coverage in dense millimeter wave cellular networks · t. bai, v. desai, and r. w. heath jr....

Coverage in Dense Millimeter Wave Cellular Networks

Tianyang Bai and Robert W. Heath Jr. The University of Texas at Austin

www.profheath.org

http://www.profheath.org

(c) Robert W. Heath Jr. 2013

Why mmWave for Cellular?

Microwave

3 GHz 300 GHz

�2

28 GHz 38-49 GHz 70-90 GHz300 MHz

Huge amount of spectrum available in mmWave bands* Cellular systems live with limited microwave spectrum ~ 600MHz

29GHz possibly available in 23GHz, LMDS, 38, 40, 46, 47, 49, and E-band

Technology advances make mmWave possible Silicon-based technology enables low-cost highly-packed mmWave RFIC**

Commercial products already available (or soon) for PAN and LAN

Already deployed for backhaul in commercial products

1G-4G cellular 5G cellular

m i l l i m e t e r w a v e

* Z. Pi,, and F. Khan. "An introduction to millimeter-wave mobile broadband systems." IEEE Communications Magazine, vol. 49, no. 6, pp.101-107, Jun. 2011. ** T.S. Rappaport, J. N. Murdock, and F. Gutierrez. "State of the art in 60-GHz integrated circuits and systems for wireless communications." Proceedings of the IEEE, vol. 99, no. 8, pp:1390-1436, 2011


Antenna Arrays are Important

�3

Narrow beams are a new feature of mmWave Reduces fading, multi-path, and interference

Implemented in analog due to hardware constraints

Baseband Processing

~100 antennas

Baseband Processing

antennas are small (mm)

highly directional MIMO transmission

used at TX and RX

Arrays will change system design principles


Sensitivity to Blockages

Signal and interference may be either LOS or NLOS

Users may connect to a further unblocked base station

Strong interferers may blocked

�4

Need to include propagation models in the analysis

T. Bai, V. Desai, and R. W. Heath Jr. “Millimeter Wave Cellular Channel Models for System Evaluation,” to appear in the Proc. of ICIN, Hawaii, Feb. 2014.

LOS channel Path loss exponent 2

NLOS channel Path loss exponent 4

Additional loss


Contributions

�5

LOS & non-LOS linksDirectional Beamforming (BF)

Incorporate beamforming & LOS/NLOS into coverage analysis RX and TX communicate via steered directional beams

Steering directions at interfering BSs are random

Paths may be LOS or NLOS depending on blockage density

!

Approach is to leverage stochastic geometric analysis of cellular Extends work by [AndrewsGantiBaccelli2011] to include mmWave features

Extends our work [BaiHeath2013] with simplified expressions for dense networks

System Model


Stochastic Geometry for Cellular

Stochastic geometry is a tool for analyzing microwave cellular Reasonable fit with real deployments

Closed form solutions for coverage probability available

Provides a system-wide performance characterization

�7

J. G. Andrews, F. Baccelli, and R. K. Ganti, "A Tractable Approach to Coverage and Rate in Cellular Networks", IEEE Transactions on Communications, November 2011.!T. X. Brown, "Cellular performance bounds via shotgun cellular systems," IEEE JSAC, vol.18, no.11, pp.2443,2455, Nov. 2000.

base station locations distributed as a Poisson point

process (PPP)performance analyzed for a typical user

Need to incorporate LOS/non-LOS links and directional antennas

Baccelli


Accounting for Beamforming

Each base station is marked with a directional antenna Antenna directions of interferers are uniformly distributed

Use “sector” pattern in analysis for simplicity Antenna pattern fully characterized by , M and m

�8

1

Summary of Cellular Millimeter Wave Channels

I. MEASUREMENT RESULTS

−3 −2 −1 0 1 2 30

1

2

3

4

5

6

7

8

9

10

Angel in Rad

Ante

nna

Gai

n

Exact antenna pattern"Sector" approximation

Fig. 1.

