Interference-Nulling Time-Reversal Beamforming for mm-Wave. Massive MIMO Systems. Carlos A. Viteri-Mera¦X, Fernando L. Teixeira¦, and Kamalesh Sainath¦.
Interference-Nulling Time-Reversal Beamforming for mm-Wave Massive MIMO Systems Carlos A. Viteri-Mera: , Fernando L. Teixeira , and Kamalesh Sainath ElectroScience Laboratory, The Ohio State University, Columbus, OH 43212, USA : Universidad de Nari˜ no, Pasto, Colombia {viteri.5, teixeira.5, sainath.1}@osu.edu Abstract—We present a novel beamforming technique for multi-user frequency-selective indoor channels called interference-nulling time-reversal (INTR). The beamformer is implemented with a pre-filter designed to minimize the interuser interference present in conventional time-reversal (TR). Moreover, our technique relies on a large number of antennas at the transmitter to mitigate inter-symbol interference, enabling low computational complexity receivers. We demonstrate that INTR outperforms previous TR techniques with respect to average bit error rate per user and achievable sum rate. Thus, INTR can be used for space division multiple access in no-line-ofsight mm-wave massive MIMO, providing remarkable diversity and multiplexing gains. Index Terms—Time-reversal, beamforming, mm-wave, massive MIMO, interference mitigation, green communications.
I. I NTRODUCTION Millimeter wave (mm-wave) and massive MIMO have been proposed in tandem for next generation wireless systems [1], [2]. A large number of antennas operating at mm-wave frequencies (e.g. 28, 38, 60 and 73 GHz) can be used in compact devices due to the small wavelength (4 to 10 mm approx.) and (hence) small antenna sizes. Also, it has been shown that mm-wave networks are suitable for dense small cell deployments, as inter-cell interference is naturally mitigated due to high propagation losses at those frequencies [3], [4]. Nevertheless, a number of challenges exist associated with mm-wave massive MIMO; in particular, beamforming is required because of large propagation losses. Traditionally, antenna arrays use beamsteering techniques in order to increase the transmitted power in specific directions. However, this is not always appropriate in multipath no-line-of-sight (NLoS) channels, such as those found in indoor environments. In such cases, more sophisticated techniques are required to take full advantage of the number of elements in the array and also exploit multipath propagation. In such scenarios, beamforming based on time-reversal (TR) techniques has been shown suitable to provide multiple access and diversity gains [5], [6]. In this paper we propose a novel TR-based solution to the multi-user beamforming problem for indoor scenarios in mm-wave massive MIMO, which provides both diversity and multiplexing gains. TR is a transmission technique [7]–[11] that enables spatial focusing of the signal at the receiver by using the time-reversed channel impulse response (CIR) as a linear filter applied to the transmitted signal [12]–[14].
TR is considered a promising technique for future massive MIMO systems [15]. By using TR, all multipath components are added in phase at the receiver at a specific instant [16], [17] providing i) an increase in the signal power in the surroundings of the receiver (commonly referred to as spatial focusing [18]), and ii) a partial equalization effect (commonly known as time focusing) that reduces inter-symbol interference (ISI) caused by the channel’s frequency selectivity [19], [20]. This features enable low computational complexity receivers, which is a key advantage of TR with respect to multicarrier (OFDM-like) systems [21]. Moreover, multipath components add incoherently when the distance from the intended receiver increases, mitigating interference to other users [22], [23]. The potential of TR beamforming and massive MIMO for green wireless communications has been also analyzed independently in [22] and [24], and they can work synergistically in future wireless system, as proposed in [15]. Thus, these technologies may play an important role on improving the energy efficiency [25] of current wireless networks, with expected gains in this parameter of more than three orders of magnitude according to the aforementioned references. Our technique, called interference-nulling time-reversal (INTR), minimizes the inter-user interference (IUI), and is enabled by a large number of antennas at the transmitter to provide ISI mitigation. This allows for low-complexity receivers, and provides both diversity and multiplexing gains. We validate this concept through Monte Carlo simulations using an IEEE TGad based 60 GHz NLoS MIMO channel model [26] that includes spatial correlation. We compare our approach against conventional TR and equalized TR (ETR) techniques [23] in terms of average bit error rate (BER) per user and achievable sum rate, and demonstrate its improved performance for the scenarios considered. II. S YSTEM M ODEL A. General Time-Reversal Signal Model Consider a digital baseband downlink wireless system, consisting of one Access Point (AP) with M transmit antennas and N single-antenna user terminals, as shown in Fig. 1. The AP has M " N and transmits simultaneously an independent data stream to each user. Let sn ptq be the complex random signal transmitted to user n, where t P Z is the discrete time index. These transmitted signals have unit average power, i.e.
