enhanced cellular coverage and throughput using rateless codes · 2017. 9. 13. · amogh rajanna...

Amogh Rajanna andMartin Haenggi

Introduction

System Model

Analytical Results

Performance Comparison

Numerical Results

Continuous Transmission

Non CooperativeTransmission

Coordinated Transmission

Enhanced Cellular Coverage andThroughput using Rateless Codes

Amogh Rajanna and Martin Haenggi

Wireless Institute, University of Notre Dame, USA.

2017


Introduction

System Model

Analytical Results


Numerical Results




Papers

I A. Rajanna and M. Haenggi, Enhanced Cellular Coverageand Throughput using Rateless Codes, IEEE Transactions onCommunications, vol. 65, no. 5, pp. 1899 -1912, May 2017.

I A. Rajanna and M. Haenggi, Downlink Coordinated JointTransmission for Mutual Information Accumulation, IEEEWireless Communications Letters, vol. 6, no. 2, pp.198-201,Apr 2017.


Introduction

System Model

Analytical Results


Numerical Results




Motivation

MAC

RLC

PDCP

IP Multicast

UDP

APP FEC – Raptor Codes

PHY FEC – Fixed Rate Codes

Latency/Throughput Benefit ?

Coverage Benefit ?

Green Benefit ?

Rateless Codes

ReplaceInsert

Figure: 3GPP MBMS Protocol Stack.


Introduction

System Model

Analytical Results


Numerical Results




Motivation

Figure: Block Diagram View


Introduction

System Model

Analytical Results


Numerical Results




Motivation

Property IR-HARQ/ Punctured Fixed-Rate Codes

Rateless/ Fountain Codes

Initial codeword Generate a low rate mother codeword of the information (info) packet.

No predetermined codeword of info packet. Info bits are selected adaptively.

Parity generation and transmission

Puncture the codeword into multiple codeblocks and transmit blocks incrementally.

Fountain generation of parity bits, i.e., incrementally obtain any number of parity bits.

Adaptive nature No adaptive generation of parity bits to channel variations.

Adapt the degree distribution (for parity bit generation) to channel variations.

Selection of info bits No adaptive selection of info bits.

Adaptively select the info bits to generate parity bits through non-uniform selection.

Rates Can realize finite discrete rates.

Can realize rates truly matched to the instantaneous channel.

Figure: Punctured Fixed-Rate Codes v/s Rateless Codes.


Introduction

System Model

Analytical Results


Numerical Results




Rateless Transmission

K=75

b1b

2b

3......b

74b

75

Encoder

Channel

y1

y100

y200

y300

…......

1st 2nd 3rd

Decoding attempt

UE

BS

Figure: Rateless Transmission of K -bit packet.


Introduction

System Model

Analytical Results


Numerical Results




Outline

I Introduction

I System Model

I Fixed Information Transmission

I Analytical Results

I Performance Comparison

I Numerical Results

I Continuous Transmission

I Non Cooperative Transmission

I Coordinated Transmission (CoMP)


Introduction

System Model

Analytical Results


Numerical Results




System Model

I BS process is a homogeneous Poisson point process (PPP)Φ of intensity λ.

I Each BS Xi ∈ Φ communicates a K -bit packet to a user Yi

in its Voronoi cell.

I Encoding - Decoding operations

I BS: Encodes K bits with a rateless code and sends Gaussiansymbols incrementally over the channel.

I User: For every L channel uses, makes an attempt todecode a subset of K bits.

I Process continues until user decodes K bits, and sends anACK to BS.


Introduction

System Model

Analytical Results


Numerical Results




Time-varying Interference

I The interference power at user Yi at time t is given by

Ii (t) =∑k 6=i

|hki |2|Xk − Yi |−αek(t). (1)

|hki |2: Fading from BS Xk to user Yi

α : Path loss exponentek(t): MAC state of k th BS.

I MAC state of BS Xk is

ek(t) = 1 (0 < t ≤ Tk) . (2)

Tk : packet transmission time of BS Xk

I Thinning of the BS PPP Φ with time t.


