Compressed Sensing Based Multi-User Millimeter Wave Systems: How Many Measurements Are Needed?
CCOMPRESSED SENSING BASED MULTI-USER MILLIMETER WAVE SYSTEMS:HOW MANY MEASUREMENTS ARE NEEDED?
Ahmed Alkhateeb † , Geert Leus ‡ , and Robert W. Heath Jr. †† The University of Texas at Austin, TX, USA, Email: { aalkhateeb, rheath } ,@utexas.edu ‡ Delft University of Technology, The Netherlands, Email: [email protected]
ABSTRACT
Millimeter wave (mmWave) systems will likely employ di-rectional beamforming with large antenna arrays at both thetransmitters and receivers. Acquiring channel knowledge todesign these beamformers, however, is challenging due to thelarge antenna arrays and small signal-to-noise ratio beforebeamforming. In this paper, we propose and evaluate a down-link system operation for multi-user mmWave systems basedon compressed sensing channel estimation and conjugateanalog beamforming. Adopting the achievable sum-rate as aperformance metric, we show how many compressed sensingmeasurements are needed to approach the perfect channelknowledge performance. The results illustrate that the pro-posed algorithm requires an order of magnitude less trainingoverhead compared with traditional lower-frequency solu-tions, while employing mmWave-suitable hardware. Theyalso show that the number of measurements need to be opti-mized to handle the trade-off between the channel estimatequality and the training overhead.
Index Terms — Millimeter wave communication, com-pressed sensing, achievable rates.
1. INTRODUCTION
Millimeter wave (mmWave) communication is a promisingtechnology for future cellular systems [1–4]. Directional pre-coding with large antenna arrays appears to be inevitable tosupport longer outdoor links and to provide sufficient receivedsignal power. The design of precoding matrices, though, isusually based on complete channel state information, whichis difficult to achieve in mmWave due to (i) the large num-ber of antennas, (ii) the small signal-to-noise ratio (SNR) be-fore beamforming, and (iii) the different hardware constraintswhich impacts the training signal design [5]. Therefore, de-veloping low-complexity mmWave channel estimation tech-niques is crucial for the mmWave system operation.To overcome the hardware limitations, analog beamform-ing solutions were proposed in [6–10], which rely on control-
This work is supported in part by the National Science Foundation underGrant No. 1218338 and 1319556, and by a gift from Huawei Technologies,Inc. ling the phase of the signal transmitted by each antenna via anetwork of analog phase shifters. To avoid the need for ex-plicit channel knowledge, the algorithms in [6–10] dependon beam training which iteratively designs the analog beam-forming coefficients at the transmitter and receiver. These so-lutions, however, have two main disadvantages: (i) they sup-port only single-stream transmissions (2) their training over-head scales linearly with the number of users. To supportmulti-stream transmission, [5] proposes an efficient mmWavechannel estimation algorithm by leveraging the sparse natureof the channel and adaptive compressed sensing tools. Thissolution, however, did not solve the overhead scaling issue.In this paper, we propose a simple downlink system oper-ation for multi-user mmWave systems based on compressedsensing channel estimation and conjugate analog beamform-ing. Contrary to prior work in [5–10], the training overheadof the proposed channel estimation solution does not scalewith the number of users. Hence, it is of special interestfor multi-user mmWave systems. Under certain assumptions,we characterize a lower bound on the achievable rate of theproposed system operation as a function of the compressedsensing measurements. Simulation results show that the pro-posed algorithm requires much less training overhead com-pared with traditional lower-frequency solutions, while em-ploying mmWave-suitable hardware.We use the following notation: A is a matrix, a is a vector,and a is a scalar. (cid:107) A (cid:107) F is the Frobenius norm of A , whereas A T , A ∗ , A − , are its transpose, Hermitian, and inverse, re-spectively. E [ · ] denotes expectation and { . } is an indicator.
