Achievable Rates of Multi-User Millimeter Wave Systems with...

© Robert W. Heath Jr. (2015)

Achievable Rates of Multi-User Millimeter Wave Systems with Hybrid Precoding

Ahmed Alkhateeb*, Geert Leus#, and Robert W. Heath Jr.* * Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin # Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology (TU Delft)

www.profheath.org


2

Key ingredients of mmWave MIMO for 5G

more spectrum

more channels bandwidth for Gpbs rates

large antenna arrays

better SINR & area spectral efficiency

spatial multiplexing

multiuser MIMO

Shu Sun, T. Rappapport, R. W. Heath, Jr., A. Nix, and S. Rangan, `` MIMO for Millimeter Wave Wireless Communications: Beamforming, Spatial Multiplexing, or Both?,'' IEEE Communications Magazine, vol. 52, no. 12, pp. 110-121, Dec. 2014. Ahmed Alkhateeb, Jianhua Mo, N. González Prelcic and R. W. Heath, Jr., `` MIMO Precoding and Combining Solutions for Millimeter Wave Systems,'' IEEE Communications Magazine, vol. 52, no. 12, pp. 122-131, Dec. 2014.


MmWave MIMO with analog beamforming

u  Low power consumption (1 RF chain) u  Beamforming gain achieved using narrow beams u  MmWave specific constraints

ª  Constant gains: Only phases are typically adjusted ª Quantized phases: Fixed set of steering directions is allowed

3

De-facto approach inIEEE 802.11ad / WiGig

and Wireless HD

RF Chain

Phase shifters

DAC Baseband Baseband RF Chain ADC

Only provides single stream MIMO beamforming


MmWave MIMO with the hybrid architecture

u  Compromise on power consumption & complexity (# ADCs << # Antennas) u  Enables spatial multiplexing and multi-user MIMO u  Digital can correct for analog limitations [Aya’14]

4

Baseband Precoding

Baseband Combining

1-bit ADC ADC

1-bit ADC ADC

RF Chain

RF Chain

RF Combining

RF Combining

Baseband Precoding

Baseband Precoding

1-bit ADC DAC

1-bit ADC DAC

RF Chain

RF Chain

+

+

+

RF Beam-forming

RF Beam-forming

o  [Aya’14] O. El Ayach, S. Rajagopal, S. Abu-Surra, Z. Pi, and R. Heath, “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Transactions on Wireless Communications, vol. 13, no. 3, pp. 1499–1513, March 2014

Provides enough flexibility to support multiuser MIMO

~ 4 ~ 64


Proposed mmWave MU MIMO system model

u  Assumptions ª  BS employs hybrid analog/digital precoders ª MS has has an array with analog-only combiners ª  Limited feedback exists between BS and MS’s ª  Channels are modeled as geometric sparse mmWave channels

5 o  [Alk’14] A. Alkhateeb, G. Leus, and R. W. Heath Jr, “Limited feedback hybrid precoding for multi-user millimeter wave systems,” submitted to IEEE Transactions

on Wireless Communications, arXiv preprint arXiv:1409.5162, 2014.

FBB FRF

w1

wU

Limited Feedback

Baseband precoder

RF precoder RF

combiner

Huwu


Channel model

u  MmWave channel assumptions ª  Single-path channels (more general channels was investigated in [Alk’14-1]) ª  Array response vector is arbitrary but known ª  Paths may be LOS/NLOS

6

✓1 ✓2 ✓3�3�2�1

path gain(includes path-loss)

array response vectors

angles of arrival/departure (AoA/AoD)

+

+

+

F

RF

Beamformers

wu

RF

combiner

RF

Chain

NBSNRFNMS

Base station uth mobile station

RF

Chain

RF

Chain

s1

sN

RF

Fig. 1. A BS with RF beamformers and NRF RF chains com-municating with the uth MS that employs RF combining.

tiplexing gain of the described multi-user precoding system,which is limited by min (NRF, U) for NBS > NRF. For sim-plicity, we will also assume that the BS will use U out of theNRF available RF chains to serve the U users.

