Coverage and Capacity Analysis of mmWave Cellular...

Coverage and Capacity Analysis of mmWave Cellular Systems

Robert W. Heath Jr. The University of Texas at Austin

Joint work with Tianyang Bai

www.profheath.orgSaturday, June 15, 13

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 possibleSilicon-based technology enables low-cost highly-packed mmWave RFIC**

Hybrid precoding solves RF limitations

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

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

Saturday, June 15, 13


Smaller wavelength means smaller captured energy at antenna3GHz->30GHz gives 20dB extra path loss due to aperture

Larger bandwidth means higher noise power and lower SNR50MHz -> 500MHz bandwidth gives 10dB extra noise power

The Need for Gain

3

Solution: Exploit array gain from large antenna arrays

microwave noise bandwidth

mmWave noise bandwidth

microwave aperture

mmWave aperture

TXRX

�2

4⇡



Antenna Arrays are Important

4

Narrow beams are a new feature of mmWaveReduces 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

O. El Ayach, S. Abu-Surra, S. Rajagopal, Z. Pi, and R. W. Heath, Jr., `` Spatially Sparse Precoding in Millimeter Wave MIMO Systems,'' submitted to IEEE Trans. on Wireless, May 2013. Available on ArXiv.



Different Propagation Behavior

Buildings block Line-of-sight (LOS) pathsmmWave signal suffer from high penetration losses

Reflections can establish non-LOS linksBest non-LOS links still tens of dB weaker than LOS signals

Different characteristics for LOS & non-LOSPath loss exponent is around 2 for LOS, and 4 for non-LOS

5

Line-of-sight linkReflection

Blockages

Signal blocked by buildings

Need to distinguish between LOS and non-LOS paths

T. S. Rappaport, et al. , "Millimeter Wave Mobile Communications for 5G Cellular: It Will Work!." IEEE Access, vol.1, no.1, pp.335-349, 2013



Challenges of mmWave Analysis

6

LOS & non-LOS linksDirectional Beamforming (BF)

How to including beamforming + blockages in mmWave cellular analysis?

Need to incorporate directional beamformingRX and TX communicate via main lobes to achieve array gain

Steering directions at interfering BSs are random

Need to distinguish LOS and non-LOS pathsDramatically different characteristics in LOS & non-LOS channels

Better characterize blockages


Stochastic Geometry for mmWave Cellular

System Analysis



Stochastic Geometry for Cellular

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

Closed form solutions for coverage probability available

Provides a system-wide performance characterization

8

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 (usually) as a

Poisson point process (PPP)performance analyzed for a typical user

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

Baccelli



Poisson Point Processes

Poisson point process (PPP): the simplest point process# of points is a Poisson variable with mean λS

Given N points in certain area, locations independent

Useful results like Campbell’s Theorem & Displacement Theorem apply

Assigning each point an i.i.d. random variable forms a marked PPP

9

14

Fig. 7. .

Antenna steering orientations as marks of the

BS PPP



Blockages in mmWave

Use random shape theory to model buildingsModel random buildings as rectangle 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

10

Boolean scheme of rectanglesK: # 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.



Directional Transmission at the BS

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

Use “sector” pattern in analysis for simplicityEquivalent to uniform linear array of antennas with spacing

11

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✓

Nt �/2

Main lobe beamwidth:

Main lobe array gain:M = Nt

Front-back ratio:

Back lobe gain:

3

✓ = 2arcsin

✓2.782

⇡Nt

◆

FBR = sin

✓3⇡

2Nt

◆

m = Nt ⇥ FBR

3

✓ = 2arcsin

✓2.782

⇡Nt

◆

FBR = sin

✓3⇡

2Nt

◆

m = Nt ⇥ FBR

3

✓ = 2arcsin

✓2.782

⇡Nt

◆

FBR = sin

✓3⇡

2Nt

◆

m = Nt ⇥ FBR

# antennas



15

Fig. 8. .

