Constellation Design in Noncoherent Massive ... - Semantic Scholar
Constellation Design in Noncoherent Massive SIMO Systems Alexandros Manolakos
Mainak Chowdhury
Andrea J. Goldsmith
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Abstract—An uplink system with a single antenna transmitter and a single receiver with a large number of antennas is considered. For this system we propose an energydetection-based noncoherent communication scheme which does not use the instantaneous channel state information at either the transmitter or the receiver. We provide a constellation design that is asymptotically optimal with an increasing number of antennas and constellation size. We also present numerical results on how it performs in non-asymptotic regimes. Since the channel statistics may not be precisely known, we present another scheme which takes into account possible uncertainty in the large scale statistics and compare numerically its performance with the constellation design assuming perfect knowledge of channel and noise statistics. This constellation design is shown to perform almost as well as the scheme designed with perfectly known statistics yet is far more robust to channel statistics mismatch. Index Terms—Massive MIMO, noncoherent Communications, mmWave communication, Energy Receiver
I. I NTRODUCTION Millimeter wave communications and massive MIMO systems have recently emerged as a new paradigm with huge potential gains due to the large number of transmit or receive antennas [1]. The large number of antennas lead to higher beamforming gain and energy efficiency, higher spectral efficiency and lower interference due to the extremely high directivities possible [2]. One of the biggest challenges in designing and implementing such a system, however, is the need for accurate channel state information (CSI) [3]. For example, even in today’s multiantenna systems (such as LTE-A), channel estimation and pilot overhead occupies a significant amount of time and frequency slots (≈ 15% in [4]). In a massive MIMO cellular scenario as proposed in [5], the base station has many more antennas than the user, i.e., only a TDD system seems a plausible solution since in an FDD system the number of pilot signals scales linearly with the number of antennas. However, in a TDD system the problem of pilot contamination has the potential to
significantly decrease the gains from large numbers of antennas [5]. Designing a noncoherent system, i.e., a system that does not exploit the exact or approximate CSI either at the receiver or at the transmitter, could potentially fit well inside the new millimeter wave paradigm. A related design simplification for massive MIMO systems can be found in [6] where the authors compare the instantaneous versus long-term transmit beamforming, an idea initially presented in [7]. A fundamental contribution towards the understanding of communication in noncoherent systems is the notion of unitarily invariant codes proposed in [8] and [9], which perform space-time coding over the Grassman manifold associated with the channel matrix. Multiuser counterparts of these ideas can be found in [10], [11]. In our work we consider the regime of a large number of receiver antennas and assume that only the largescale channel and noise statistics are known to the transmitter. Furthermore, we assume that that the channel realizations are independent and identically distributed. The same system model has been used in [12] where we considered a simple energy-based single shot transmission and decoding scheme and showed that, for a single user setting, achievable rates are no different from coherent schemes (i.e., schemes with perfect CSIT and CSIR) in a scaling law sense with increasing numbers of antennas. In the encoding scheme presented in that work, we needed more than 103 receive antennas to get a low probability of symbol error performance. The same problem was considered for a multiuser setting in [13]. In this work, we investigate whether the number of receive antennas required for a certain system performance can be brought down further by optimizing the constellation design. We keep the same system model as in [12] and present constellation designs which bring down the number of receive antennas significantly (from several hundreds down to less than 150). This is especially promising because with current technology, one
PL power constraint L1 i=1 pi ≤ 1. Note that the phase of the particular symbols that the transmitter chooses to transmit does not matter, since the receiver uses energy measurements. In order to detect the transmitted constellation point, the decoder computes the following statistic Pn |yi |2 ∈ R+ , (3) ||y||2 /n = k=1 n i.e., it estimates the average received power across all its antennas. Based on its knowledge of the statistics of the channels and of the constellation P, it then divides the positive real line into non-intersecting intervals or decoding regions {Ii }L , corresponding to each pi ∈ P, i=1 2 ∈ Ii }. Then, the probability of and returns pˆ ∈ {p : ||y|| n error given that p∗ is transmitted, is given by Pe (p∗ ) , P r{p∗ 6= pˆ}, andPthe average probability of symbol error is Pe , L1 p∈P Pe (p), assuming equiprobable signaling.
can support antenna arrays with 64 elements [13], [14]. Specifically, assuming that the receiver knows the largescale channel and noise statistics, we design a simple energy-based encoding and decoding scheme which is asymptotically optimal, with respect to the error exponent as the number of receive antennas increases. Based on this design, we also present a robust constellation design which takes into account uncertainty in the largescale statistics. The rest of the paper is organized as follows. We present the system model in Section II and describe an upper bound on the probability of symbol error in Section III. Then, Section IV presents the constellation design for the case of perfectly known large-scale channel statistics and Section V describes the construction of the robust counterpart. Finally, in Section VI, we present plots showing the numerical performance of the suggested schemes with representative statistics. II. S YSTEM M ODEL
III. E RROR P ROBABILITY U PPER B OUND AND R ATE F UNCTION √ Fix any transmitted pi ∈ P, observe yj = hj pi + νj and define as r(pi ) the value of the average receiver energy (statistic defined in ((3))) in the limit of infinite receive antennas, i.e.,
In this work we keep the same system model initially described in [12]. Specifically, we consider one single antenna transmitter and one receiver with n antennas. The system is represented as y = hx + ν
(1)
with y ∈ C n×1 , ν ∈ C n×1 , h ∈ C n×1 , each νi ∼ CN (0, σ 2 ), and hi ∼ f (h),
r(pi ) , pi + σ 2 .
Then, for a finite number of receive antennas, the devi||y||2 ation of n from r(pi ) is Pn Pn 2 j=1 |yj | − r(pi ) j=1 uj = , n n
K , E[|hi − such that E[hi ] = √µ2 + j √µ2 with µ2 , K+1 1 2 E[hi ]| ] = K+1 , for some known K > 0. Assume that the moment generating function of |yi |2 , i.e., 2
M (θ) , E[eθ|yi | ],
(4)
(2)
2
where uj = |yj | − r(pi ) are independent realizations of the same random variable U ∼ g(u) whose m.g.f.
exists and is twice differentiable in an interval around θ = 0. Many fading distributions fall within this model, e.g., Rayleigh and Rician fading [15], in which case hi ∼ CN (µ, σ 2 ). We assume that the instantaneous channel realization is unknown to both the transmitters and the receiver and investigate energy-based detection schemes for recovering transmitter data, based only on the knowledge of the statistics of the system, i.e., the parameters K and σ 2 . We also assume that every transmission is associated with an independent channel realization. To be more specific, if Tc is the coherence time of the channel, we assume that Ts = Tc , where Ts is the symbol time. We focus on the following encoding and decoding procedure. The transmitter transmits symbols from the constellation whose energies are specified by P = {p1 , p2 , · · · , pL }. The transmitter is subject to an average
Mi (θ) , E[eθUi ] = M (θ)e−r(pi )
(5)
exists and is twice differentiable around θ = 0 from (2). An upper bound on Pe is given by [16] 1 X −nIR,i (dR,i ) e + e−nIL,i (dL,i ) , (6) PU , L p∈P
where IL,i (d) , sup (θd − log(Mi (−θ))) ,
(7)
IR,i (d) , sup (θd − log(Mi (θ))) ,
(8)
θ>0
θ>0
and the decoding regions Ii = (r(pi ) − dL,i , r(pi ) + dR,i ] 2
are chosen by the decoder. In other words, the values dR,i and dL,i define the maximum distance from r(pi ) the received statistic should be in order to decide that the value pi was transmitted. In [12] we proved the following behavior of the rate functions IR,i (d), IL,i (d) for small values of d:
the above constellation design depends only on the first, second and forth moments of the channel statistics and leads to an easily-interpretable robust constellation design that holds for very general channel models. The latter is especially important since determining the exact small-scale fading channel models in millimeter wave communication systems is still ongoing research, and may not be reliably known beyond the first few moments.