On the downlink, the BS applies an NBS×U RF precoder,F = [f1, f2, ..., fU ]. The sampled transmitted signal is there-fore x = Fs, where s = [s1, s2, ..., sU ]T is the U×1 vector oftransmitted symbols, such that E [ss∗] = PT

UIU , and PT is the

average total transmitted power. Since F is implemented us-ing quantized analog phase shifters, [F]m,n = 1√

NBS

ejφm,n ,

where φm.n is a quantized angle, and the factor of 1√NBS

is

for power normalization.For simplicity, we adopt a narrowband block-fading chan-

nel model [5,11,12], by which the uth MS receives the signal

ru = Hu

U!

r=1

frsr + nu, (1)

where Hu is the NMS × NBS matrix that represents themmWave channel between the BS and the uth MS, andnu ∼ N (0,σ2I) is a Gaussian noise vector.

At the uth MS, the RF combiner wu is used to process thereceived signal ru to produce the scalar

yu = w∗uHu

U!

r=1

frsr +w∗unu, (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 Hu can be expressed as

Hu ="

NBSNMSαuaMS (θu)a∗BS (φu) , (3)

where αu is the complex path gain, including the path-loss,with E

#

|αu|2$

= α. The variables θu, and φu ∈ [0, 2π]are the angles of arrival and departure (AoA/AoD) respec-tively. Finally, aBS (φu) and aMS (θu) are the antenna arrayresponse vectors of the BS and uth 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.

3.1. Compressed Sensing Based 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 pm, and the uth MS employs atraining combining vector qn to combine the received signal,the resulting signal can be written as

yn,m = qHn Hupmsm + qH

n nn,m, (4)

where sm is the training symbol on the beamforming vec-tor pm, and we use sm =

√P , with P the average power

used per transmission in the training phase. If the BS em-ploys MBS such beamforming vectors pm,m = 1, ...,MBS,at MBS successive time slots, and the MS uses MMS mea-surement vectors qn, n = 1, 2, ...,MMS at MMS successiveinstants to detect the signal transmitted over each of the beam-forming vectors, the resulting received matrix will be [5]

YMS =√PQHHuP+N, (5)

where Q = [q1,q2, ...,qMMS] is the NMS ×MMS measure-

ment matrix, P = [p1,p2, ...,pMBS] is the BS NBS ×MBS

beamforming matrix, and N is an MMS ×MBS noise matrix.

o  [Alk’14] A. Alkhateeb, G. Leus, and R. W. Heath Jr, “Limited feedback hybrid precoding for multi-user millimeter wave systems,” submitted to IEEE Transactions on Wireless Communications, arXiv preprint arXiv:1409.5162, 2014.


Design objective u  Find hybrid precoders/combiners to maximize sum-rate

u  Solution requires a search over BS/MS analog codebooks

ª  High complexity ª  Significant training and feedback overhead

7

Develop low-complexity approach for finding precoders

Analog codebook due to phase quantization

Power constraint

8

the vectors aBS

⇣

2⇡kQ

NQ

⌘

, for the variable k

Q

taking the values 0, 1, 2, and N

Q

� 1. The RF

combining vectors codebook W can be similarly defined.

Motivated by the good performance of single-user hybrid precoding algorithms [4], [5] which

relied on RF beamsteering vectors, and by the relatively small size of these codebooks which

depend on single parameter quantization, we will adopt the beamsteering codebooks for the

analog beamforming vectors. While the problem formulation and proposed algorithm in this paper

are general for any codebook, the performance evaluation of the proposed algorithm done in

Sections V-VI depends on the selected codebook. For future work, it is of interest to evaluate the

performance of the proposed hybrid precoding algorithm with other RF beamforming codebooks.

