REAL WORLD FEASIBILITY OF INTERFERENCE ALIGNMENT USING MIMO-OFDM CHANNEL MEASUREMENTS

Transcription

1 REAL WORLD FEASIBILITY OF INTERFERENCE ALIGNMENT USING MIMO-OFDM CHANNEL MEASUREMENTS Omar El Ayach, Steven W. Peters, and Robert W. Heath Jr. Wireless Networking and Communications Group The University of Texas at Austin, Austin, Texas {omarayach, speters, ABSTRACT Interference alignment (IA) has been shown to achieve linear sum capacity growth, at high SNR, with the number of users in the interference channel by cooperatively precoding transmitted signals to align interference subspaces at the receivers. The theory of IA was derived under assumptions about the richness of the propagation channel; practical channels do not guarantee such ideal decorrelation. This paper presents the first experimental study of IA in measured interference channel and shows that IA achieves the claimed scaling factors in a variety of measured channel settings for a 3 user, 2 antennas per node setup. INTRODUCTION Recent information-theoretic work has provided a characterization of the sum capacity of the interference channel. Under certain assumptions, the sum capacity has been shown to scale linearly with the number of users at high signal-to-noise ratios [1]. This scaling can be achieved by employing the concept of interference alignment [2]. The key idea of interference alignment (IA) is to design transmit precoders such that interference aligns at the receivers, effectively reducing the number of distinguishable interferers. While these results hold in theory, and algorithmic IA solutions have been proposed in [3], [4], the viability and performance of IA in measured channels has not yet been fully established [5] [7]. Theoretical work on IA has all been done using a baseband model assuming channel coefficients are drawn independently from a continuous distribution [1], [8]; an assumption not guaranteed in practical channels with nonideal characteristics such as line-of-sight channels or channels with closely spaced antennas. Moreover, there are no comprehensive measurements of the interference channel suitable for studying IA in practice. The only comparable results on multiuser MIMO channel measurements, which are not directly related to IA, target simpler setups consisting of a single base station and several receivers, This work was supported by the DARPA IT-MANET program, Grant W911NF and thus does not provide the required data on measured interference channels [9], []. The authors of [11], for example, present multiuser measurements formed by concatenating separate single user measurements, claiming that the static measurement environment ensures validity of the results. Related work on demonstrating IA in practice is limited to [7], which tested their own version of IA coupled with interference cancellation and successive decoding in a single carrier narrowband MIMO wireless local area network. The work in [7] does not provide insight into the performance of the original MIMO IA solutions in realistic wideband channels. Consequently, the viability of IA in measured propagation channels has not been evaluated. In this paper, we establish the feasibility of IA in real world propagation channels. To acquire suitable channel measurements, we implemented a MIMO-OFDM measurement testbed for the 3-user 2 2 MIMO interference channel, using a software defined radio platform, generalizing our prior testbed in [12] to multiple interference channels. We gave special attention to the proper implementation of a synchronized MIMO-OFDM physical layer to guarantee the validity of our measurements; a consideration that was not emphasized in previous work [7], [9] [11]. We perform indoor channel measurements for a variety of static node deployments. We summarize the data collection and use it to establish the true feasibility of IA in realistic channels. We examine the average sum rate achieved versus signalto-noise ratio and show that, as predicted in theory, IA outperforms time division multiple access (TDMA) and achieves the maximum degrees of freedom, the ratio of the achieved sum rate to log 2 (SNR), in our setup. Finally, we comment on the effect of channel spatial correlation on the performance of IA. MIMO INTERFERENCE CHANNEL MODEL Consider the K-user narrowband version of the interference channel shown in Figure 1 with M i transmit antennas and N i receive antennas at node i. In this channel, each transmitter i communicates with its corresponding designated receiver i and interferes with all other receivers, l i. In this section we explain the MIMO interference model in 1 of 6

