Dynamic Subarray Architecture for Wideband Hybrid Precoding in Millimeter Wave MIMO Systems

Sungwoo Park, Ahmed Alkhateeb, and Robert W. Heath Jr. Wireless Networking and Communications Group Department of Electrical and Computer Engineering The University of Texas at Austin

This work is supported in part by the National Science Foundation under Grant No. 1319556, and by a gift from Huawei Technologies.

2

Digital precoding vs. Hybrid precoding

Digitalbasebandprecoding(Conven4onalMIMO)

Analog RF

precoder

Digital baseband precoder

RF chain

RF chain

RF chain

. . .

Hybridanalog/digitalprecoding

NRF < NTX

FRF FBB

. . .

[1] O. El Ayach et el., “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. on Wireless Comm., Mar. 2014. [2] A. Alkhateeb et el, “MIMO precoding and combining solutions for millimeter-wave systems,” IEEE Comm. Mag., Dec. 2014. [3] W. Ni et el., “Hybrid block diagonalization for massive multiuser MIMO systems,” IEEE Trans. on Comm., Jan 2016.

. . .

Digital baseband precoder

RF chain

RF chain

RF chain

RF chain

RF chain

. . .

. . .

FFD

. . .

RF chains (r=1,…,NRF)

antennas (n=1,…,NTX)

streams (s=1,…,S)

FHB = FRF FBB

3

Digital precoding vs. Hybrid precoding

Digitalbasebandprecoding(Conven4onalMIMO)

Analog RF

precoder

Digital baseband precoder

RF chain

RF chain

RF chain

. . .

Hybridanalog/digitalprecoding

NRF < NTX

FRF FBB

. . .

[1] O. El Ayach et el., “Spatially sparse precoding in millimeter wave MIMO systems,” IEEE Trans. on Wireless Comm., Mar. 2014. [2] A. Alkhateeb et el, “MIMO precoding and combining solutions for millimeter-wave systems,” IEEE Comm. Mag., Dec. 2014. [2] W. Ni et el., “Hybrid block diagonalization for massive multiuser MIMO systems,” IEEE Trans. on Comm., Jan 2016.

. . .

Digital baseband precoder

RF chain

RF chain

RF chain

RF chain

RF chain

. . .

. . .

FFD

. . .

RF chains (r=1,…,NRF)

antennas (n=1,…,NRF)

streams (s=1,…,S)

FHB = FRF FBB

4

Hybrid precoding: Narrowband vs. Wideband

Narrowbandhybridprecoding

Analog RF

precoder

Digital baseband precoder

RF chain

RF chain

RF chain

. . .

. . .

WidebandHybridprecoding(MIMO-OFDM)

. . .

. . .

. . .

IFFT & RF chain

IFFT & RF chain

IFFT & RF chain . . .

. . .

. . .

. . .

Analog RF

precoder

subcarrier (k=1,…,K)

FBB[2]

FBB[K]

FBB FRF FRF

. . .

Digital baseband precoder

FBB[1]

FHB = FRF FBB FHB[k]= FRF FBB[k] for k=1,…,K per-subcarrier & frequency-domain wideband & time-domain

5

Hybrid precoding: Narrowband vs. Wideband

Narrowbandhybridprecoding

Analog RF

precoder

Digital baseband precoder

RF chain

RF chain

RF chain

. . .

. . .

WidebandHybridprecoding(MIMO-OFDM)

. . .

. . .

. . .

IFFT & RF chain

IFFT & RF chain

IFFT & RF chain . . .

. . .

. . .

. . .

Analog RF

precoder

subcarrier (k=1,…,K)

FBB[2]

FBB[K]

FBB FRF FRF

. . .

Digital baseband precoder

FBB[1]

FHB = FRF FBB FHB[k]= FRF FBB[k] for k=1,…,K

Constraints on FRF

1) NRF < NTX 2) Composed of phase shifters 3) Common for all subcarriers

Constraints on FRF

1) NRF < NTX 2) Composed of phase shifters

per-subcarrier & frequency-domain wideband & time-domain

�[FRF]m,n = ej✓m,n

�

6

Hybrid structure – Fully-connected vs. Partially-connected

Fully-connected structure Partially-connected (subarray) structure

FRF =⇥fRF,1 fRF,2 · · · fRF,NRF

⇤

Analog RF precoder

RF chain

. . . . . .

. . . . . .

. . . . . .

RF chain

NRF NTX

. . . . . .

NTX

NTX

FRF =

2

66664

fRF,S1 0 · · · 0

0 fRF,S2 0...

... 0. . . 0

0 · · · 0 fRF,SNRF

3

77775

SNRF

S1

. . .

Analog RF precoder

RF chain

. . .

