Noncoherent compressive channel estimation for mm-wave massive MIMO

Noncoherent compressi v e channel estima tion for mm-w a ve massi v e MIMO Maryam Eslami Rasekh ∗ , Upamanyu Madhow † Departmen t of Electrical and Computer Eng ineering University of California Santa Barba r a Email: ∗ rasekh@ucsb.edu, † madhow@ece.ucsb.edu Abstract —Millimeter (mm) wa ve massive MIMO has the potential fo r delivering o rders of magnitude increases in mobile data ra tes, with compact antenna arrays prov iding narrow steerable beams fo r unprecedented le vels of spatial reuse. A fundamental techn ical bottleneck, howev er , i s rapid spatial channel estimation and bea m adaptation in th e face of mobility and blockage. Recently proposed compressi ve techniq u es which exploit the sp arsity of mm wave channels are a promising ap- proach to this p roblem, with ov erhead scaling linearly with the number of dominant paths and logarithmically with th e nu mber of array elements. Further , they can be implemented with RF beamf orming wi th low-precision phase control. Howev er , these methods make i mplicit assumpti ons on long-term p h ase coher- ence that ar e not satisfied by existing hardwar e. In th is paper , we propose and evaluate a noncoher ent compressiv e channel es- timation techniqu e which can estimate a sparse spatial chann el based on received signal strength (RSS ) alone, and is compatible with off-the-shelf hardwar e. The approach is based on cascading phase retriev al (i.e., recov ery of complex-valued measurements from RSS measure ments, up to a scalar multiple) with coherent compressi ve estimation. While a conv enti onal cascade scheme would multiply two measurement matrices t o obtain an overa ll matrix whose entries are in a continuum, a key novelty in our scheme is that we constrain the overa ll measureme nt matrix to be implementable using coarsely quantized pseudorandom phases, employing a virtua l decomposition of the matrix into a p roduct of measureme nt matrices for ph ase retriev al and compressi ve estimation. Theoretical and simulation results show that our noncoherent meth od scales almost as well with array size as its coherent counterpart, th us inheriting the scalability and low overhead of th e latter . Index T erms —Milli meter W av e, Channel Esti mation, Sparse Multipath Channel, Noncoherent M easurement, Compressiv e Estimation, Phase Retrieval. I . I N T RO D U C T I O N Emerging mm wave mobile n etworks have the po tential to deliver several orders of m a gnitude in c r eases in both per-user and network cap acity , using a den se deploymen t of small cell base stations and a ggressive spatial reu se. The small wa velengths allo ws scaling to massiv e MIMO with a very large number of elements within a form factor compa tib le with compact base stations dep loyable, for example, o n lampposts. The pencil b eams en abled by such array s enab les drastically increased sp atial r e use compared to existing ce llu - lar n etworks. A critical challenge in r ealizing such mm wav e mobile links, h owe ver, is the agile adaptation o f such large arrays to tra ck mobile u sers, while ac c ounting for fr equently occurrin g blo ckages. As array sizes g row , simp lification o f fron t-ends is cr itical. Existing mm wa ve transceivers employ RF b eamform ing in place of costly p er-element baseband contro l. This m e ans a single I/Q stream is upconverted at the transmitter and dis- tributed to all arr ay elements, an d the phase of e ach element is tu ned with RF phase shifte r s. At the recei ver , the phase of each elem ent is m a n ipulated at RF and the combinatio n of element outputs provides a single I/Q stream for signal processing. Thus, classical least squares tech niques, which require individual access to the I/Q sign al at e ach antenna element, can not b e applied for adaptive beamfo rming. Con ventional techniques fo r discovering spatial paths wh en constrained to RF beamfo rming include exhausti ve and hi- erarchical scan. In exhaustive scan, the transmitter scans the entire angular dom ain with its