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IEEE TRANSACTIONS ON COMMUNICATIONS 1

Channel Estimation Strategies for Multi-user mmWave Systems

Manoj A and Arun Pachai Kannu.E-mail: {ee14d210, arunpachai}@ee.iitm.ac.in

Abstract—In a multi-user millimeter (mm) wave communica-tion system, we consider the problem of estimating the channelresponse between the base station (BS) and each of the userequipments (UE). We propose three different pilot trainingschemes/strategies, namely, 1) Estimating mm wave channelsseparately at the UE’s, 2) Joint estimation of mm wave channelsof all the UE’s simultaneously at the BS, and 3) Two stageprocess involving estimation at both the UE’s and the BS.We propose two random constructions for the pilot/trainingsignals/beamforming weights and analytically characterize themutual coherence parameter of the resulting sensing matrices forone of the constructions. By exploiting the structure in mm wavechannels, we develop a generalized block orthogonal matchingpursuit (G-BOMP) algorithm for channel estimation in all thethree strategies. Further, we establish sufficient conditions forexact support recovery by the proposed G-BOMP algorithm andderive an upper bound on the norm of the channel estimationerror. We also study the performance of our G-BOMP algorithmin terms of the average beamforming gain and average spec-tral efficiency achieved via simulations. Our results show thatthe proposed G-BOMP algorithm outperforms the conventionalOMP algorithm and other existing multiuser mm wave channelestimation techniques. G-BOMP also performs comparable tothe computationally expensive compressive estimation using New-ton’s refinement technique, for the first pilot training strategy inwhich the UE’s estimate the channels separately.

Index Terms—mm Wave Beamforming, Channel Estimation,Beamforming weights, Block Sparse Matrices, Block OMP Algo-rithm, Average Spectral Efficiency

I. INTRODUCTIONMillimeter (mm) wave communication systems, initially

employed mainly for indoor [1]–[3] and fixed outdoor shortrange LOS communications, are being considered for wirelesscellular type scenarios, due to the spectrum gridlock at the sub-6GHz frequencies and the continuous increase in the demandfor high data rate services. The recent advancements in antennadesign [4] facilitate the deployment of large uniform lineararray (ULA) of antennas with small form factors in mm wavesystems, thanks to the small carrier wavelength. AppropriateULA steering (beamforming) ensures highly directional signaltransmission, thereby, providing the gains needed to combatthe high path losses suffered by mm waves. Further, studiesin [5]–[8] substantiate the ability of mm wave systems tosupport large network capacity and coverage, and entitles thesesystems to be a viable candidate for the 5G technology.

Beamforming, which is crucial in mm wave systems, is doneusing various architectures: analog, digital and hybrid. Analog

Manoj A and Arun Pachai Kannu are with Electrical Engineering Depart-ment, Indian Institute of Technology Madras, Chennai - 600036, India. Partsof the work were published in SPAWC 2017 conference.

beamforming steers the ULA output using single RF chain andphase shifters. Though simple, analog method does not providemultiplexing facility. On the other hand, digital beamformingoffers the flexibility to support multi-stream data transmission,but the hardware proves to be expensive and power consumingas it consists of separate RF chains (with ADC/DAC) for everyantenna element in the ULA. The hybrid structure combinesboth the analog and digital beamforming architecture together,and provides trade-off between cost/complexity and spectralefficiency [9], [10]. However, the throughput performance ofthe mm wave hybrid model depends on the design of thebeamforming weights, which in turn completely depends onthe channel state information [9], [11]–[14]. Thus, mm wavechannel estimation problem has garnered a lot of attention inthe recent times.

The mm wave channel with ULA at both the transmitterand receiver is modeled as a weighted sum of array responsesfor each path [9]. Each path is composed of two spatialfrequencies which depend on the angle of departure (AoD)at the transmitter and angle of arrival (AoA) at the receiver.Since the number of paths is small compared to the dimensionof the ULA [15], [16], several compressive sensing (CS)based channel estimation schemes were developed for single-user mm wave systems, exploiting the sparse nature of mmwave channels [17]–[23]. More specifically, [17] discussed anexpectation-maximization algorithm to estimate the mm wavechannel, [18]–[20] developed procedures to design complexcodebooks for pilot training and described algorithms toestimate the channel parameters, [21] proposed a two-stagecompressive sensing method for channel estimation assumingan analog beamforming model, and [22] devised a pilot train-ing scheme and employed orthogonal matching pursuit (OMP)to solve the problem. An echoing process based scheme toestimate the subspaces spanned by the singular vectors of themm wave channel is proposed in [23]. Performance of the CSalgorithms such as OMP, typically depend on the parameterssuch as mutual coherence of the sensing matrix [24], resultingfrom the training signals.

Many of the above works were extended to multi-userchannel estimation problem considering a single cell mm wavenetwork consisting of a base station (BS) and many userequipments (UE’s). A technique to estimate all mm wavechannels at the BS using fast iterative shrinkage thresholdingalgorithm (FISTA) was proposed in [25]. In [26], a trainingscheme is proposed where UE’s estimate their channels usingOMP and an asymmetric channel estimation method wasproposed in [27], where the UE’s and the BS estimate the


channels individually with the aim to avoid feedback (but de-tails of estimation algorithms are lacking). Moreover, all thesemethods assume that the spatial frequencies corresponding tothe AoA and AoD of each path fall exactly in the grid pointsof DFT matrices of ULA sizes.

In our work, we consider the practical case where thespatial frequencies may not fall exactly in the DFT bins, andhence, spectral leakage occurs in the frequency domain. Asa consequence, the mm wave channels exhibit block sparsestructure in the Fourier domain, where the non-zero blocksare centered around the spatial frequencies corresponding tothe AoA and AoD of the paths. Our main contributions are:

(a) We propose three different pilot training and channel es-timation schemes/strategies, namely, 1) Estimation of channelsat each UE separately, 2) Joint estimation of channels of all theUE’s simultaneously at the BS, and 3) Two stage estimationprocess where both the UE’s and the BS individually estimatetheir beamforming weights.

(b) Conventional block sparse recovery algorithms such asblock OMP [28] require the apriori knowledge of partitioningthe unknown vector into disjoint sub-blocks, with each sub-block being either zero or non-zero. Such partitioning is notpossible for mm wave channels, as the spatial frequencies arerandom and can take continuous values. Hence, we developa generalized block orthogonal matching pursuit (G-BOMP)algorithm to estimate block sparse matrices that contain non-zero blocks at arbitrary starting locations, which is applicablefor all of the above three estimation strategies. We establishsufficient conditions for exact support recovery for G-BOMPalgorithm under bounded noise scenario, in terms of themutual coherence of the sensing matrix. We also derive anupper bound on the channel estimation error based on thesupport set recovered by the algorithm.

(c) We propose two random constructions for the pilotsignals and training beamforming vectors, and analyticallycharacterize the mutual coherence parameter of the resultingsensing matrix for one of the constructions. Such analysis ofmutual coherence of sensing matrices in mm wave channelestimation problem has not been done previously in theliterature.

(d) Finally, we show via simulations that, the performance ofour G-BOMP algorithm, in terms of the average beamforminggain and average spectral efficiency achieved by the mm wavesystems, is superior to the existing methods such as OMP[26] in the literature. On the other hand, computationally moreexpensive Newton refinement technique in [29] performs betterthan G-BOMP, for the separate channel estimation at the UE’sstrategy.

The paper is organized as follows: We present the mm wavechannel model and its block sparse structure in Section II. InSection III, we present the three training and channel estima-tion strategies and discuss the details of G-BOMP algorithmin Section IV. Simulation results are given in Section V andconclusions in Section VI.