Sector antenna approximation

Half-Power BW✓ Main lobe beamwidth

Main lobe array gain M

Back lobe gain m

✓

✓


Incorporating Blockages

Use random shape theory to model buildings Model random buildings as a rectangular Boolean scheme

Buildings distributed as PPP with independent sizes & orientations

Compute the LOS probability based on the building model # of blockages on a link is a Poisson random variable

The LOS probability that no blockage on a link of length R is

�9

Boolean scheme of rectangles K: # of blockages on a link

Randomly located buildings

e��R

LOS: K=0non-LOS

K>0

Tianyang Bai and R. W. Heath, Jr., ``Using Random Shape Theory to Model Blockage in Random Cellular Networks,'' Proc. of the International Conf. on Signal Processing and Communications, Bangalore, India, July 22-25, 2012.


15

Fig. 8. .

Proposed mmWave Model

Use stochastic geometry to model BSs as marked PPP Model the steering directions as independent marks of the BSs

Use random shape theory to model buildings Model the building as rectangle Boolean schemes

Different path loss exponents for LOS and NLOS paths

�10

PPP Interfering BSs

Serving BS

Buildings

Typical User

Reflections


Different path loss model for LOS and nonLOS links Line-of-sight with probability : average LOS range is

!

!

LOS path Loss in dB:

Non-LOS path loss in dB:

28GHz system: let C=70 dB, K=10 dB

General small scale fading No fading case: small scaling fading is minor in mmWave [RapSun]

Link budget Tx antenna input power: 30dBm

Signal bandwidth: 500 MHz (Noise: -87 dBm)

Noise figure: 5dB

System Parameters

�11

e��R

PL2 = C +K + 40 logR(m)

PL1 = C + 20 logR(m)

h

Tempting file for making slidesTianyang Bai

I. INTRODUCTION

⇢ =Average size of LOS region

Average cell size

1

�

2⇡

�2

P(SINR > T ) ⇡ ⇢e�⇢NX

n=1

(�1)n+1

Z 1

0

4Y

k=1

e⇢ak(e�nbkt�te�nbk)✓1� e�n⌘bkt

1� e�n⌘bk

◆n⇢akbkt

dt,

b̂k = (N !)1N bk

lim⇢!1

SINRp.= 0


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt

� =2⌘([L] + [W ])

⇡[W ][L]

1/�The fraction of land covered by buildings

Average building length and width

Coverage Results


SINR Expressions

�13

where

7

P(SINR > T ) =

Z 1

�1

Z 1

0

E[e�tP

k

Lk |L⇤

= x ]fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt,

E[e�tP

k

Lk |L⇤

= x ] = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u

+ (1� pt)pre� sxmt

uMt+ pt(1� pr)e

� sxmruMr

+(1� pt)(1� pr)e� sxmtmr

uMrMt � 1

⌘⇤(du)

i,

Li = r↵i

i ,

L⇤= max

i{Li} ,

SINR =

MrMtH0`(r0)

N0/Pt +P

k>0 AkBkHk`(rk),

`(x) =

8<

:Cx�2 w. p. e��x

C Kx�4 w. p. 1� e

��x,

Ak =

8<

:Mt w. p. ✓t

2⇡

mt w. p. 1� ✓t2⇡

,

Bk =

8<

:Mr w. p. ✓r

2⇡

mr w. p. 1� ✓r2⇡

,

H0`(r0) = min

k>0{Hk`(rk)} ,

7

P(SINR > T ) =

Z 1

�1

Z 1

0

E[e�tP

k

Lk |L⇤

= x ]fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt,

E[e�tP

k

Lk |L⇤

= x ] = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i,

Li = r↵i

i ,

L⇤= max

i{Li} ,

SINR =

MrMtH0`(r0)

N0/Pt +P

k>0 AkBkHk`(rk),

`(x) =

8<

:Cx�2 w. p. e��x

C Kx�4 w. p. 1� e

��x.