If the system sum-rate is adopted as a performance metric, the precoding design problem is

then to find F?RF

,�

f?BB

u

U

u=1

and {w?u}Uu=1

that solve

n

F?RF

,

�

f?BB

u

U

u=1

, {w?u}Uu=1

o

=argmax

UX

u=1

log

2

1 +

PU

�

�w⇤uHuFRF

fBB

u

�

�

2

PU

P

n 6=u |w⇤uHuFRF

fBB

n |2 + �

2

!

s.t. [FRF

]

:,u 2 F , u = 1, 2, ..., U,

wu 2 W , u = 1, 2, ..., U,

kFRF

⇥

fBB

1

, fBB

2

, ..., fBB

U

⇤ k2F = U.

(7)

The problem in (7) is a mixed integer programming problem. Its solution requires a search over

the entire FU ⇥WU space of all possible FRF

and {wu}Uu=1

combinations. Further, the digital

precoder FBB

needs to be jointly designed with the analog beamforming/combining vectors.

In practice, this may require the feedback of the channel matrices Hu, u = 1, 2, ..., U , or the

effective channels, w⇤uHuFRF

. Therefore, the solution of (7) requires large training and feedback

overhead. Moreover, the optimal digital linear precoder is not known in general even without

the RF constraints, and only iterative solutions exist [25], [26]. Hence, the direct solution of this

sum-rate maximization problem is neither practical nor tractable.

Similar problems to (7) have been studied before in literature, but with baseband (not hybrid)

precoding and combining [25]–[31]. The main directions of designing the precoders/combiners

in [25]–[28], [30], [31] can be summarized as follows.

• Iterative Coordinated Beamforming Designs The general idea of these algorithms is to

iterate between the design of the precoder and combiners in multi-user MIMO downlink

systems, with the aim of converging to a good solution [25], [26]. These algorithms,

due to hybrid assumption


Related work on MU MIMO (not at mmWave) u  MU MIMO with limited feedback/feedforward

ª  Coordinated beamforming with iterations or limited feedforward [Cha’08] ª  Receive combining solutions with limited feedback [Jin’08], [Tri’07]

u  Limitations ª  Consider digital (not hybrid) precoders/combiners design [Cha’08]-[Tr’07] ª  Requires channel at transmitter/receiver (difficult in mmWave)[Jin’08],[Tri’07] ª  Convergence of iterative algorithms with hybrid precoding is not studied ª  RF constraints impose different limitations on combining filter design

8

Need to develop mmWave-suitable low-complexity solutions

o  [Cha’08] C.-B. Chae, D. Mazzarese, T. Inoue, and R. Heath, “Coordinated beamforming for the multiuser mimo broadcast channel with limited feedforward,” IEEE Transactions on Signal Processing, vol. 56, no. 12, pp. 6044–6056, Dec 2008.

o  [Jin’08] N. Jindal, “Antenna combining for the MIMO downlink channel,” IEEE Transactions on Wireless Communications, vol. 7, no. 10, pp. 3834–3844, October 2008.

o  [Tri’07] M. Trivellato, H. Huang, and F. Boccardi, “Antenna combining and codebook design for the MIMO broadcast channel with limited feedback,” in Proc. of the Forty-First Asilomar Conference on Signals, Systems and Computers, Nov 2007, pp. 302–308.