Notations: The following notations are used throughoutthe paper: F denotes a set. Fc means complement of Fand |F| stands for the cardinality of F . f and F both referto a scalar, while f and F denote a vector and a matrix

respectively. [F](q,q̄) refers to the (q, q̄)th element of F, f(q)the qth element of the vector f, |f | the absolute value of thescalar f . FF is a sub-matrix of F that contains those columnsof F specified by the index set F . And, fF is a sub-vector off containing those entries of f specified by F . λmax(F) andλmin(F) represent the maximum and minimum eigen valuesof the matrix F respectively. IF is an identity matrix of orderF×F . Let (.)∗, (.)T and (.)H represent conjugation, transposeand hermitian operations respectively. Finally, ||.||2, ⊗ and Eindicate l2-norm, kronecker product and expectation operatorrespectively.

II. MM WAVE HARDWARE MODEL AND MM WAVECHANNEL MODEL

A. System Model

Consider a mm wave communication system consisting ofa single base station (BS) and L user equipments (UEs). Weassume hybrid beamforming system architecture for both theBS and all the L UEs. With Nb antennas and NRFb RF chainsat the BS, let U ∈ CNb×NRFb be the effective (combinedbaseband and RF) precoding/beamforming matrix employedat the BS. Similarly, with Nu antennas and NRFu RF chains,let Vl ∈ CNu×N

RFu be the effective (combined baseband and

RF) precoding/beamforming matrix employed at the lth UE.With s ∈ CNRFb ×1 being the vector of transmit symbols, theobservation at the lth UE is given by

yl = VHl H

Hl Us + nl, (1)

where HHl is the Nu ×Nb mm wave channel seen from thelth UE to the BS and nl is Nu × 1 complex noise vectordistributed as CN (0, σ2uINu).

For developing channel estimation strategies, we assumethat both the BS and the UEs have only one RF chain. Welater discuss the impact of multiple RF chains in Section V.In this regard, if the BS transmits data symbol s using a unitnorm beamforming vector u (of size Nb×1) in the downlink,then the final output at the lth UE, which receives the datausing a unit norm beamforming vector vl (of size Nu× 1), isgiven by,

yl = vHl H

Hl us+ v

Hl nl = v

Hl H

Hl us+ nl, l = 1, 2, ..., L,

(2)

where nl is the complex additive white Gaussian noise(AWGN) with variance σ2u, ∀l.

Similarly, in the uplink, if wul is the unit norm beam-forming vector assigned to the ULA at the lth UE, wherel = 1, 2, ..., L, then the signal observed at the BS, whichapplies a unit norm beamforming weight wb to its ULA, isgiven by,

y = wHb

L∑l=1

Hlwulxl + n, (3)

where Hl is the Nb ×Nu mm wave channel from the BS tothe lth UE, xl is the data symbol sent by the lth UE and n iscomplex AWGN with variance σ2b .


B. Channel Model

We adopt the geometric channel model used in [9], [20],[22], [23] and define the channel from the BS to the lth UEas,

Hl =

√NuNbKl

Kl∑k=1

αl(k)abl(k)aul(k)H , (4)

where Kl is the total number of multi-paths that exists betweenthe BS and the lth UE with each multi-path being a clusterof paths produced by a scatterer in the environment, αl(k)is the small-scale fading complex channel gain of kth multi-path, and abl(k) and aul(k) are the ULA responses at the BSand the lth UE respectively for the kth multi-path. We assumeRayleigh fading, and hence, model αl(k), ∀k = 1, ...,Kl and∀l = 1, 2, ..., L as i.i.d. circular Gaussian random variablewith zero mean and variance equal to σ2α. The ULA responsevectors are given by,

abl(k) =1√Nb

[1 e−jΩbl (k) ... e−j(Nb−1)Ωbl (k)]T , (5)

aul(k) =1√Nu

[1 e−jΩul (k) ... e−j(Nu−1)Ωul (k)]T , (6)

where Ωul(k) = 2πdλ sin(φul(k)), Ωbl(k) = 2π

dλ sin(φbl(k)),

d is the spacing between the antenna elements in the ULA,λ is the operating carrier wavelength, φul(k) and φbl(k) arethe angles of departure (AoD) and arrival (AoA) respectively,for the kth multi-path. In a typical mm wave communicationsystem, since the number of scattering clusters is very small[15], [16], [30], we assume that Kl ≤ Kmax, for all l =1, ..., L, where Kmax is some positive integer.

C. Block Sparse Structure

Each path in (4) has the array responses (5) and (6), whichare complex exponentials with spatial frequencies Ωbl(k) andΩul(k). Hence, the channel matrix Hl in (4) is sparse inFourier domain. To understand the structure, let us define the2D DFT of Hl as,

Hωl = FHb HlFu, (7)

where Fb and Fu are unitary DFT matrices of size Nb and Nurespectively. Since AoA φbl(k) and AoD φul(k) in (5) and (6)are, in general, uniformly distributed in a subset of [−π, π],the spatial frequencies will not fall exactly into the DFT bins(i.e., the spatial frequencies will not be integer multiples of2πNb

or 2πNu ). Hence, we encounter spectral leakage in the 2DDFT of Hl, which is concentrated around the exact spatialfrequencies.

In Figure 1, we illustrate the sparse structure of Hωl , byshading each square based on the sum of magnitude of theDFT grid points which enclose that square. Since the spectralleakage is negligible for the grid points which are far from theactual spatial frequencies, Hωl can be approximated as a 2Dblock sparse matrix, with each path contributing to a non-zerosquare block, say of size (b × b). Depending on the actualvalues of the spatial frequencies, the non-zero square blocksin Hωl may or may not be overlapping.

0 5 10 15 20 25 30 35

DFT index points

0

5

10

15

20

25

30

35

DFT

index

points

50

100

150

200

250

300

350

400

450

Fig. 1: Magnitude plot of Hωl with Nu = Nb = 32, Kl = 4.

III. CHANNEL ESTIMATION STRATEGIES

A. UE-Method: Channel Estimation at individual UE’s

In this method, each UE estimates its channel Hl based onthe training signals sent by the BS. We consider a trainingphase of duration M , where the UE’s make measurementsof the form (2), with the mth measurement at the lth UEdenoted by, y(m)l = v

(m)Hl H

Hl u

(m)s(m) +n(m)l , l = 1, ..., L.

Here, u(m) and v(m)l denote the beamforming weights used bythe BS and the lth UE, respectively, during the mth measure-ment. We assume that the training symbol s(m) = 1, ∀m ∈{1, 2, ...,M}. With hl = vec(Hl), the above equation canbe re-written as, y(m)Hl = (v

(m)Tl ⊗ u(m)H)hl + n

(m)Hl .

From (7), we have hl = Ψhωl , with Ψ = F∗u ⊗ Fb and

hωl = vec(Hωl ). Collecting the M observations into a vector,

yUE = [y(1)Hl ... y

(M)Hl ]

T , we get,

yUE = ĀlΨ︸︷︷︸Al

hωl + nUE , l = 1, 2, ..., L, (8)

where nUE = [n(1)Hl ... n

(M)Hl ]

T and the mth row of Āl isv

(m)Tl ⊗u(m)H , m = 1, ...,M . For the observation model (8),

we present a generalized block OMP framework in Section IV,to reconstruct hωl , which is the vectorized version of a 2-D block sparse matrix Hωl . Once the UE’s estimate theirchannels, they also compute the optimal precoding weightsto maximize the beamforming gain using singular value de-composition, and feed the beamforming weights back to theBS individually.