2. SYSTEM MODEL
Consider a mmWave system with a base station (BS) hav-ing N BS antennas and N RF RF chains as shown in Fig. 1.The BS is assumed to communicate with U mobile stations(MS’s), and each MS is equipped with N MS antennas. Wefocus on the multi-user beamforming case in which the BScommunicates with every MS via only one stream . Further,we assume that the maximum number of users that can be si-multaneously served by the BS equals the number of BS RFchains, i.e., U ≤ N RF . This is motivated by the spatial mul- a r X i v : . [ c s . I T ] M a y ++ F RFBeamformers w u RFcombinerRF Chain N BS N RF N MS Base station uth mobile station
RF Chain RF Chain s s N RF Fig. 1 . A BS with RF beamformers and N RF RF chains com-municating with the u th MS that employs RF combining.tiplexing gain of the described multi-user precoding system,which is limited by min ( N RF , U ) for N BS > N RF . For sim-plicity, we will also assume that the BS will use U out of the N RF available RF chains to serve the U users.On the downlink, the BS applies an N BS × U RF precoder, F = [ f , f , ..., f U ] . The sampled transmitted signal is there-fore x = Fs , where s = [ s , s , ..., s U ] T is the U × vector oftransmitted symbols, such that E [ ss ∗ ] = P T U I U , and P T is theaverage total transmitted power. Since F is implemented us-ing quantized analog phase shifters, [ F ] m,n = √ N BS e j φ m,n ,where φ m.n is a quantized angle, and the factor of √ N BS isfor power normalization.For simplicity, we adopt a narrowband block-fading chan-nel model [5,11,12], by which the u th MS receives the signal r u = H u U (cid:88) r =1 f r s r + n u , (1)where H u is the N MS × N BS matrix that represents themmWave channel between the BS and the u th MS, and n u ∼ N ( , σ I ) is a Gaussian noise vector.At the u th MS, the RF combiner w u is used to process thereceived signal r u to produce the scalar y u = w ∗ u H u U (cid:88) r =1 f r s r + w ∗ u n u , (2)MmWave channels are expected to have limited scatter-ing [2]. Therefore, and to simplify the analysis, we will as-sume a single-path geometric channel model [9, 13]. Underthis model, the channel H u can be expressed as H u = (cid:112) N BS N MS α u a MS ( θ u ) a ∗ BS ( φ u ) , (3)where α u is the complex path gain, including the path-loss,with E (cid:2) | α u | (cid:3) = ¯ α . The variables θ u , and φ u ∈ [0 , π ] are the angles of arrival and departure (AoA/AoD) respec-tively. Finally, a BS ( φ u ) and a MS ( θ u ) are the antenna arrayresponse vectors of the BS and u th MS respectively. The BS and each MS are assumed to know the geometry of their an-tenna arrays. While the results and insights developed in thepaper can be generalized to arbitrary antenna arrays, we willassume uniform arrays in the simulations of Section 5.
3. PROPOSED DOWNLINK SYSTEM OPERATION
The proposed downlink operation for multi-user mmWavesystems consists of two phases: (i) compressed sensing baseddownlink channel estimation and (ii) conjugate analog beam-forming/combining. For the downlink channel training, ran-dom beamforming and projections are used to efficientlyestimate the mmWave channel with relatively low trainingoverhead thanks to the sparse nature of the channel. Onemain advantage of this technique is that all the MS’s can si-multaneously estimate their channels. Therefore, the trainingoverhead does not scale with the number of users. This iscontrary to the adaptive channel estimation and beamformingdesign techniques in [5, 6, 9], which are user-specific. Theestimated channels are then used to build the analog beam-formers and combiners. Extensions to hybrid analog/digitalprecoders are also possible [14], but our focus in this paper ison the evaluation of compressed sensing channel estimation.
Given the geometric mmWave channel model in (3), estimat-ing the channel is equivalent to estimating the different pa-rameters of the channel path; namely its AoA, AoD, and thecomplex gain. In this section, we exploit this poor scatteringnature of the mmWave channel, and formulate the channel es-timation problem as a sparse problem. We then briefly showhow compressed sensing can be used to estimate the channel.
A sparse formulation:
Consider the system and mmWavechannel models described in Section 2. If the BS uses a train-ing beamforming vector p m , and the u th MS employs atraining combining vector q n to combine the received signal,the resulting signal can be written as y n,m = q Hn H u p m s m + q Hn n n,m , (4)where s m is the training symbol on the beamforming vec-tor p m , and we use s m = √ P , with P the average powerused per transmission in the training phase. If the BS em-ploys M BS such beamforming vectors p m , m = 1 , ..., M BS ,at M BS successive time slots, and the MS uses M MS mea-surement vectors q n , n = 1 , , ..., M MS at M MS successiveinstants to detect the signal transmitted over each of the beam-forming vectors, the resulting received matrix will be [5] Y MS = √ P Q H H u P + N , (5)where Q = [ q , q , ..., q M MS ] is the N MS × M MS measure-ment matrix, P = [ p , p , ..., p M BS ] is the BS N BS × M BS beamforming matrix, and N is an M MS × M BS noise matrix.o exploit the sparse nature of the channel, we first vec-torize the resultant matrix Y MS as in [5] to get y MS = √ P (cid:0) P T ⊗ Q H (cid:1) ( a ∗ BS ( φ u ) ⊗ a MS ( θ u )) α u + v , (6)To complete the problem formulation, we assume that theAoDs, and AoAs are taken from a grid of G BS and G MS points, respectively. By neglecting the grid quantization er-ror, we can approximate y MS in (6) as y MS = √ P (cid:0) P T ⊗ Q H (cid:1) (cid:16) A ∗ BS ⊗ A MS (cid:17) z u + v , (7)where the N BS × G BS matrix A BS and N MS × G MS matrix A MS are the dictionary matrices that consist of the columnvectors a BS (cid:0) ¯ φ k (cid:1) , and a MS (cid:0) ¯ θ (cid:96) (cid:1) , respectively, with ¯ φ k , and ¯ θ (cid:96) the k th, and (cid:96) th points of the angle grids. z u is a G BS G MS × vector which carries the path gains of the correspondingquantized directions. Hence, z u is a sparse vector with only non-zero element. Note that detecting the column of A BS and A MS that corresponds to this non-zero element directlyimplies the detection of the AoA and AoD of the channel path. Compressed sensing measurements:
Thanks to thesparse formulation of the mmWave channel estimation prob-lem in (7), compressed sensing tools can be leveraged todesign efficient training beamforming/combining matrices[5,15]. Considering the measurement matrix Φ = P T ⊗ Q H ,and the dictionary Ψ = A ∗ BS ⊗ A MS , one interesting researchdirection is to study the conditions on Φ , Ψ under which thesupport of the sparse vector z u can be recovered with highprobability and with low training overhead. Leaving this ob-jective for future work, we will try in this paper to get someinsights into a sufficient (not necessarily the minimum) num-ber of measurements, and the relation between this numberand the achievable rate of mmWave systems. For that, weconsider the following measurement matrix.The BS will design its training beamforming matrix P , such that [ P ] m,n = e j φ m,n where φ m,n is randomlyand uniformly selected from the set of quantized angles { , πN BS Q , ..., ( N BS Q − ) πN BS Q } . Each MS similarly designs itstraining combining matrix Q , with N MS Q angle quantizationbits. Note that for this design, each entry of the measurementmatrix Φ will also be equal to e j ζ , with the angle ζ selectedrandomly from a certain quantized angle set. AoA/AoD estimation:
To estimate its channel AoA/AoD,each MS u needs to recover the support of its sparse vector z u . As the measurement and dictionary matrices Φ , Ψ areknown for all MS’s, each MS can use sparse recovery algo-rithms (such as LASSO [16], Orthogonal Matching Pursuit(OMP) [17], etc.) to estimate its channel AoA/AoD. In thesimulations of Section 5, we adopt OMP for low-complexity.In this case, the support of z u , supp( z u ), is determined bysolving supp( z u ) = arg max Ψ H Φ H y MS , (8)which directly determines the estimated AoA/AoD , ˆ θ u , ˆ φ u . Given the estimated AoA, each MS u will design its analogcombining vector such that w u = a MS (cid:16) ˆ θ u (cid:17) . Each MS u will also feed the index of its estimated AoD back to the BSwhich needs log G BS bits. Finally, the BS designs its analogbeamforming matrix F to match the effective channels (in-cluding the effect of the combining vectors), i.e., the BS sets F = (cid:104) a BS (cid:16) ˆ φ (cid:17) , a BS (cid:16) ˆ φ (cid:17) , ..., a BS (cid:16) ˆ φ U (cid:17)(cid:105) .
4. ACHIEVABLE AND EFFECTIVE RATES
In this section, we evaluate the achievable rate of the proposeddownlink mmWave system operation in a special case and tryto get some insights into the relation between the performanceof mmWave systems and compressed sensing measurementsin more general cases.Consider the system model in Section 2 and the proposedconjugate analog beamforming/combining in Section 3.2. Fortractability, we make the following assumptions
Assumption 1
All path gains are constants. This is relevantto mmWave LOS paths, which are dominant in dense mmWavenetworks [18].
Assumption 2
The BS and MS’s employ uniform arrays.