Introduction

System Model

Analytical Results


Numerical Results




Achievable Rate

I Nearest-Neighbor Decoder: Performs minimum Euclideandistance decoding for non-Gaussian noise. (only CSIR)

I Achievable rate at user Yi is

Ci (t) = log2

(1 +|hii |2D−αi

Ii (t)

)(3)

Ii (t)- 2nd moment of non-Gaussian noise.

I Time averaged interference up to time t

Ii (t) =1

t

∫ t

0

Ii (τ)dτ. (4)


Introduction

System Model

Analytical Results


Numerical Results




Packet Transmission Time

I The time to decode K information bits

Ti = min {t : K < t · Ci (t)} (5)

I Each packet transmission of K bits is subject to a delayconstraint of N.

I Packet transmission time of user Yi is

Ti = min(N, Ti ) (6)

I CCDFs of T and T are key −→

Quantify the performance advantages of rateless codes forPHY-FEC.


Introduction

System Model

Analytical Results


Numerical Results




Analytical Model

I CCDF of T

P(T > t

)= P

(K

t≥ log2

(1 +|h|2D−α

I (t)

))(7)

We let θt = 2K/t − 1, then (7) can be written out as

P(T > t

)= E

[1− P

(|h|2D−α

I (t)≥ θt

∣∣∣D)](a)= E

[1− E

[exp

(− θtDα I (t)

)∣∣D]]= E

[1− LI (t) (θtD

α)], (8)

where (a) follows from evaluating the tail of |h|2 ∼ Exp(1)at θtD

α I (t).


Introduction

System Model

Analytical Results


Numerical Results




Analytical Model

I Interference

I (t) =∑k 6=0

|hk |2|Xk |−α min (1,Tk/t) (9)

Marks Tk are correlated for different k −→ No characteristicfunction.

I Approximation:I Replace Tk in (9) by i.i.d. Tk .

I Tk : Transmission duration of interferer Xk .

I T : Distribution of Interferer transmission duration

P(T > t

)= 1− 2F1 ([1, δ] ; 1 + δ;−θt) , (10)

where δ = 2/α and θt = 2K/t − 1.


Introduction

System Model

Analytical Results


Numerical Results




Analytical Model

I Independent Thinning Model- Let I (t) be the interferencein (9) with Tk in place of Tk .

I (t) =∑k 6=0

|hk |2|Xk |−α min(1, Tk/t

)(11)

⇒ Closed form characteristic function.

I Typical user’s packet transmission time

T = min

{t : K < t · log2

(1 +|h|2D−α

I (t)

)}T = min(N, T ). (12)

Analytical Study Feasible.


Introduction

System Model

Analytical Results


Numerical Results




Distribution of T

TheoremAn upper bound on the CCDF of the typical user packettransmission time under the independent thinning model, T in(12), is given by

P (T > t) ≤

{Pub(t) t < N

0 t ≥ N,(13)

where

Pub(t) = 1− 1

2F1 ([1,−δ] ; 1− δ;−θt min (1, µ/t)), (14)

δ = 2/α, θt = 2K/t − 1, and

µ = E[T]

=

∫ N

0

(1− 2F1

([1, δ] ; 1 + δ; 1− 2K/t

))dt. (15)


Introduction

System Model

Analytical Results


Numerical Results




Comparison

I Fixed-Rate Coding:I K info bits −→ Transmit codeword of N parity symbols.I Rate of K/N if success or 0 if outage.

ps(N) , P(SIR > 2K/N − 1

)(16)

RN , ps(N)K

N. (17)

I Rateless Coding:I Incrementally transmit up to N parity symbols.I Multiple decoding attempts.I K bits are decoded by variable number of parity symbols.

ps(N) , 1− P(T > N) (18)

RN ,Kps(N)

E [T ]. (19)


Introduction

System Model

Analytical Results


Numerical Results




SIR Gain

I SIR Gain (Horizontal Gap): F1 and F2 are SIR CCDFs oftwo schemes. If

F2(θ) ∼ F1 (θ/Γ) , θ → 0, (20)

then scheme 2 provides a SIR gain Γ over scheme 1.

I (20) ⇒ F2(θ) ≈ F1 (θ/Γ) for all θ.

I Scheme 1: Fixed-Rate Coding & Scheme 2: Rateless Coding

I θ = 2K/N − 1 with N →∞ as θ → 0.

I Compare coverage probability ps(N) for schemes 1 & 2.