. . . . . .

. . .

RF chain

NRF

. . . . . .

. . . . . .

Nsub =NTX

NRF

Nsub =NTX

NRF

NRFNTX phase shifters and NTX adders are necessary. NTX phase shifters are necessary.

7

arg max

{FRF,FBB(1),··· ,FBB(K)}

KX

k=1

log det

✓I+

1

N0

H(k)FRF

FBB

(k)F⇤BB

(k)F⇤RF

H⇤(k)

◆

s.t.KX

k=1

||FRF

FBB

(k)||2F

Ptot

(Eq. 1)

(Eq. 2)

arg max

{FRF,ˆFBB(1),··· ,ˆFBB(K)}

KX

k=1

log det

✓I+

1

N0

He↵

(k)ˆFBB

(k)ˆF⇤BB

(k)H⇤e↵

(k)

◆

s.t.KX

k=1

||ˆFBB

(k)||2F

Ptot

He↵(k) = H(k)FRF(F⇤RFFRF)

� 12

F̂BB(k) = (F⇤RFFRF)

12 FBB(k)

Using

If FRF is given, the optimum solution of can be found by using a conventional SVD scheme with respect to the effective channel at each subcarrier.

F̂BB 1( ),!, F̂BB K( ){ }

equivalent

Goal: Maximize the sum of per-subcarrier mutual information Constraints

on FRF

1) NRF < NTX 2) Composed of phase shifters 3) Common for all subcarriers

WB hybrid precoding design - Problem formulation (1/2)

8

The above problem is a non-convex optimization problem. Instead, let us maximize the sum of squared singular values of effective channels ( = maximize the sum of effective SNR per subcarrier & stream)

argmax

FRF

KX

k=1

SX

s=1

�2s

�H (k)FRF(F

⇤RFFRF)

� 12�

(Eq. 3)

(Eq. 4)

Once the optimum FBB(k) for each k is determined given FRF, the optimization problem is only with respect to FRF.

argmax

FRF

KX

k=1

SX

s=1

log

1 +

�2s

�H (k)FRF(F

⇤RFFRF)

� 12

�ps,k

N0

!

equivalent

approximated (relaxation)

ps,k: water-filling power control

WB hybrid precoding design - Problem formulation (2/2)

Solution

9

Hybrid precoding solution – Fully-connected case

F?RF = argmax

FRF

KX

k=1

SX

s=1

�2s

�H[k]FRF(F

⇤RFFRF)

� 12�

Optimization problem

F?RF = [VR]1:NRF

A

where A is an arbitrary invertible matrix and

R =1

K

KX

k=1

H⇤[k]H[k] = VR⇤RV⇤R

Solution w/ phase shifter constraint

F̂?RF = arg min

X,|[X]m,n|=1kX� F?

RFk2F = ]F?

RF

where is a matrix with ]F?RF [F̂?

RF]m,n = ej]([F?RF]m,n)

Analog RF precoder

RF chain

. . . . . .

. . . . . .

. . . . . .

RF chain

NRF NTX

. . . . . .

NTX

NTX

Solution

10

Hybrid precoding solution – Partially-connected case

H[k] =⇥HS1 [k] HS2 [k] · · · HSNRF

[k]⇤

RSr =

1

K

KX

k=1

H⇤Sr[k]HSr [k], for r = 1, · · · , NRF

S1 = {1, · · · , Nsub}S2 = {Nsub + 1, · · · , 2Nsub}

...

SNRF = {(NRF � 1)Nsub + 1, · · · , NRFNsub}

F?RF = argmax

FRF

KX

k=1

SX

s=1

�2s

�H[k]FRF(F

⇤RFFRF)

� 12�

Optimization problem

FRF =

2

64fRF,S1 · · · 0

.... . .

...0 · · · fRF,SNRF

3

75

where αr is an arbitrary nonzero complex number, is the dominant eigenvector of , and Subarray partitioning vRSr ,1 RSr

f?RF,Sr= ↵rvRSr ,1, for r = 1, · · · , NRF

Solution w/ phase shifter constraint à the same as the fully-connected case

SNRF

S1

. . .

Analog RF precoder

RF chain

. . .

. . . . . .

. . .

RF chain

NRF

. . . . . .

. . . . . .

Nsub =NTX

NRF

Nsub =NTX

NRF

11

Pre-codin

g (BB) (freq-

domain)

Streams Antennas

Pre-codin

g (BB) (freq-

domain)

FFT & DAC

FFT & DAC

FFT & DAC

FFT & DAC

RF

. . .