Proposed MU hybrid precoding design

u  Two-stage multi-user hybrid precoding algorithm ª  1st stage: Single-user analog beamforming design for max. desired power ª  2nd stage: Multi-user interference management

9

Low-complexity near-optimal algorithm

FBB FRF

w1

w2

wU

Limited Feedback

Baseband precoder

RF precoder RF

combiner


Required training and feedback u  1st stage

ª  Training: Per-user beam training (no need for explicit channel estimation) ª  Feedback: Only the index of the quantized beamsteering angle

u  2nd stage

ª  Training: Effective channel training (much reduced dimensions) ª  Feedback: Index of the quantized effective channel vectors

10

FRF

w1

wU

RF precoder

FBB FRF

w1

wUBaseband precoder

Effective channel (after RF beamforming/combining)


Achievable rates u  Achievable rate of user u is lower bounded by

with and gathers the user array response vectors u  As # BS antennas increase, and u  Asymptotic optimality 11

−20 −15 −10 −5 0 5 101

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Spec

tral E

ffici

ency

(bps

/ Hz)

Single−user (No interference)Proposed Hybrid PrecodingDerived Lower BoundAnalog−only Beamsteering

Fig. 3. Achievable rates using the hybrid precoding and beamsteeringalgorithms with perfect channel knowledge.

we can bound the achievable rate of user u in (12) as

Ru � log

2

✓

1 +

SNR

UN

BS

NMS

|↵u|2G⇣

{�u}Uu=1

⌘

◆

, (15)

In addition to characterizing a lower bound on the ratesachieved by the proposed hybrid analog/digital precodingalgorithm, the bound in (15) separates the dependence on thechannel gains ↵u, and the AoDs �u, u = 1, 2, ..., U which canbe used to claim the optimality of the proposed algorithm insome cases and to give useful insights into the gain of theproposed algorithm over analog-only beamsteering solutions.This is illustrated in the following results.

Proposition 6: Let ˚Ru = log

2

⇣

1 +

SNRU N

BS

NMS

|↵u|2⌘

denote the single-user rate, and let RBS

u denote the rateachieved by user u when the BS employs analog-only beam-steering designed based on the first stage of Algorithm 1,i.e., RBS

u = log

2

⇣

1 +

SNRU NBSNMS|↵u|2

SNRU NBSNMS|↵u|2

Pn 6=u|�u,n|2+1

⌘

, with�u,n = a⇤

BS

(�u)aBS

(�n). When Algorithm 1 is used todesign the hybrid precoders and RF combiners described inSection II, and given the Assumptions 1-3, the achievable rateby any user u satisfies

1) Eh

˚Ru �Ru

i

K (NBS

, U),

2) limNBS!1 Ru =

˚Ru with probability one,3) limNMS!1 E

⇥

Ru �RBS

u

⇤

= 1,where K (N

BS

, U) is a constant whose value depends only onN

BS

and U .Proof: Please refer to Appendices B and C in [8].

Proposition 6 indicates that the average achievable rateof any user u using the proposed low-complexity precod-ing/combining algorithm grows with the same slope of thatof the single-user rate at high SNR, and stays within aconstant gap from it. This gap, K (N

BS

, U), depends onlyon the number of users and the number of BS antennas. As

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9

Number of BS Antennas NBS

Spec

tral E

ffici

ency

(bps

/ Hz)

Single−user (No interference)Hybrid Precoding with (BRF

BS=5, BRFMS=4)

Hybrid Precoding with (BRFBS=4, BRF

MS=3)

Analog−only Beamsteering with (BRFBS=5, BRF

MS=4)


MS=3)

Fig. 4. Achievable rates using the hybrid precoding and beamsteeringalgorithms for different numbers of RF beamforming quantization bits

the number of BS antennas increases, the gap between theachievable rate using Algorithm 1 and the single-user ratedecreases, and approaches zero at infinite antenna numbers.One important note here is that this gap does not depend onthe number of MS antennas, which is contrary to the analog-only beamsteering, given by the first stage only of Algorithm1. This leads to the third part of the proposition. This impliesthat multi-user interference management is still important atmmWave systems even when large numbers of antennas areused at the BS and MS’s, and perfect alignment is possible.Note also that this is not the case when the number of BSantennas goes to infinity as it can be easily shown that theperformance of RF beamsteering alone becomes optimal inthis case.

VI. SIMULATION RESULTS

In this section, we evaluate the performance of the proposedhybrid analog/digital precoding algorithm and the presentedbounds using numerical simulations.