B. BS-Method: Joint Channel Estimation at BS

Here we propose a pilot training scheme where the BSestimates all the channels, for which all the UE’s transmitpilot data simultaneously (we assume that all the UE’s aresynchronized before transmission). Assuming that the pilotdata symbols sent are all equal to 1, the reception model at

the BS can be written as, y = wHbL∑l=1

Hlwul +n =[(wTu1 ⊗

wHb ) ... (wTuL ⊗ w

Hb )]h + n, where h = [hT1 h

T2 ...h

TL]T .

Now, h = Φhω , where hω = [(hω1 )T (hω2 )

T ... (hωL)T ]T and


Φ = IL ⊗ Ψ. Suppose M measurements are obtained, i.e.,yBS = [y

(1) ... y(M)]T , then we get,

yBS = B̄Φ︸︷︷︸B

hω + nBS , (9)

where mth row of B̄ is[(w

(m)Tu1 ⊗ w

(m)Hb ) ... (w

(m)TuL ⊗

w(m)Hb )

]and nBS = [n

(1) ... n(M)]T . Since hω in the obser-vation model (9) is concatenation of vectors from 2D blocksparse matrices, we use G-BOMP framework from Section IVto recover the channels of all the UE’s jointly. Once thechannels are estimated and the optimal weights are computed,the BS informs all the UE’s about their corresponding beam-forming vectors via feedback.

C. Joint-Method: Two Stage Channel Estimation Strategy

In this method, the estimation process is done in two stages.In the first phase, UE’s will estimate the channel using M1pilots sent by the BS (Method 1). In the second phase, theUE’s will assign the estimated optimal beamforming vectorto the ULA and transmit M2 pilots to the BS. BS will usethese pilots and determine its optimal beamforming weightscorresponding to each UE. For this method, the total trainingduration is M = M1+M2 and we do not require any feedbackmechanism to convey the optimal weights.

First phase proceeds as per Method 1 from Section III-Awith M1 measurements. Now, let u

[l]opt be the estimated optimal

beamforming vector for the lth UE. In the second phase, BSobtains M2 measurements with all the UE’s choosing theirestimated weights for beamforming. The mth (m = 1, ...,M2)measurement at the BS will then be,

y(m) = w(m)Hb

( L∑l=1

Hlu[l]opt x

(m)l

)+ n(m). (10)

In order to discuss the estimation process at the BS,we make an approximation that u[l]opt will be orientedalong the path corresponding to the largest gainαl(k) in (4). Assuming that |αl(1)| is the largest, weapproximate that u[l]opt ≈ aul(1) (Note that we haveu

[l]opt = aul(1) for a single rank channel matrix). With this

approximation, we get, Hlu[l]opt ≈

√NuNbKl

[αl(1)abl(1) +

Kl∑k=2

αl(k)abl(k)aul(k)Hu

[l]opt

], and equation (10) can be

re-formulated as, y(m) ≈ w(m)HbL∑l=1

√NuNbKl

αl(1)x(m)l abl(1) + ñ

(m), where ñ(m) =

w(m)Hb

[ L∑l=1

Kl∑k=2

αl(k)abl(k)aul(k)Hu

[l]optx

(m)l

]+ n(m).

In the above equation, each abl(1) is a complex sinusoidwith spatial frequency Ω(1)bl . Denoting

√NuNbKl

αl(1)abl(1) as

cl, we get y(m) = w(m)Hb [c1 c2 ... cL]x

(m) + ñ(m), wherex(m) = [x

(m)1 x

(m)2 ... x

(m)L ]

T . Suppose cl = Fbcωl , where cωl

is a 1-D block sparse vector with spectral leakage concentrated

around the frequency Ω(1)bl and yJoint = [y(1) ... y(M2)]T , then,

yJoint = D̄Υ︸︷︷︸D

cω + nJoint , (11)

where the mth row of D̄ is (x(m)T ⊗ w(m)Hb ), m =1, ...,M2, nJoint = [ñ

(1) ... ñ(M2)]T , Υ = IL ⊗ Fb andcω = [(cω1 )

T ... (cωL)T ]T . Observation model (11) can again

be solved using generalized block OMP framework discussedin Section IV by specializing it to the 1−D case. Here,we directly obtain the optimal beamforming weights abl(1)without estimating the entire channel at the BS. Note thateffective noise term in (11) includes contributions from allmulti-path components except the strongest one. This multi-path interference will become the limiting factor when noisepower σ2u is small in the estimation model in (11).

D. Training Beamforming vectors and Signals

The training symbols needed in Joint-Method are generatedas i.i.d. {±1} with equal probability. We propose two differentdesigns for the training beamforming vectors in the following:

Rademacher Design: For each measurement, the entries oftraining beamforming vectors used at the BS can be generatedas i.i.d.

{± 1√

Nb

}with equal probability while those used at

the UE’s as i.i.d.{± 1√

Nu

}with equal probability.

Fourier Design: For each measurement, the Nb-point DFTof vectors with entries being i.i.d.

{± 1√

Nb

}equally likely

can be employed as the training beamforming vectors at theBS. Similarly at the UE’s, the Nu-point DFT of vectors withentries being i.i.d.

{± 1√

Nu

}equally likely can be used as

beamforming vectors during pilot training. In this case, thestructure of the sensing matrices gets simplified. In order tounderstand it better, recall the final measurement model of BS-Method given in equation (9). Suppose for every 1 ≤ m ≤M ,

w(m)ul = Fuw̃(m)ul

, 1 ≤ l ≤ L; w(m)b = Fbw̃(m)b , (12)

where entries of w̃(m)ul and w̃(m)b are i.i.d. over

{± 1√

Nu

}and

{± 1√

Nb

}with equal probability, respectively, then for

any l = 1, ..., L and m = 1, ...,M , we obtain,

(w(m)Tul ⊗w(m)Hb ) = (w̃

(m)Tul

FTu ⊗ w̃(m)Hb F

Hb )

= (w̃(m)Tul ⊗ w̃(m)Hb )(F

Tu ⊗ Fb

H)

= (w̃(m)Tul ⊗ w̃(m)Hb )Ψ

H .

As a result, the mth row of B is given by,

[(w(m)Tu1 ⊗w(m)Hb ) . . . (w

(m)TuL ⊗w

(m)Hb )]Φ

= [(w̃(m)Tu1 ⊗ w̃(m)Hb )Ψ

H . . . (w̃(m)TuL ⊗ w̃(m)Hb )Ψ

H ]Φ

= [(w̃(m)Tu1 ⊗ w̃(m)Hb ) . . . (w̃

(m)TuL ⊗ w̃

(m)Hb )].

Thus, the sensing matrix B has entries{± 1√

NuNb

}but are

not i.i.d. The same holds for the sensing matrices Al and Dstated in equations (8) and (11) respectively.


E. Characterization of the mutual coherence parameter

Mutual coherence of a matrix C, denoted as µC, is definedas, µC = max

i1 6=i2

|cHi1ci2 |||ci1 ||2||ci2 ||2

, where ck refers to kth column

of C.

Theorem 1. Suppose the sensing matrix B used in equation(9) is constructed using the Fourier Design method discussedabove, then the mutual coherence parameter of B, µB, is lessthan any β ∈ (0, 1) with probability at least equal to 1− δ, ifM > 8β2 loge

(2L2N2uN

2b

δ

).

Proof: Proof is given in Appendix B.The mutual coherence parameter of the sensing matrices Al

and D in equations (8) and (11) can be characterized in thesame way and the corresponding condition on the number ofmeasurements can be obtained from theorem 1 by substitutingL = 1 and Nu = 1, respectively.