Assumption 3
The sizes of the grids in (7) are G BS = N BS , G MS = N MS , and the angle grid points ¯ φ k , ¯ θ (cid:96) , of the dictio-nary Ψ are the virtual directions satisfying πdλ sin (cid:0) ¯ φ k (cid:1) = πkN BS and πdλ sin (cid:0) ¯ θ (cid:96) (cid:1) = π(cid:96)N MS [19]. Under the assumptions 2-3, the matrices A BS and A MS become DFT matrices [19]. Further, using the virtual chan-nel model transformation in [19], we note that the steeringvectors of both the actual and estimated AoAs/AoDs (of thedictionary Ψ ) are columns of these matrices. Therefore, therate of user u , R u , can be written as log { ˆ θ u = θ u ,φ u = ˆ φ u } { ˆ θ u = θ u } (cid:80) Ur =1 { φ u = ˆ φ r } + SNR U N BS N MS | α u | . (9)Denoting ˚ R u = log (cid:16) SNR U N BS N MS | α u | (cid:17) as thesingle-user rate (without interference), the average achievablerate of user u , ¯ R u , can be then lower bounded as ¯ R u ≥ E (cid:104) ˚ R u { (cid:84) r (cid:54) = u ( φ u (cid:54) = ˆ φ r ) } { ˆ θ u = θ u ,φ u = ˆ φ u } (cid:105) , (10) ( a ) = ˚ R u P { (cid:84) r (cid:54) = u ( φ u (cid:54) = ˆ φ r ) } ) P { θ u = θ u ,φ u = ˆ φ u } , (11) ≥ ˚ R u (cid:18) − UN BS (cid:19) P { ˆ θ u = θ u ,φ u = ˆ φ u } , (12)where (a) follows from the independence between the estima-tion success event { ˆ θ u = θ u , φ u = ˆ φ u } and the single-user
50 100 150 200 250 300 350 400 450 5000123456789
Number of Measurements (M BS x M MS ) A c h i e v ab l e R a t e ( bp s / H z ) Analog Beamforming − Perfect Channel KnowledgeAnalog Beamforming − CS−Based Channel EstimationLower Bound in (12) with 0.95 recovery success probability
Fig. 2 . Achievable rates using the proposed system operationfor different numbers of compressed sensing measurementsrate given assumption 1. Now, we note that P { ˆ θ u = θ u ,φ u = ˆ φ u } is the probability of correct support recovery of the sparsevector z u . This directly relates the achievable rate to the com-pressed sensing literature. For a general relation, assume thechannel fading coherence equals L C symbols, the effectiveachievable rate of user u (considering the training overhead)can be written as ¯ R u, eff ≥ ˚ R u (cid:18) − UN BS (cid:19) (cid:18) − M (cid:15) L C (cid:19) (1 − (cid:15) ) , (13)where M (cid:15) equals the number of measurements needed toguarantee the support recovery of the sparse vector z u withprobability at least − (cid:15) . Given this relation in (13), it isinteresting to design the measurement matrix Φ to maximizethis lower bound on the achievable rate. It is also of interest tooptimize the number of measurements to handle the tradeoffshown between the accurate channel estimate (small (cid:15) ) andlarge training overhead.
5. SIMULATION RESULTS
In this section, we evaluate the performance of the proposeddownlink mmWave system operation. We consider the sys-tem model in Section 2 with the BS employing a ULA of antennas and MS’s, each with antennas. The system op-erates at GHz with a bandwidth of MHz. The BS-MSdistance is m, and all channels are LOS and single-path.In the channel estimation phase, the BS and MS’s apply therandom beamforming and measurement procedure describedin Section 3.1 with N BS Q = N MS Q = 4 , and with average trans-mit power equal to dBm. The beamformers/combiners arethen built as described in Section 3.2.In Fig. 2, the per-user achievable rate of the proposedsystem operation is shown versus the number of compressed Number of Measurements (M BS x M MS ) E ff e c t i v e A c h i e v ab l e R a t e ( bp s / H z ) L C = 600 symbolsL C = 400 symbolsL C = 200 symbols Training that maximizesthe achievable rate
Fig. 3 . Achievable rates using the proposed system operationfor different values of channel fading coherence.sensing measurements. This figure illustrates that ∼ − measurements are needed to approach the achievable ratewith perfect channel knowledge. While this number may looklarge, it is actually an order of magnitude less than what is re-quired by traditional lower frequency solutions, which is ∼ N BS N MS = 2048 symbols. Note also that this training over-head does not depend on the number of users, which makescompressed sensing of special interest to multi-user mmWavesystems. Compared with adaptive compressed sensing tech-niques [5], they may require less training to estimate each userchannel, e.g., ∼ − in [5]. This overhead, however,scales with the number of users, which means − for users, and larger if more users are served. Finally, note thatthese results just illustrate a sufficient number of measure-ments given the proposed design of the measurement matrix Φ in Section 3.1. Therefore, optimizing this design may leadto even smaller training overhead.In Fig. 3, the effective achievable rate is shown for dif-ferent numbers of fading coherence values, L C . This figureindicates that it is important to compromise between the ac-curacy of the channel estimate and the training overhead tomaximize the achievable rate, especially with fast channels.
6. CONCLUSION
In this paper, we proposed and evaluated a low-complexitydownlink system operation for multi-user mmWave systemsbased on compressed sensing channel estimation. Simulationresults showed that the proposed system operation requiresa relatively small training overhead (w.r.t. the channel matrixdimensions) to achieve a very good performance. Results alsoindicated that the number of measurements need to be wiselyselected to maximize the effective system sum-rate. . REFERENCES [1] Z. Pi and F. Khan, “An introduction to millimeter-wave mobilebroadband systems,”
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