Lemma 1. The rate functions IR,i (d), IL,i (d) satisfy lim
d→0
IL,i (d) 1 IR,i (d) = lim = . 2 2 d→0 d d 2E[Ui2 ]
IV. C ONSTELLATION D ESIGN FOR P ERFECT K NOWLEDGE OF LARGE - SCALE STATISTICS
In that work, using just a typical minimum distance 2i decoder with transmit powers pi = (L−1) , it was demonstrated that the scaling law behavior of the proposed noncoherent system is the same as that of a coherent system. An optimized constellation design for a finite number of receive antennas was not considered. In this work, we propose a simple, asymptotically optimal constellation design based on Lemma 1 and then show a robust variant of it that allows us to accommodate uncertainty in the channel and noise statistics at the receiver.
We first discuss the constellation design with exact knowledge of the large-scale fading and additive noise statistics and then in Section V presents a robust constellation design which mitigates the effects of this uncertainty on performance. Assume the user transmits pi ∈ P. Recall that uj = √ |hj pi + νj |2 − pi − σ 2 and denote h = hre + jhim , where hre , him ∈ R. Then, E[Ui2 ] , s(pi ) = α1 p2i + α2 pi + α3 ,
A. Probability of Symbol Error Minimization where
The constellation design problem for minimum average symbol error is minimize
1 L
P,{Ii }L i=1 PL i=1 pi ≤1, 0≤pi ,
Pe .
α1 = E[h4re ] + E[h4im ] + 2E[h2re ]E[h2im ] − 1,
(9)
α2 = 2σ 2 , α3 = σ 4 ,
since E[Im(νi )] = E[Re(νi )] = E[Im(νi )3 ] = E[Re(νi )3 ] = 0, E[(Re(νi )4 )] = E[(Im(νi )4 )] = 3σ 4 /4 and the noise and the channel state are independent random variables. For example, in the case of Rician fading with K-factor equals to K it follows that α1 = 1+2K (1+K)2 . Consider the case of known function s(p); i.e., the receiver knows the first, second and forth moment of the channel distribution f (h). The constellation design problem then results in the decoding regions
This problem is in general hard, but we relax it by focusing on minimizing the upper bound (6) for the case of small d; i.e., we approximate IR,i (d) = IL,i (d) ≈ I¯i (d) ,
d2 2E[Ui2 ]
(10)
and solve the following problem L
minimize
P,{Ip }p∈P
d2
d2
R,i −n L,i2 1 X −n 2E[U 2] i e + e 2E[Ui ] L i=1
L
!
Ii = (pi + σ 2 − dL,i , pi + σ 2 + dR,i ],
(11)
1X pi ≤ 1, 0 ≤ pi . L i=1
where dL,1 = dR,L = ∞. Figure 1 shows a graphical representation of the regions. Fix P and assume that
The motivation behind the above relaxation is twofold. First of all, problem (11) is an approximation of an asymptotically (with respect to n) tight upper bound on the objective function in (9) which becomes more and more exact as the constellation size L increases. This is because, as L increases, the transmitted powers will be packed closer together, and the decoding regions will be smaller, i.e., dR,i , dL,i will be small and, based on Lemma 1, the quadratic approximation in (10) will be better. Secondly, as we will see in the next section,
dR,1& dL,2& dR,2& c1& p1+σ2&
dL,3&
dR,3&
c2& p2+σ2&
dL,4& c3&
p3+σ2&
p4+σ2&
Fig. 1: Example of the decoding regions using L = 4. we want to solve (11) over only the decoding regions {Ii }, i.e., over {dR,i , dL,i }L i=1 . Then, the optimization 3
solutions calculated by iteratively fixing pi and solving for pi+1 in (16). Then, if
problem is 1 X L
minimize
{dL,i ,dR,i }L i=1
d2
R,i −n 2s(p )
e
i
d2
+e
L,i −n 2s(p )
!
i
p∈P
dL,i+1 + dR,i = pi+1 − pi , ∀i,
subject to
L
1X ∗ p ≤ 1, L i=1 i
(12)
dL,i ≥ 0, dR,i ≥ 0, ∀i.
(15) is feasible for t∗ . Identifying the largest t∗ for which (15) is feasible, solves (12). Then, the approximate upper bound on the probability of error that results from this constellation design is ∗ 2 2 e−0.5n(t ) , (18) Pe ≤ PU ≈ 2 − L
The objective function of (12) is separable to L functions, each one of which is minimized for d2R,i , d2L,i+1 such that d2
d2
e
R,i −n 2s(p ) i
=e
L,i+1 −n 2s(p ) i+1
.
Then, since dL,i+1 + d1,i = pi+1 − pi , it follows that d2L,i+1 d2R,i (pi+1 − pi )2 = = p 2 . p s(pi+1 ) s(pi ) s(pi+1 ) + s(pi )
where the approximation is because we use (10) to simplify the error rate expressions.