Ak =

8<

:Mt w. p. ✓t

2⇡

mt w. p. 1� ✓t2⇡

,

Bk =

8<

:Mr w. p. ✓r

2⇡

mr w. p. 1� ✓r2⇡

,

H0`(r0) = min

k>0{Hk`(rk)} ,

Connecting to the strongest signal before BF

Array gain of the TX antenna

Array gain of the RX antenna

Path Loss of LOS or non-LOS

Serving BS and User connect via main lobe

Use stochastic geometry to compute SINR distribution

General small-scale fading

7

P(SINR > T ) =

Z 1

�1

Z 1

0

E[e�tP

k

Lk |L⇤

= x ]fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt,

E[e�tP

k

Lk |L⇤

= x ] = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i,

Li = r↵i

i ,

L⇤= max

i{Li} ,

SINR =

MrMtH0`(r0)

N0/Pt +P

k>0 AkBkHk`(rk),

`(x) =

8<

:Cx�2 w. p. e��x

C Kx�4 w. p. 1� e

��x.

Ak =

8<

:Mt w. p. ✓t

2⇡

mt w. p. 1� ✓t2⇡

,

Bk =

8<

:Mr w. p. ✓r

2⇡

mr w. p. 1� ✓r2⇡

,

H0`(r0) = min

k>0{Hk`(rk)} ,

7

P(SINR > T ) =

Z 1

�1

Z 1

0

E[e�tP

k

Lk |L⇤

= x ]fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt,

E[e�tP

k

Lk |L⇤

= x ] = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i,

Li = r↵i

i ,

L⇤= max

i{Li} ,

SINR =

MrMtH0`(r0)

N0/Pt +P

k>0 AkBkHk`(rk),

`(x) =

8<

:Cx�2 w. p. e��x

C Kx�4 w. p. 1� e

��x.

Ak =

8<

:Mt w. p. ✓t

2⇡

mt w. p. 1� ✓t2⇡

,

Bk =

8<

:Mr w. p. ✓r

2⇡

mr w. p. 1� ✓r2⇡

,

H0`(r0) = min

k>0{Hk`(rk)} ,

7

P(SINR > T ) =

Z 1

�1

Z 1

0

E[e�tP

k

Lk |L⇤

= x ]fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt,

E[e�tP

k

Lk |L⇤

= x ] = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i,

Li = r↵i

i ,

L⇤= max

i{Li} ,

SINR =

MrMtH0`(r0)

N0/Pt +P

k>0 AkBkHk`(rk),

`(x) =

8<

:Cx�2 w. p. e��x

C Kx�4 w. p. 1� e

��x.

Ak =

8<

:Mt w. p. ✓t

2⇡

mt w. p. 1� ✓t2⇡

,

Bk =

8<

:Mr w. p. ✓r

2⇡

mr w. p. 1� ✓r2⇡

,

H0`(r0) = min

k>0{Hk`(rk)} ,


Coverage Probability of mmWaveTheorem1 [mmWave Coverage probability] The coverage probability can be computed as

!!!where

,

! ,

.

�14

P[SINR > T ]

4

P(SINR > T ) =

Z 1

�1

Z 1

0

U(x, t)fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt

U(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i

⇤(x) = 2⇡�

0

@Z x

14

0

t�1� e

��t�dt+

Z px

0

te��tdt

1

A

⇤L(x) = 2⇡�

Z px

0

te��tdt

⇤NL(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt

fL⇤(x) = � d

dxe

�⇤(x)

Li = r↵i

i

L⇤= max

i{Li}

F =

1

SINR

E⇥e

�sF |L⇤= x

⇤= U(x, s)

E⇥e

�sF⇤=

Z 1

0

U(x, s)fL⇤(x)dx

4

P(SINR > T ) =

Z 1

�1

Z 1

0

U(x, t)fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt

U(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i

⇤(x) = 2⇡�

0

@Z x

14

0

t�1� e

��t�dt+

Z px

0

te��tdt

1

A

⇤L(x) = 2⇡�

Z px

0

te��tdt

⇤NL(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt

fL⇤(x) = � d

dxe

�⇤(x)

