Towards a novel communication theory for wireless networks

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Towards a novel communication theory for wireless networks

Third Nordic Workshop on System & Network Optimization for Wireless

Towards a novel communication theory for

wireless networks

(Interference alignment, lattices, integer forcing, compute and forward and scheduling)

Giuseppe Caire

EE Department, University of Southern California, Los Angeles CA

Trysil, Norway, April 10-12, 2012


The conventional approach to wireless networks

Methodology:

• Make users as orthogonal as possible: TDMA, CSMA, CDMA, OFDM/TDMA,

frequency reuse.

• Treat the residual interference as noise: heavy coding, tight power control,

target system design able to work at SNR ≈ 0 dB.

Successes:

• 2G (GSM), WLANs (802.11), 3G (WCDMA), 4G (LTE, mobile WiMax).

Failures:

• Very dense user-deployed small cells (femtocells, HetNets).

1


Theoretical foundations of the conventional approach

• Random coding, superposition coding, successive decoding (essentially

single-user).

• Information theoretic G-MAC and scalar (degraded) G-BC.

• With fading, CSIT and multiuser scheduling, channel-driven TDMA is optimal

(e.g., high-data rate 3G systems such as Ev-Do and HSDPA-HSUPA).

• Vector G-BC is different, but eventually linear precoding (e.g., ZF) is “good

enough” (orthogonalization in the antenna domain).

3


The “modern” approach

Methodology:

• Focus on signal interaction.

• Do not treat interference as noise (unless it is optimal to do so).

• Target high-SNR design: lightweight coding, almost uncoded modulation is

good enough.

Theoretical breakthroughs:

• Quantize Map and Forward (QMF), a.k.a. Noisy Network Coding.

• Compute and Forward (CoF). .... quench the noise before forwarding.

• Interference Alignment (IA).

Successes:

• ????

4


Quantize Map and Forward: main ideas

• Each intermediate node quantizes what it receives and encodes the

quantization bits into a downstream codeword.

• At each destination, a typical set decoder perform joint typicality test across

the whole network.

• No explicit decoding of the quantization indices.

• Optimality for linear deterministic finite-field networks for the multi source

multicast problem.

• Approximate optimality for multi-source multicast problem Gaussian relay

networks (within a constant gap from cut-set bound).

5


A simple example

• The relay applies quantization and binning in the complex plain, using 2 R 0

bins.

6


0

1

0

1 0 1

1 0

1

0

1 0

0

1

0

1

0

1

1

0

1

0

1

0

0

1 0

1

0

1

1

0

1

0

1 0

7


A complicated decoder

• Given y 1 and x r , the destination computes λ i , the LLR corresponding to the

i’th bit position, for i = 1, . . . , k:

λ i = log


{x 1 ∈X 1 ,x 2 ∈X 2 |A i (x 1 )=1} p(y 1, x r |x 1 , x 2 )

∑{x ′ 1 ∈X 1,x ′ 2 ∈X 2|A i (x ′ 1 )=0} p(y 1, x r |x ′ 1 , x′ 2 ),

where X 1 , X 2 are the two constellations.

• To compute p(y 1 , x r |x 1 , x 2 ), we have:

p(y 1 , x r |x 1 , x 2 ) = p(y 1 |x 1 , x 2 )p(x r |x 1 , x 2 )

8


Compute and Forward

• A key source of complexity in QMF is that the end-to-end typical set decoder

needs to search over all possible “intermediate” quantization messages.

• The corresponding end-to-end ML decoder has to marginalize out the

“intermediate” quantization messages.

• An alternative consists of using Belief Propagation to perform approximate

marginalization (on-going work by Fragouli et al.).

• Another alternative consists of quenching the noise at each node, and

forwarding a deterministic function of the information messages.

• Compute and Forward: decode (with arbitrarily high reliability) a function of

the information messages.

• Nazer and Gastpar CoF: use lattice codes, and decode integer linear

combinations.

10


Compute and Forward: main ideas

• G-MAC:

y =

K∑

h k x k + z

k=1

• Lattice coding: L k = Λ c ∩ V, where V is the Voronoi cell of Λ ⊆ Λ c .