(13)
V. ROBUST C ONSTELLATION D ESIGN
Thus, for any fixed P, the optimal dR,i , dL,i can be calculated by (13). Then, in order to find the optimal P we can substitute (13) into (12) and get maximize
t
subject to
p
p1 ,··· ,pL ,t
Perfect knowledge of large-scale channel statistics is often impossible due to changing propagation environments associated with user mobility. This motivates the need for designs which can take into account uncertainties in the channel statistics. We build upon the design principles laid out in the previous sub-section in order to come up with a scheme that performs well even in the face of bounded uncertainties in the channel statistics. More precisely, we consider the definition of E[U 2 ] = s(p) = α1 p2 + α2 p + α3 , where α1 depends on the channel statistics and α2 , α3 depends on the noise variance. Thus for a fixed p, E[U 2 ], and hence the rate function approximation, depends on the channel and noise statistics only through α1 and σ. We define the following set
pi+1 − pi p ≥t s(pi+1 ) + s(pi ) (14) L X 1 pi = 1, 0 ≤ pi ≤ pi+1 , ∀i; L i=1
the solution of which is the largest fixed t∗ such the following problem is feasible: find
{pi }L i=1
subject to pi+1 − pi ≥ t∗
p
s(pi+1 ) +
p
(17)
s(pi )
L
1X pi ≤ 1, 0 ≤ p1 . L i=1
F = {(α1 , σ) : αmin < α1 < αmax , σmin < σ < σmax )},
(15) Observe that for t∗ = 0 the above problem is always feasible, and for t∗ = ∞ it is always infeasible due to the power constraint. Note that the feasibility problem can be efficiently solved if t∗ is fixed. To see this, consider the following constellation construction. Fix t∗ and choose p1 = 0. Then, iteratively choose pi+1 for i = 1, 2 · · · , L − 1, such that p
pi+1 − pi p = t∗ . s(pi )
s(pi+1 ) +
and note that for each f = (α˜1 , σ ˜ ) ∈ F, we can define sf (p) , α ˜ 1 p2 + α ˜2p + α ˜3, where α ˜ 2 = 2˜ σ2 , α ˜3 = σ ˜ 4 . We now design a constellation which maximizes the worst case error exponent for all statistics in F. Specifically, the problem is simplified L−1 if we denote a constellation using {pi }L i=1 and {ci }i=1 such that
(16)
dR,i = ci − (pi + σ 2 ), dL,i = pi + σ 2 − ci−1 , (19)
It is easy to see that the above equation can have at most one solution since the left hand side is an increasing function of pi+1 . Denote as p = [p∗1 , p∗2 , · · · , p∗L ] the
as shown in Figure 1. Then, using (10) the problem of maximizing the worst case error exponent is expressed 4
Note, that chosing a higher value for c1 would only make p2 larger and thus, use more transmit power than necessary. Finally, choose c2 such that q 2 ∗ c2 = sup t sf (p2 ) + σ ,
as follows: maximize
L−1 t,{pi }L i=1 ,{ci }i=1
t
q 2 subject to ci − pi ≥ sup σ + t sf (pi ) f ∈F f ∈F q 2 pi+1 − ci ≥ sup t sf (pi+1 ) − σ since then I¯2 (dR,2 ) ≥ 0.5(t∗ )2 . f ∈F Using the same procedure we can sequentially specify PL L L−1 1 1X p ≤ 1, the all {pi }L and {c } . Then, if i i=1 i=1 i i L pi = 1, pi ≥ 0, designed constellation will lead to a worst error exponent L i=1 ∗ 2 (20) of 0.5(t ) . However, if the average power constraint is i.e., I¯i (dR,i ) ≥ 0.5(t∗ )2 and I¯i (dL,i ) ≥ 0.5(t∗ )2 for all not satisfied, it is not possible to guarantee this error channel statistics in F. This problem is equivalent to exponent for all channels in F, since in our construction, finding the largest t∗ > 0 which gives a feasible point we pack the decoding regions as closely as possible. Note that for t∗ = 0 problem (21) always has in this formulation: a feasible assignment {dR,i , dL,i , pi }L L−1 i=1 satisfying the find {pi }L , {c } i i=1 i=1 constraints. However, it may not be possible to have a q ∗ 2 ∗ feasible positive t if F is very large. Such an extreme subject to ci − pi ≥ sup σ + t sf (pi ) example is presented in Subsection V-A. f ∈F q pi+1 − ci ≥ sup t∗ sf (pi+1 ) − σ 2 A. Channel Uncertainty and Existence of robust constelf ∈F lation design L X 1 In this section we present a simple example that shows pi = 1, pi ≥ 0. L i=1 that, for a fixed average power constraint P , a very (21) high uncertainty on the channel statistics could lead to Solving the above feasibility problem can be done effi- infeasibility in the robust constellation design problem. ciently by using the following simple procedure: Initially Consider the case of constructing a constellation of fix a small t∗ > 0 and choose p1 = 0 and dL,1 = ∞ L = 2 with uncertainty region for σ 2 ∈ (, 1 ) for some so that I¯i (dL,1 ) = ∞. Using p1 > 0 would lead to a > 0 and perfectly known α1 = 1 for simplicity1 . Fix ∗ sub-optimal solution since the transmitter has an average t∗ > 0. Then, p1 = 0 and c1 = 1 + t , and therefore power constraint and thus, higher transmit power leads we should choose as p2 the smallest p > 0 that satisfies to lower error rate (sf (p) is an increasing function of p p 1 + t∗ p− − sup t∗ p2 + 2σ 2 p + σ 4 − σ 2 ≥ 0 ⇔ for every f ∈ F). Then, choose c1 that satisfies σ 2 ∈(, 1 ) q 2 ∗ 1 + t∗ c1 = sup σ + t sf (0) , − sup t∗ p + σ 2 (t∗ − 1) ≥ 0. p− f ∈F σ 2 ∈(, 1 ) and as p2 , the minimum p that satisfies ∗ If t∗ ≥ 1, then p ≥ t∗ p + 2 t which is impossible, and q ∗ 1+t 1 if t∗ < 1, then p ≥ 1−t ∗ − . Therefore, the choice of p − c1 − sup t∗ sf (p) − σ 2 ≥ 0. f ∈F p2 that guarantees a worst case exponent of 0.5(t∗ )2 is Note that for 0 < t∗ ≤ inf f ∈F √1α1 there always exists a p ≥ c1 − inf f ∈F σ 2 that satisfies the above equation. To see this, define the following auxiliary function q wf (p) = p − c1 − t∗ sf (p) + σ 2 , (22)
p2 =
1 + t∗ 1 − . 1 − t∗
In this case though, since
p1 + p2 ≤ 2P ⇒
for which, for any fixed f ∈ F, it holds that wf (c1 − √ w (p ) σ 2 ) < 0 and lim fp2 2 = 1 − t∗ α1 > 0. In other
1 + t∗ 1 − < 2P, 1 − t∗
it follows that, no matter how small t∗ is, if the uncertainty is so large such that 2P < 1 −, the robust design problem will be infeasible.
p2 →∞
words, these choices of c1 and p2 guarantee that
I¯1 (dR,1 ) ≥ 0.5(t∗ )2 , I¯2 (dL,2 ) ≥ 0.5(t∗ )2 , ∀f ∈ F.
1 e.g.
5
this is the case of Rayleigh fading.
3-bit Constellation
0 −1
In other words, it is not possible to guarantee any finite error exponent as the uncertainty grows without increasing the average power constraint concurrently.
log10(S E R)
−2 C 0,10
−3
(2)
C 0,10 C 10,10
−4
(2)
VI. N UMERICAL RESULTS
C 10,10 −5
C 20,10 (2)
C 20,10
This section contains three simulation studies that demonstrate the performance gains of the robust design compared to the optimal design and the minimumdistance constellation design described in [12]. For demonstration purposes we use a channel matrix h 1 ), i.e., that follows the Gaussian distribution CN (µ, 1+K q √ K we assume Rician fading, with µ = 1+K / 2 + q √ K j 1+K / 2. For simplicity of the notation in the figures, we refer to a (K, σ 2 ) channel as the channel with Rician fading (K−factor in dB, variance unity) and additive Gaussian noise of σ 2 dB. We then denote as CK,σ2 the constellation code designed for the (K, σ 2 ) (a) channel (Section IV) and as CK,σ2 (Section V) the robust constellation designed with a dB of uncertainty for both K and σ 2 around the nominal channel characteristics (K, σ 2 ).