nar row beam to iden tify the strongest p a th (s) to the user . The numb er of measur ements scales linearly with array size, hence the overhead becomes prohib iti ve for large arrays. In hierarchical scan, the en tire an - gular space is initially scanned with a small number of b road beams, with fe e dback from the receiver u sed to successively narrow the search space. Th e nu m ber of m easurements scales logarithm ic a lly w ith array size, but waiting for feedb a c k f rom the rec ei ver be f ore each scan can be high ly time- consuming and impractical u p on implem entation. T h is m ethod does no t scale well with the numb er of users, since each user may require a different beacon sequence, dependin g on its location and feed back. Neither of these metho ds is theref ore suitab le for lo w-overhead trac king for mm wa ve massi ve MI M O. A promising alternative, which is ou r startin g point in this paper, is to employ comp r essive tech niques, introd uced in [1], [2] an d discussed in de tail in the context of pico cellular networks in [3 ], which leverage the inh erent sparsity o f the mm wav e ch annel to track u sers with a small numb er o f measuremen ts. In this a p proach , the transmitter broadc a sts beacons using p seudoran d om phases, and uses f eedback fro m the receiver regarding the complex g ain observed fo r each beacon to estimate the spatial chann e l by identify ing the dominan t path s. Such schemes can be implemented usin g RF beamf o rming (i.e., a single RF ch ain, r ather than on e RF chain pe r elemen t) with sev erely q uantized phase contro l, which allo ws simplification of the RF f r ont end. W e term the app roach in [1]–[3] coh er en t compressive estimation, since th e receiver must m aintain phase cohe r- ence acro ss suc c essi ve measur ements in order to p rovide the desired c o mplex g ains as feed back to the transmitter . Unfortu n ately , such an approach does n ot work with co m - modity hardware, since current mm wave systems such as the 802 .11ad standar d are n ot d esigned to maintain phase coheren ce across p ackets, and the oscillator offset and drift between th e transmitter a n d receiver can alter the p hase of each channe l measurement randomly . This moti vates us to develop the nonco her ent co mpressive estimation approach presented in this paper . Contributions: W e use the same compressive b e acon strat- egy as in prior work [1]–[3], but provide an algo rithm that can estimate a sparse spatial channel fro m RSS measurem e n ts alone. It do es not re q uire ph ase coherence acro ss b eacons, and can there f ore be realized with co m modity hard ware. 1) W e pro pose an d ev a lu ate a two-stage alg orithm. The first stage is phase retriev al, in which the phase of the RSS measuremen ts is rec overed up to a constant ph ase o ffset. The second stage is co herent co mpressive estimation on the the output o f the phase retriev al stage. 2) Th e two-stage appro ach r equires decom position of the measuremen t matrix in to a prod uct o f a phase retr ie val matrix and a com p ressi ve measure m ent matrix. It is known that matrices with independ ent and identically distributed (i.i.d.) complex Gaussian entries are effecti ve for this p urpose, but the p roduct of such matrice s has entries with comp lex values lying in a continu um, which cannot be realized with coarse phase con trol. A key innovation in our propo sed appro ach is to constrain the pr od uct of the two ma trices (i.e., the actual measuremen t matrix) to be implementab le with coarse phase control, and to decom pose it into two v ir tual matrices: an inner matr ix that is used fo r cohere n t co mpressive estimation, and an o uter matrix that is u sed fo r ph ase r etriev al. T hat is, we choose one of the matrices, and infer th e other via a p seu- doinv erse. W e pr ovid e design guide lines f or th e (non- unique) virtual decomp osition, and demon strate its p erform a nce an d scalability th r ough simu lations. 