Introduction

System Model

Analytical Results


Numerical Results




SIR Gain

PropositionRateless coding in cellular downlink leads to a SIR gain of Γ = N

µrelative to fixed-rate coding under the independent thinningmodel, where µ = E

[T]

is the mean interferer transmissionduration given in (15).

I Fixed-rate coding

ps(N) =1

2F1

([1,−δ] ; 1− δ; 1− 2K/N

)I Rateless coding

ps(N) ≥ 1

2F1

([1,−δ] ; 1− δ; (1− 2K/N) min (1, µ/N)

) .I µ decreases with path loss exponent α and monotonic with

delay constraint N.


Introduction

System Model

Analytical Results


Numerical Results




Rate Gain

PropositionIn cellular downlink, the rate gain gr of rateless codes relative tofixed-rate codes is

gr = gsN

E [T ]. (21)

I Success probability gain:

gs ≥2F1

([1,−δ] ; 1− δ; 1− 2K/N

)2F1

([1,−δ] ; 1− δ;

(1− 2K/N

)µ/N

) ≥ 1. (22)

I Transmission time gain:

N

E [T ]≥ N∫ N

0Pub(t)dt

≥ 1. (23)

I ⇒ gr ≥ 1.


Introduction

System Model

Analytical Results


Numerical Results




Per-User Gain

I Typical User Performance −→ Spatial Average

Per-User Downlink Performance −→ Conditioned on Φ.

I Coverage and Rate are random variables (RVs), achieved byany BS-UE pair in a given Φ realization.

I Fixed-rate coding:

RN ,K

NP(SIR > 2K/N − 1 | Φ

). (24)

I Rateless coding:

RN ,K[1− P(T > N | Φ)

]E [T | Φ]

. (25)


Introduction

System Model

Analytical Results


Numerical Results




Per-User Gain

I Rate gain conditioned on Φ: Ratio of random rates ofrateless and fixed-rate coding.

GR = GSN

E [T | Φ](26)

GS =1− P(T > N | Φ)

P(SIR > 2K/N − 1 | Φ

) . (27)

PropositionEvery BS-UE pair in a cellular downlink with PPP Φ realizationexperiences a throughput gain due to rateless code PHY-FECrelative to fixed-rate codes, i.e., GR ≥ 1.

I Can show GS ≥ 1.


Introduction

System Model

Analytical Results


Numerical Results




Coverage Probability

0 5 10 15 20 25 30

10 log10 N

0

0.2

0.4

0.6

0.8

1

ps(N

)

Fixed Rate Code

Rateless Code-Simulation

Rateless Code-Analysis

α=4

α=3

Figure: Success Probability ps(N) v/s Delay constraint N.


Introduction

System Model

Analytical Results


Numerical Results




Rate

5 10 15 20 25 30

10 log10 N

0

0.2

0.4

0.6

0.8

1

1.2

RN

Analysis

Simulation

Fixed Rate

Code

α=4α=3

Figure: Rate RN v/s Delay constraint N.


Introduction

System Model

Analytical Results


Numerical Results




Rate

2.5 3 3.5 4

α

0.2

0.4

0.6

0.8

1

1.2

RN

Fixed Rate Code

Rateless-Simulation

Rateless-Analysis N = Nr

N = Nf

Figure: Rate RN v/s path loss exponent α.


Introduction

System Model

Analytical Results


Numerical Results




Per-User Case

-3 -2 -1 0 1 2 3

-3

-2

-1

0

1

2

3x-BS o-Users

8.5

1.5

1.13

0.69

1.13

0.74

0.63

0.3

4.09

1.45

3.87

1.43

1.12

0.760.58

0.36

1.02

0.67

0.04

0.01

0.95

0.69

0.16

0.09

3.36

1.4

0.27

0.13

1.61

1.060.09

0.03

3.64

1.42

2.7

1.27

6.44

1.49

3.6

1.4

17

1.5

1.78

1.02

0.83

0.58

0.6

0.42

0.51

0.27

0.41

0.166.04

1.48

1

0.481.92

1.18

1.78

1.06

3.42

1.41

Figure: Ratio of Rates conditioned on Φ.


