2, 8−QAM, Nr=2
b)
Nt=4, QPSK, Nr=2 N =2, 8−QAM, N =3 t
Relative EE Gain (%)
80
r
N =4, QPSK, N =3 t
r
60
40
20
0 0
0.1
0.2
0.3
0.4 r
0.5
0.6
0.7
0.8
Fig. 2. SM and single antenna/MRC comparison in Rayleigh channels in terms of: a) ABEP vs. Es /N0 and b) Relative energy gain vs. correlation factor. 4 bits/s/HZ
a)
0
10
Single−antenna/MRC, 16−QAM SM, Nt=2, 8−QAM, r=0
−1
10
Nr=2
SM, N =4, QPSK, r=0 t
SM, N =2, 8−QAM, r=0.8 t
−2
ABEP
remains unaffected of course from the channel correlation. As we see, in the high correlation case of r = 0.8 SM is outperformed with respect to single antenna/MRC. Regarding the energy efficiency comparison, Fig. 2b) shows the relative energy efficiency gain of SM over single antenna/MRC versus the correlation factor. We observe that much higher gains are expected for increasing number of receive antennas. In addition, we see that also in the high correlation case SM is more energy efficient than the single antenna/MRC although it is outperformed in the ABEP comparison. This is due to the lower PAPR of SM due to the use of a smaller modulation order. In particular, we have: ξQP SK = 0 dB, ξ8−QAM = 1.98 dB (we used a circular constellation diagram for 8-QAM), and ξ16−QAM = 2.55 dB. This clearly shows that the ABEP trends do not necessarily fall in line with the energy efficiency trends, which makes the energy efficiency analysis essential.
25
0
SM, Nt=4, QPSK, r=0.8
10
−3
10
N =3 r
−4
10
B. Nakagami-m
−5
10
C. Weibull Finally, we examine how both methods compare in performance and energy efficiency in the case of Weibull fading. Fig. 5 shows the b = 3 case (fading less severe than Rayleigh). The trends fall in line with the corresponding trends of the m = 2 case in Nakagami-m fading channels. A higher diversity order can be observed for the single antenna/MRC case. Fig. 6 shows the results for the b = 1.5 case (fading more severe than Rayleigh). Again SM outperforms single
0
100
5
10
15 20 Es/N0 [dB]
25
30
35
b)
0
Relative EE Gain (%)
For Nakagami-m fading channels, Fig. 3a) shows the performance comparison for the two methods when m = 2 (fading less severe than Rayleigh). From that Figure, it is clear that single-antenna/MRC significantly outperforms SM due to a higher diversity gain, which comes in agreement with the analysis in [3]. For increasing number of receive antennas, this gap tends to be smaller. As far as the relative energy gain is concerned, we see in Fig. 3b) that SM is much more energy inefficient than the single-antenna/MRC for m = 2. This energy inefficiency becomes smaller for Nr = 3 and for Nt = 4 diminishes due to the PAPR reduction. Fig. 4a) shows the performance comparison for the case of m = 0.7 (fading more severe than Rayleigh). As we can see, the situation totally changes with respect to the m = 2 case. SM outperforms single-antenna/MRC for all the setups, even when there is high correlation among the transmit antennas. This outperformance is also verified in Fig. 4b) in the relative energy gain results.
−100 −200 N =2, 8−QAM, N =2 −300
t
r
N =4, QPSK, N =2 t
−400
r
N =2, 8−QAM, N =3 t
r
N =4, QPSK, N =3 −500 −600 0
t
r
0.1
0.2
0.3
0.4 r
0.5
0.6
0.7
0.8
Fig. 3. SM and single antenna/MRC comparison in Nakagami-m channels with m = 2 in terms of: a) ABEP vs. Es /N0 and b) Relative energy gain vs. correlation factor.
antenna/MRC in all the setups, like in the m = 0.7 case for Nakagami-m case. The same trends that are observed for the cases of less severe and more severe fading in Nakagami-m and Weibull fading channels, clearly indicate that for rural or semi-urban areas, where fading is likely to be less severe than Rayleigh due to the existence of strong signal components, conventional single-antenna is the preferred transmission method for a small number of receive antennas. On the other hand, for urban areas (fading is Rayleigh or more severe than Rayleigh) SM is the preferred choice. Another interesting trend is that the more severe the fading is, the more the Nt = 4 scenario for SM outperforms the corresponding one for Nt = 2.