3) Using a central limit theorem argument, we show that the nu m ber of measurements fo r sparse ch annel recovery with non c oherent co mpressive estimation scales only slightly worse than for coherent estimation . Related work: As m entioned, our star tin g p o int he re is the work on coheren t comp ressi ve estimation in [1]–[3]. In our own prior work [4], non c oherent compre ssi ve chan nel estimation on the c ontinuum is pe r formed using non c oherent template matc h ing and Newton refinement. This ap proach , while o ptimal f or a single path channel, cannot be ap plied to a multipath chann e l with paths of comp a r able stren gth d ue to the n o nlinear co mbination of non coheren t beaco n respon ses. In [5], nonco h erent tracking of multipath chann els is attempted by designin g bea c on pattern s that illumin ate care- fully chosen in tervals of the angular space. Measur ement of se veral such beacons can be used to identify the strongest paths in the c hannel. This method suffers fro m pattern imperfectio ns caused by stron g sidelobe s that distort mea- base station base station Picocell Picocell Fig. 1: Base station to mo bile commu nication usin g narrow pencil beams in the dense pico cellular network. surements. The sensing proc e d ure is also disrupted b y the possibility of destructive co mbination of paths that fall in sid e different bins in on e beaco n. Our ap p roach in this paper is insp ired by r ecent work on comp r essi ve pha se retr ie val [6], which cascades phase retriev al with cohere n t co mpressed sensing to reconstruct a sparse signal fro m non coheren t com pressiv e projection s. The pr e sent paper differs from [6] in two key r espects. First, rather than mu ltiplying ma tr ices k nown to be e ffective for phase retr ie val and c o mpressive sen sing to obtain th e measuremen t matr ix, we provid e a virtu al deco m position th at enables use of a m easurement matrix that can be re a lized with co arse ph ase-only con trol. Second, we are interested in contin u ous-valued (“off-grid”) param eter estimation rather than e stimation o f a signal that is sparse in a discr e te basis, hence we replace the co h erent com pressiv e sensing stage by coheren t co mpressive estimatio n. Compressive r ecovery of sparse sign a ls with incomp lete measuremen ts has been studie d in several previous w orks. In [ 7], [8], the pro blem of spar se sign al recovery f r om 1-b it quantized pr ojections is co nsidered. In this case th e sign of compressive pro jections is measured by the receiver a nd a m - plitude infor mation is effectiv ely lo st. Such a prob lem can be formu late d as a constrained ℓ 1 norm minimization and solved efficiently . The lo ss of ph ase inform a tio n, howe ver , is more challengin g . Se veral in teresting approaches to sparse p hase retriev al hav e been prop osed in recent literature, includin g [9]–[11]. All of these methods are design ed to recover a sparse vector a n d, when applied to p ath recovery o n the contin - uum of spatial frequ encies, su ffer fro m gr id mismatch error . This err or can only be re duced b y oversampling the spa tial frequen cy co ntinuum, which re su lts in a larger “sp a r se” vec- tor that requires m ore mea su rements to recover , effecti vely defeating the pur p ose of compressive estimation. I I . S Y S T E M M O D E L W e c onsider the link be twe e n a directional transm itter using a linear ph ased arr ay antenn a of size N (o ur app r oach applies directly to two-dimensiona l arr a ys, but we restrict attention to linear arra y s for simplicity of exposition) and one or mor e rece ivers. T he wire le ss chann el of each link con sists of a n umber of paths fr om the transm itter to the receiver . The c h annel response is the sum o f the respon ses o f each of the pa ths o n the array weighted by their respectiv e comp lex amplitude. The ar r ay response of a path at angle of departur e θ is equ a l to: a ( θ ) = h e j 1 2 π λ d sin θ , e j 2 2 π λ d sin θ , ..., e j N 2 π λ d sin θ i T where d is the inter -element spacing of the array and λ is the wa veleng th. Defining the spatial fr equen cy ω = 2 π λ d sin θ , the array response of a p ath at ang le θ can b e described in term s of the correspon ding spatial frequen cy ω by a ( ω ) = [ e j 1 ω , e j 2 ω , ..., e j N ω ] T In the remaind er o f this pap er p aths will be ch aracterized by their spatial freque n cy in stead of ang le of depa rture. Th e net channel r esponse on th e N d im ensional array is conseq u ently equal to: h = K X k =1 α k a ( ω k ) (1) where α k and ω k are the comp lex amplitu de and spatial frequen cy of the k ’th path. W e assume that the transmitter bro a d casts a series of M beacons, and eac h receiver o bserves the strength of each beacon and provides an M × 1 me asurement vector as feedback to the tran smitter at th e end of the beaco ning inter- val, wh ich the transmitter em p loys to estimate the do minant spatial frequ encies to the r eceiv er . T his pro cedure is depicted in Fig . 2. Beacon b excites the ar ray with th e ra ndomly generated weight vector w b = [ w b 1 , w b 2 , ..., w b N ] T which sprays the emitted power differently in d ifferent d irections. The respo n se of b eacon b in the d irection of spatial freq uency ω is den oted by : f b ( ω ) = w b T a ( ω ) . (2) The measurement made at the recei ver will be a combi- nation of the beacon respo n se of all paths weig hted b y their correspo n ding co mplex path amplitude: y b = w T b h = K X k =1 α k w b T a ( ω k ) = K X k =1 α k f b ( ω k ) (3) Therefo re each angle of dep arture (A OD) will have its own M -dimension al beacon response “sign ature” and the weighted sum of the signatu res of the paths in th e ch annel arrives at the receiver . Based on the feedb ack from th e receiver , the tra n smitter estimates th e A OD of the stronge st path(s) and beamfor ms in tha t direction fo r co m munication . Although the transmitter will gene rally beamfor m toward the strong est path in the ch a n nel, maintainin g a libr ary of multiple strong paths is useful for rapid recovery fro m sudd en blockag e of the strong e st path, as well as f o r usin g altern ate paths to mana g e inter-cell interferen ce. Due to time-variant channel con ditions and mob ility of the receiver , channe l estimation is repeate d period ically to track the channel. This tracking takes pla c e at a hig h enou gh r ate to ensure th e time 2‐ bit be a c ons Fig. 2: Radiation pattern of comp ressi ve be acons on a 16-elemen t array excited by weights from distribution U ( {± 1 , ± j } ) , and feed back of RSS mea su rements m ade by mobile users. between consecu ti ve mea su rements is shorter than ch annel coheren ce time. For simplicity , we assum e omnid ir ectional rec e p tion, as in the early work on coheren t compressive estimation [1]. Includin g the effect of rece i ve arrays in n oncoher ent com- pressiv e estimatio n is an impo rtant topic for future work . In our prop osed system, comp r essi ve beacons are used fo r channel discovery with certain limitations on array weig hts. First, we assume no c o ntrol over the amplitud es o f the sens- ing matrix and confine the system to unitary weights. Seco nd, we assume limited control over element ph a ses to simplify the f ront en d hard ware; to th is en d, we allow 2 - bit phase control fo r the sensing m atrix so that array phases are selected from the set of {± 1 , ± j } . The se limitation s are crucial for minimizing hardware c omplexity as ar ray sizes grow . As shown in [1], the degrad a tio n in beamfo rming perf o rmance caused by severe phase q u antization is negligible as array size gro ws, m e aning sacrificin g p hase granularity for array size ca n be an advantageous tr adeoff. Another impor tant limitation is the loss of coherence from one be acon to the next. Due to oscillator design and dynamics, there is an u nknown freque ncy offset between the local oscillators at the tran smitter an d recei ver , that is constan tly ch anging at an unknown rate. This ran dom offset tran slates into a r a ndom offset in the p hase of the channel measured b y each beaco n, corrupting the phase informa tio n o f beacon measurements. For this reason , we assume RSS-on ly measure ments are made at the receiver , effecti vely c hanging the cohe r ent measurem ent mode l o f (3) to no ncohere nt inten sity measure ments formu lated b y (4 ) . y b =   w T b h   2 =      K X k =1 α k f b ( ω k )      2 (4) The next section presen ts o u r propo sed algor ith m fo r com- pressiv e estimation of sparse ch annels under the nonlin e ar measuremen t m odel of (4). I I I . N O N C O H E R E N T C O M P R E S S I V E E S T I M AT I O N While se veral algor ithms have been propo sed for compres- si ve estimation o f spar se vectors, we foc us on the Newtonized orthog onal matching p u rsuit m ethod (NOMP) which is a robust, computation ally ef ficient, and scalable estimation framework suited for sparse estimation o n a contin uum [1 2]. NOMP operates o n the basis that th e com pressiv e projectio ns of dif ferent spatial frequenc ies are almost orthogonal so strong paths are detected one by one via temp late ma tch ing o f the co mpressive measuremen ts against a diction ary of beaco n responses of spatial f requenc ie s on an oversampled grid. The response of each detected fre q uency is subtracted from the measuremen ts an d Newton descent is u sed to fine tu ne the extracted paths on the continu u m at ea c h stage. When pha se informa tio n is unavailable, h owe ver, quasi-orth ogonality o f the p aths is diminished and NOM P , or other standard com- pressiv e estimation techn iques, cannot b e app lied. Therefo re, we p r opose a hy b rid scheme fo r n oncohe r ent co mpressive estimation a s fo llows. A. The noncoh er en t channe l estimation algorithm In order to maintain the benefits of NOMP and insp ir ed by the appro ach put fo rth in [6 ], our method poses the estimation problem as a concatenation of p hase retriev al and cohe rent compressive e stima tio n v ia NOMP . The measure m ent matrix A is de fin ed as th e pro duct of a ph ase retriev al matrix, A P R of size M × M C S , and a compressive sen sing matrix, A C S of size M C S × N : A = A P R A C S The m easurement vector is then den oted as y = | Ah | 2 = | A P R A C S h | 2 where the chan n el h is d efined in (1). W e introduce the auxiliary me a surement vector y C S = A C S h . Th is vector , if retrieved, effectiv ely provides (up to an unknown, and irr elev ant, co mplex gain) coh erent compr essi ve measuremen ts of the target ch annel h by sensing matr ix A C S . The algorithm is thus divided into two stage s: the complex auxiliary measurem ent vector y C S is first estimated fro m RSS observations y = | A P R y C S | 2 using a phase retrie val algorithm , and th e resulting estimate is subsequen tly used for compr essi ve estimatio n of h b y sensing matrix A C S . The two stages of the algo rithm are summar ized as follows. Stage 1: Phase retrieval. While several different alg orithms have bee n prop osed f or ph a se retrie val, eac h with different specifications and con d itions, we consider the method pro- posed in [6] fo r our b aseline analysis, since it has provable perfor mance boun ds. This method is shown to obtain the target vector up to a constan t pha se amb iguity with high probab ility un der th e con d ition th at elemen ts of the M × N sensing matrix A P R are i.i.d. Gau ssian and the number of RSS mea su rements, or M , scales as n log n f or target vector size o f n . For evaluating the alg orithm perfor m ance on simp lified front end s with q uantized beamfor ming ph ases, we use the more robust W irtin ger Flow algorithm [13] that utilizes gradient descent w ith careful initialization of the target vector to insur e convergence to the g lobal m inimum. T h e read er is referred to these texts fo r a detailed description of the two methods. Stage 2: Compressive estimation. Th e com plex valued estimates ob tained fo r y C S are used in this step to solve for h the compre ssive estimation prob le m , y C S = A C S h in which h is a sparse com bination of K co ntinuou s-valued spatial frequen cies as expressed in (1). For self-containe d exposition, we b riefly revie w the N O M P algorithm