Introduction

System Model

Analytical Results


Numerical Results




Per-User Rates

0 0.2 0.4 0.6 0.8 1D

0

2

4

6

RN

(b)

Rateless-Curve Fit

Rateless-Simulation Data

Fixed Rate-Simulation Data

Fixed Rate-Curve Fit

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10(a)

Figure: Per-User Rates v/s BS-UE Distance D.


Introduction

System Model

Analytical Results


Numerical Results




Rate Gain

0 0.2 0.4 0.6 0.8 10

5

10

15

(a)

0 0.2 0.4 0.6 0.8 1D

0

5

10

15

20

Ratio o

f R

ate

s

(b)

Simulation Data

Curve Fit

Figure: Rate Gain v/s BS-UE Distance D.


Introduction

System Model

Analytical Results


Numerical Results




Outline

I Introduction

I System Model

I Fixed Information Transmission

I Analytical Results

I Performance Comparison

I Numerical Results

I Continuous Transmission

I Non Cooperative Transmission

I Coordinated Transmission (CoMP)


Introduction

System Model

Analytical Results


Numerical Results




Assumptions

I Interfering BSs transmit continuously. MAC state ofinterfering BS Xk at time t is ek(t) = 1, t ≥ 0.

I Typical user receives a K -bit packet via a rateless code byone BS.

I Interference at typical user

I =∑k 6=0

|hk |2|Xk |−α. (28)

Simple characteristic function.

I CCDF of the typical user packet transmission time T

P (T > t) = 1− 1

2F1 ([1,−δ] ; 1− δ;−θt), t < N. (29)


Introduction

System Model

Analytical Results


Numerical Results





I No Coverage gain: ps(N) same for both rateless coding andfixed-rate coding. gs = 1

I Rate gain: Ratio of rates

gr =N

E [T ]. (30)

I The rate gain of rateless codes in the cellular downlinkunder the continuous transmission case is given by

gr =

[1− 1

N

∫ N

0

1

2F1 ([1,−δ] ; 1− δ;−θt)dt

]−1. (31)


Introduction

System Model

Analytical Results


Numerical Results




Rate Gain

I Rate Gain due to Interferer Activity:

The rate gains gr and gr in the cellular downlink by usingrateless codes for PHY-FEC satisfy the relation

1 ≤ gr ≤ gr. (32)

I gr: Monotonically decreasing interference due to thinningof interfering BSs.

I gr: Constant interference due to continuous transmission ofinterfering BSs.

I Rate gain of a practical user:I gr - Lower boundI gr - Upper bound.


Introduction

System Model

Analytical Results


Numerical Results




Rate Gain

I Rate Gain due to User Location:

I General User: close to only one BS or

I Edge User: equidistant from two BSs or

I Vertex User: equidistant from three BSs.

I General user was the focus till now.

I Vertex User: Resides on a vertex of the Voronoi tessellationof Φ.

I CCDF of the packet transmission time of the vertex user

P (T > t) = 1−[

1/ (1 + θt)

2F1 ([1,−δ] ; 1− δ;−θt)

]2, t < N.

(33)


Introduction

System Model

Analytical Results


Numerical Results




Rate Gain

I Rate gain of the typical vertex user is

grv =

[1− 1

N

∫ N

0

( 1/ (1 + θt)

2F1 ([1,−δ] ; 1− δ;−θt)

)2dt

]−1.

(34)

I Rate gains of the vertex user and general user satisfy

1 ≤ grv ≤ gr. (35)

I gr: Interfering BSs are all further away than serving BS.

I grv: Two interfering BSs at same distance as serving BSwhile remaining are further away.

Rateless code adapts to changing channel conditions ⇒ Variedgains over fixed-rate code.


Introduction

System Model

Analytical Results


Numerical Results




Coordinated Transmission (CoMP)

Cloud/ BBU pool

Figure: CoMP for Mutual Information (MI) Accumulation


Introduction

System Model

Analytical Results


Numerical Results




MI Accumulation

I Cooperating BSs: Joint Transmission of K -bit packet touser.

I BSs access: K bits - X2/S1 interface of backhaul to cloud.

I Each BS uses a unique rateless code of K -bit packet.

I NOMA schemes resolve codewords.

I Multiple codewords input to iterative decoder at user.