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4 bits/s/Hz
a)
0
10
4 bits/s/Hz
a)
0
10
Single−antenna/MRC, 16−QAM SM, N =2, 8−QAM, r=0
Single−antenna/MRC, 16−QAM SM, N =2, 8−QAM, r=0 t
−1
10
t
−1
10
SM, Nt=4, QPSK, r=0
SM, Nt=4, QPSK, r=0 SM, Nt=2, 8−QAM, r=0.8
SM, Nt=2, 8−QAM, r=0.8 −2
SM, Nt=4, QPSK, r=0.8
ABEP
ABEP
−2
10
N =2 −3
r
N =3
10
r
r
10
−5
−5
0
5
10
15 20 E /N [dB] s
25
30
10
35
0
5
10
15
20 E /N [dB]
25
30
35
40
0.4 r
0.5
0.6
0.7
0.8
s
0
b)
100
0
b)
80
Relative EE Gain (%)
80
Relative EE Gain (%)
N =2
N =3
−4
−4
10
100
t
−3
10
r
10
SM, N =4, QPSK, r=0.8
10
60
40
N =2, 8−QAM, N =2 t
r
N =4, QPSK, N =2 t
20
r
40
r
t
0.2
r
r
Nt=2, 8−QAM, Nr=3 N =4, QPSK, N =3 t
r
0.1
t t
N =4, QPSK, N =3 0 0
N =2, 8−QAM, N =2 N =4, QPSK, N =2
20
N =2, 8−QAM, N =3 t
60
0.3
0.4 r
0.5
0.6
0.7
0 0
0.8
Fig. 4. SM and single antenna/MRC comparison in Nakagami-m channels with m = 0.7 in terms of: a) ABEP vs. Es /N0 and b) Relative energy gain vs. correlation factor.
0.1
r
0.2
0.3
Fig. 6. SM and single antenna/MRC comparison in Weibull channels with b = 1.5 in terms of: a) ABEP vs. Es /N0 and b) Relative energy gain vs. correlation factor.
4 bits/s/Hz
0
10
Single−antenna/MRC, 16−QAM SM, N =2, 8−QAM, r=0
setups with higher number of transmit antennas outperform those with a smaller number.
t
−1
10
SM, N =4, QPSK, r=0 t
SM, Nt=2, 8−QAM, r=0.8
ABEP
−2
SM, N =4, QPSK, r=0.8
10
t
ACKNOWLEDGEMENTS
N =2
−3
r
N =3
10
r
The research leading to these results has received funding from the Spanish Ministry of Science and Innovation (Ministerio de Ciencia e Innovacion) under the project GRE3NLINK-MAC (TEC2011-29006-C03-02), the Catalan Government (2009SGR0891), and the research projects: CO2GREEN (TEC2010-20823), GREENET (PITN-GA-2010-264759), and GREEN-T (CP8-006).
−4
10
−5
10
0
5
10
15 20 E /N [dB] s
100
25
30
35
0
b)
Relative EE Gain (%)
0 −100 −200
R EFERENCES
N =2, 8−QAM, N =2 −300
t
r
N =4, QPSK, N =2 t
−400
r
N =2, 8−QAM, N =3 t
r
N =4, QPSK, N =3 −500 −600 0
t
r
0.1
0.2
0.3
0.4 r
0.5
0.6
0.7
0.8
Fig. 5. SM and single antenna/MRC comparison in Weibull channels with b = 3 in terms of: a) ABEP vs. Es /N0 and b) Relative energy gain vs. correlation factor.
V. C ONCLUSIONS In this paper, we provided a performance and energy efficiency comparison for a target low ABEP of SM with its competitor, single-antenna/MRC, for well known fading channels and under different fading conditions and MIMO setups. The main conclusions based on the energy efficiency comparison are: i) For fading conditions less severe than Rayleigh, it is preferred to use the conventional single-antenna transmission method when the number of receive antennas is low. On the other hand, for Rayleigh fading conditions or more severe than Rayleigh, SM is the preferred choice even when the number of receive antennas is low, e.g. equal to two, and even for a high correlation value among the transmit antennas. ii) the more severe the fading conditions are, the more SM
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