First, we compare the achievable rates without quantizationloss in Fig. 3, where we consider the system model inSection II with a BS employing an 8 ⇥ 8 UPA with 4 MS’s,each having a 4 ⇥ 4 UPA. The channels are single-path andRayleigh distributed, the azimuth and elevation AoAs/AoDsare assumed to be uniformly distributed in [0, 2⇡] and [�⇡

2

, ⇡2

],respectively. The rate achieved by the proposed algorithm iscompared with single-user and beamsteering rates. The figureshows that the performance of hybrid precoding is very closeto the single-user rate thanks to canceling the residual multi-user interference. The gain over beamsteering increases withSNR as the beamsteering rate starts to be interference limited.The tightness of the derived lower bound is also shown.

To illustrate the impact of RF quantization, the performanceof hybrid precoding and analog-only beamforming is evaluatedin Fig. 4 with different numbers of beamsteering quantization

single-user rate MU penalty

In the second stage: The BS trains the effective channels,hu = w⇤

uHuFRF

, u = 1, 2, ..., U , with the MS’s. Note thatthe dimension of each effective channel vector is U ⇥1 whichis much less than the original channel matrix. Then, each MSu feeds its effective channel back to the BS, which designs itszero-forcing digital precoder based on these effective channels.Thanks to the narrow beamforming and the sparse mmWavechannels, the effective channels are expected to be well-conditioned which makes adopting a simple multi-user digitalbeamforming strategy like zero-forcing capable of achievingnear-optimal performance as will be shown in Section V.

V. ACHIEVABLE RATE WITH SINGLE-PATH CHANNELS

The analysis of hybrid precoding is non-trivial due to thecoupling between analog and digital precoders. Therefore, wewill study the performance of the proposed algorithm in thecase of single-path channels. This case is of special interestas mmWave channels are likely to be sparse, i.e., only afew paths exist [3]. Further, the analysis of this special casewill give useful insights into the performance of the proposedalgorithms in more general settings.

Next, we will characterize a lower bound on the achievablerate by each MS when Algorithm 1 is used to design the hybridprecoders at the BS and RF combiners at the MS’s. Considerthe BS and MS’s with the system and channels described inSection II with the following assumptions:

Assumption 1: All channels are single-path, i.e., Lu = 1,u = 1, 2, ..., U . For ease of exposition, we will omit thesubscript ` in the definition of the channel parameters in (4).

Assumption 2: The RF beamforming and combining vec-tors fRF

u and wu, u = 1, 2, ..., U , u = 1, 2, ..., U are beam-steering vectors with continuous angles.

Assumption 3: The BS perfectly knows the effective chan-nels hu, u = 1, 2, ..., U .

In the first stage of Algorithm 1, the BS and each MS ufind v?

u and g?u that solve

{g?u,v

?u} = argmax

8gu2W8vu2F

kg⇤uHuvuk. (7)

As the channel Hu has only one path, and given thecontinuous beamsteering capability assumption, the optimalRF precoding and combining vectors will be g?

u = aMS

(✓u),and v?

u = aBS

(�u). Consequently, the MS sets wu = aMS

(✓u)and the BS takes fRF

u = aBS

(�u). If we let the NBS

⇥U matrixA

BS

gather the BS array response vectors associated with theU AoDs, i.e., A

BS

= [aBS

(�1

) ,aBS

(�2

) , ...,aBS

(�U )], wecan then write the BS RF beamforming matrix, including thebeamforming vectors of the U users as F

RF

= ABS

.The effective channel for user u after designing the RF

precoders and combiners is

hu = wuHuFRF

=

p

NBS

NMS

↵ua⇤BS

(�u)FRF

.(8)