• Dithered encoding: send x k = [c k + d k ] mod Λ.

• Scaling, dithering removal and modulo Λ:

[

]

K∑

ỹ = αy − a k d k mod Λ = [c + z eff (h, a, α)] mod Λ

k=1

where c = ∑ K

k=1 a kc k , a ∈ Z K and

z eff (h, a, α) =

K∑

(αh k − a k )x k + αz

k=1

11


• Main result: for given h, α and any ɛ > 0 there exists n large enough and

lattice codes L k with rate R k such that

P

(

ĉ ≠

)

K∑

a k c k ≤ ɛ

k=1

if

where

R k < R comp (h, a, α) = 1 (

)

2 log SNR

σeff 2 (h, a, α)

σ 2 eff(h, a, α) = SNR ‖αh − a‖ 2 + α 2 12


i =1,...,n

• Deterministic function of the information messages: using the lattice

encoding linearity (construction A: Λ c = C + pZ 2 p), there exists an

isomorphism:

K∑

K⊕

c = a k c k ↔ u = q k w k

k=1

where w k is the information message corresponding to c k and q k =

a k mod pZ.

k=1

u 1 =

4M

q 1k w k

Enc 1

Enc 2

Enc 3

Enc 4

x 1

x 2

x 3

yi = Hxi + zi

y 1

y 2

y 3

x 4

y 4

Dec 1

Dec 2

Dec 3

Dec 4

u 2 =

u 3 =

k=1

4M

q 2k w k

k=1

4M

q 3k w k

k=1

4M

u 4 = q 4k w k

k=1

message recovered

by Gaussian elimination

Q =

2

6

4

if

3

q 11 ··· q 14

7

. . 5

q 41 ··· q 44

is full rank mod p

Independent decoding

13


Quantized CoF (QCoF)

• We let Λ = κpZ, Λ c = κZ and C a linear code over Z p of rate k/n.

• All users make use of the same code C: let c k = w k G be the codeword of C

sent by UT k.

• Symbol by symbol demodulation: pick α ∈ R M and a ∈ Z K and compute the

quantized output:

u i = κ −1 [

Q Λc

(αy i −

)

K∑

a k x k,i

k=1

mod Λ

]

scalar quantizer

y(i)

α

remove dither

× +

u(i)

sawtooth transformation

14


• It is immediate to show that

v =

(

⊕ K

)

q k c k

k=1

⊕ e

where q k = [a k ] mod pZ and where

[ ( K

)


e i = κ −1 Q Λc (αh k − a k )x k,i + αz i

k=1

mod Λ

]

• Probability of error P(ĉ ≠ c) → 0 if

R < R QCoF = log p − H(e i )

15


Code design for QCoF: very easy

4.5

4

3.5

3

CoF

QCoF, p=251

QCoF, p=17

QCoF, p=7

QCoF, p=3

RA codes over Z 17

RA codes over Z 7

R=4/5

Computation Rates

2.5

2

RA codes over Z 3

IRA codes over Z 7

R=2/3

R=4/5

R=1/2

R=2/3

1.5

1

R=2/3

R=4/5

R=1/2

0.5

R=1/2

0

15 20 25 30 35 40 45 50

SNR [dB]

Fig. 2. Computation rates for three-user Gaussian MAC with coefficients h =[1, 0.75, − √ 2] and ADC level, p =3, p =7,

p =17, or 255.

16


Code design for CoF: challenges

• Using Λ = κpZ incurs in the shaping loss.

• We would like to use very large p, but LDPC decoding for p-ary codes is

computationally expensive.

• We wish codes for the complex-valued channel, without breaking real and

imaginary parts.

• Feng, Silva and Kschischang has recently provided a framework to build

nested Z[j]-lattices (Narayanan and Boutros, extended this to Z[e j2π/3 ]-

lattices).

• Signal codes, low-density lattice codes, .... possible ways.

• Polar codes for large p ....since the QCoF channel is additive discrete noise

modulo p.