IV. GENERALIZED BLOCK OMP FRAMEWORK

A. Algorithm Description

We consider the block sparse signal recovery framework forthe estimation of channel matrices Hwl from our measurementmodels. However, the block OMP algorithm from [28] isdeveloped for the case when the block sparse vector is aprioripartitioned into disjoint sub-blocks, out of which few are non-zero. In our model, such disjoint apriori partitioning is notpossible since the active blocks (squares) in Hwl depend onthe actual spatial frequencies of each path. Hence, we proposea generalized version of block OMP algorithm (G-BOMP),which will be applied to solve the mm wave channel estimationproblem, for all the three strategies discussed in Section III.

Consider a g1 × g2 matrix, G = S1ES2, where E (of sizeV × V̄ ) is a block sparse matrix with atmost K non-zero(active) blocks of order b × b at arbitrary locations, with allremaining entries being zero, and matrices S1 and S2 formthe sparsifying basis for G. Consider the noisy observationmodel,

y = Pe + n, (13)

where P = TS, S = (ST2 ⊗S1), T is the measurement matrixof size m′ × V V̄ and e = vec(E).

First we define sub-blocks of the matrix E. The illustrationis given below for b = 2. Define sets V = {1, ..., V } andV̄ = {1, ..., V̄ }. Let Ev,v̄ with v ∈ V , v̄ ∈ V̄ , represent a sub-block of size b×b with top left entry being ev,v̄ . For example,in the below picture when b = 2, E1,2 is the block containingentries {e1,2, e1,3, e2,2, e2,3} and EV,4 is the one that containsthe entries {eV,4, eV,5, e1,4, e1,5}. Let the index set Jv,v̄ denotethe locations of entries of Ev,v̄ in the vector e. From theillustration below, J1,1 = {1, 2, V + 1, V + 2}. For a givenv and v̄, Jv,v̄ is obtained as follows: Let rk1,k2 = v + k1 +(k2 + v̄−1)V , where k1, k2 ∈ {0, 1, ..., b−1}. Suppose rk1,k2exceeds (v̄ + k2)V , then modify rk1,k2 as rk1,k2 − V . Refine

rk1,k2 = rk1,k2 mod (V V̄ ). Then, Jv,v̄ =b−1⋃k1=0

b−1⋃k2=0

{rk1,k2}.

e1,1 e1,2 e1,3 e1,4 e1,5 ... e1,V̄e2,1 e2,2 e2,3 e2,4 e2,5 ... e2,V̄e3,1 e3,2 e3,3 e3,4 e3,5 ... e3,V̄e4,1 e4,2 e4,3 e4,4 e4,5 ... e4,V̄... ... ... ... ... ... ...eV,1 eV,2 eV,3 eV,4 eV,5 ... eV,V̄

Let E = {Ev,v̄}v∈V,v̄∈V̄ be the set of all the possible

non-zero blocks in E and J = {Jv,v̄}v∈V,v̄∈V̄ denote thecorresponding collection of index sets. The inputs to the G-BOMP algorithm are y, P, the index set collection J , and astopping criterion.

Algorithm 1 Generalized Block OMP (G-BOMP) Algorithm

1: Initialize: r = y, M = 0V×V̄ (all zero matrix), Î = ∅.2: Compute: b← PHr.3: Assign: [M]q,q̄ ← ||bJq,q̄ ||2, for every Jq,q̄ ∈ J .4: Enumerate: (λr(t), λc(t))← arg max

(v,v̄)∈V×V̄[M]v,v̄

5: Store: Î ← Î ∪ Jλr(t),λc(t).6: Evaluate: xt ← arg min

x||y−PÎx||2, where PÎ is a sub-

matrix of P, containing those columns of Ā indexed byÎ.

7: Update Residue: r = y −PÎxt.8: Increment t by 1. If the stopping criterion described below

is satisfied, then stop. Else, goto Step 2. The stoppingcriterion we use is: t ≤ K and ||PHr||2∞ ≤ τ , where thethreshold τ is appropriately chosen based on the operatingSNR and the number of measurements used.

9: Estimate: The estimate ê is given by,

êÎ = (PHÎ PÎ)

−1PHÎ y; êÎc = 0. (14)

After initializing the required variables in Step 1, the G-BOMP algorithm computes the inner products between thecolumns of P and the residue r in Step 2. It then searches forthe set of columns in P that highly correlate with r in Steps3 and 4. In doing so, the location of the active blocks areidentified. Since y is a weighted linear combination of certaincolumns of P, the weights of those columns identified in Step4 are estimated using least squares method in Step 6. Finally,the residual vector is updated by removing the contributionof the chosen columns from the observation vector in Step7. The entire procedure is repeated until a given stoppingcriterion is met. The threshold τ used in the stopping criterionis appropriately chosen based on the operating SNR andthe number of measurements. When the algorithm stops, theestimate of e, denoted as ê, is obtained using the recoveredindex set Î via least squares as mentioned in equation (14).

The above algorithm reduces to the conventional BOMPalgorithm if V = Ub, V̄ = Ūb, V = {1, ..., U}, V̄ ={1, ..., Ū} for some positive integers U , Ū , and for every

v ∈ V, v̄ ∈ V̄ , we assign Jv,v̄ =b−1⋃k1=0

b−1⋃k2=0

{rk1,k2}, where

rk1,k2 = (v − 1)b + k1 + 1 + ((v̄ − 1)b + k2)V . And, thealgorithm reduces to the well-known OMP algorithm if b = 1.


B. Algorithm Usage in the context of mm wave channelestimation problem

The measurement models derived in Section III can be di-rectly related to the G-BOMP framework specified in equation(13) as described below:

1) From equations (8) and (13), we see that P = Al, E =Hωl , y = yUE and n = nUE . Let J (l) be the collection of indexsets corresponding to all possible non-zero blocks of matrixHωl . Then J = J (l) and the inputs to the G-BOMP algorithmwill be yUE , Al and J (l). Here, K = Kmax.

2) Similarly, on comparing equations (9) and (13), we getP = B, E = [Hω1 H

ω2 ...H

ωL], y = yBS , n = nBS , J =⋃

1≤l≤LJ (l) and K = LKmax.

3) Finally in Joint-Method, the G-BOMP framework forstage 1 is same as that for the UE-Method. For the secondstage, we needed 1−D version of the G-BOMP algorithmwhich is obtained by setting V̄ = 1. In this case, P = D,e = cω , y = yJoint , n = nJoint and J =

⋃1≤l≤L

J (l) with

V̄ = 1.In all these cases, the set of all the possible non-zero blocks

can be specified by considering the range of spatial frequenciesin the channel model (and by neglecting the leakage outsideb× b squares centered around the spatial frequencies).

C. Analysis on G-BOMP Algorithm

Recall the observation model y = Pe. For a given (v, v̄) ∈V × V̄ , denote P(v,v̄) = PJv,v̄ and e(v,v̄) = eJv,v̄ . Assumethat the active blocks of E be located at (r1, c1), (r2, c2), ...,(rK , cK), where rk̄ ∈ V and ck̄ ∈ V̄ , for all k̄ = 1, 2, ...,K.Let the corresponding sub-matrices P(rk̄,ck̄), ∀k̄ = 1, 2, ...,K,be termed as the active sub-matrices of P. With these nota-tions, we re-write the measurement model in (13) as,

y =

K∑k̄=1

P(rk̄,ck̄)e(rk̄,ck̄) + n. (15)

Further, if I denotes the support set of e, then,

y =∑i∈I

pie(i) + n, (16)

where |I| = b2K and pi is the ith column of P. Letpmax = max

i||pi||22, pmin = min

i||pi||22 and µP be the mutual

coherence parameter of P. Then in what follows, we derivethe sufficient condition for exact recovery by the proposed G-BOMP algorithm. For simplicity, we consider the case wherethere exists no block in E that overlaps with two or moreactive blocks, i.e., for every k̄, l̄ ∈ {1, ...,K} such that k̄ 6= l̄,we have |rk̄ − rl̄| > b and |ck̄ − cl̄| > b. When this conditionholds, the active blocks of E will be well separated and weterm such a matrix as Sufficiently Sparse Matrix (SSM).

Theorem 2. Consider the G-BOMP observation model givenin equation (13). Suppose E is a SSM, then under boundednoise condition (i.e., ||n||2 ≤ ε), the G-BOMP algorithmrecovers the locations of active blocks in K iterations with

b > 1 if,

K <

pminXµP

+ Xpmax + pmin + pmaxµP − 2εpminµP√pmaxemaxb2(

3pmaxµP + 2pmin + Xpmax) ,

(17)where X = pmineminpmaxemax , emin = minrk̄,ck̄;1≤k̄≤K

min1≤l̄≤b2

|e(rk̄,ck̄)(l̄)|

and emax = maxrk̄,ck̄;1≤k̄≤K

max1≤l̄≤b2

|e(rk̄,ck̄)(l̄)|.

Proof: Refer to Appendix C for the proof.

D. Guarantees for Other Sparse Recovery Algorithms: Or-thogonal Matching Pursuit (OMP):

Recall that e is a sparse vector that contains atmost b2Knon-zero entries. If OMP algorithm is employed to recovere, then it requires b2K iterations to recover all the non-zeroelements in e. Therefore, the sufficient condition for the OMPalgorithm to recover e is as follows.

Theorem 3. Under bounded noise condition, i.e., ||n||2 < ε,the OMP algorithm recovers locations of the non-zero ele-ments of e from the observation vector y, given in equation(13), in b2K iterations if,

K <1

3b2

( pminpmaxµP

+ 2 + pmaxµPpmin −2ε√

pmaxµPemin

1 + pmaxµPpmin

). (18)

Proof: Proof is similar to that of Theorem 2.Irrespective of the greedy algorithm used, the error in the

estimate completely depends on the support set recovered bythe algorithm. Hence, we obtain the following result.

Theorem 4. Suppose Î is the recovered support set of e andê is the estimate of e obtained using equation (14), then theestimation error ||ê− e||2 is bounded above as,

||ê− e||2 ≤ ||eÎc ||2

+

(√|Î||Îc|pmaxµP||eÎc ||2 + �

√|Î|pÎmax

)λmin(PHÎ PÎ)

,

where pÎmax = maxi∈Î||pi||22.

Proof: Proof is given in Appendix D.

V. SIMULATION RESULTS

In this section, we present our simulation results comparingthe performance of the proposed G-BOMP algorithm withthe OMP algorithm, method discussed in [26], [25] (FISTA),and the 2D-compressive estimation via Newton’s refinement(Newton-Ref.) method proposed in [29], in terms of aver-age beamforming gain (γ) and average spectral efficiencyachieved. We refer to the mm wave channel estimation tech-nique proposed in [26] as Random phase (RP) method becauseit designs the pilot training beamforming vectors using randomphase terms of the form ejφ. As mentioned in Section I, the RPmethod assumes that the spatial frequencies that characterizethe mm wave channels fall exactly into the DFT bins andemploys OMP to estimate the same. Thus, in RP method, a


K-rank mm wave channel is treated as K-sparse matrix (notas a K-block sparse matrix) and is recovered by the OMPalgorithm in just K iterations. The parameters considered forsimulation are: Nb = 32, Nu = 16, dλ =

12 , b = 2, block size

for 1−D G-BOMP algorithm is 2, φbl(k), φul(k) are i.i.d.random variables uniform in [−π2 ,

π2 ], ∀k = 1, 2, ...,Kl and

l = 1, ..., L, σ2α = 1. We assume that Kl = K̄, ∀l = 1, 2, ..., Land σ2u = σ

2b = σ

2. In our simulations, we assume the mmwave channels, Hl, ∀l = 1, ..., L, to be, Hl = FbbHωl FHuu,where Fbb and Fuu are Nb × 2Nb and Nu × 2Nu Fouriermatrices respectively, i.e., we sample the channel in thefrequency domain with a finer DFT grid that has spacing equalto 2π2Nb and

2π2Nu

respectively. Average beamforming gain, γ,is defined as,

γ =1

L

L∑l=1

|ŵ(l)Hopt Hl f̂(l)opt |2, (19)

where ŵ(l)opt and f̂(l)opt are the left and the right singular vectors

of Ĥl (the estimate of Hl). And, average spectral efficiency(ASE) is given by,

ASE =(1− α)L

L∑l=1

E[

log2

(1 +|ŵ(l)Hopt Hl f̂

(l)opt |2

σ2

)], (20)

where α = MT denotes the fraction of time devoted for pilottraining and T the coherence time of the mm wave channel.We fix T = 1000 in all simulation results. The G-BOMP algo-rithm is labeled as ”G-BOMP-UE”, ”G-BOMP-BS” and ”G-BOMP-Joint” when implemented for UE-Method, BS-Methodand Joint-Method respectively, in the legends of all figures.OMP is labeled the same way. The results are averaged over1000 channel realizations and are given for the RademacherDesign of the training beamforming vectors/symbols as iteasy to implement with phase shifters (we observed that theperformance of the Fourier design is nearly same).

Beamforming gain γ and ASE Vs 1σ2 (dB): Fig. 2 plotsγ and ASE of different techniques as a function of noisevariance. We fix K̄ = 2,Kmax = 3,M = 225,M1 = 150and L = 4. As expected, γ increases with decrease in σ2

for all schemes. At the same time, the rate of increase ofγ decays with 1σ2 . Secondly, we observe that the proposedG-BOMP algorithm outperforms the OMP algorithm and RPmethod in all the three pilot training schemes. Further, ”G-BOMP-BS” is better than the FISTA technique [25] whenimplemented for the BS-Method. Newton’s refinement methodperforms better as it not just produces the coarse estimatesof the spatial frequencies but also refines it (refer to [29] formore details). However, this good performance is achieved atthe cost of high computational complexity (see Fig. 5).

ASE Vs Number of measurements: Fig. 3 shows thevariation of ASE w.r.t M for 1σ2 = −10dB and −5dB. M1for Joint-Method is chosen such that it is approximately equalto 23M . Here we set K̄ = 2,Kmax = 3 and L = 4. Atrend similar to that observed in Fig. 2 can be noticed here.We see that the G-BOMP algorithm, in general, outperformsthe OMP algorithm and RP method in all the beamformingtraining strategies. We also observe that the average ASE of

all the methods increases gradually with M and reaches asaturation point at certain value of M . At the same time,the fraction of training period (α) increases linearly with M .As a consequence, from equation (20), we notice that theASE curve for all the methods decays beyond certain valueof M . Since the average ASE increases with the number ofmeasurements initially and then decreases due to α variation,the average ASE of all the algorithms attains a maximum value(at different values of M for different techniques), which areindicated within parenthesis in the figure legends.

Comparing G-BOMP-UE, G-BOMP-BS and G-BOMP-Joint in terms of γ Vs 1σ2 : Fig. 4 analyzes the performance ofthe G-BOMP algorithm in terms of γ Vs noise variance whenapplied to each of the pilot training strategies and for differentvalues of K̄ with M = 225 (M1 = 150 for Joint-Method). Wenotice that when K̄ = Kmax = 1, the approximation used inJoint-Method (equation (11)) becomes exact and it performsbetter compared to the other two training schemes. We also no-tice that γ of ”G-BOMP-UE” is higher than that of ”G-BOMP-BS”. This is because in the UE-Method each UE estimates itschannel without any interference from the other UEs, whereas”G-BOMP-BS” jointly re-constructs all the L channels. As aconsequence, the entity to be estimated in the BS-Method is oflarger dimension, and hence, it requires larger M to achievethe same level of performance as that of ”G-BOMP-UE”. ForK̄ > 1, ”G-BOMP-Joint” decays at a faster rate than ”G-BOMP-UE” and ”G-BOMP-BS” algorithms, mainly becausethe direction with the largest gain sees increased interferencefrom the paths corresponding to other spatial frequencies inthe Joint-Method, as per our remarks from eqn. (11).