Now, defining H = [hT

1

,hT

2

, ...,hT

U ]T, and given the design

of FRF

, we can write the effective channel matrix H as

H = DA⇤BS

ABS

, (9)

where D is a U⇥U diagonal matrix, [D]u,u =

pN

BS

NMS

↵u.Based on this effective channel, the BS zero-forcing digital

precoder is defined as

FBB

= H⇤ ⇣

HH⇤⌘�1

⇤, (10)

where ⇤ is a diagonal matrix with the diagonal ele-ments adjusted to satisfy the precoding power constraints�

�FRF

fBB

u

�

�

2

= 1, u = 1, 2, ..., U . The diagonal elements of⇤ are then equal to [See Appendix A in [8] for a derivation]

⇤u,u =

s

NBS

NMS

(A⇤BS

ABS

)

�1

u,u

|↵u| , u = 1, 2, ..., U. (11)

Note that this ⇤ is different than the traditional digital zero-forcing precoder due to the different power constraints in thehybrid analog/digital architecture.

The achievable rate for user u is then

Ru = log

2

✓

1 +

SNR

U

�

�

�

h⇤uf

BB

u

�

�

�

2

◆

,

= log

2

1 +

SNR

U

NBS

NMS

|↵u|2

(A⇤BS

ABS

)

�1

u,u

!

.(12)

To bound this rate, the following lemma, which character-izes a useful property of the matrix A⇤

BS

ABS

, can be used.Lemma 4: Let A

BS

= [aBS

(�1

) ,aBS

(�2

) , ...,aBS

(�U )],with the angles �u, u = 1, 2, ..., U taking continuous valuesin [0, 2⇡], then the matrix P = A⇤

BS

ABS

is positive definitealmost surely.

Proof: Let the matrix P = A⇤BS

ABS

, then for any non-zero complex vector z 2 CNBS , it follows that z⇤Pz =

kABS

zk22

� 0. Hence, the matrix P is positive semi-definite.Further, if the vectors a

BS

(�1

) ,aBS

(�2

), ...,aBS

(�U ) arelinearly independent, then for any non-zero complex vector z,A

BS

z 6= 0, and the matrix P is positive definite. To show that,consider any two vectors a

BS

(�u) ,aBS

(�n). These vectorsare linearly dependent if and only if �u = �n. As theprobability of this event equals zero when the AoDs �u and�n are selected independently from a continuous distribution,the matrix P is positive definite with probability one.

Now, using the Kantorovich inequality [12], we can boundthe diagonal entries of the matrix (A⇤

BS

ABS

)

�1 using thefollowing lemma from [13].

Lemma 5: For any n ⇥ n Hermitian and positive definitematrix P with the ordered eigenvalues satisfying 0 < �

min

�2

... �max

, the element (P)

�1

u,u , u = 1, 2, ..., n satisfies

(P)

�1

u,u 1

4[P]u,u

✓

�max

(P)

�min

(P)

+

�min

(P)

�max

(P)

+ 2

◆

. (13)

We also note that for the matrix A⇤BS

ABS

, we have(A⇤

BS

ABS

)u,u = 1, �min

(A⇤BS

ABS

) = �2

min

(ABS

), and�max

(A⇤BS

ABS

) = �2

max

(ABS

), where �max

(ABS

) and�min

(ABS

) are the maximum and minimum singular values,respectively. Using this note and Lemma 5, and defining

G⇣

{�u}Uu=1

⌘

= 4

✓

�2

max

(ABS

)

�2

min

(ABS

)

+

�2

min

(ABS

)

�2

max

(ABS

)

+ 2

◆�1

,

(14)

depends only on the user AoDs and # of BS antennas

−20 −15 −10 −5 0 5 101

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Spec

tral E

ffici

ency

(bps

/ Hz)

Single−user (No interference)Proposed Hybrid PrecodingDerived Lower BoundAnalog−only Beamsteering

Fig. 3. Achievable rates using the hybrid precoding and beamsteeringalgorithms with perfect channel knowledge.