2 the general case, though in the remainder of this paper we specialize to the K = 3 user 2 2 interference channel. The received signal at node i, in the absence of timing and frequency offset impairments, is given by y i = H i,i F i s i + l i H i,l F l s l + v i (1) where y i is the N i 1 received signal vector, H i,l is the N i M l channel matrix from transmitter l to receiver i with elements drawn from an i.i.d continuous zero mean distribution, F i is the precoding matrix used at node i, s i is the transmitted QAM or PSK symbol at node i, and v i is a complex vector of i.i.d circularly symmetric white Gaussian noise with covariance matrix σ 2 I. For frequency selective channels, we assume that MIMO- OFDM modulation is employed with N subcarriers. The received signal at node i for subcarrier n at the input of the signal decoder is then given by y i [n] = H i,i [n]f i [n]s i [n] + l i H i,l [n]f l [n]s l [n] + v i [n] (2) where we use the same notation as in the narrowband system model. In this signal model, we assume perfect carrier recovery and symbol timing estimation at the output of our synchronization modules, and that the impulse response of the channel is shorter than the cyclic prefix used, thus allowing us to write the received signal as in (2). INTERFERENCE ALIGNMENT IA, using an adequate number of antennas per node, aims to choose the precoding matrices {F i } to force the received interference at each of the K receivers to lie within a lower dimensional subspace. Specifically, if receiver i intends on decoding S i independent data streams with no interference using a linear receiver, it must restrict interference to a N i S i dimensional subspace of the receive signal space, C Ni. Let W i [n] be the N i S i matrix describing the orthonormal basis for the interference free subspace used at node i and subcarrier n. Projecting the received signal on the basis of the interference free subspace. Ignoring the AWGN noise term, this yields z i [n] = W i [n] (H i,i [n]f i [n]s i [n] + l i H i,l [n]f l [n]s l [n]) (3) where W i [n] is the conjugate transpose of W i [n]. For interference alignment, the received interference must lie in the nullspace of W i [n], which gives W i [n] H i,k [n]f k [n]s k [n] =, k i (4) Effectively, this restricts all interference to the chosen N i S i dimensional nullspace, thus satisfying span(h i,k [n]f k [n]) null(w i [n] ), k i (5) where span(a) denotes the spaced spanned by the column vectors of A, and null(a) denotes the space spanned by all vectors x such that Ax =. In addition to satisfying (4), the interference alignment solution must satisfy rank(u i [n] H i,i [n]f i [n]) = S i (6) to successfully decode all S i streams with a linear receiver. This spatial alignment approach, with possible symbol extensions that are not considered in this paper, guarantees the maximum number of degrees of freedom as shown by [1] and is optimal in the high SNR regime. Interference alignment does not claim optimality at low-to-moderate SNR where the optimal multiple access strategy is unknown. Our work focuses on the three user case with M i = N i = 2, proven to provide each user with one interference free spatial stream, i.e. S i = 1. Although closed form interference alignment solutions do not yet exist for networks with more than three users, we test the performance of the existing 3 user solution presented in [1]. SYSTEM IMPLEMENTATION AND MEASUREMENT SETUP In this section, we present the software and hardware details of our measurement testbed. We discuss briefly the main concepts in our MIMO-OFDM system implementation such as training, channel estimation, carrier recovery. We also introduce the system parameters used in our experiments to collect channel measurements. A. Software Implementation Our MIMO-OFDM testbed software, using National Instruments (NI) hardware with software implemented implemented in LabVIEW. For the experiments the testbed uses the parameter values indicated in Table I for all communicating users in the network. We use OFDM modulation with an FFT size of 256 and a 64 sample guard interval. The total signal bandwidth used is 16 MHz, resulting in an OFDM symbol time of 2µs. Communication is done in the 2.4GHz industrial, scientific, and medical (ISM) band. We use periodic frequency domain pilot symbols to estimate the channel and equalize received data. Pilot symbols are shared among transmit antennas, i.e different subcarriers are assigned to different transmit antennas to achieve orthogonal training for each antenna. To obtain a full frequency response for each antenna, we transmit pilot symbols in pairs, with opposite subcarrier multiplexing. 2 of 6

