Optimal Power Allocation for Pilot-Assisted Interference Alignment in MIMO Interference Networks: Test-bed Results Nima N. Moghadam
Hamed Farhadi
Per Zetterberg
Signal Processing Department KTH Royal Institute of Technology Stockholm, Sweden
[email protected]
Department of Signals and Systems Chalmers University of Technology Gothenburg, Sweden
[email protected]
Signal Processing Department KTH Royal Institute of Technology Stockholm, Sweden
[email protected]
Abstract—This paper addresses channel training and data communication over multi-input multi-input (MIMO) interference networks. We consider a pilot-assisted interference alignment scheme in which part of radio resources are allocated to channel training and the remaining resources are used for data transmission. A more accurate channel estimation can be obtained by increasing pilot transmission power. Since each transmitter has limited energy budget, this implies that less power is available for data transmission. Clearly, there is a trade off between the allocated power for channel training and the one for data communication. In order to investigate this trade off, first we compute an achievable sum-rate, and next we find the optimum power allocation to pilot transmission and data transmission. Finally, we verify these theoretical results with experimental measurements on USRP-based test-bed. Index Terms—Interference alignment; pilot-assisted channel training; OFDM; USRP test-bed; power allocation.
I. I NTRODUCTION Multi-input multi-output (MIMO) interference networks, in which multiple transmitter-receiver pairs share transmission medium for communication, appear in several existing [1] and emerging wireless communication scenarios [2]. In these networks, each transmitter intends to communicate to an intended receiver while it may cause interference to nonintended receivers. The inter-user interference makes quality of each transmitter-receiver link interrelated to the other users’ transmission strategies. Therefore, terminals require to coordinate their transmission in order to provide a reliable communication. Interference alignment technique, in which transmitters coordinate their transmissions such that interference signals at each receiver be aligned into a subspace of the received signal space and desired signal be received into an independent subspace, is an effective approach for data transmission over interference networks [3]. In order to implement interference alignment scheme, terminals require to know channel state information (CSI). However, CSI is not a priori available at terminals and certain mechanisms should be designed to acquire CSI at terminals. In order to obtain CSI estimation at receivers, pilot-assisted channel training schemes have been developed for single-input multi-output (SIMO) point-to-point
communication systems [4], single-input single-output (SISO) interference networks [5], and MIMO interference networks [6]. Also, digital/analog channel state feedback schemes have been proposed in the literature to acquire CSI estimation at transmitters (see e.g. [7], [8], [9]). The pilot-assisted channel training schemes require each transmitter to allocate a portion of total radio resources (time/frequency/power) for pilot transmission; consequently, only the remaining resources can be used for data transmission. Clearly, there is a trade-off for resource allocation between pilot transmission and data transmission: A more accurate channel estimation can be acquired by allocating more radio resources for channel training which implies that less radio resources are left for data transmission. The optimum length of channel training phase for MIMO systems has been investigated in [6], [10], and the optimum power allocation for SISO interference networks is addressed in [5]. In this paper, we study the problem of power allocation to pilot transmission in MIMO interference networks. In order to address this problem, first we compute the achievable rate of a pilot-assisted interference alignment scheme. Next, we find the optimum power allocation to pilot transmission. Finally, we verify the theoretical results with experimental measurements on a USRP-based wireless test-bed composed of three transmitters and three receivers. II. S IGNAL
AND
S YSTEM M ODEL
In this paper, we consider an OFDM-MIMO interference network with K source-destination pairs. The sources and destinations are denoted by Sj and Dk (j, k ∈ {1, . . . , K}) each equipped with Ns and Nd antennas, respectively. The signals transmitted from the sources are modulated according to OFDM modulation. In each OFDM symbol there are N subcarriers, where αN subcarriers are allocated for pilot transmission and (1 − α)N subcarriers are used for data transmission. The parameter α (1/N < α < 1) is a design parameter. We refer to these subcarriers as pilot subcarriers and data subcarriers, respectively. Each source-destination link conveys d ≤ min{Ns , Nd } signal streams per OFDM