FBB[2]

BB

FBB[1]

RF chains

FBB[K ]RFF

Comparison between fully-connected vs. partially-connected Fully-connected structure Partially-connected (subarray) structure

Pre-codin

g (BB) (freq-

domain)

Streams Antennas

Pre-codin

g (BB) (freq-

domain)

FFT & DAC

FFT & DAC

FFT & DAC

FFT & DAC

. . .

FBB[2]

BB

FBB[1]

RF chains

FBB[K ]AF

RF

RF

RF

RF

RFF

Analog RF precoder

RF chain

. . . . . .

. . . . . .

. . . . . .

RF chain

NRF NTX

. . . . . .

NTX

NTX

max

FRF

KX

k=1

SX

s=1

�2s

�H [k]FRF(F

⇤RFFRF)

� 12�= K

NRFX

r=1

�r (R) max

FRF

KX

k=1

SX

s=1

�2s

�H [k]FRF(F

⇤RFFRF)

� 12�= K

NRFX

r=1

�1 (RSr )

λi(A) : i-th largest singular value of A

Analog RF precoder

RF chain

. . .

. . . . . .

. . .

RF chain

NRF

. . . . . .

. . . . . .

Nsub =NTX

NRF

Nsub =NTX

NRF

12

Pre-codin

g (BB) (freq-

domain)

Streams Antennas

Pre-codin

g (BB) (freq-

domain)

FFT & DAC

FFT & DAC

FFT & DAC

FFT & DAC

RF

. . .

FBB[2]

BB

FBB[1]

RF chains

FBB[K ]RFF

Comparison between fully-connected vs. partially-connected Fully-connected structure Partially-connected (subarray) structure

Analog RF precoder

RF chain

. . . . . .

. . . . . .

. . . . . .

RF chain

NRF NTX

. . . . . .

NTX

NTX

max

FRF

KX

k=1

SX

s=1

�2s

�H [k]FRF(F

⇤RFFRF)

� 12�= K

NRFX

r=1

�r (R) max

FRF

KX

k=1

SX

s=1

�2s

�H [k]FRF(F

⇤RFFRF)

� 12�= K

NRFX

r=1

�1 (RSr )

Depends on and partitioning Depends on R =

1

K

KX

k=1

H⇤[k]H[k]

!(S1, ...,SNRF)R

λs(A) : s-th largest singular value of A

Pre-codin

g (BB) (freq-

domain)

Streams Antennas

Pre-codin

g (BB) (freq-

domain)

FFT & DAC

FFT & DAC

FFT & DAC

FFT & DAC

. . .

FBB[2]

BB

FBB[1]

RF chains

FBB[K ]AF

RF

RF

RF

RF

RFF

Analog RF precoder

RF chain

. . .

. . . . . .

. . .

RF chain

NRF

. . . . . .

. . . . . .

Nsub =NTX

NRF

Nsub =NTX

NRF

Which is the best sub-array structure (best partitioning) ?

13

Adjacent type Interlaced type AF

Analogprecoding

Analogprecoding

Analogprecoding

Analogprecoding Analog

recodingAnalogrecodingAnalog

recodingAnalogprecoding

Analogrecoding

Analogprecoding

Analogrecoding

Analogprecoding

Uniform Linear Array (ULA) case

The best subarray structure depends on the R matrix. è We propose a dynamic subarray structure that adapts to the R matrix.

S1 = {1, 2, 3}S2 = {4, 5, 6}S3 = {7, 8, 9}

S1 = {1, 4, 7}S2 = {2, 5, 8}S3 = {3, 6, 9}

1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9

S1 = {1, 2, 3, 4}S2 = {5, 6, 7, 8}S3 = {9, 10, 11, 12}S4 = {13, 14, 15, 16}

S1 = {1, 5, 9, 13}S2 = {2, 6, 10, 14}S3 = {3, 7, 11, 15}S4 = {4, 8, 12, 16}

S1 = {1, 2, 5, 6}S2 = {3, 4, 7, 8}S3 = {9, 10, 13, 14}S4 = {11, 12, 15, 16}

Horizontal type Vertical type Squared type

Uniform Planar Array (UPA) case

Dynamic subarray - Problem formulation

14

Partition a set of NTX antennas into NRF subsets - cardinality of each subset : variable (non-empty)

What to do

Goal (optimum criterion)

Optimal solution

* Optimum solution: Exhaustive search - # of total cases: Stirling number of the second kind

1

(NRF)!

NRFX

k=0

(�1)NRF�k

✓NRF

k

◆kNTX

Ex) {NTX=16, NRF=4} à 1.7×108

{S?r }

NRF

r=1 =arg max

S1,...,SNRF

NRFX

r=1

�1 (RSr )

s.t.

NRF[

r=1

Sr = {1, · · · , NTX} , Si \ Sj = ; for i 6= j, |Sr| > 0 8r

Analog RF precoder

RF chain

. . . . . .