3 NI PXI-567 Signal Generator H11 H21 NI PXI-566 Signal Analyzer H31 NI PXI-567 Signal Generator H12 H22 H23 NI PXI-566 Signal Analyzer H13 H23 NI PXI-567 Signal Generator H33 NI PXI-566 Signal Analyzer NI PXI-6653 Synchronization Modules Fig. 1. Simplified hardware block diagram (Illustration of alignment adapted from Fig. 1 in [1]). Moreover, training from the three different users is orthogonal in time. Assuming perfect time and frequency synchronization at the output of the synchronization software blocks, and assuming that all training sequences are known to all users, with all antennas sending sufficiently cyclically prefixed training data, orthogonal in time or frequency, we estimate the channel using least-squares frequency domain estimation and apply a frequency domain zero forcing equalizer to any sent data, a common method similar to narrowband equalization [13]. The ratio of pilot-to-data symbols is variable and enables accommodating channels with short coherence times and can emulate channel estimation frequencies used in emerging MIMO wireless standards. To provide valid and extensive measurements we send pilot symbols from all users with minimal data for verification in between, thus keeping the measurement time in the microsecond range. Sending data in the measurement exercise is recommended to verify correct reception and decoding, ensuring that the recorded channel measurements correspond to successful transmissions. Pilot symbols are also used to estimate frequency offsets between each transmit-receive pair. For typical MIMO communication, transmit chains corresponding to the 2 transmit antennas per user are synchronized to justify assuming of a single frequency offset per user. Since we want to test the performance of IA in the absence of such impairments, however, we synchronize all users transmit chains. B. Hardware Description Our hardware setup consists of four NI PXI-45 chassis connected to 3 s. The first controls 2 PXI Chassis, containing the three users transmit chains, and one user s receive chains. The remaining two PXI-45 chassis house the receive chains of the remaining two users and are each connected to a separate. In addition to the RF hardware TABLE I MIMO-OFDM SYSTEM PARAMETERS Carrier Freq. 2.4GHz Transmit Power 6 dbm Bandwidth 16MHz FFT Size 256 Subcarrier Spacing 6.25kHz Guard Interval 64 samples Total Symbol Duration 2µs MIMO Scheme Alamouti installed, each PXI-45 chassis holds a NI PXI-6653 timing and synchronization module. A simplified hardware block diagram is shown in Figure 1. Each transmitter, or RF signal generator, NI PXI-567, consists of two physical units, an arbitrary waveform generator, named NI PXI-5421, and an upconverter, named NI PXI-56. The arbitrary waveform generator writes an intermediate frequency signal which is later modulated to RF via the upconverter. Each receiver, or RF signal analyzer, NI PXI-566,constitutes a downconverter, named NI PXI- 56, and a digitizer, named NI PXI-562. On the receive side, the downconverter downconverts the signal to an intermediate frequency after which the digitizer samples the waveform which is then sent to the for processing using the LabView software blocks. Note that each user consists of two transmit and two receive chains, totaling six transmit chains and six receive chains for our overall network setup. This RF setup is not entirely new, and has been used in papers such as [12] to implement single user MIMO communication. Our setup, however, is more complex to support multiple users whose hardware components are housed in different chassis and controlled by different s. Moreover, software implementation differs greatly in the methods used for training and channel estimation as well as carrier recovery. The software and hardware has been augmented to support proper concurrent transmission for 3 of 6