Computational Complexity of different algorithms:The computational complexity of the proposed G-BOMPscheme,the OMP algorithm, the RP method and the New-ton’s refinement technique are evaluated in terms of the totalnumber of complex multiplications required to estimate allthe L channels and are plotted w.r.t M in Fig. 5. We fixL = 4, b = 2, Nb = 32, Nu = 16 and Kmax = 3.For calculation purposes, we assume that the thresholdingcondition in the stopping criterion does not influence thetermination of any of the algorithms. Further, as explainedin detail before, if the G-BOMP algorithm (described asAlgorithm 1 in Section IV) runs for K iterations, then theOMP algorithm and the algorithm in RP method run for b2Kand K iterations respectively. As a consequence, in everypilot training scheme the complexity of the OMP algorithmhas the highest computational complexity and the RP methodhas the least. We observe that the complexity of all thealgorithms is maximum when implemented for BS-Method,and also notice that the Newton’s refinement technique ismore complex than ”G-BOMP-UE”. This is mainly because ofthe additional refinement stage implemented in the Newton’srefinement method.

MSE Vs 1σ2 and K̄: Fig. 6 plots channel estimation errorw.r.t SNR (low values) for different values of sparsity orderwith Nu = Nb = 16, L = 1, b = 2 and M = 180. Inthis figure, we also verify the validity of the upper boundderived in Section IV for the estimation error. From the figure,we observe that the estimation error decreases with increase


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5

10

15

20

25

30

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18

20

22

24

26

28

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18

20

22

24

26

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2

4

6

8

10

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2

4

6

8

10

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2

4

6

8

10

Fig. 2: (Top): γ (dB) Vs 1σ2

(dB); (Bottom): ASE Vs 1σ2

(dB).

100 200 300 4000

1

2

3

4

5

100 200 300 4000

1

2

3

4

5

6

100 200 300 4000

1

2

3

4

5

Fig. 3: ASE Vs Number of measurements (M ) for different values of 1σ2

(dB).

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24

24.5

25

25.5

26

26.5

27

27.5

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14

16

18

20

22

24

26

Fig. 4: γ (dB) Vs 1σ2

for G-BOMP algorithms Nu = 16, Nb = 32.


0 100 200 300 400 5000

0.5

1

1.5

2

2.5108

0 100 200 300 400 5000

1

2

3

4

5107

0 100 200 300 400 5000

0.5

1

1.5

2

2.5

3

3.5

4107

Fig. 5: Computational complexity in terms of total number of complex multiplications Vs M .

-15 -10 -5 05

10

15

20

25

30

35

Fig. 6: MSE Vs 1σ2n

(dB) for G-BOMP algorithm with L = 1, Nu = Nb = 16.

in SNR and the derived upper bound for estimation error islegitimate. Finally, we notice that for a given SNR estimationerror is high for higher values of K̄ as more number ofmeasurements are needed to recover more number of non-zeroentries to attain the same level of error performance.

Impact of using more RF chains: From the system modelin (1), with NRFb = 1 and sending constant symbol s = 1, themeasurement vector at the mth time instant at the lth UE canbe written as, y(m)l = V

(m)Hl H

Hl u

(m)+n(m)l . Since each time

instant gives NRFu measurements at the UE, the total trainingduration needed to get M observations is MNRFu . Similarly, thetraining duration gets reduced for the BS-Method and theJoint-Method. We also note that, increasing the number of RFchains at the transmitter does not reduce the training overhead.In the simulations, we generate the entries in the trainingbeamforming vectors/matrices (such as u(m) and V(m)l ) inan i.i.d. manner with values equally likely from

{± 1√

Nb

}or{

± 1√Nu

}. For the Joint-Method, we set NRFb = N

RFu = NRF.

Results shown in the Fig. 7 indicate that ASE increases withthe number of RF chains.

ASE performance for channels obtained using NYUSIM:We verify the applicability of our proposed methods in esti-mating the channels generated by NYUSIM, an open-sourcemm wave channel simulator [31]. We set the parameters inthe simulator as given below: Array type is ULA, Nb = 32,Nu = 16, carrier frequency = 28 GHz, bandwidth (B) = 800MHz. We consider a single cell mm wave system and the cell

radius denotes the maximum possible distance between BSand UE. We fix L = 3, place the UEs randomly inside the celland average the performance over 500 channel realizations foreach UE. Thermal noise with variance equal to kTB, wherek = Boltzman’s constant and T = 300 kelvin (operatingtemperature) is added. As expected, ASE performance shownin Fig. 8 decreases with increase in cell radius. We alsonote that G-BOMP algorithm is slightly better than OMP,outperforms RP method, and is comparable with the Newton’srefinement method.

VI. CONCLUSION

In this article we presented three different strategies of es-timating mm wave channels in a multi-user scenario, namely:1) UE’s estimating their channels separately, 2) BS jointlyestimating all the channels, and 3) Two stage process whereboth UE’s and the BS estimate the channels. We exploitedthe sparse nature of mm wave channels and proposed ageneralized BOMP algorithm to estimate the same in all thethree strategies mentioned above. Our simulation results showthat our G-BOMP algorithm performs better compared to theOMP algorithm and other prior works, and is comparablewith the 2D-compressive estimation via Newton’s refinementmethod in terms of the average beamforming gain and averagespectral efficiency achieved.


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10

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2

4

6

8

10

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2

4

6

8

10

Fig. 7: ASE Vs 1σ2

with multiple RF chains at the receiver.

500 1000 1500

Cell Radius (m)

2

4

6

8

10

12

14

16

ASE

(bps/Hz)

BS-Method, L = 3, Kmax = 5

GenieG-BOMP-BSOMP-BSRP-BS

500 1000 1500

Cell Radius (m)

2

4

6

8

10

12

14

16

ASE

(bps/Hz)

Joint-Method, L = 3, Kmax = 5

GenieG-BOMP JointOMP-JointRP-Joint

500 1000 1500

Cell Radius (m)

4

6

8

10

12

14

ASE

(bps/Hz)

UE-Method, L = 3, Kmax = 5

GenieG-BOMP-UEOMP-UERP-UENewton’s Ref.

Fig. 8: ASE Vs cell radius with channels generated using NYUSIM.

APPENDIX

A. An useful Lemma:

Lemma 1. Let PP1 and PP2 be two sub-matrices of P, andX = PHP1PP2 .

1. If PP1 is a tall sub-matrix then,

λmin(PHP1PP1) ≥ pmin − pmax(|P1| − 1)µP. (21)

2. Suppose P1 ∩ P2 = ∅ and |PP1 | = |PP2 | = b2, then,

λmax(XHX) ≤

(pmaxb

2µP

)2. (22)

3. If n is a vector such that ||n||2 ≤ ε and the productPP1n is well defined, then

||PHP1n||2 ≤ ε√|P1|pP1max ≤ ε

√|P1|pmax, (23)

where pP1max = maxi||p(1)i ||22 and p

(1)i is the i

th column ofPP1 .