that we use to sequen tially extract the spatial f r equency comp o nents. Iteration t consists of th e following steps: • Subtract the response of all paths extracted so far fro m the measuremen t vector to obtain the residual vector y r ; y r = y − t − 1 X k =1 ˆ α k a ( ˆ ω k ) • By match ing the r esidual observation vector with th e oversampled r e sponse d ictionary , iden tify the strongest frequen cy in the rema in ing m ixture and its c o rrespond - ing complex a m plitude on the ov ersampled grid of spatial fre q uencies, Ω ; ˆ ω t = arg max ω ∈ Ω G r ( ω ) , G r ( ω ) = | f ( ω ) H y r | 2 / k f ( ω ) k 2 ˆ α t = f ( ˆ ω t ) H y r / k f ( ˆ ω t ) k Where the notatio n f ( ω ) rep resents the vector [ f 1 ( ω ) , f 2 ( ω ) , ..., f M ( ω )] o f respon ses to the M bea- cons as describe d in (2). • Use Newton refin ement to fine-tune the identified fre- quency with respect to the residual. Add the resulting frequen cy an d its corre sp onding com plex amplitude to the set of estimated paths; • Use cyclic Newton refineme nt steps on all paths ex- tracted so far to fine-tune fre quencies and amplitudes with resp e c t to the measuremen t vector y . The algo rithm proceed s un til the residual en ergy is lower than a thresh old, ind ic a tin g all do minant p aths have b een extracted. The reader is referred to [12] for a detailed discussion of implementatio n and perfor m ance guaran tees of the alg o rithm. T o en su re successful chan nel recovery of a K -spar se channel, the sensing matrix A C S must satisfy the restricted isometry p r operty (RIP), i.e. provide sufficient sep aration between any two K -sparse vectors in th e observation sp a c e. Random m atrices have been shown to pr ovide this con d i- tion with high probability under suitable circumstances. I n particular, it has been shown pre viously in [14] that if the elements o f the M C S × N matr ix A C S are generated from an i.i. d . Gaussian distribution, the n A C S will satisfy the above pr operty with high probab ility wh e n M C S (the num ber of compressi ve measurements) scales as K log N . Th ese prediction s are usef ul gu idelines in designin g the a lgorithm parameters and creating the measurement matrices, as well as op tim ally allo cating resources to the two stage s of the process. B. Generating the sensing matrices The compressi ve sensing and p h ase retr ie val matrices, A C S and A P R , must satisfy certain requireme n ts to ensure the p e rforman ce of each stage. A t the same tim e, in o rder to be compatible with the har dware requirem ents descr ib ed in Section II, the comb ined m atrix A = A P R A C S , mu st take values in the set {± 1 , ± j } . Generatin g the req u ired matr ic e s for hardware imp lementation and processing is therefo re n ot trivial. In fact, identifyin g an M × M C S and M C S × N pair of matrices whose produc t, an M × N matrix , is in the quantized space is an overcomp lete problem th a t can not b e solved exactly . W e use the following p rocedu r e to minimize the d istance between the prod uct matrix an d its quan tized version, and treat this distance as a measur ement erro r that can be compen sated for by increasing th e nu mber of mea- surements and u sing stable phase r etriev al and compressive sensing a lg orithms. As men tio ned in Section III-A, i.i.d . comp lex Gaussian matrices have the ne c essary projection characteristics for phase retr ieval and compressiv e sensing and are a go od choice fo r A P R and A C S . In the first step, the elemen ts of A are gen erated ind ependen tly from a unifor m distribution over the allowed values, i.e., A ( i, j ) ∼ U ( {± 1 , ± j } ) , and A P R and A C S are then defined so th at 1) their matrix produ ct is eq ual to A , and 2) they are i.i.d . com plex Gaussian (exactly f o r one of the matrices, and app roximately for the other). After produ cing th e net matrix A , we independ ently g en- erate A C S from an i.i.d . comp lex Gau ssian d istribution: A C S ( i, j ) ∼ C N (0 , 2 σ 2 ) , σ 2 = 1 2 N The p hase retr ie val matrix is then defined