4 12 11 Channel Amplitude (db) Fig. 2. Picture of the measurement testbed implemented Subcarrier all users in the network. To support the cross chassis synchronization needed for this multi-user prototype, we install NI PXI-6653 timing and synchronization modules in each chassis. This module has a high stability oven-controlled-crystal-oscillator (OCXO) which can be exported to other chassis, thus enabling synchronization. Locking all the transmitters phase locked loops to this OCXO ensures a unique carrier frequency offset at each receiver for all transmitters. Although frequency offset correction is implemented in software, this synchronization further strengthens the validity of the obtained measurements. Our measurements indicate that the difference in frequency offsets between transmitters, when locked onto the reference signal from the PXI-6653, is below Hz. This remaining frequency offset is then estimated and corrected in software [14], and made possible by tight hardware synchronization. Without tight hardware synchronization, large frequency offsets in interference channels would be much more challenging to correct. In addition to synchronizing clocks, the PXI-6653 allows us to export the trigger generated when the master user begins signal generation. This digital signal is then used to trigger generation at the other transmitters and can also be used to trigger acquisition at the receivers. We discard any measurement that has been corrupted by the ambient interference in the 2.4 GHz ISM band resulting in incorrectly demodulated OFDM data symbols, and thus retain only valid interference free measurements. This triggered acquisition and clock synchronization ensure that our measurements include only channel effects, and are thus free of any timing impairments. The use of digital triggers also organizes the acquisition process over all receivers and thus guarantees that all receivers process the same symbols, without missing transmissions done while processing previous data. Amplitude Frequency x 6 (a) H 11 frequency plot Packet Number (b) The temporal evolution of H 21 Fig. 3. Channel response plots. RESULTS In this section we present the results collected using our measurement setup. We then use the measured data to evaluate the performance gains of using IA. We conducted experiments in the Wireless Laboratory, a classroom in the Engineering Science Building room 113 at The University of Texas at Austin. Transmitters and receivers are placed at varying distances, ranging from 1 meter, to approximately 6 meters apart in both line-of-sight and non-line-of-sight arrangements and are immobile during each measurement scenario. Figure 3(a) is an example frequency plot of H 11 showing limited frequency selectivity as it was generated using a strict lineof-sight arrangement. Moreover, to visualize the temporal evolution of the channel experienced in relatively static indoor environments, we plot the evolution of H 21 over 25 packet transmissions. Inspecting Figure 3(b) reveals that the observed channel variation over successive packet transmissions is minimal. In fact, our measurements indicate 5 4 of 6

5 Sum Rate b/s/hz IA d=.5λ IA d=1λ IA d=2λ IA d=3λ IA d=4λ IA d=5λ IA Rayleigh TDMA d=.5λ TDMA d=4λ Sum Rate b/s/hz SNR (db) Distance (in λ) Fig. 4. Network sum rate for varying antenna spacing. Fig. 5. Network sum rate versus antenna spacing d. that the channel correlation after 2ms for immobile users remains above 97%. We also study the effect of antenna spacing, and thus channel spatial correlation, on IA s performance. IA bases its logic on the fact that equations (4) and (6) can be simultaneously satisfied, almost surely, when all the elements of the channel matrices are drawn independently at random from a continuous distribution with zero mean. This independence assumption, however, is not likely satisfied in practical node deployments that exhibit varying degrees of channel spatial correlation. Therefore, feasibility, in general, remains an open problem, in the sense that it is still unknown whether one can usually find a set of precoders, {F}, and combiners, {W}, that in reality achieve a theoretically achievable degree of freedom allocation in the network. Our measurement results, therefore, give insight into the actual feasibility and performance of this theoretically optimal transmit strategy in realistic channels with actual complexities that are never entirely captured in simulations and models. We conducted measurements for variable antenna separation d {λ/2, λ, 2λ, 3λ, 4λ, 5λ} to properly study the effect of spatial correlation. We also conduct measurements in a configuration with co-located transmitters. In this last arrangement, we position transmit-receive pairs at a distance 1m apart, and place antennas of the same node once at a distance of λ/2 and then at 3λ. We collect channel measurements with significantly high SNR to ensure that