data subcarrier, where d is an achievable degrees of freedom s +Nd in the considered network. In [11], it is shown that d ≤ NK+1 is achievable using interference alignment technique. It is assumed that the cyclic prefix of the OFDM symbols are long enough to avoid the interference between the subcarriers. Therefore, we study the subchannels independently and omit the subcarrier indexing for the sake of notational simplicity. In the remaining parts of this section, the signal transmission within data and pilot subcarriers are presented separately. A. Data Transmission Within the data subcarriers, all the sources transmit simultaneously. The transmitted signal from Sj is xj = Vj bj ∈ CNs ×1 where bj = [bj,1 , . . . , bj,d ]T is the vector of transmitted data streams and Vj = [vj,1 , . . . , vj,d ] ∈ CNs ×d is the precoding matrix corresponding to Sj (i.e. vj,m is the beamforming filter corresponding to bj,m ). The data streams are zero-mean Gaussian distributed with unit variance. The observed signal in a data subcarrier at Dk is K r X Pd Hkj xj + zk (1) yk = d j=1 where zk ∼ CN (0, N0 INd ) is the thermal noise received at Dk and the power used for data transmission by each user is denoted by Pd . The matrix Hkj ∈ CNd ×Ns denotes the channel matrix between Sj and Dk . The entries of Hkj are independent and distributed as CN (0, 1). The receive combining filter uk,m ∈ CNd ×1 is employed at Dk to extract the mth stream yielding K X d r X Pd H uk,m Hkj vj,m bj,m + uH (2) yk,m = k,m zk d j=1 m=1 where (.)H denotes the Hermitian transpose. Both beamforming and receive combining filters are unit vectors. B. Pilot Transmission and Channel Estimation The channels are estimated by the help of known transmitted pilot symbols. In each OFDM symbol, equally separated subcarriers with distance L, are allocated for pilot transmission. The channels at pilot subcarriers are estimated using least square (LS) estimator and channels at data subcarriers are estimated using frequency-domain piecewise-linear interpoolation [12]. In the following, the channel estimation at pilot and data subcarriers is explained and the average channel estimation error is computed. The pilot symbols are transmitted in an orthogonal fashion, with power Pτ , from all the transmitting antennas in pilot subcarriers. The orthogonality is obtained by coding the pilot signals across KNs successive OFDM symbols. The decoded pilot signal transmitted from the lth antenna of Sj to the qth antenna of Dk (l ∈ {1, . . . , Ns } and q ∈ {1, . . . , Nd } ) is p rkq ,jl = Pτ tTq Hkj tl + tTq zk (3) where tq , with appropriate dimension, is antenna selection vector which picks the channel coefficients corresponding to
the qth antenna (i.e. the qth element of tq is equal to one while all the other elements are equal to zero). By utilizing an LS estimator, the corresponding estimated channel coefficient is p p 1/Pτ rkq ,jl = tTq Hkj tl + 1/Pτ tTq zk . (4)
Note that, in this case the estimation error is a zero-mean 0 Gaussian distributed random variable with variance σz2 = N Pτ and is independent from the channel coefficient. In [12], the mean square error (MSE) of channel estimations at different subcarriers using frequency-domain piecewiselinear interpolation is computed as a function of frequencydomain correlation of channel responses. It can be observed that when the root mean square (RMS) delay-spread of channel is small (i.e. frequency-domain correlation of channel responses goes to one), the MSE is independent from the channel and is computed as σ ¯h˜2 = λσz2 , (5) By estimating the channel coefficients where λ = using the aforementioned method, the channel matrix between Sj and Dk can be represented as 1 2 3 (2+1/L ).
ˆ kj = Hkj + H ˜ kj , H (6) ˜ kj ∼ CN (0, σ where H ¯h˜2 I) is the channel estimation error 1 which is independent from Hkj . C. Interference Alignment In order to increase the spectral efficiency, interference alignment is used to design the beamforming and receive combining filters. Since the channels are not perfectly known, the perfect alignment of the interference signals is not possible. However, replacing the channels with their corresponding estimates in the interference alignment conditions [13] leads to alignment of part of interference signals. The unaligned interference power will leak to the signal subspace and will be treated as noise. In this case, the following conditions needs to be satisfied for all j ∈ {1, . . . , K} and m ∈ {1, . . . , d} ˆ uH k,m Hkj vj,m = 0, ∀j 6= k, ˆ uH k,m Hkk vk,l 6= 0. If the filters satisfy (7), the equation (2) turns into r Pd H ˆ yk,m = u Hkk vk,m bk,l + d k,m K X d r X Pd H ˜ uk,m Hkj vj,m bj,m + uH k,m zk . d m=1 j=1
(7) (8)
(9)
III. O PTIMAL P OWER A LLOCATION The allocated power for data and pilot transmission can be different in general as long as the average transmitted power by each source remains constant. In this section, we find the optimal power allocation for data and pilot transmission to maximize the achievable sum-rate of the users in the network. 1 Note that the estimation error variance can generally be different for different subcarriers. Here, the average error variance, i.e. MSE, is considered since we are interested in improving the average performance over all the bandwidth.