where zn ptq represents additive white Gaussian noise (AWGN) with variance σz2 . The general idea behind TR is to use the time-reversed CIR from every antenna to the receiver as a pre-filter for the transmitted signal. Such pre-filter acts as a beamformer in the spatial domain, focusing the RF signal around the receiver. In conventional TR, assuming perfect channel state information (CSI) at the transmitter, the prefilter vector is ptr n ptq
hn pL 1 a tr Ph
tq
,
(3)
where Phtr is the power normalization factor given by Fig. 1: System model in which an AP with M transmit antennas sends simultaneously independent data streams to N single antenna users using time-reversed pre-filters pm,n ptq.
E |sn ptq| 1, @n, t, regardless of the modulation. The baseband transmitted signal from antenna m is 2
N ?ρ ¸ sn ptq b pm,n ptq,
xm ptq
?ρ s ptq b n
M ¸ pm,n ptq b hm,n ptq m1 looooooooooooooooooooooomooooooooooooooooooooooon signal directed to the n-th user
?ρ
N M ¸ ¸
sn ptq b pm,n ptq b hm,n ptq 1
1
n n loooooooooooooooooooooooooomoooooooooooooooooooooooooon 1
1
1m 1 n
ptoqn, lozonmo noise
N L¸ 1 ¸
}hn plq}2 .
(4)
Note that, in this case, the pre-filter’s length is equal to the CIR length, i.e. Ltr p L. The time domain received signal in conventional TR is
ptq
(1)
where ρ is the total average transmitted power in the AP, pm,n ptq is the power-normalized pre-filter from the m-th transmit antenna to the n-th user (with a duration of Lp samples, i.e. t 0, . . . , Lp 1), and hm,n ptq is the random channel impulse response (CIR) from the m-th transmit antenna to the n-th user (with a length of L samples). Note that the power normalization on the pre-filters ensures that the transmitted power is kept constant regardless of the number of users and the number of antennas. The random CIR vector to the n-th user is defined as hn ptq rh1,n ptq, . . . , hM,n ptqsT P CM . Let Hm,n pf q be the discrete Fourier transform (DFT) of hm,n ptq. In an analogous way to the time domain representation, the steering vector to the nT th user is Hn pf q rH1,n pf q, . . . , HM,n pf qs P CM . The selection of pm,n ptq depends on the particular TR technique, as discussed later in this section. We define the pre-filter T vector to the n-th user as pn ptq rp1,n ptq, . . . , pM,n ptqs P CM . Let Pm,n pf q be the DFT of pm,n ptq. Then, we define the frequency domain pre-filter vector to the n-th user as T Pn pf q rP1,n pf q, . . . , PM,n pf qs P CM . The received baseband signal at user n is
n 1 l 0
yntr
n 1
yn ptq
Phtr
c
c
L1 ρ ¸ }hn plq}2 sn pt L Ph l 0
1q
loooooooooooooooooooomoooooooooooooooooooon desired symbol directed to the n-th user
ρ Ph
M L¸ 1 ¸ ¸
2L 2
hm,n pl1 qhm,n pL 1 l
l1 qsn pt lq
l0 m1 l 0 lL1 loooooooooooooooooooooooooooooooooooooomoooooooooooooooooooooooooooooooooooooon 1
ISI directed to the n-th user
c
N M ρ ¸ ¸ hm,n pL 1 tq b hm,n ptq b s1n ptq Ph n 1 m1 n n looooooooooooooooooooooooooooooooomooooooooooooooooooooooooooooooooon 1
1
1
ptoqn, lozonmo
IUI
(5)
noise