. . .

RF chain

NRF

. . . . . .

. . . . . .

NTX

Dynamic (mapper)

1)  Needs to check too many cases 2)  Needs to calculate the singular value at every case

Analog RF precoder

RF chain

. . . . . .

. . .

RF chain

NRF

. . . . . .

NTX

NRF

NTX

NRF

Fixed

15

Approximation of the largest singular value

Definition of the approximate largest singular value

�̂1 (RS) ,1

|S|

|S|X

i=1

|S|X

j=1

|[RS ]i,j | =1

|S|X

i2S

X

j2S|[R]i,j |

�1,LB (RS) �1 (RS) �1,UB (RS)

�1,LB (RS) = m+s

(|S|� 1)12

�1,UB (RS) = m+ s(|S|� 1)12

m =Tr(RS)

|S| , s =

✓Tr(R2

S)

|S| �m2

◆ 12

[4] H. Wolkowicz and G. P. Styan, “Bounds for eigenvalues using traces,” Linear Algebra and Its Applications, pp. 471-506, 1980

�1,LB (RS) �̂1 (RS) �1,UB (RS)

* Lower & upper bound on the largest singular value (exact value)

Property of the approximate largest singular value

16

Dynamic subarray partitioning algorithm (proposed)

f (S, n, r) ,(

0, if |S| = 0 or {n = NRF and r = 0}1|S|

Pi2S

Pj2S |[R]i,j |, otherwise.

Complexity: O(NTX2 logNTX)

Complexity: O(NTX2NRF)

Total complexity: O( NTX

2 max(NRF,logNTX) )

Complexity: O(NTX2NRF)

è Polynomial time !

Dynamic subarray

partitioning algorithm

R, NRF, NTX inpu

t ou

tput

S1, ...,SNRF

Channel model for simulations

17

Azimuth domain Elevation domain

range of cluster angles [-ΔAZ, ΔAZ]

bore- sight

range of cluster angles [-ΔEL, ΔEL]

bore- sight

… …

… …

Cluster 1

Cluster Ncluster

…

Nsubray subrays (w/ angle spread 2o)

a (�p, ✓p) =h1 ej

2⇡d� sin(�p) · · · ej

2⇡d� (NANT�1) sin(�p)

iT

a (�p, ✓p) =h1 · · · ej

2⇡d� (m sin(�p) sin(✓p)+n cos(✓p)) · · · ej

2⇡d� ((NW�1) sin(�p) sin(✓p)+(NH�1) cos(✓p))

iTfor ULA (uniform linear array) for UPA (uniform planar array)

H[k] =NclusterX

c=1

NsubrayX

r=1

↵c,r!⌧c,r [k]aR(�R,c,r, ✓R,c,r)a⇤T(�T,c,r, ✓T,c,r) !⌧p [k] =

D�1X

d=0

p(dTs � ⌧p)e� j2⇡kd

K

* Distribution of AoAs and AoDs : Uniform in a confined range TX RX

18

Simulation results

-10 -5 0 5 10 15 20SNR (dB)

0

5

10

15

20

25

30

35

40

Spec

tral e

ffici

ency

(bps

/Hz)

Digital basebandHybrid, fully-connectedHybrid, subarray, dynamic (proposed)Hybrid, subarray, fixed (vertical)Hybrid, subarray, fixed (horizontal)Hybrid, subarray, fixed (squared)Hybrid, subarray, fixed (interlaced)

4.5 5 5.518

18.5

19

- TX: 64 antennas (UPA), 8 RF chains - RX: 4 antennas (UPA), 4 RF chains - CH: 8 clusters 10 subrays (ΔAZ=180°, ΔEL=90°)

16 64 144 256Number of transmit antennas

0

5

10

15

20

25

30

35

Spect

ral e

ffic

iency

(bps/

Hz)

Digital basebandHybrid, fully-connectedHybrid, subarray, dynamic (proposed)Hybrid, subarray, fixed (vertical)Hybrid, subarray, fixed (horizontal)Hybrid, subarray, fixed (squared)Hybrid, subarray, fixed (interlaced)

- TX: NTX antennas (UPA), 4 RF chains - RX: 4 antennas (UPA), 4 RF chains - CH: 8 clusters 10 subrays (ΔAZ=180°, ΔEL=90°), SNR 10 dB

- solid: w/o phase shifter constraint ( ) - dashed: w/ phase shifter constraint ( )

F?RF

]F?RF

Conclusions

19

We derived closed-form solutions for wideband hybrid precoding.

Fully connected structure Partially-connected structure (Subarray structure)

We proposed a dynamic subarray structure based on spatial channel covariance.

20

Thank you !