as: A P R = AA C S + A C S + = A C S H  A C S A C S H  − 1 (5) where A + C S is th e pseu d oinv erse of A C S . W e now argue tha t the elements of A P R are also approx imately i.i.d. with com- plex Gaussian distribution. Since the elements of A C S are i.i.d. and zer o mean, its rows ar e appro ximately or thonorm al, which motiv ates the following app roximation : A C S A C S H ≈ I M C S . Substituting in (5) we find that A C S + is a pproxim ately equal to A C S H , so that the elem ents of th e latter c a n be approx imated as i.i.d. complex Gaussian. Element ( i, j ) of A P R is the refore eq ual to: A P R ( i, j ) = M C S X k =1 A ( i, k ) A C S + ( k , j ) ≈ M C S X k =1 A ( i, k ) A C S ( j, k ) It is easy to see that the elements of A P R are ze ro m ean and uncorr elated. Since each is the sum of a modera tely large number o f zero m ean, ind e p endent random variables, we can 10 15 20 25 30 M CS 0 0.5 1 beamforming loss (frac tion) K=2, M=25K=50 median 95th percentile (a) K=2 15 20 25 30 35 40 45 50 55 60 M CS 0 0.5 1 beamforming loss (frac tion) K=4, M=25K=100 median 95th percentile (b) K=4 Fig. 3 : Beamfo rming erro r of estimated pa th s as a fu nction of M C S for a fixed observation size M . 0 500 1000 1500 2000 2500 3000 array length (N) 0 50 100 150 200 250 300 350 400 # measurements (M) K=3, NC quantized K=2, NC quantized K=3, noncoherent K=2, noncoherent K=3, coherent K=2, coherent Fig. 4: Required number o f measuremen ts M for 99% probab ility of accu rate chann el recovery (b eamform ing lo ss ≤ 1d B). in voke the cen tral limit theorem to argue that the elements of A P R are join tly com plex Gaussian, an d the r efore i.i.d . The produ ct A = A P R × A C S is th en r eev alu ated and quantized to obtain the fin al sen sing matr ix. I V . S I M U L A T I O N R E S U LT S Both stages o f the algor ithm m ust succeed in order to accurately recover all stron g paths in the channel. This requires approp riate ch oices for M and M C S . The num ber of measur ements required fo r co h erent compr essi ve sensing of an N -dimen sional K -sparse vector is known to scale as O ( K log N ) [1 4]. The co nditions for co mpressive estimation of a sp atial channel with K significant compon ents are more complex [15], but when the compon ents are separated “well enou gh, ” the numb er of measur e ments scales with O ( K log N ) as we ll (Theorem 4 in [15 ]). On the other hand, successful p hase retriev al also poses restrictions on the relation b etween M and M C S . One f u ndamen tal lower bound f or the n umber of observations require d to identify M C S complex values is M ≥ 2 M C S since each complex variable is defined by tw o real v alues. The best algorithms av ailab le until now perf o rm th is estimation using M = O ( M C S log M C S ) real-valued o b servations [6]. Since M C S scales as K log N and M scales as M C S log M C S (with the current state of the art), we con clude that the number of measur ements required fo r the pro posed algorithm scales with array size N and nu m ber of path s K as: M = O ( K log N lo g( K log N )) . This is only slightly m o re costly than the O ( K log N ) measuremen t complexity of coherent compressive estimation. The proposed algo rithm thus scales we ll with arra y size for spar se channels with a small num ber , K , of domin ant compon ents. Fig. 4 d epicts simulatio n results for the required number of measurem e nts for accurate chann el recovery as a function of array size for different v alues of K . These results are in agr eement with the preceding th eoretical pred ictions. For a sufficiently large number o f measurements M , the choice of M C S must p rovide an op timum trad eoff between phase retriev al and comp ressi ve sensing accuracy . This trade - off is d emonstrated in Fig. 3, wher e the overall per formanc e of the a lgorithm is q uantified for a fixed sensing configu ration ( N = 1000 , M = 25 K, K = 2 , 4 ) as a fu nction o f M C S . The b est c h oice for M C S , giv en by the lowest point of the curve, cor responds to the tradeo ff between the accu