Proposed mmWave Model

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

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

Different path loss exponents for LOS and non-LOS paths

12

PPP Interfering BSs

Serving BS

Buildings

Typical User

Reflections



System Parameters Different path loss model for LOS and nonLOS links

Line-of-sight with probability

LOS path Loss in dB:

Non-LOS path loss in dB:

28Ghz system: let C=50 dB, K=10 dB

General small scale fadingNo fading case: small scaling fading is minor in mmWave [RapSun]

Link budgetTx antenna input power: 30dBm

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

Noise figure: 5dB

13

e��R

PL2 = C +K + 40 logR(m)

PL1 = C + 20 logR(m)

h


Results on Coverage



Coverage Analysis

15

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 =

NrNtH0`(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<

:Nt w. p. ✓t

2⇡

NtFBRt w. p. 1� ✓t2⇡

,

Bk =

8<

:Nr w. p. ✓r

2⇡

NrFBRr 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 =

NrNtH0`(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<

:Nt w. p. ✓t

2⇡


,

Bk =

8<

:Nr w. p. ✓r

2⇡


,

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 =

NrNtH0`(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<

:Nt w. p. ✓t

2⇡


,

Bk =

8<

:Nr w. p. ✓r

2⇡


,

H0`(r0) = min

k>0{Hk`(rk)} ,



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

where

,

,

.

16

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

Transform interference field into 1D space by Displacement Thm

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



10

!10 !8 !6 !4 !2 0 2 4 6 8 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR Threshold

Co

ve

rag

e P

rob

ab

ilit

y

Nt=64, !=1.6o

Nt=32, !=3.2O

Nt=16, !=6.5o

Microwave: SU MIMO 4X4

Fig. 6.

Coverage Gain from Large Arrays

17

Mobile user 16 antennasBS density Rc=200 mBuildings cover 5% of the landAverage building size 15 m by 15 m

Large arrays provide better coverage probabilitythe more antennas, the smaller beamwidth, the larges array gain

Smaller beamwidth provides better coverage

mmWave coverage probability comparable to microwave

Gain from directional antenna array



10

!10 !8 !6 !4 !2 0 2 4 6 8 10

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR Threshold

Co

ve

rag

e P

rob

ab

ilit

y

rc=50 m

rc=100 m

rc=200 m

rc=300 m

Microwave: 4X4 SU MIMO

Fig. 6.

Coverage Gain from Higher Density

18

Higher density can also increase coverage probabilityCoverage probability no longer invariant with BS density

Become interference-limited when coverage probability is good

32 antennas at BSBlockages covers 10% land

Noise-limited region due to high path loss

Interference-limited region

Gain over microwave



LOS & non-LOS Path Loss

Coverage probability differs in LOS and non-LOS regionNeed to incorporate blockage model & differentiate LOS and nonLOS

Non-LOS coverage probability generally provides a lower bound

Buildings may improve coverage by blocking more interference19

7

!10 !8 !6 !4 !2 0 2 4 6 8 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR Threshold

Co

ve

rag

e P

rob

ab

ilit

y

Proposed model

LOS path loss only

non!LOS path loss only

Fig. 2. .

32 antennas at BSsBlockages covers 10% landRc=100 m

Gain from blocking more interference

pure LOS (no buildings)

pure NLOS

proposed model



LOS & Reflection CoverageCorollary 1 [Coverage probability by LOS BSs]

The coverage probability provided by LOS BSs is

,

where

20

6

P(SINR > T ) =

Z 1

�1

Z 1

0

U1(x, t)U2(t)fL⇤(x)

e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i

⇤1(x) = 2⇡�

Z px

0

te��tdt

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt

fL⇤(x) = � d

dxe

�⇤1(x)

6

P(SINR > T ) =

Z 1

�1

Z 1

0


e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i,

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i,

⇤1(x) = 2⇡�

Z px

0

te��tdt,

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt,

fL⇤(x) = � d

dxe

�⇤1(x).

6

P(SINR > T ) =

Z 1

�1

Z 1

0


e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i,

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i,

⇤1(x) = 2⇡�

Z px

0

te��tdt,

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt,

fL⇤(x) = � d

dxe

�⇤1(x).

6

P(SINR > T ) =

Z 1

�1

Z 1

0


e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i,

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i,

⇤1(x) = 2⇡�

Z px

0

te��tdt,

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt,

fL⇤(x) = � d

dxe

�⇤1(x).

6

P(SINR > T ) =

Z 1

�1

Z 1

0


e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i,

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i,

⇤1(x) = 2⇡�

Z px

0

te��tdt,

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt,

fL⇤(x) = � d

dxe

�⇤1(x).

6

P(SINR > T ) =

Z 1

�1

Z 1

0


e

j2⇡t/T � 1

j2⇡tdxdt

U2(s) = exp

Z 1

0

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤2(du)

i,

U1(x, s) = exp

� sx

MrMt⇢+

Z 1

x

⇣ptpre

� sx

u


uMt+ pt(1� pr)e

� sxmruMr


uMrMt � 1

⌘⇤1(du)

i,

⇤1(x) = 2⇡�

Z px

0

te��tdt,

⇤2(x) = 2⇡�

Z(

x

K

)

1/4

0

t�1� e

��t�dt,

fL⇤(x) = � d

dxe

�⇤1(x).