Proof:First, we prove equation (21) .Denote the matrix product

PHP1PP1 as Q. From Gershgorin’s circle theorem [32], we get

|λ−[Q](q,q)| ≤|P1|∑

q̄ 6=q,q̄=1|[Q](q,q̄)|, where λ is some eigen value

of Q. On further simplification, we obtain λpmax ≥[Q](q,q)pmax

−

1pmax

( |P1|∑q̄ 6=q,q̄=1

|[Q](q,q̄)|)

. Since [Q](q,q) is the l2-norm square

of some column of P, [Q](q,q) ≥ pmin. Also,[Q]q,q̄pmax

≤ µP,∀q̄ ∈ {1, ..., |P1|} \ {q}. This implies,

λ ≥ pmax( pminpmax

− (|P1| − 1)µP)≥ pmin − pmax(|P1| − 1)µP.

Since the above is true for any q = 1, 2, ..., |P|1, we getλmin(Q) ≥ pmin − pmax

((|P| − 1)µP

).

Now, we proceed to prove equation (22). Given X =PHP1PP2 . Since P1 ∩P2 = ∅, [X](i1,i2) is inner product sometwo different columns of P. Thus, |[X](i1,i2)| ≤ pmaxµP,∀i1, i2 = 1, ..., b2. Let Y = XHX. Then, [Y](i1,i2) =b2∑k=1

[X]∗(k,i1)[X](k,i2) for some i1, i2 = 1, ..., b2. We will now

obtain a bound on the absolute value of each and every elementof Y.

|[Y](i1,i2)| =∣∣∣ b2∑k=1

[X]∗(k,i1)[X](k,i2)

∣∣∣ ≤ b2∑k=1

|[X]∗(k,i1)||[X](k,i2)|

≤ b2p2maxµ2P.

In general, for any i1, i2 = 1, ..., b2, we have |[Y](i1,i2)| ≤

b2(

maxk||pk||22

)2µ2P. Gershgorin’s circle theorem [32]


bounds the maximum eigen value of Y, λmax(Y), as,

λmax(Y) ≤ p2max(b2µP

)2.

Finally, consider the following:

||PHP1n||2 =

√√√√|P1|∑i=1

|(p(1)i )Hn|2 ≤

√√√√|P1|∑i=1

||p(1)i ||22||n||22

≤ ε√|P1|pP1max ≤ ε

√|P1|pmax.

Here, the 3rd inequality is due to Cauchy Schwarz’s In-equality of norms. Hence, proved.

B. Proof of Theorem 1:

Let B = [b1 b2 . . . bLNuNb ]. Since [B](q,q̄) =± 1√

NuNb, ∀q = 1, 2, . . . ,M, ∀q̄ = 1, 2, . . . , LNuNb, the

norm of ith column of B, for every i = 1, 2, . . . , LNuNb,

is given by, ||bi||2 =M∑k=1

|bi(k)|2 = MNuNb . Thus, the mutualcoherence of B simplifies to,

µB = maxi1 6=i2

|bHi1bi2 |||bi1 ||2||bi2 ||2

= maxi1 6=i2

|bHi1bi2 |M

NuNb

.

Then, µB < β ⇒ maxi1 6=i2

|bHi1bi2 | <MβNuNb

. We now upper

bound the term maxi1 6=i2

|bHi1bi2 | using Mcdiarmid’s concentrationinequality, which states that if f(X1, . . . , XN ) is an arbitraryfunction of N independent random variables {Xi}Ni=1, eachdefined over a discrete set, such that ∀i, ∀X̃i, X1, ..., XN ,|f(X1, . . . , Xi, . . . , XN )−f(X1, . . . , X̃i, . . . , XN )| < di, forsome di > 0, then P(|Z − E[Z]| > t) ≤ 2 exp

(− 2t

2

d

), with

d =N∑i=1

d2i and Z = f(X1, . . . , XN ). The constant di is called

the Mcdiarmid’s constant of the random variable Xi.Consider the columns bi1 and bi2 . Since each row of B

contains kronecker product of vectors, each element in bi1and bi2 is actually a product of two different entities. Inwhat follows, we closely study the nature of these entities,the relation between entries of bi1 and bi2 and then derivethe concentration inequality.

First, we assume bi1 = [x1z1, x2z2, . . . , xMzM ]T , with

xi’s being i.i.d. over {± 1√Nu } and zi’s over {±1√Nb} with

equal probability each. Also, xi’s and zk’s are independent.Note that E[xi] = E[zi] = 0, ∀i = 1, . . . ,M . Now we havethe following cases:

Case 1: Let bi2 be a column such that |i1 − i2| > NuNb.For example, bi1 = b1 and bi2 = bNuNb+2. In this case,the elements of bi1 are products of first entries in w̃

(m)u1 ’s

and w̃(m)b ’s. Whereas the elements of bi2 are products offirst entries in w̃(m)u2 ’s and second entries in w̃

(m)b ’s, which

are completely independent of the entities that constitute theelements of bi1 .

We can, therefore, assume bi2 = [x̃1z̃1, . . . , x̃M z̃M ]T ,

where x̃i’s and xi’s are i.i.d., and z̃k’s and zk’s are i.i.d.Also, x̃k’s and z̃k’s are independent. Then bHi1bi2 is a functionof 4M independent random variables and so we obtain,

E[bHi1bi2 ] =M∑k=1

E[xkzkx̃kz̃k] = 0. It can be verified that

the Mcdiarmid’s constants for xi, x̃i, zi and z̃i, for all i, areall equal to 2NuNb . Thus, Mcdiarmid’s inequality for b

Hi1bi2

is given by, P(|bHi1bi2 | > t) < 2 exp(− 2t

2

4M 4N2

bN2u

)=

2e−t2N2uN

2b

8M .On the other hand, the two columns can also be such

that bi1 = b1 and bi2 = bNuNb+1. In this case, thefirst entries in w̃(m)b ’s contribute to the elements of bothbi1 and bi2 . Then, the other way to model the structure ofbi2 can be bi2 = [x̃1z1, . . . , x̃MzM ]

T . Here bHi1bi2 is afunction of 3M independent random variables and we have,

E[bHi1bi2 ] =M∑k=1

E[xkx̃kz2k] =1Nb

M∑k=1

E[xkx̃k] = 0. Also,

for every i = 1, . . . ,M , the Mcdiarmid’s constant for ziis zero and that of xi and x̃i are 2NuNb . Hence, we get,

P(|bHi1bi2 | > t) < 2 exp(− 2t

2

2M 4N2

bN2u

+(M)(0)

)= 2e−

t2N2uN2b

4M .

On combining all the inequalities stated above, we get,

P(|bHi1bi2 | > t) < 2 exp(− t

2N2uN2b

8M

). (24)

Case 2: Considering the other case when |i1− i2| < NuNb,we see that bi2 can be once again [x̃1z̃1, . . . , x̃M z̃M ]

T or[x̃1z1, . . . , x̃MzM ]

T or additionally it can take the form[x1z̃1, . . . , xM z̃M ]

T . One can verify that the concentrationinequality for this case will be same as equation (24).

Lastly, by union bound property in probability theory, weobtain,

P(maxi1 6=i2|bHi1bi2 | > t) (25)

< 2LNuNb(LNuNb − 1) exp(− t

2N2uN2b

8M

)(26)

≤ 2L2N2uN2b exp(− t

2N2uN2b

8M

). (27)

Moving back to our problem, from above equation, we get,

P(maxi1 6=i2|bHi1bi2 | <

Mβ

NuNb)

≥ 1− 2L2N2uN2b exp(− M

2β2N2uN2b

8MN2uN2b

)= 1− 2L2N2uN2b exp

(− Mβ

2

8

).