−6 20
40
60
80 100 120 Number of Receive Antennas
In the second numerical example (Fig. 3), we compare the SER performance results in three different channels of the 3−bit nominal and robust design assuming the channel was actually the one corresponding to the nominal constellation design. This plot shows the performance loss of being conservative in choosing a robust constellation against the nominal. Figure 3 shows that in all three channels, the maximum performance loss due to the robust design, is tolerable, especially because being aggressive could lead to significant performance deterioration as presented in Fig. 2. P s = 10 −4
250
Min Dist, (10, 10) channel Min Dist, (10, 20) channel
Number of Receive Antennas
−1.5
log10(S E R)
−2 −2.5 −3 −3.5
C −20,20 in a (−20, 20) channel
−4
C −20,20 in a (−20, 20) channel
−4.5 −5 20
(2)
C −20,20 in a (−18, 18) channel (2)
C −20,20 in a (−18, 18) channel 30
40
50
60
70
80
90
Number of Receive Antennas
100
110
160
Fig. 3: SER performance Curves
3-bit Constellation
−1
140
120
200
(2)
C 10,10, (10, 10) channel 150
C 10,20, (10, 20) channel (2)
C 10,20, (10, 20) channel 100 50 0 3
Fig. 2: SER performance Curves
C 10,10, (10, 10) channel
4
5
6
7
8 9 10 11 Constellation Size
12
13
14
15
16
Fig. 4: Minimum number of receive antennas for a 10−4 SER performance for three different constellation designs for the (0, 10) and (0, 20) channels.
In the first numerical example (Fig. 2), we demonstrate the inefficiency of the CK,σ2 constellation in a (2) mismatched channel and the ability of CK,σ2 to sustain good performance even with this statistics mismatch. Specifically, consider the case of a user with a 2−dB uncertainty in both the K and σ 2 values and that the center of the uncertainty interval corresponds to the (−20, 20) channel; approximately Rayleigh fading (K is very low) with a high SNR value. Then, using L = 8 (3−bit constellation) and the C−20,20 design on the (−18, 18) channel leads to a huge SER performance loss, as seen in Figure 2. On the other hand, using the (2) C−20,20 constellation leads to good performance for both the nominal channel statistics (−20, 20) and the channel with statistics on the edge of the uncertainty interval.
Fig. 4 and 5 summarize the minimum number of antennas needed in order to achieve a SER of at least 10−4 using different constellation designs for (10, 10), (10, 20) channels and (0, 10), (0, 20) channels respectively. Specifically, we compare the minimum distance constellation design presented in [12], with the nominal and 2-dB robust constellation design when the channel statistics have the nominal values. These figures show a very interesting byproduct of the nominal constellation design. For small constellation sizes, the approximation (10) does not hold, and the robust constellation design achieves a better performance. As the constellation size 6
ACKNOWLEDGEMENTS increases though, the nominal constellation design is asymptotically optimal and will eventually get better performance results. Comparing Fig. 4 and 5 we observe that the crossing point after which the nominal constellation design is better than the robust design is attained at lower constellation sizes as the channel gets worse (smaller SNR, smaller K).
This work was supported by a 3Com Corporation and an Alcatel-Lucent Stanford Graduate Fellowship, an A.G. Leventis Foundation Scholarship, the NSF Center for Science of Information (CSoI): NSF-CCF-0939370, and a research funding from CableLabs. R EFERENCES [1] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access, vol. 1, pp. 335–349, 2013. [2] J. Brady, N. Behdad, and A. Sayeed, “Beamspace MIMO for millimeter-wave communications: System architecture, modeling, analysis, and measurements,” IEEE Transactions on Antennas and Propagation, vol. 61, pp. 3814–3827, 2013. [3] T. L. Marzetta, “How much training is required for multiuser mimo?” in IEEE 40th Asilomar Conference on Signals, Systems and Computers (ACSSC), 2006, pp. 359–363. [4] 3GPP, “6.10.1.2 mapping to resource elements,” ETSI, Tech. Rep., Dec. 2010. [Online]. Available: http://www.3gpp.org/ftp/ Specs/html-info/36211.htm [5] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Transactions on Wireless Communications, vol. 9, no. 11, pp. 3590–3600, 2010. [6] M. R. Akdeniz, Y. Liu, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” arXiv preprint arXiv:1312.4921, 2013. [7] A. Lozano, “Long-term transmit beamforming for wireless multicasting,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 3, 2007, pp. III–417. [8] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 543–564, 2000. [9] L. Zheng and D. N. C. Tse, “Communication on the grassmann manifold: A geometric approach to the noncoherent multipleantenna channel,” IEEE Transactions on Information Theory, vol. 48, no. 2, pp. 359–383, 2002. [10] S. Shamai and T. L. Marzetta, “Multiuser capacity in block fading with no channel state information,” IEEE Transactions on Information Theory, vol. 48, no. 4, pp. 938–942, 2002. [11] S. Murugesan, E. Uysal-Biyikoglu, and P. Schniter, “Optimization of training and scheduling in the non-coherent SIMO multiple access channel,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 7, pp. 1446–1456, 2007. [12] M. Chowdhury, A. Manolakos, and A. J. Goldsmith, “Design and performance of non coherent massive SIMO systems,” in IEEE 48th Annual Conference on Information Sciences and Systems (CISS), 2014. [13] C. H. Doan, S. Emami, D. A. Sobel, A. M. Niknejad, and R. W. Brodersen, “Design considerations for 60 Ghz CMOS radios,” IEEE Communications Magazine, vol. 42, no. 12, pp. 132–140, 2004. [14] S. Rajagopal, S. Abu-Surra, Z. Pi, and F. Khan, “Antenna array design for multi-Gbps mmwave mobile broadband communication,” in IEEE Global Telecommunications Conference (GLOBECOM), 2011, pp. 1–6. [15] A. Goldsmith, Wireless communications. Cambridge University Press, 2005. [16] A. Manolakos, M. Chowdhury, and A. J. Goldsmith, “CSI is not needed for optimal scaling in multiuser massive SIMO systems,” in IEEE International Symposium on Information Theory, 2014.
Number of Receive Antennas
P s = 10 −4 250
Min Dist, (0, 10) channel Min Dist, (0, 20) channel
200
C 0,10, (0, 10) channel (2)
C 0,10, (0, 10) channel 150
C 0,20, (0, 20) channel (2)
C 0,20, (0, 20) channel 100 50 0 3
4
5 6 Constellation size
7
8
Fig. 5: Minimum number of receive antennas for a 10−4 SER performance for three different constellation designs for the (0, 10) and (0, 20) channels.
VII. C ONCLUSIONS AND F UTURE W ORK In this work we formulated the problem of constellation design for a noncoherent SIMO system with a large number of antennas at the receiver which uses only energy measurements. We presented an asymptotically optimal constellation design with respect to the achieved error exponent when the receiver has a perfect knowledge of the large-scale statistics of the channel. The proposed design requires only the knowledge of the first, second and forth moments of the fading statistics, along with the second moment of the zero mean additive Gaussian noise. We also presented a robust constellation design which takes into account the uncertainty in the large scale statistics. Via numerical simulations, we presented the symbol error performance results for typical scenarios and demonstrated the performance gains of the robust design against the nominal design when channel statistics are not perfectly known. The proposed system asks for very simple encoding and decoding and for a receiver which only senses the received energy. In the future we will investigate whether additional performance gains could be achieved by employing space-time coding schemes to reduce the minimum number of antennas needed at the receiver to achieve an acceptable symbol error rate.