Let us show the average transmitting power of each user per subcarrier by P . Adapting the convention in [5], we assume that Pd = βP , where 0 ≤ β ≤ 1/(1−α) is a power allocation factor. Energy conservation for each OFDM symbol dictates αN Pτ + (1 − α)N Pd = N P,
(10)
P . The filters and therefore the pilot power is Pτ = (1−(1−α)β) α designed by helps of the equations (7) and (8) are independent from direct channel estimates and the estimation errors [14]. Therefore the achievable rate of user k is derived in [6] as Rk =
K X d X
E
j=1 n=1
2 ˆ uH k,n Hkk vk,n log 1 + K d (11) 2 P P Pd H ˜ d E uk,m Hkj vj,m +N0 Pd d
j=1 m=1
ˆ kk . Considering where the outer expectation is taken over H that the filters are unit vectors designed independently from ˜ kj , the achievable rate can be channel estimation error, H rewritten as !# " K X d 2 Pd X d |heq,k,n | Rk = E log2 1 + KPd σ ¯h˜2 + N0 j=1 n=1 =Kd log2 (e) exp(1/SNReq )E1 (1/SNReq )
(12)
ˆ where heq,k,n = uH k,n Hkk vk,n is the equivalent desired channel of the nth stream, distributed as heq,k,n ∼ CN (0, 1 − σ ¯h˜2 ). The expectation in (12) Ris calculated using the equation (34) ∞ in [15] where E1 (x) = 1 1t e−xt dt, x > 0 and Pd 2 1 − σ ¯ ˜ d h SNReq = . (13) KPd σ ¯h˜2 + N0 As it is proposed in [5], the rate Rk is a monotonically increasing function of SNReq and therefore to maximize the sum-rate it is sufficient to maximize SNReq . Substituting σ ¯h˜2 from (5) in (13) and replacing Pd and Pτ by their corresponding values as functions of P leads to, P 2 P 1 −β N0 (1 − α) + β( N0 − λα) SNReq = d β [λKα − (1 − α)] + 1
(14)
which is a strictly concave function of β. By solving the KKT conditions [16] for the SNReq maximization problem the optimal powers will be computed as Pd = βopt P,
Pτ =
1 − (1 − α)βopt P α
(15)
where βopt =
1 1 − (1 + λK)α v u u (λα − × 1 − t1 +
P N0 ) [1 − (1 + P N0 (1 − α)
λK)α]
. (16)
IV. T EST- BED I MPLEMENTATION The implementation was performed on KTH four-multi test-bed. KTH four-multi is a software defined radio (SDR) test-bed with multiple users and multiple antennas (http:// fourmulti.sourceforge.net/). The hardware and software structure of the test-bed is presented in the following. A. Hardware Platform The test-bed consists of six nodes, each equipped with two vertically polarized antennas. Three of the nodes are fixed and transmitting while the rest are movable and receiving. Two universal software radio peripheral (USRP) transceivers are employed in each transmitting or receiving node to convert based-band to RF or RF to base-band signals, respectively. Transmitting and receiving USRPs are using custom frontends which have better noise figure compared to Ettus standard daughter-boards. The RF signal generated by each USRP is amplified before transmission using a 30 dB gain ZRL2400LN power amplifier. The baseband processing is performed in KTH four-multi software framework installed on two computers. One computer is controlling all the transmitting nodes and the other one controls all the receiving nodes. B. Software Platform The software platform is implemented based on KTH four-multi framework, which is developed using UHD driver toolkit. Similar to [17], modulation and coding, interleaving and OFDM modulation of the signals are performed using AMC and modem_OFDM1 toolboxes of KTH four-multi. Minimum interference leakage (MinIL) and maximum signalto-interference-and-noise-ratio (Max-SINR) algorithms [13] are implemented as beamforming functions and the ZF and MMSE receivers are exploited to filter the received signals at the destinations. The MinIL and Max-SINR are two iterative representations of interference alignment. In MinIL, the objective is to minimize the interference leakage power after filtering at the destinations while in Max-SINR the received SINR is maximized at the destinations. C. Frame Structure Two types of cell-specific downlink reference signals (CRS) are transmitted within each frame, namely channel state information reference signal (CSI-RS) and demodulation reference signal (DM-RS). CSI-RS is used at the receivers to acquire the channel state information. These channel states are fed back to one of the sources (master node) in order to design the precoders. DM-RS helps the destinations to perceive the effective channels. The effective channels encompass the precoders as well as the physical channels between the sources and destinations. The CSI-RS are the equivalent of the pilot symbols in the previous sections while the DM-RS are introduced as a practical adaptation. Each frame consists of 33 OFDM symbols. Three symbols are dedicated for DM-RS transmission (DM-RS OFDM symbol), while the rest are used for CSI-RS and payload transmission (payload OFDM symbol). A DM-RS OFDM
Destination
Baseband Processing
Source
Time
10 m
Frequency Subcarriers
Fig. 1. Frame structure: 33 OFDM symbols with 38 subcarriers are transmitted within each frame. The light gray, dark gray and white circles are representing DM-RS, CSI-RS and data symbols, respectively.