Li = r↵i

i

L⇤= max

i{Li}

F =

1

SINR

E⇥e

�sF |L⇤= x

⇤= U(x, s)

E⇥e

�sF⇤=

Z 1

0

U(x, s)fL⇤(x)dx

4

P(SINR > T ) =

Z 1

�1

Z 1

0

U(x, t)fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt

U(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i

⇤(x) = 2⇡�

0

@Z x

14

0

t�1� e

��t�dt+

Z px

0

te��tdt

1

A

⇤L(x) = 2⇡�

Z px

0

te��tdt

⇤NL(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt

fL⇤(x) = � d

dxe

�⇤(x)

Li = r↵i

i

L⇤= max

i{Li}

F =

1

SINR

E⇥e

�sF |L⇤= x

⇤= U(x, s)

E⇥e

�sF⇤=

Z 1

0

U(x, s)fL⇤(x)dx

4

P(SINR > T ) =

Z 1

�1

Z 1

0

U(x, t)fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt

U(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤(du)

i

⇤(x) = 2⇡�Eh

"Z(

xh

K

)

0.25

0

t�1� e

��t�dt+

Z pxh

0

te��tdt

#

⇤L(x) = 2⇡�

Z px

0

te��tdt

⇤NL(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt

fL⇤(x) = � d

dxe

�⇤(x)

Li = r↵i

i

L⇤= max

i{Li}

F =

1

SINR

E⇥e

�sF |L⇤= x

⇤= U(x, s)

E⇥e

�sF⇤=

Z 1

0

U(x, s)fL⇤(x)dx

T. Bai and R. W. Heath Jr., ``Coverage analysis of millimeter cellular networks with blockage effects (invited)", to appear in Proc. of IEEE GlobalSIP, Austin, TX, Dec. 2013.


Dense Network Analysis (1/3)

Good coverage requires dense BS deployments LOS BSs exist with high probability in dense networks

Noise and NLOS interference become much weaker than LOS interf.

Theorem 1 sometimes inefficient to compute The underlying reason is that LOS region is very irregular

Need to simplify expressions in Theorem 1

�15

LOS Region of the typical user

LOS BS

Typical User

Approximate LOS region & neglect NLOS contributions

Equivalent LOS ball


Derive approximate bounds on SINR distribution in dense network Approximate LOS region as a ball with equal average size

Ignore noise, approximate LOS fading with high order gamma distribution

Use Alzer’s inequality to further simplify to single-finite integral

Provides approximate upper and lower bounds on SINR

Can be extended to other LOS exponents

Dense Network Analysis (2/3)

�16

Theorem 2 [Coverage probability with dense BSs] In dense networks with a LOS path loss exponent of 2

where and are constants determined by RX and TX antenna patterns.

ak bk

How accurate is the approximations?


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

✓N

`

◆Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,

b̂k = (N !)1N bk

lim⇢!1

SINRp.= 0


n=1

(�1)n+1

✓N

`

◆Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt

� =2⌘([L] + [W ])

⇡[W ][L]

R =1

ln(2)

Z C

0

Pc(T )

1 + TdT


−10 −5 0 5 10 15 20 25 30 35 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR in dB

Cov

erag

e Pr

obab

ility

SimulationsUpper bound: N=5Lower bound: N=5Lower bound: N=3Lower bound: N=1

Fig. 1.

Dense Network Analysis(3/3)

Larger N (more terms) increases accuracy

N=5 terms generally provides acceptable accuracy

�17

Tx directivity gain: 20 dB Tx beamwidth: 30 degree Rx directivity gain: 10 dB Rx beamwidth: 90 degree Rc=100 m =141 m1/�

Becomes better as N increases

What is the design insight from coverage analysis?


Insights for Dense Networks

Given antenna patterns, SINR only depends on

!