7
Mainak Chowdhury
Andrea J. Goldsmith
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Dep. of Electrical Engineering Stanford University, Stanford, CA Email: [email protected]
Abstract—An uplink system with a single antenna transmitter and a single receiver with a large number of antennas is considered. For this system we propose an energydetection-based noncoherent communication scheme which does not use the instantaneous channel state information at either the transmitter or the receiver. We provide a constellation design that is asymptotically optimal with an increasing number of antennas and constellation size. We also present numerical results on how it performs in non-asymptotic regimes. Since the channel statistics may not be precisely known, we present another scheme which takes into account possible uncertainty in the large scale statistics and compare numerically its performance with the constellation design assuming perfect knowledge of channel and noise statistics. This constellation design is shown to perform almost as well as the scheme designed with perfectly known statistics yet is far more robust to channel statistics mismatch. Index Terms—Massive MIMO, noncoherent Communications, mmWave communication, Energy Receiver
I. I NTRODUCTION Millimeter wave communications and massive MIMO systems have recently emerged as a new paradigm with huge potential gains due to the large number of transmit or receive antennas [1]. The large number of antennas lead to higher beamforming gain and energy efficiency, higher spectral efficiency and lower interference due to the extremely high directivities possible [2]. One of the biggest challenges in designing and implementing such a system, however, is the need for accurate channel state information (CSI) [3]. For example, even in today’s multiantenna systems (such as LTE-A), channel estimation and pilot overhead occupies a significant amount of time and frequency slots (≈ 15% in [4]). In a massive MIMO cellular scenario as proposed in [5], the base station has many more antennas than the user, i.e., only a TDD system seems a plausible solution since in an FDD system the number of pilot signals scales linearly with the number of antennas. However, in a TDD system the problem of pilot contamination has the potential to
significantly decrease the gains from large numbers of antennas [5]. Designing a noncoherent system, i.e., a system that does not exploit the exact or approximate CSI either at the receiver or at the transmitter, could potentially fit well inside the new millimeter wave paradigm. A related design simplification for massive MIMO systems can be found in [6] where the authors compare the instantaneous versus long-term transmit beamforming, an idea initially presented in [7]. A fundamental contribution towards the understanding of communication in noncoherent systems is the notion of unitarily invariant codes proposed in [8] and [9], which perform space-time coding over the Grassman manifold associated with the channel matrix. Multiuser counterparts of these ideas can be found in [10], [11]. In our work we consider the regime of a large number of receiver antennas and assume that only the largescale channel and noise statistics are known to the transmitter. Furthermore, we assume that that the channel realizations are independent and identically distributed. The same system model has been used in [12] where we considered a simple energy-based single shot transmission and decoding scheme and showed that, for a single user setting, achievable rates are no different from coherent schemes (i.e., schemes with perfect CSIT and CSIR) in a scaling law sense with increasing numbers of antennas. In the encoding scheme presented in that work, we needed more than 103 receive antennas to get a low probability of symbol error performance. The same problem was considered for a multiuser setting in [13]. In this work, we investigate whether the number of receive antennas required for a certain system performance can be brought down further by optimizing the constellation design. We keep the same system model as in [12] and present constellation designs which bring down the number of receive antennas significantly (from several hundreds down to less than 150). This is especially promising because with current technology, one
PL power constraint L1 i=1 pi ≤ 1. Note that the phase of the particular symbols that the transmitter chooses to transmit does not matter, since the receiver uses energy measurements. In order to detect the transmitted constellation point, the decoder computes the following statistic Pn |yi |2 ∈ R+ , (3) ||y||2 /n = k=1 n i.e., it estimates the average received power across all its antennas. Based on its knowledge of the statistics of the channels and of the constellation P, it then divides the positive real line into non-intersecting intervals or decoding regions {Ii }L , corresponding to each pi ∈ P, i=1 2 ∈ Ii }. Then, the probability of and returns pˆ ∈ {p : ||y|| n error given that p∗ is transmitted, is given by Pe (p∗ ) , P r{p∗ 6= pˆ}, andPthe average probability of symbol error is Pe , L1 p∈P Pe (p), assuming equiprobable signaling.
can support antenna arrays with 64 elements [13], [14]. Specifically, assuming that the receiver knows the largescale channel and noise statistics, we design a simple energy-based encoding and decoding scheme which is asymptotically optimal, with respect to the error exponent as the number of receive antennas increases. Based on this design, we also present a robust constellation design which takes into account uncertainty in the largescale statistics. The rest of the paper is organized as follows. We present the system model in Section II and describe an upper bound on the probability of symbol error in Section III. Then, Section IV presents the constellation design for the case of perfectly known large-scale channel statistics and Section V describes the construction of the robust counterpart. Finally, in Section VI, we present plots showing the numerical performance of the suggested schemes with representative statistics. II. S YSTEM M ODEL
III. E RROR P ROBABILITY U PPER B OUND AND R ATE F UNCTION √ Fix any transmitted pi ∈ P, observe yj = hj pi + νj and define as r(pi ) the value of the average receiver energy (statistic defined in ((3))) in the limit of infinite receive antennas, i.e.,
In this work we keep the same system model initially described in [12]. Specifically, we consider one single antenna transmitter and one receiver with n antennas. The system is represented as y = hx + ν
(1)
with y ∈ C n×1 , ν ∈ C n×1 , h ∈ C n×1 , each νi ∼ CN (0, σ 2 ), and hi ∼ f (h),
r(pi ) , pi + σ 2 .
Then, for a finite number of receive antennas, the devi||y||2 ation of n from r(pi ) is Pn Pn 2 j=1 |yj | − r(pi ) j=1 uj = , n n
K , E[|hi − such that E[hi ] = √µ2 + j √µ2 with µ2 , K+1 1 2 E[hi ]| ] = K+1 , for some known K > 0. Assume that the moment generating function of |yi |2 , i.e., 2
M (θ) , E[eθ|yi | ],
(4)
(2)
2
where uj = |yj | − r(pi ) are independent realizations of the same random variable U ∼ g(u) whose m.g.f.
exists and is twice differentiable in an interval around θ = 0. Many fading distributions fall within this model, e.g., Rayleigh and Rician fading [15], in which case hi ∼ CN (µ, σ 2 ). We assume that the instantaneous channel realization is unknown to both the transmitters and the receiver and investigate energy-based detection schemes for recovering transmitter data, based only on the knowledge of the statistics of the system, i.e., the parameters K and σ 2 . We also assume that every transmission is associated with an independent channel realization. To be more specific, if Tc is the coherence time of the channel, we assume that Ts = Tc , where Ts is the symbol time. We focus on the following encoding and decoding procedure. The transmitter transmits symbols from the constellation whose energies are specified by P = {p1 , p2 , · · · , pL }. The transmitter is subject to an average
Mi (θ) , E[eθUi ] = M (θ)e−r(pi )
(5)
exists and is twice differentiable around θ = 0 from (2). An upper bound on Pe is given by [16] 1 X −nIR,i (dR,i ) e + e−nIL,i (dL,i ) , (6) PU , L p∈P
where IL,i (d) , sup (θd − log(Mi (−θ))) ,
(7)
IR,i (d) , sup (θd − log(Mi (θ))) ,
(8)
θ>0
θ>0
and the decoding regions Ii = (r(pi ) − dL,i , r(pi ) + dR,i ] 2
are chosen by the decoder. In other words, the values dR,i and dL,i define the maximum distance from r(pi ) the received statistic should be in order to decide that the value pi was transmitted. In [12] we proved the following behavior of the rate functions IR,i (d), IL,i (d) for small values of d:
the above constellation design depends only on the first, second and forth moments of the channel statistics and leads to an easily-interpretable robust constellation design that holds for very general channel models. The latter is especially important since determining the exact small-scale fading channel models in millimeter wave communication systems is still ongoing research, and may not be reliably known beyond the first few moments.