symbol is transmitted for each stream in the network in a TDMA fashion. During DM-RS transmission, the corresponding transmitter fills all the subcarriers with beamformed pilots while the other transmitters remain silent. Two subcarriers in each payload OFDM symbol are allocated for CSI-RS transmission while the rest of the subcarriers are allocated for payload transmission. The payload is modulated according to 16 QAM modulation while the pilots are QPSK modulated symbols. As suggested in IEEE802.11ac standard, and also confirmed by measurement in [18], frequency domain granularity of feedback of every fourth subcarrier (i.e. L = 4) does not incur any performance loss in our measurements. Hence 10 ≈ 38/4 subcarriers were dedicated for pilot transmission. During transmission of a frame, channels remain constant. Therefore to further decrease the overhead, the pilot subcarriers were spread over 5 different OFDM symbols with only two pilot subcarriers per each symbol (i.e. α = 2/38). To maintain the orthogonality of the pilots transmitted from each transmitting antenna, the pilots are coded across the same subcarriers of every six successive OFDM symbols. Figure 1 shows the frame structure used in our measurements. V. M EASUREMENT M ETHODOLOGY The measurements were performed in an indoor office environment. The floor-map of the measurement environment is depicted in Fig. 2. Each source is represented by a colored square in the figure. A circle with the same color as the source node’s color shows the area in which the corresponding destination node is moving. In each batch of measurements, the destination nodes are placed in a random position inside the circles and remain still until the end of the measurement. During the measurements, the channels are kept relatively static as we made sure that nobody is moving through the corridor or working in the nearby offices. During the measurement 50 arrangements of destinations were measured. In each batch of measurements, two bursts of frames, each including nine frames with nine different β values, were transmitted successively. The source and destination filters of the first and the second bursts were designed based on MinIL and Max-SINR algorithms, respectively. During the measurements, the maximum number of iterations in both algorithms was set to 10 and the received SNR was 25 dB.
Fig. 2. KTH four-multi test-bed and the Measurement environment. 200
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VI. M EASUREMENT R ESULTS The performance metric used for measurements is error vector magnitude (EVM). An error vector is a vector in I-Q plane between the actual constellation point transmitted from a source and the received point at the corresponding destination. The magnitude of error vector represents the amplitude of the received noise (and interference) after decoding the signal. The EVM is usually defined as s Perror EVM = × 100 (17) Pref erence where Perror is the average power of error vectors and Pref erence is the average power of the constellation. Therefore EVM can beprelated to the measured SINR in the symbol level as EVM = 1/SINR × 100. The measured EVM averaged over three users for MinIL and Max-SINR algorithms are presented in Fig. 3. The simulation result is also shown in this figure. The minimum EVM (equivalently, maximum SINR) for MinIL, Max-SINR and the simulation curves is achieved at β = 0.75, β = 0.8 and β = 0.79, respectively. Although the optimal values of β found in the measurement and simulation are close but there is a gap between the EVM values in the measurement and simulation. This gap is mainly due to the leakage of interference power to the signal subspace. Limited number of iterations in the considered iterative interference alignment algorithms, channel model mismatch and hardware imperfections cause interference leak to the signal subspace.
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