racy of phase retrieval an d compressive sensing th at yield s the best beamfor ming per forman c e . V . C O N C L U S I O N S W e have shown that it is possible to estimate a sparse mul- tipath c hannel with RSS measure m ents that c an be realized using RF beamf orming with coarse phase-on ly con trol. Th e approa c h inh erits the desirab le theor etical pro perties of phase retriev al and coheren t com p ressi ve estimation, and is shown to be effective v ia simulations. In future work , we plan to validate the p roposed approach on our mm wave testbed , a nd to explo re techniq u es f or r educing com putational com plexity of th e ph ase retrieval com ponent of o ur a lgorithm. Ano ther importan t to pic for fu ture investigation is to account for the receive antenn a array in mo re detail. A C K N O W L E D G M E N T S This work was suppo rted by NSF grants CNS-1317 1 53 and CNS-1518 812, and ComSenT er, one of six c e n ters in JUMP , a Semiconducto r Research Corporatio n ( SRC) p ro- gram sponsored by D A RP A. R E F E R E N C E S [1] D. Ramasamy , S. V enkateswara n, and U. Madho w , “Compressi ve adapta tion of large steerable arrays, ” in Information Theory and Applicat ions W orkshop (ITA), 2012 . IEEE, 2012, pp. 234–239. [2] ——, “Compressi ve tracking with 1000-element arrays: A frame work for multi-gbps mm wave cellul ar downlinks, ” in 2012 50th Annual Allerton Confer ence on Communicati on, Contr ol, and Computing (Allerton) , Oct 2012, pp. 690–697. [3] Z. Marzi, D. Ramasamy , and U. Madho w , “Compressi ve channel estimati on and trac king for large arrays in mm-wave picocell s, ” IEEE J ournal of Selec ted T opics in Signal Proc essing , vol. 10, no. 3, pp. 514–527, 2016. [4] M. E. Rasekh, Z. Marzi, Y . Z hu, U. Madho w, and H. Zheng, “Non- coheren t mmwav e path tracking, ” in P r oceedings of the 18th Inter- national W orkshop on Mobile Computing Systems and Applicati ons . A CM, 2017, pp. 13–18. [5] O. Abari, H. Hassanieh, M. Rodriguez, and D. Katabi , “Millime ter wa ve communicatio ns: From point-to-poi nt links to agile network connec tions. ” in HotNets , 2016, pp. 169–175. [6] S. Bahmani and J. Romber g, “Ef ficient compressiv e pha se retri ev al with constrain ed sensing vector s, ” in Advances in Neural Information Pr ocessing Systems , 2015, pp. 523–531. [7] C. Rusu, R. Mendez -Rial, N. Gonza lez-Prel cic, and R. W . Heath, “ Adapti ve one -bit co mpressi ve sensing wit h applicati on to lo w- precisio n recei vers at mmwa ve, ” in 2015 IEEE Global Co mmunicatio ns Confer ence (GLOBECOM) , Dec 2015, pp. 1–6. [8] P . T . Boufounos a nd R. G. Ba raniuk, “1 -bit compressiv e sensing, ” in Information Science s and Systems, 2008. CISS 2008. 42nd Annual Confer ence on . IEEE, 2008, pp. 16–21. [9] P . Schnite r and S. Rangan, “Compressi ve phase retrie val via gener- alize d approximate message passing, ” IEEE T ransacti ons on Signal Pr ocessing , vol. 63, no. 4, pp. 1043–1055, 2015. [10] R. Pedarsa ni, D. Y in, K. Lee, and K. Ramchandran, “Phasecode : Fast and ef fi cient compressi ve phase retrie va l based on sparse-graph codes, ” IEEE T ransactions on Information Theory , 2017. [11] H. Ohlsson, A. Y ang, R. Dong, and S. Sastry , “Cprl–an exte nsion of compressi ve sensing to the phase retrie val problem, ” in Advances in Neural Informati on Pr ocessing Systems , 2012, pp. 1367–1375. [12] B. Mamandipoor , D. Ramasamy , and U. Madho w , “Ne wtonize d or- thogonal matchi ng pursuit: Frequency estimati on over the continuum, ” arXiv prepri nt arXiv:1509.01942 , 2015. [13] E. J. Candes, X. L i, and M. Soltano lkota bi, “Phase retrie v al via wirtinge r flow: Theory and algorithms, ” IEEE Tr ansactions on Infor- mation Theory , vol. 61, no. 4, pp. 1985–2007, 2015. [14] E. J. Candes, “The restricted isometry property and its implications for compressed sensing, ” Comptes Rendus Mathemat ique , vol. 346, no. 9, pp. 589–592, 2008. [15] D. Ramasamy , S. V enkate swaran, and U. Madho w , “Compressiv e paramete r estimat ion in A W GN, ” IEEE T rans. Signal Proc essing , vol. 62, no. 8, pp. 2012–2027, 2014.