I Achievable rate at user

C =M∑i=1

log2 (1 + SIRi ) . (36)


Introduction

System Model

Analytical Results


Numerical Results




System Model

I Two independent PPPs Φ1 and Φ2 of intensity λ/2.

I BSs ∈ Φk : Use spreading code k ∈ {1, 2}.

I BSs in cellular downlink Φ = Φ1 ∪Φ2 = {Xi}, i = 1, 2, · · · .

I Typical user receives a codeword from nearest BS in bothΦ1 and Φ2.

I Time to decode K -bit packet and packet transmission time

T = min {t : K < t · C} (37)

T = min(N, T ). (38)


Introduction

System Model

Analytical Results


Numerical Results




General User

TheoremThe CCDF of the general user packet transmission time with MIaccumulation, T in (38), is lower bounded as

P (T > t) ≥∫ γ

0

(G (γ − y)− 1)G ′(y)dy , (39)

G (ν) =1

2F1 ([1,−δ] ; 1− δ;−ν). (40)

where γ = 2(2K/2t − 1

)and δ = 2/α.

Success Probability and Rate

ps(N) , 1− P(T > N

)(41)

RN ,Kps(N)

E [T ]. (42)


Introduction

System Model

Analytical Results


Numerical Results




Diversity Gain

I For rateless coding, the diversity gain is

gd , limN→∞

log (1− ps(N))

− logN. (43)

I No Cooperation

1− ps(N) ∼ K log 2

N

δ

1− δ, N →∞. (44)

gd = 1.

I MI Accumulation

1− ps(N) ∼ 1

2

(δ

1− δ

)2(K log 2

N

)2

, N →∞, (45)

gd = 2.


Introduction

System Model

Analytical Results


Numerical Results




Vertex User

Vertex user is served by the two equidistant BSs with uniquespreading codes (M = 2) with the third equidistant BS asinterferer.

TheoremThe CCDF of the vertex user packet transmission time with MIaccumulation, T in (38), is lower bounded as

P (T > t) ≥∫ ∞0

∫ γ

0

(U (γ − y)− 1) G ′(y)fD(r)dy dr (46)

G (y) = exp

(−πλ

2r2 (2F1 ([1,−δ] ; 1− δ;−y)− 1)

)(47)

U(y) =G (y)

1 + y, γ = 2

(2K/2t − 1

). (48)


Introduction

System Model

Analytical Results


Numerical Results




Diversity Gain

I Vertex User: CoMP achieves diversity gain - Analysis trickythough

I No Cooperation

1− ps(N) ∼ K log 2

N

(2 +

2δ

1− δ

), N →∞. (49)

gd = 1.

I MI Accumulation

1− ps(N) ∼(K log 2

N√

2

)2 ∫ (πλ

2r2

δ

1− δ

)2fD (r) dr , N →∞.

(50)

gd = 2.


Introduction

System Model

Analytical Results


Numerical Results




Coverage Probability

5 10 15 20 25 30

10 log10 N

0

0.2

0.4

0.6

0.8

1

ps(N

)

GU M = 2

WU M = 2

GU M = 1

WU M = 1

Figure: Success Probability ps(N) v/s Delay constraint N.


Introduction

System Model

Analytical Results


Numerical Results




Rate

5 10 15 20 25 30

10 log10 N

0

0.5

1

1.5

RN

GU M = 2

WU M = 2

GU M = 1

WU M = 1

Figure: Rate RN v/s Delay constraint N. Rate gain of 2.6 and 6.12 forgeneral and vertex user.


Introduction

System Model

Analytical Results


Numerical Results




Conclusion

I Stochastic geometry model for downlink:

Rateless codes for PHY-FEC =⇒ Enhanced cellularcoverage and throughput.

I Typical user: Rateless PHY relative to fixed-rate PHY ⇒I SIR gain (Horizontal Gap): Coverage improvementI Rate gain: Throughput improvement

I Per-user caseI Every BS-UE pair in a cellular network realization has a

throughput gain ≥ 1 by rateless codes.I Achieve per-user rates ⇒ Efficient network operation.

I CoMP with Rateless codes:I Achieves coverage and rate improvement relative to

no-cooperation.I Vertex user benefits more from CoMP.

enhanced cellular coverage and throughput using rateless codes · 2017. 9. 13. · amogh rajanna...

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