Lemma 1. The rate functions IR,i (d), IL,i (d) satisfy lim
d→0
IL,i (d) 1 IR,i (d) = lim = . 2 2 d→0 d d 2E[Ui2 ]
IV. C ONSTELLATION D ESIGN FOR P ERFECT K NOWLEDGE OF LARGE - SCALE STATISTICS
In that work, using just a typical minimum distance 2i decoder with transmit powers pi = (L−1) , it was demonstrated that the scaling law behavior of the proposed noncoherent system is the same as that of a coherent system. An optimized constellation design for a finite number of receive antennas was not considered. In this work, we propose a simple, asymptotically optimal constellation design based on Lemma 1 and then show a robust variant of it that allows us to accommodate uncertainty in the channel and noise statistics at the receiver.
We first discuss the constellation design with exact knowledge of the large-scale fading and additive noise statistics and then in Section V presents a robust constellation design which mitigates the effects of this uncertainty on performance. Assume the user transmits pi ∈ P. Recall that uj = √ |hj pi + νj |2 − pi − σ 2 and denote h = hre + jhim , where hre , him ∈ R. Then, E[Ui2 ] , s(pi ) = α1 p2i + α2 pi + α3 ,
A. Probability of Symbol Error Minimization where
The constellation design problem for minimum average symbol error is minimize
1 L
P,{Ii }L i=1 PL i=1 pi ≤1, 0≤pi ,
Pe .
α1 = E[h4re ] + E[h4im ] + 2E[h2re ]E[h2im ] − 1,
(9)
α2 = 2σ 2 , α3 = σ 4 ,
since E[Im(νi )] = E[Re(νi )] = E[Im(νi )3 ] = E[Re(νi )3 ] = 0, E[(Re(νi )4 )] = E[(Im(νi )4 )] = 3σ 4 /4 and the noise and the channel state are independent random variables. For example, in the case of Rician fading with K-factor equals to K it follows that α1 = 1+2K (1+K)2 . Consider the case of known function s(p); i.e., the receiver knows the first, second and forth moment of the channel distribution f (h). The constellation design problem then results in the decoding regions
This problem is in general hard, but we relax it by focusing on minimizing the upper bound (6) for the case of small d; i.e., we approximate IR,i (d) = IL,i (d) ≈ I¯i (d) ,
d2 2E[Ui2 ]
(10)
and solve the following problem L
minimize
P,{Ip }p∈P
d2
d2
R,i −n L,i2 1 X −n 2E[U 2] i e + e 2E[Ui ] L i=1
L
!
Ii = (pi + σ 2 − dL,i , pi + σ 2 + dR,i ],
(11)
1X pi ≤ 1, 0 ≤ pi . L i=1
where dL,1 = dR,L = ∞. Figure 1 shows a graphical representation of the regions. Fix P and assume that
The motivation behind the above relaxation is twofold. First of all, problem (11) is an approximation of an asymptotically (with respect to n) tight upper bound on the objective function in (9) which becomes more and more exact as the constellation size L increases. This is because, as L increases, the transmitted powers will be packed closer together, and the decoding regions will be smaller, i.e., dR,i , dL,i will be small and, based on Lemma 1, the quadratic approximation in (10) will be better. Secondly, as we will see in the next section,
dR,1& dL,2& dR,2& c1& p1+σ2&
dL,3&
dR,3&
c2& p2+σ2&
dL,4& c3&
p3+σ2&
p4+σ2&
Fig. 1: Example of the decoding regions using L = 4. we want to solve (11) over only the decoding regions {Ii }, i.e., over {dR,i , dL,i }L i=1 . Then, the optimization 3
solutions calculated by iteratively fixing pi and solving for pi+1 in (16). Then, if
problem is 1 X L
minimize
{dL,i ,dR,i }L i=1
d2
R,i −n 2s(p )
e
i
d2
+e
L,i −n 2s(p )
!
i
p∈P
dL,i+1 + dR,i = pi+1 − pi , ∀i,
subject to
L
1X ∗ p ≤ 1, L i=1 i
(12)
dL,i ≥ 0, dR,i ≥ 0, ∀i.
(15) is feasible for t∗ . Identifying the largest t∗ for which (15) is feasible, solves (12). Then, the approximate upper bound on the probability of error that results from this constellation design is ∗ 2 2 e−0.5n(t ) , (18) Pe ≤ PU ≈ 2 − L
The objective function of (12) is separable to L functions, each one of which is minimized for d2R,i , d2L,i+1 such that d2
d2
e
R,i −n 2s(p ) i
=e
L,i+1 −n 2s(p ) i+1
.
Then, since dL,i+1 + d1,i = pi+1 − pi , it follows that d2L,i+1 d2R,i (pi+1 − pi )2 = = p 2 . p s(pi+1 ) s(pi ) s(pi+1 ) + s(pi )
where the approximation is because we use (10) to simplify the error rate expressions.