If 2L2N2uN2b exp

(− Mβ

2

8

)< δ, for some δ ∈ (0, 1), then

it implies that M > 8β2 loge(

2L2N2uN2b

δ

). Thus, P(µB <

β) = P(maxi1 6=i2

|bHi1bi2 | <MβNuNb

) ≥ 1 − δ if M >8β2 loge

(2L2N2uN

2b

δ

). Hence, proved.

C. Proof of Theorem 2:

Using mathematical induction, assuming that the GBOMPalgorithm identifies the active blocks correctly in the firstt iterations, we obtain a condition for identifying an activeblock correctly in the (t + 1)th iteration. This condition isderived to be independent of the iteration index t and hence


provides the sufficient condition for recovery of active blocksby GBOMP. To proceed further, without loss of generality,at the end of t iterations, let the active blocks identifiedbe {e(r1,c1) . . . e(rt,ct)}. Denote e(1) = [e(r1,c1) . . . e(rt,ct)]and e(2) = [e(rt+1,ct+1) . . . e(rK ,cK)] and the correspondingactive sub-matrices of P be Q1 = [P(r1,c1) . . . P(rt,ct)] =[q

(1)1 . . . q

(1)tb2 ] and Q2 = [P(rt+1,ct+1) . . . P(rK ,cK)] =

[q(2)1 . . . q

(2)(K−t)b2 ]. The residue rt, after t iterations, is given

by, rt = y − PQ1y = Q2e(2) − PQ1Q2e(2) + P⊥Q1n, wherePQ1 = Q1(QH1 Q1)−1QH1 is the projection matrix onto col-umn space of Q1 and P⊥Q1 = I−PQ1 is the projection matrixof the corresponding orthogonal complement. At (t + 1)th

iteration, GBOMP identifies an active block from e(2) if

||PH(rt+1,ct+1)rt||2 > maxPz||PHz rt||2, (28)

where Pz is a non-active sub-matrix of P, i.e., it has atleastone column which does not contribute to y. This impliesthat Pz may overlap with P(rt+1,ct+1). Assume that p ∈{0, 1, · · · , b2 − b} columns are same between P(rt+1,ct+1)and Pz . Thus, denote, P(rt+1,ct+1) = [p̃1 p̃2 . . . p̃b2 ],Pz =

[q̃1 . . . q̃b2−p p̃1 . . . p̃p]. Here,{q̃i

}b2−pi=1

are some non-active columns of P, i.e., these columns of P do not con-

tribute to y. Now, equation (28) implies,

√b2∑i=1

|p̃Hi rt|2 >√b2−p∑i=1

|q̃Hi rt|2 +p∑l=1

|p̃Hl rt|2. Since each term inside the

square root is positive on LHS and RHS, the above conditionimplies

b2∑i=1

|p̃Hi rt|2 >b2−p∑i=1

|q̃Hi rt|2 +p∑l=1

|p̃Hl rt|2

⇒b2∑

i=p+1

|p̃Hi rt|2 >b2−p∑i=1

|q̃Hi rt|2.

Note that (28) is satisfied if

mini=1,...,(K−t)b2

|(q

(2)i

)Hrt| > max

q̃|q̃Hrt|, (29)

where q̃ is a non-active column. To guarantee (29), we

lower bound the LHS as, |(q

(2)i

)Hrt| ≥ |

(q

(2)i

)HQ2e

(2)| −

|(q

(2)i

)HPQ1Q2e(2)| − |

(q

(2)i

)HP⊥Q1n|. Now, we get,

|(q

(2)i

)HP⊥Q1n|

(a)≤ ||q(2)i ||2||P

⊥Q1n||2

(b)≤ √pmax

√λmax

[(P⊥Q1

)HP⊥Q1

]||n||2

(c)≤ √pmaxε.

In the above, inequalities (a), (b) and (c) are due to Cauchy-Schwarz’s inequality, Rayleigh-Ritz theorem, the facts that the

eigen values of P⊥Q1 are either zero or one and ||n|| ≤ ε. Now,

|(q

(2)i

)HPQ1Q2e(2)|

= |(q

(2)i

)HQ1(Q

H1 Q1)

−1QH1 Q2e(2)|

(a)≤ ||Q

H1 q

(2)i ||2

λmin(QH1 Q1)||QH1 Q2e(2)||2

(b)≤

√tb2∑l=1

|(q

(1)l

)Hq

(2)i |2

pmin − pmax(tb2 − 1

)µP||QH1

(K−t)b2∑l=1

q(2)l e

(2)(l)||2

≤√tb2pmaxµP


)µP

(K−t)b2∑l=1

√tb2pmaxµPemax

<

(Kb2pmaxµP

)2emax


)µP

(c)≤

(Kb2pmaxµP

)2emax

pmin − pmax(Kb2 − 1

)µP

.

First, (a) is due to Cauchy-Schwarz’s inequality and Rayleigh-Ritz theorem, then (b) is obtained as a consequence of equation(21). In order to get (c), we need a condition that pmin −pmax

(Kb2−1

)µP > 0⇒ K < 1b2

(1+ pminpmaxµP

). Finally, we

have,

|(q

(2)i

)HQ2e

(2)| ≥ |(q

(2)i

)Hq

(2)i e

(2)(i)|

−(K−t)b2∑l 6=i;l=1

|(q

(2)i

)Hq

(2)l e

(2)(l)|

≥ pmin|e(2)(i)|− (Kb2 − tb2 − 1)pmaxµPemax> pminemin − (Kb2 − 1)pmaxµPemax.

Putting these together, we have mini=1,...,(K−t)b2

|(q

(2)i

)Hrt| ≥

pminemin − (Kb2 − 1)pmaxµPemax −

(Kb2pmaxµP

)2emax

pmin−pmax(Kb2−1)µP −√pmaxε. Proceeding in the similar way, we have

maxq̃|q̃Hrt| ≤ Kb2µPpmaxemax +

(Kb2pmaxµP

)2emax

pmin−pmax

(Kb2−1

)µP

+

√pmaxε. Using these bounds, the condition (29) holds true if,

pmineminpmaxemax

+ µP − 2Kb2µP−2K2b4µ2Ppmax

pmin − pmax(Kb2 − 1

)µP

>2ε

√pmaxemax

.

Denoting X = pmineminpmaxemax , and using the fact that pmin −pmax

(Kb2−1

)µP < pmin, we obtain the sufficient condition

for recovery in (17).


D. Proof of Theorem 4:

Given that Î is the estimated support set, ê is the estimatedsparse vector and êÎ = (P

HÎ PÎ)

−1PHÎ y. W.l.o.g let us as-sume that ê = [êTÎ ê

TÎc ]

T and e = [eÎ eÎc ]T . We, therefore, get

||ê−e||2 ≤ ||êÎ−eÎ ||2+||êÎc−eÎc ||2 = ||êÎ−eÎ ||2+||eÎc ||2.The last step is due to the fact that êÎc = 0. Now,

||êÎ − eÎ ||2 = ||(PHÎ PÎ)

−1PHÎ y − eÎ ||2= ||(PHÎ PÎ)

−1PHÎ PÎceÎc + (PHÎ PÎ)

−1PHÎ n||2

≤ 1λmin(PHÎ PÎ)

(||PHÎ PÎceÎc ||2 + ||P

HÎ n||2

),

where the last inequality is due to Rayleigh Ritz theorem [32].

From equation (23), we obtain, ||PHÎ n||2 ≤ �√|Î|pÎmax. Also,

||PHÎ PÎceÎc ||2 ≤√λmax(PHÎcPÎP

HÎPÎc)||eÎc ||2

≤√|Î||Îc|pmaxµP||eÎc ||2.

Once again the first inequality is because of Rayleigh Ritztheorem [32] and the last inequality can be obtained byfollowing the proof of equation (22). Hence, derived thedesired result.

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