which is composed of four terms: i) the desired symbol multiplied by a real factor resulting from coherent combination of multipath components in the CIR, ii) ISI caused by incoherent addition of CIR components, iii) IUI caused by the signals directed to other users (whose TR pre-filters do not match the CIR to the n-th user), and iv) AWGN. Thus, in a conventional multiuser TR beamforming system, ISI and IUI are important problems that hamper detection. In the single-user scenario, ETR was proposed before as a solution to mitigate the ISI component in the received signal [23]. Moreover, in the same work it was also demonstrated that the ratio between the desired signal power and ISI power grows linearly with the number of antennas. Thus, in massive MIMO systems ISI is automatically mitigated by the large number of antennas. Given this situation, we now present INTR as a technique to reduce IUI in the multiuser case, which is the main detection impairment in a massive MIMO TR beamforming system. B. Interference-Nulling Time-Reversal
IUI
(2)
We are now concerned with the design of pre-filter vectors that combine the spatial focusing properties of conventional
TR, while also providing additional IUI mitigation. We start from the frequency representation of the received signal, and formulate an optimization problem for the design of the prefilters. The frequency domain equivalent of (2) is
?
N ¸
xHn pf q, Pn pf qy Sn pf q 1
Uncorrelated
BER
signal directed to the n-th user
?ρ
10
1
-4
n 1 n n loooooooooooooooooooomoooooooooooooooooooon 1
1
IUI
Zonmo pfonq, lo
(6)
where Sn pf q is the DFT of sn ptq, Zn pf q is the DFT of zn ptq, and the operator x, y denotes complex inner product between two vectors. Appropriate zero padding is used in the time domain in order to represent linear convolution as a product in the frequency domain. The complex inner product defined above allows a convenient simplification in (6) with respect to (2), which is useful for the problem formulation. Let Hpf q rH1 pf q . . . HN pf qs P CM N be the matrix with columns given by the steering vectors to all users. Also, let Hn pf q P CM N 1 be the matrix formed by removing the n-th column from Hpf q, i.e. removing the steering vector to user n. For notational simplicity, we drop the frequency dependence in the remainder of this ° section. Note that the IUI power in (6) is proportional to n n | xHn , Pn y |2 . Thus, our objective is to find the pre-filter P,n which is closest to the conventional TR solution in the frequency domain (Ptr n Hn ), and such that the IUI is set to zero. Formally, this optimization problem can be formulated as 1
1
}Hn Pn }2 arg min P subject to HH n Pn 0, @n P N , @f P r0, . . . , L Lp 1s, n
whose solution is P,n
1
HHn H n
(7)
Hn Hn , @n, f. M M identity matrix. Thus, we
IM
HHn
(8)
where IM is the call P,n the interference-nulling time-reversal (INTR) pre-filter in the frequency domain. Geometrically, the !constraint ) in the problem ensures that the vector P,n P null HH n , and the solution is the projection of Hn into that null space. III. S IMULATION R ESULTS AND D ISCUSSION The proposed technique was simulated using a 60 GHz MIMO channel model with spatial correlation (CIRs are correlated across transmit antennas but uncorrelated across users) based on the IEEE TGad [26] model1 . We compared these 1 The
10 -6 -20
Uncorrelated
TR Lp = 60 ETR Lp = 90 ETR Lp = 120 INTR Lp = 60 INTR Lp = 90 INTR Lp = 120
-10
0
Correlated 10
20
SNR [dB]
noise
P,n
Correlated
10 -2
ρ xHn pf q, Pn pf qy Sn pf q Yn pf q loooooooooooooooomoooooooooooooooon
interest reader can find a channel model description in [27].