(13)
V. ROBUST C ONSTELLATION D ESIGN
Thus, for any fixed P, the optimal dR,i , dL,i can be calculated by (13). Then, in order to find the optimal P we can substitute (13) into (12) and get maximize
t
subject to
p
p1 ,··· ,pL ,t
Perfect knowledge of large-scale channel statistics is often impossible due to changing propagation environments associated with user mobility. This motivates the need for designs which can take into account uncertainties in the channel statistics. We build upon the design principles laid out in the previous sub-section in order to come up with a scheme that performs well even in the face of bounded uncertainties in the channel statistics. More precisely, we consider the definition of E[U 2 ] = s(p) = α1 p2 + α2 p + α3 , where α1 depends on the channel statistics and α2 , α3 depends on the noise variance. Thus for a fixed p, E[U 2 ], and hence the rate function approximation, depends on the channel and noise statistics only through α1 and σ. We define the following set
pi+1 − pi p ≥t s(pi+1 ) + s(pi ) (14) L X 1 pi = 1, 0 ≤ pi ≤ pi+1 , ∀i; L i=1
the solution of which is the largest fixed t∗ such the following problem is feasible: find
{pi }L i=1
subject to pi+1 − pi ≥ t∗
p
s(pi+1 ) +
p
(17)
s(pi )
L
1X pi ≤ 1, 0 ≤ p1 . L i=1
F = {(α1 , σ) : αmin < α1 < αmax , σmin < σ < σmax )},
(15) Observe that for t∗ = 0 the above problem is always feasible, and for t∗ = ∞ it is always infeasible due to the power constraint. Note that the feasibility problem can be efficiently solved if t∗ is fixed. To see this, consider the following constellation construction. Fix t∗ and choose p1 = 0. Then, iteratively choose pi+1 for i = 1, 2 · · · , L − 1, such that p
pi+1 − pi p = t∗ . s(pi )
s(pi+1 ) +
and note that for each f = (α˜1 , σ ˜ ) ∈ F, we can define sf (p) , α ˜ 1 p2 + α ˜2p + α ˜3, where α ˜ 2 = 2˜ σ2 , α ˜3 = σ ˜ 4 . We now design a constellation which maximizes the worst case error exponent for all statistics in F. Specifically, the problem is simplified L−1 if we denote a constellation using {pi }L i=1 and {ci }i=1 such that
(16)
dR,i = ci − (pi + σ 2 ), dL,i = pi + σ 2 − ci−1 , (19)
It is easy to see that the above equation can have at most one solution since the left hand side is an increasing function of pi+1 . Denote as p = [p∗1 , p∗2 , · · · , p∗L ] the
as shown in Figure 1. Then, using (10) the problem of maximizing the worst case error exponent is expressed 4
Note, that chosing a higher value for c1 would only make p2 larger and thus, use more transmit power than necessary. Finally, choose c2 such that q 2 ∗ c2 = sup t sf (p2 ) + σ ,
as follows: maximize
L−1 t,{pi }L i=1 ,{ci }i=1
t
q 2 subject to ci − pi ≥ sup σ + t sf (pi ) f ∈F f ∈F q 2 pi+1 − ci ≥ sup t sf (pi+1 ) − σ since then I¯2 (dR,2 ) ≥ 0.5(t∗ )2 . f ∈F Using the same procedure we can sequentially specify PL L L−1 1 1X p ≤ 1, the all {pi }L and {c } . Then, if i i=1 i=1 i i L pi = 1, pi ≥ 0, designed constellation will lead to a worst error exponent L i=1 ∗ 2 (20) of 0.5(t ) . However, if the average power constraint is i.e., I¯i (dR,i ) ≥ 0.5(t∗ )2 and I¯i (dL,i ) ≥ 0.5(t∗ )2 for all not satisfied, it is not possible to guarantee this error channel statistics in F. This problem is equivalent to exponent for all channels in F, since in our construction, finding the largest t∗ > 0 which gives a feasible point we pack the decoding regions as closely as possible. Note that for t∗ = 0 problem (21) always has in this formulation: a feasible assignment {dR,i , dL,i , pi }L L−1 i=1 satisfying the find {pi }L , {c } i i=1 i=1 constraints. However, it may not be possible to have a q ∗ 2 ∗ feasible positive t if F is very large. Such an extreme subject to ci − pi ≥ sup σ + t sf (pi ) example is presented in Subsection V-A. f ∈F q pi+1 − ci ≥ sup t∗ sf (pi+1 ) − σ 2 A. Channel Uncertainty and Existence of robust constelf ∈F lation design L X 1 In this section we present a simple example that shows pi = 1, pi ≥ 0. L i=1 that, for a fixed average power constraint P , a very (21) high uncertainty on the channel statistics could lead to Solving the above feasibility problem can be done effi- infeasibility in the robust constellation design problem. ciently by using the following simple procedure: Initially Consider the case of constructing a constellation of fix a small t∗ > 0 and choose p1 = 0 and dL,1 = ∞ L = 2 with uncertainty region for σ 2 ∈ (, 1 ) for some so that I¯i (dL,1 ) = ∞. Using p1 > 0 would lead to a > 0 and perfectly known α1 = 1 for simplicity1 . Fix ∗ sub-optimal solution since the transmitter has an average t∗ > 0. Then, p1 = 0 and c1 = 1 + t , and therefore power constraint and thus, higher transmit power leads we should choose as p2 the smallest p > 0 that satisfies to lower error rate (sf (p) is an increasing function of p p 1 + t∗ p− − sup t∗ p2 + 2σ 2 p + σ 4 − σ 2 ≥ 0 ⇔ for every f ∈ F). Then, choose c1 that satisfies σ 2 ∈(, 1 ) q 2 ∗ 1 + t∗ c1 = sup σ + t sf (0) , − sup t∗ p + σ 2 (t∗ − 1) ≥ 0. p− f ∈F σ 2 ∈(, 1 ) and as p2 , the minimum p that satisfies ∗ If t∗ ≥ 1, then p ≥ t∗ p + 2 t which is impossible, and q ∗ 1+t 1 if t∗ < 1, then p ≥ 1−t ∗ − . Therefore, the choice of p − c1 − sup t∗ sf (p) − σ 2 ≥ 0. f ∈F p2 that guarantees a worst case exponent of 0.5(t∗ )2 is Note that for 0 < t∗ ≤ inf f ∈F √1α1 there always exists a p ≥ c1 − inf f ∈F σ 2 that satisfies the above equation. To see this, define the following auxiliary function q wf (p) = p − c1 − t∗ sf (p) + σ 2 , (22)
p2 =
1 + t∗ 1 − . 1 − t∗
In this case though, since
p1 + p2 ≤ 2P ⇒
for which, for any fixed f ∈ F, it holds that wf (c1 − √ w (p ) σ 2 ) < 0 and lim fp2 2 = 1 − t∗ α1 > 0. In other
1 + t∗ 1 − < 2P, 1 − t∗
it follows that, no matter how small t∗ is, if the uncertainty is so large such that 2P < 1 −, the robust design problem will be infeasible.
p2 →∞
words, these choices of c1 and p2 guarantee that
I¯1 (dR,1 ) ≥ 0.5(t∗ )2 , I¯2 (dL,2 ) ≥ 0.5(t∗ )2 , ∀f ∈ F.
1 e.g.
5
this is the case of Rayleigh fading.
3-bit Constellation
0 −1
In other words, it is not possible to guarantee any finite error exponent as the uncertainty grows without increasing the average power constraint concurrently.
log10(S E R)
−2 C 0,10
−3
(2)
C 0,10 C 10,10
−4
(2)
VI. N UMERICAL RESULTS
C 10,10 −5
C 20,10 (2)
C 20,10
This section contains three simulation studies that demonstrate the performance gains of the robust design compared to the optimal design and the minimumdistance constellation design described in [12]. For demonstration purposes we use a channel matrix h 1 ), i.e., that follows the Gaussian distribution CN (µ, 1+K q √ K we assume Rician fading, with µ = 1+K / 2 + q √ K j 1+K / 2. For simplicity of the notation in the figures, we refer to a (K, σ 2 ) channel as the channel with Rician fading (K−factor in dB, variance unity) and additive Gaussian noise of σ 2 dB. We then denote as CK,σ2 the constellation code designed for the (K, σ 2 ) (a) channel (Section IV) and as CK,σ2 (Section V) the robust constellation designed with a dB of uncertainty for both K and σ 2 around the nominal channel characteristics (K, σ 2 ).