Coverage probability by reflections derived in a similar way



10

!10 !8 !6 !4 !2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR Threshold

Co

ve

rag

e P

rob

ab

ilit

y

Overall coverage probability

Covergage probability by LOS BSs

Coverage probability by reflections

Fig. 6.

Reflections Improve Coverage

Reflections can establish links in the shadowed areasWith dense blockages, most users are served by reflected links

Non-LOS links improve the coverage probability of mmWave

21

128 antennas at BSsBlockages covers 30% land(Heavy shadowing case)Rc=200 m

With dense blockages, a large fraction of users is non-LOS


Results on Rate



Data Rate ComparisonGiven coverage probability, the achievable rate is

Microwave network 4X4 MU MIMO with bandwidth 50MHz: Spectrum efficiency is 4.95 bps/ Hz

Data rate is 248 Mbps (invariant with the cell size Rc)

mmWave network with bandwidth 500MHz:

23

3

A. Distribution of L0

Lemma 8: The probability density function (pdf) of L0 is fL0(x) = ⇤

0(x)(e)⇤x, where ⇤

0(x) =

d⇤(x)dx , ⇤(x) = ⇤1(x) + ⇤2(x), and

⇤1(x) = 2⇡�EH1

"Z pxH1

0

te��tdt

#, (3)

⇤2(x) = 2⇡�EH2

"Z pxH2

0

t�1� e

��t�dt

#. (4)

Remark 9: Note that if there is no fading in both LOS and non-LOS links, then

⇤1(x) = 2⇡�

Z px

0

te��tdt, (5)

⇤2(x) = 2⇡�

Z px

0

t�1� e

��t�dt. (6)

B. Laplace Transform of F

First, we compute the conditional Laplace transform of F, given that L0 = x, in the following

lemma.

Lemma 10: Given that L0 = x, the conditional Laplace transform of F is

E⇥e

�sF |L0 = x⇤= e

�sx

M⇢

exp

✓Z 1

x

⇣e

�Msx

u � 1

⌘ ✓

2⇡⇤(u)du

◆exp

✓Z 1

x

�e

�msx

u � 1

�2⇡ � ✓

2⇡⇤(u)du

◆

(7)The unconditional Laplace transform of F is

E⇥e

�sF⇤=

Z 1

0

E⇥e

�sF |L0 = x⇤fL0(x)dx (8)

C. Computation of Coverage probability

Using Parseval’s Theorem, we can obtain the coverage probability as

Pc(T ) = P(SINR > T ) (9)

= P(F < 1/T ) (10)

=

Z 1

0

E⇥e

�sF⇤e

j2⇡/T � 1

j2⇡dt (11)

D. Rate

As we did previously, the achievable rate ⌘ can be computed as

⌘ =

Z 1

0

Pc(T )

1 + TdT (12)

100m 200m

32 3.4Gbps 3.25Gbps

64 3.8Gbps 3.45Gbps

NtRc

mmWave achieves high gain in average rate

16 antenna at MSBlockages covers 10% land

# of antenna at BS

Average cell radius



7

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Cell Throughput in Mbps

Rate

Co

vera

ge P

rob

ab

ilit

y

CoMP: Nt=4, 2 Users, 3 BSs/ Cluster

Massive MIMO: Nt=!

mmWave: Nt=64, R

c=100m

SU MIMO: 4X4

Fig. 3.

Cell Throughput Comparison

24

Gain from larger bandwidth

Gain from serving multiple users

mmWave can support much higher data rate

* for more information on this setup refer to: Robert W. Heath Jr., “ Role of MIMO Beyond LTE: Massive? Coordinated? mmWave?”, Workshop on Beyond 3GPP LTE-A ICC 2013.


Conclusions



Going Forward with mmWave

mmWave coverage probability and rateNeed 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 aboundAnalog 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

More advanced stochastic geometry models including multi-tier

26


Questions?Robert W. Heath Jr.

The University of Texas at Austin

www.profheath.org

See forthcoming book:

Millimeter Wave Wireless

Communication

Theodore Rappaport, Robert W. Heath Jr.,

Robert C. Daniels, and James N. Murdock




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