we can bound the achievable rate of user u in (12) as

Ru � log

2

✓

1 +

SNR

UN

BS

NMS

|↵u|2G⇣

{�u}Uu=1

⌘

◆

, (15)

In addition to characterizing a lower bound on the ratesachieved by the proposed hybrid analog/digital precodingalgorithm, the bound in (15) separates the dependence on thechannel gains ↵u, and the AoDs �u, u = 1, 2, ..., U which canbe used to claim the optimality of the proposed algorithm insome cases and to give useful insights into the gain of theproposed algorithm over analog-only beamsteering solutions.This is illustrated in the following results.

Proposition 6: Let ˚Ru = log

2

⇣

1 +

SNRU N

BS

NMS

|↵u|2⌘

denote the single-user rate, and let RBS

u denote the rateachieved by user u when the BS employs analog-only beam-steering designed based on the first stage of Algorithm 1,i.e., RBS

u = log

2

⇣

1 +

SNRU NBSNMS|↵u|2

SNRU NBSNMS|↵u|2

Pn 6=u|�u,n|2+1

⌘

, with�u,n = a⇤

BS

(�u)aBS

(�n). When Algorithm 1 is used todesign the hybrid precoders and RF combiners described inSection II, and given the Assumptions 1-3, the achievable rateby any user u satisfies

1) Eh

˚Ru �Ru

i

K (NBS

, U),

2) limNBS!1 Ru =

˚Ru with probability one,3) limNMS!1 E

⇥

Ru �RBS

u

⇤

= 1,where K (N

BS

, U) is a constant whose value depends only onN

BS

and U .Proof: Please refer to Appendices B and C in [8].

Proposition 6 indicates that the average achievable rateof any user u using the proposed low-complexity precod-ing/combining algorithm grows with the same slope of thatof the single-user rate at high SNR, and stays within aconstant gap from it. This gap, K (N

BS

, U), depends onlyon the number of users and the number of BS antennas. As

0 20 40 60 80 100 120 140 160 180 2000

1

2

3

4

5

6

7

8

9

Number of BS Antennas NBS

Spec

tral E

ffici

ency

(bps

/ Hz)

Single−user (No interference)Hybrid Precoding with (BRF

BS=5, BRFMS=4)

Hybrid Precoding with (BRFBS=4, BRF

MS=3)


MS=4)


MS=3)

Fig. 4. Achievable rates using the hybrid precoding and beamsteeringalgorithms for different numbers of RF beamforming quantization bits

the number of BS antennas increases, the gap between theachievable rate using Algorithm 1 and the single-user ratedecreases, and approaches zero at infinite antenna numbers.One important note here is that this gap does not depend onthe number of MS antennas, which is contrary to the analog-only beamsteering, given by the first stage only of Algorithm1. This leads to the third part of the proposition. This impliesthat multi-user interference management is still important atmmWave systems even when large numbers of antennas areused at the BS and MS’s, and perfect alignment is possible.Note also that this is not the case when the number of BSantennas goes to infinity as it can be easily shown that theperformance of RF beamsteering alone becomes optimal inthis case.

VI. SIMULATION RESULTS

In this section, we evaluate the performance of the proposedhybrid analog/digital precoding algorithm and the presentedbounds using numerical simulations.

First, we compare the achievable rates without quantizationloss in Fig. 3, where we consider the system model inSection II with a BS employing an 8 ⇥ 8 UPA with 4 MS’s,each having a 4 ⇥ 4 UPA. The channels are single-path andRayleigh distributed, the azimuth and elevation AoAs/AoDsare assumed to be uniformly distributed in [0, 2⇡] and [�⇡

2

, ⇡2

],respectively. The rate achieved by the proposed algorithm iscompared with single-user and beamsteering rates. The figureshows that the performance of hybrid precoding is very closeto the single-user rate thanks to canceling the residual multi-user interference. The gain over beamsteering increases withSNR as the beamsteering rate starts to be interference limited.The tightness of the derived lower bound is also shown.