Fig. 2: Average BER per user comparison of TR, ETR, and INTR, under correlated and uncorrelated channels (across antenna elements) with M 32, and N 5. Results are shown for different pilot lengths (Lp ).
results with conventional TR and ETR [23]. The simulation consisted on the transmission of 106 complex symbols with the system model presented above, with different number of antennas M and users N . We used BPSK and computed the average BER per user and the achievable sum rate over 1000 realizations with different SN R ρΓ{σz2 , where σz2 is the noise power. We observe the impact of signal power components over the BER performance in Fig. 2, where the influence of channel spatial correlation is also shown. These results were obtained for 5 users and 32 antennas in a cubicle office scenario. Performance of both TR and ETR is limited by IUI, which causes a lower bound in the BER. We notice that ETR does not provide a significant improvement over conventional TR in the case of multi-user massive MIMO systems. Thus, ETR does not offer any advantage, given its greater computational complexity with respect to TR. In the case of INTR, IUI is successfully mitigated and hence INTR outperforms the other techniques. We also observe that channel correlation degrades system performance in all cases. Fig. 3 shows the average BER performance results per user for varying number of antennas and users. These results where obtained with the transmission of 106 BPSK symbols over 1000 spatially correlated channel realizations. We used a fixed pre-filter length Lp 90 and the cubicle office scenario. The desired signal power increases linearly with the number of antennas while interference components remain constant. Thus, the minimum BER per user improves by increasing the number of antennas, providing diversity gain. On the other hand, increasing the number of users with a fixed number of antennas decreases the desired signal power and increases IUI. This is reflected in a higher BER for larger N . INTR outperforms conventional TR in every simulated scenario. Nevertheless, the performance improvement provided by INTR is more evident with a large number of users or a limited number of antennas. We also analyze the
TR INTR
M = 32 M = 64
10 -2
10 -2
BER
BER
M = 128
10 -4
TR INTR
N = 10 N =5 N =2
10 -4
10 -6 -20
-10
0
10
20
10 -6 -20
-10
0
SNR [dB]
SNR [dB]
(a)
(b)
10
20
Fig. 3: Average BER per user for TR and INTR. (a) Different number of antennas M with Lp 90 and N 5. (b) Different number of users N with Lp 90 and M 64. An important diversity gain is achieved even in spatially correlated channels.
60
60 TR INTR
50
50
TR INTR
40
R [bps/Hz]
R [bps/Hz]
N = 10
30 20 10
40 30
N = 10
20 N =5
10 N =5
0 -30
-20
-10
0 10 SNR [dB]
20
30
(a)
achievable sum rate, which measures the downlink spectral efficiency in a multiple-access system. Assuming that each user treats ISI and IUI as Gaussian interferences, the average achievable sum rate for a multi-user TR system is RE
N ¸
n 1
log2 1
Pisi,n
Ps,n Piui,n
-20
-10
0 10 SNR [dB]
20
30
(b)
Fig. 4: Achievable sum rate with (a) M
0 -30
σz2
,
(9)
where Ps,n , Pisi,n , and Piui,n are the desired signal power, ISI power, and IUI power, respectively, calculated at user n for a given realization. For simplicity, it is also assumed that the channel is used for downlink transmission all the time. A proper reduction factor can be used to account for uplink time in a TDD system or channel estimation overheads. Numerical results for the achievable sum rate are shown in Fig. 4. These results were obtained in a living room scenario with correlated channels and Lp 90. As seen, INTR offers a significant improvement over conventional TR, doubling its rate in some
32 antennas and (b) M 128 antennas. cases and providing a remarkable multiplexing gain. An spectral efficiency of almost 60 bps/Hz can be achieved with 128 antennas and 10 users. In all the simulated scenarios our proposed INTR technique outperforms conventional TR, as it can better withstand an increase in the user load. IV. C ONCLUSION We have analyzed a baseband TR beamforming system for mm-wave multi-user massive MIMO and noted that its performance is mainly IUI limited, as ISI in mitigated by the large number of antennas at the transmitter. Thus, we confirm the potential of TR as a beamforming technology for massive MIMO. We also note that equalizing solutions such as ETR are not necessary when the number of transmit antennas is large. After identifying IUI as the main detection impairment for TR systems, we propose a modified technique called INTR. This technique calculates the transmit pre-filters in the frequency domain that set the IUI to zero and are closest to
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