!The larger , the denser the BSs (and closer)

Increasing BS density need not improve SINR SINR goes to zero in a infinitely dense network

!Optimal BS density is finite

�18


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,⇢


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,

b̂k = (N !)1N bk

lim⇢!1

SINRp.= 0

What is the optimal BS density in dense networks?

⇢


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

✓N

`

◆Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,

b̂k = (N !)1N bk

lim⇢!1

SINRp.= 0


n=1

(�1)n+1

✓N

`

◆Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt

� =2⌘([L] + [W ])

⇡[W ][L]

R =1

ln(2)

Z C

0

Pc(T )

1 + TdT


Finding Optimal BS Density

Exhaustive search optimal BS density using Theorem 2 Maximize the coverage probability given a target SINR

Much efficient than simulations with minor errors

Optimal cell radius is approximately 2/3 of the avg. LOS range.�19

20 40 60 80 100 120 140 160 180 200

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Average cell radius

Cov

erag

e Pr

obab

ility

with

T=1

0 dB

Exhuastive search using SimulationsExhaustive search using Theorem 2

Fig. 2.

Tx directivity gain: 20 dB Tx beamwidth: 30 degree Rx directivity gain: 10 dB Rx beamwidth: 10 dB Avg. LOS range: =141 m Target SINR: T=10 dB

1/�

Increasing BS density need not improve SINR

Optimal BS density is finite


0 100 200 300 400 500 600 700 8000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Avg. Cell Radius Rc

Cov

erag

e Pr

obab

ility

SimulationsAnalytical Results

Fig. 7.

Finding Optimal BS Density

Exhaustive search optimal BS density using Theorem 2 Maximize the coverage probability given a target SINR

Much efficient than simulations with minor errors

Optimal cell radius is approximately 2/3 of the avg. LOS range.�20

Tx directivity gain: 20 dB Tx beamwidth: 30 degree Rx directivity gain: 10 dB Rx beamwidth: 10 dB Avg. LOS range: =141 m Target SINR: T=20 dB

1/�

Increasing BS density need not improve SINR

Optimal BS density is finite

Gap by ignoring NLOS and noise

(c) Robert W. Heath Jr. 2013(c) Robert W. Heath Jr. 2013

Given coverage probability, the achievable rate is

!

!

Microwave network 4X4 SU MIMO with bandwidth 50MHz: Spectrum efficiency is 4.56 bps/ Hz

Data rate is 228 Mbps (invariant with the cell size Rc)

mmWave network with bandwidth 500MHz:

Achievable Rate Analysis

�21

100m 200m

10 dB 2.76 Gbps 1.61 Gbps

20 dB 2.91Gbps 1.88 Gbps

Rc

mmWave achieves high gain in average rate

Tx beamforming Gain

Average cell radiusMTx beamwidth: 30 degree Rx directivity gain: 10 dB Rx beamwidth: 90 degree Avg. LOS range: =141 m1/�


I. INTRODUCTION


Average cell size

1

�

2⇡

�2


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt,

b̂k = (N !)1N bk

lim⇢!1

SINRp.= 0


n=1

(�1)n+1

Z 1

0

4Y

k=1


1� e�n⌘bk

◆n⇢akbkt

dt

� =2⌘([L] + [W ])

⇡[W ][L]

R =1

ln(2)

Z C

0

Pc(T )

1 + TdT

clipped by 6 bps/Hz (64QAM)

Conclusions


Going Forward with mmWave

mmWave coverage probability and rate Need to include both LOS and Non-LOS conditions

Interference is reduced by directional antennas and blockages

Good rates and coverage can be achieved

!

Theoretical challenges abound Analog beamforming algorithms & hybrid beamforming

Channel estimation, exploiting sparsity, incorporating robustness

Multi-user beamforming algorithms and analysis

Microwave-overlaid mmWave system a.k.a. phantom cells

Going away from cells to a more ad hoc configuration

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coverage in dense millimeter wave cellular networks · t. bai, v. desai, and r. w. heath jr....

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