To illustrate the impact of RF quantization, the performanceof hybrid precoding and analog-only beamforming is evaluatedin Fig. 4 with different numbers of beamsteering quantization

with probability onesingle-user rate

ABS = [aBS(�1), ...,aBS(�U )]

G⇣{�u}Uu=1

⌘! 1

�max

(ABS

)

�min

(ABS

)! 1


Simulation results

u  Near-optimal performance compared with single-user rate u  Simple & tight lower bounds can be derived u  Reasonable hybrid precoding gains over only beam steering u  Effective channels need to be well quantized



−20 −15 −10 −5 0 5 101

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Spec

tral

Effi

cien

cy (b

ps/ H

z)

Single−user (No Interference)Unconstrained (All Digital) Block DiagonalizationProposed Hybrid PrecodingLower Bound (Theorem 1)Analog−only Beamsteering

0 10 20 30 40 50 60 70 80 90 1000

1

2

3

4

5

6

Number of BS Antennas N BS

Spec

tral

Effi

cien

cy (b

ps/ H

z)

Hybrid Precoding − Perfect Effective Channel nowledgeHybrid Precoding − B

BB=13

Hybrid Precoding − BBB

=11

Hybrid Precoding − BBB

=8

Analog−only Beamsteering

Number of users = 4 MS has a 4x4 UPA Azimuth AoAs/AoDs are uniform. distributed [0, 2 ] Elevation AoAs/AoDs are uniform. distributed [- /2, /2]

π

π π

L=1 path, BS has a 4x4 UPA L=3 paths, Effective channels are quantized with BBB bits

Performance with quantized effective channels


Simulation results

u  Hybrid precoding gain exists even with large antenna arrays u  Hybrid precoding has coverage gain over analog beamsteering

ª Gain is mainly due to interference management capability of hybrid precoding ª Gain increases with number of users



0 20 40 60 80 100 120 140 160 180 2000

2

4

6

8

10

12

14

Number of BS and MS Antennas (N = NBS

= NMS

)

Sp

ect

ral E

ffici

en

cy (

bp

s/ H

z)

Single−user (No interference)

Proposed Hybrid Precoding

Lower Bound (Proposition 6)

Analog−only Beamsteering

3 4 5 6 7 8 9 10 11 12 13 14 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate Threshold

Co

vera

ge

Pro

bab

ility

Single−user per cell

Hybrid Precoding − 2 users




Analog−only Beamsteering − 2 users




SNR=0 dB, L=3 paths Number of users = 4 Uniform planar arrays Azimuth AoAs/AoDs are Uniform. distributed [0, 2 ] Elevation AoAs/AoDs are uniformly distributed in [ - /2, /2]

π

π πBS-MS link is single-path LOS/NLOS BS’s have 8x8 UPAs MS’s have 4x4 UPAs Elevation angles fixed at /2 PPP of BS and users

π


Conclusion u  Proposed multi-user hybrid precoding

ª  Low-complexity (requires low training and feedback overhead) ª  General for arbitrary antenna arrays ª  Achieves comparable rates to digital unconstrained precoding ª  Offers good gain over analog-only solutions in some cases ª  Asymptotic optimality was proved is some cases

u  Future work ª  Extension to uplink mmWave systems with hybrid precoding ª  Codebook design for multi-user limited feedback hybrid precoding

u  More results on limited feedback performance and general channels in A. Alkhateeb, G. Leus, and R. W. Heath Jr, “Limited feedback hybrid precoding for multi-user millimeter wave systems,” submitted to IEEE Transactions on Wireless Communications, arXiv preprint arXiv:1409.5162, 2014.

14


Questions?

15

Professor Robert W. Heath Jr.

Wireless Networking and Communications Group Department of Electrical and Computer Engineering

The University of Texas at Austin

www.profheath.org

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