−6 20
40
60
80 100 120 Number of Receive Antennas
In the second numerical example (Fig. 3), we compare the SER performance results in three different channels of the 3−bit nominal and robust design assuming the channel was actually the one corresponding to the nominal constellation design. This plot shows the performance loss of being conservative in choosing a robust constellation against the nominal. Figure 3 shows that in all three channels, the maximum performance loss due to the robust design, is tolerable, especially because being aggressive could lead to significant performance deterioration as presented in Fig. 2. P s = 10 −4
250
Min Dist, (10, 10) channel Min Dist, (10, 20) channel
Number of Receive Antennas
−1.5
log10(S E R)
−2 −2.5 −3 −3.5
C −20,20 in a (−20, 20) channel
−4
C −20,20 in a (−20, 20) channel
−4.5 −5 20
(2)
C −20,20 in a (−18, 18) channel (2)
C −20,20 in a (−18, 18) channel 30
40
50
60
70
80
90
Number of Receive Antennas
100
110
160
Fig. 3: SER performance Curves
3-bit Constellation
−1
140
120
200
(2)
C 10,10, (10, 10) channel 150
C 10,20, (10, 20) channel (2)
C 10,20, (10, 20) channel 100 50 0 3
Fig. 2: SER performance Curves
C 10,10, (10, 10) channel
4
5
6
7
8 9 10 11 Constellation Size
12
13
14
15
16
Fig. 4: Minimum number of receive antennas for a 10−4 SER performance for three different constellation designs for the (0, 10) and (0, 20) channels.
In the first numerical example (Fig. 2), we demonstrate the inefficiency of the CK,σ2 constellation in a (2) mismatched channel and the ability of CK,σ2 to sustain good performance even with this statistics mismatch. Specifically, consider the case of a user with a 2−dB uncertainty in both the K and σ 2 values and that the center of the uncertainty interval corresponds to the (−20, 20) channel; approximately Rayleigh fading (K is very low) with a high SNR value. Then, using L = 8 (3−bit constellation) and the C−20,20 design on the (−18, 18) channel leads to a huge SER performance loss, as seen in Figure 2. On the other hand, using the (2) C−20,20 constellation leads to good performance for both the nominal channel statistics (−20, 20) and the channel with statistics on the edge of the uncertainty interval.
Fig. 4 and 5 summarize the minimum number of antennas needed in order to achieve a SER of at least 10−4 using different constellation designs for (10, 10), (10, 20) channels and (0, 10), (0, 20) channels respectively. Specifically, we compare the minimum distance constellation design presented in [12], with the nominal and 2-dB robust constellation design when the channel statistics have the nominal values. These figures show a very interesting byproduct of the nominal constellation design. For small constellation sizes, the approximation (10) does not hold, and the robust constellation design achieves a better performance. As the constellation size 6
ACKNOWLEDGEMENTS increases though, the nominal constellation design is asymptotically optimal and will eventually get better performance results. Comparing Fig. 4 and 5 we observe that the crossing point after which the nominal constellation design is better than the robust design is attained at lower constellation sizes as the channel gets worse (smaller SNR, smaller K).
This work was supported by a 3Com Corporation and an Alcatel-Lucent Stanford Graduate Fellowship, an A.G. Leventis Foundation Scholarship, the NSF Center for Science of Information (CSoI): NSF-CCF-0939370, and a research funding from CableLabs. R EFERENCES [1] T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access, vol. 1, pp. 335–349, 2013. [2] J. Brady, N. Behdad, and A. Sayeed, “Beamspace MIMO for millimeter-wave communications: System architecture, modeling, analysis, and measurements,” IEEE Transactions on Antennas and Propagation, vol. 61, pp. 3814–3827, 2013. [3] T. L. Marzetta, “How much training is required for multiuser mimo?” in IEEE 40th Asilomar Conference on Signals, Systems and Computers (ACSSC), 2006, pp. 359–363. [4] 3GPP, “6.10.1.2 mapping to resource elements,” ETSI, Tech. Rep., Dec. 2010. [Online]. Available: http://www.3gpp.org/ftp/ Specs/html-info/36211.htm [5] T. L. Marzetta, “Noncooperative cellular wireless with unlimited numbers of base station antennas,” IEEE Transactions on Wireless Communications, vol. 9, no. 11, pp. 3590–3600, 2010. [6] M. R. Akdeniz, Y. Liu, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter wave channel modeling and cellular capacity evaluation,” arXiv preprint arXiv:1312.4921, 2013. [7] A. Lozano, “Long-term transmit beamforming for wireless multicasting,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), vol. 3, 2007, pp. III–417. [8] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communications in rayleigh flat fading,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 543–564, 2000. [9] L. Zheng and D. N. C. Tse, “Communication on the grassmann manifold: A geometric approach to the noncoherent multipleantenna channel,” IEEE Transactions on Information Theory, vol. 48, no. 2, pp. 359–383, 2002. [10] S. Shamai and T. L. Marzetta, “Multiuser capacity in block fading with no channel state information,” IEEE Transactions on Information Theory, vol. 48, no. 4, pp. 938–942, 2002. [11] S. Murugesan, E. Uysal-Biyikoglu, and P. Schniter, “Optimization of training and scheduling in the non-coherent SIMO multiple access channel,” IEEE Journal on Selected Areas in Communications, vol. 25, no. 7, pp. 1446–1456, 2007. [12] M. Chowdhury, A. Manolakos, and A. J. Goldsmith, “Design and performance of non coherent massive SIMO systems,” in IEEE 48th Annual Conference on Information Sciences and Systems (CISS), 2014. [13] C. H. Doan, S. Emami, D. A. Sobel, A. M. Niknejad, and R. W. Brodersen, “Design considerations for 60 Ghz CMOS radios,” IEEE Communications Magazine, vol. 42, no. 12, pp. 132–140, 2004. [14] S. Rajagopal, S. Abu-Surra, Z. Pi, and F. Khan, “Antenna array design for multi-Gbps mmwave mobile broadband communication,” in IEEE Global Telecommunications Conference (GLOBECOM), 2011, pp. 1–6. [15] A. Goldsmith, Wireless communications. Cambridge University Press, 2005. [16] A. Manolakos, M. Chowdhury, and A. J. Goldsmith, “CSI is not needed for optimal scaling in multiuser massive SIMO systems,” in IEEE International Symposium on Information Theory, 2014.
Number of Receive Antennas
P s = 10 −4 250
Min Dist, (0, 10) channel Min Dist, (0, 20) channel
200
C 0,10, (0, 10) channel (2)
C 0,10, (0, 10) channel 150
C 0,20, (0, 20) channel (2)
C 0,20, (0, 20) channel 100 50 0 3
4
5 6 Constellation size
7
8
Fig. 5: Minimum number of receive antennas for a 10−4 SER performance for three different constellation designs for the (0, 10) and (0, 20) channels.
VII. C ONCLUSIONS AND F UTURE W ORK In this work we formulated the problem of constellation design for a noncoherent SIMO system with a large number of antennas at the receiver which uses only energy measurements. We presented an asymptotically optimal constellation design with respect to the achieved error exponent when the receiver has a perfect knowledge of the large-scale statistics of the channel. The proposed design requires only the knowledge of the first, second and forth moments of the fading statistics, along with the second moment of the zero mean additive Gaussian noise. We also presented a robust constellation design which takes into account the uncertainty in the large scale statistics. Via numerical simulations, we presented the symbol error performance results for typical scenarios and demonstrated the performance gains of the robust design against the nominal design when channel statistics are not perfectly known. The proposed system asks for very simple encoding and decoding and for a receiver which only senses the received energy. In the future we will investigate whether additional performance gains could be achieved by employing space-time coding schemes to reduce the minimum number of antennas needed at the receiver to achieve an acceptable symbol error rate.
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