recorded measurements are minimally affected by noise. Before evaluating IA over real channels, we first obtain normalized channel matrices, H, for fair comparison with the Rayleigh channels generated to have matrix elements of unit variance. Therefore, we normalize the measured channels over the full data set, to have the same average Frobenius norm of 4. The set of normalized channel matrices { H} is calculated as H kl = 4 1 Ω Ω H kl ( trace ( H kl H kl )) (7) where Ω is the set of all data collected in the scenario considered, i.e. when normalizing a matrix obtained when d = 1λ, Ω would be the set of all channel measurements obtained in that configuration. Figure 4 plots the ergodic sum rate, given by C = 1 N K I ( log N 2 + σ 2 I + R k [n] ) 1 where n=1 k=1 (H kk [n]f k [n]f k [n]h kk [n]) (8) R k [n] = m k H k,m [n]f m [n]f m[n]h k,m [n] is the per-subcarrier interference covariance matrix. Figure 4 verifies the fact that IA outperforms time division multiple access (TDMA), and its equivalent resource allocation schemes in our 3-user setup in terms of ergodic sum capacity, as predicted by theory. Moreover, we note that the throughput gain observed when using IA is most pronounced in the high SNR regime; which is the case of claimed optimality. Comparing the gradients in Figure 4 we observe that interference alignment benefits more from marginal increase of SNR, and thus achieves more degrees of freedom than TDMA. The slope of the plots relative to log 2 (SNR) can be shown to be approximately 1.8 for TDMA and 2.8 for IA, confirming the fact that as SNR approaches infinity, IA is optimal in terms sum capacity. Consequently, Figure 4 illustrates the optimality in the high SNR regime and confirms that IA provides the maximum achievable degrees of freedom, which in this case is 3. 5 of 6

6 Sum Rate b/s/hz IA d=3λ IA d=.5λ TDMA d=3λ TDMA d=.5λ software defined radio base. We have presented indoor network channel measurements collected in our classroom environment, and then processed them to evaluate the performance gains of IA in such node deployments. We showed that the observed gains closely parallel those found through theory and simulation. In subsequent work we will report our outdoor experiments and comparisons with other alignment algorithms SNR (db) Fig. 6. Network sum rate with co-located transmitters We note that there is a constant difference between simulated and measured results in all plots of Figure 4. This is due, primarily, to varying degrees of spatial correlation and frequency selectivity. Measured results exhibit significantly more spatial correlation across antennas and users, and thus more aligned channels, than the simulated i.i.d Rayleigh channels, which decreases SNR after alignment. Varying spatial correlation is also responsible for the constant differences in performance among the different measurement scenarios of Figures 4. Figure 5 shows that IA benefits from increased antenna and user spacing until decreases in spatial correlation become negligible, saturating the achievable sum rate at fixed SNR. In addition to spatial correlation, the simulated i.i.d Rayleigh channel assume independent fading on each subcarrier, which corresponds to significantly more frequency selectivity than the levels present in our indoor measurement results. Frequency selectivity is a main factor aiding IA in achieving the maximum number of degrees of freedom, and therefore contributes to the gap between measured and simulated performance [1]. In addition to confirming IA s theoretical achievements, our measurements give insight into the effect of node placement on performance, and the feasibility of adopting the iterative algorithms proposed in [3] and [4] in relatively static indoor deployments. Though these algorithms may require thousands of iterations to converge, thus consuming valuable computational power, the static nature of the channels observed suggests that, once found, the precoding matrices can be used over many successive packet transmissions. This fact minimizes the relative overhead incurred by using iterative algorithms. CONCLUSION AND FUTURE WORK In this paper, we have presented the first MIMO interference channel testbed programmed using a flexible REFERENCES [1] V. Cadambe and S. Jafar, Interference alignment and degrees of freedom of the K-user interference channel, IEEE Transactions on Information Theory, vol. 54, no. 8, pp , August 28. [2] M. Maddah-Ali, A. Motahari, and A. 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