17


Compute and Forward: application to DAS

UT

UT

AT

UT

AT

AT

UT

UT

UT

AT

AT

AT

UT

UT

R 0

CP

• Uplink channel model (from UTs to ATs): y i = Hx i + z i ,

with H = [h 1 , . . . , h L ] T ∈ R L×K .

• Downlink channel model (from ATs to UTs): ỹ i = ˜H˜x i + ˜z i ,

with ˜H = [h 1 , . . . , h K ] T ∈ R K×L .

18


Alternative schemes for the uplink

UT

UT

UT

UT

z 1

h 11

h k

+ AT

R 0

z 2

+ AT R 0

z 3

+ AT

+ AT

R 0

R 0

CP

• QCoF: constrained by k n log p ≤ R 0.

• Compressed Integer-Forcing Receiver (CIFR).

• Lattice Quantize and Forward (LQF), for the A/D constrained case p fixed and

log p ≤ R 0 .

• CF (Sanderovich and Shamai), equivalent to quantize, map and forward

(QMF) or noisy network coding, in this case.

19


Alternative schemes for the downlink

UT

AT

UT

UT

+

+

z 1

h 11

z 2

R 0

AT R 0

z 3

h`k

AT

z 4

+

+

R 0

R 0

CP

UT

AT

• Reverse QCoF (RQCoF).

• Compressed Integer-Forcing Beamforming (CIBF).

• Reverse Lattice Quantize and Forward (RLQF), for the A/D constrained case

p fixed and log p ≤ R 0 .

• Compressed DPC (CDPC, Simeone and Shamai).

20


Network decomposition and relay / user selection

• Network decomposition:

blocks.

it may happen that Q decomposes in diagonal

• In this case, each diagonal block forms a subnetwork, and CoF/RCoF need

not use the same code across subnetworks.

• Relay (AT) selection: for a given subset U ⊆ [1 : K] of UTs transmitting, we

have to decide which ATs to activate.

• Goal: obtain a rank-|U| system matrix, while maximizing the sum rate.

• This problem reduces to the maximization of a linear function subject to

matroid constraints, for which greedy selection is optimal.

• Scheduling groups of UTs: unfortunately, this is a combinatorial problem

without much structure: we use heuristic greedy selection.

21


Results: IFB with RCoF vs Gaussian DPC

70

60

DPC

IFB

ZFB

50

Sum Rates

40

30

20

10

0

10 15 20 25 30 35 40

SNR [dB]

Downlink Wyner model with inter-cell interference γ = 0.7, L = K = 10

ATs/UTs and backhaul capacity R 0 = ∞.

22


Results: effect of network decomposition

30

25

q=1

q=1, ND

q=0.2

q=0.2, ND

20

Sum Rates

15

10

5

5 10 15 20 25

SNR [dB]

Uplink DAS with Bernoulli-Gaussian random connections, L = K = 20, with

“on” probability q. ND denotes that network decomposition is considered to

compute the sum-rates of QCoF.

23


Results: Wyner model downlink

4

3.5

3

Rate per User

2.5

2

1.5

Upper Bound

1

CDPC

CIFB

0.5

RCoF

RQCoF (p=251)

RCoF with PA

0

1 2 3 4 5 6 7 8 9 10

R 0

[bits]

SNR = 25 dB. Achievable rates per user as a function of finite capacity R 0 , for

the downlink Wyner model with inter-cell interference γ = 0.7.

24


Results: random versus greedy AT selection

30

25

Random, 10 ATs

Random, 20 ATs

Random, 30 ATs

Greedy

20

Sum Rates

15

10

5

0

0 5 10 15 20 25 30

SNR [dB]

Uplink DAS with Bernoulli-Gaussian random connections, K = 10, q = 0.3.

25


Interference Alignment

• Interference alignment (IA) technique was first proposed by Maddah Ali et al.

for MIMO X channel (2006).

• Cadambe and Jafar showed that in a K-user interference channel with timevarying

coefficients, K/2 sum degrees of freedom is achievable using IA

(2007).

• IA has since been studied in various setups, for example in cellular systems

(Suh and Tse 2009), downlink channel with switched-antennas (Jafar 2010),

delayed Channel State Information Feedback (Maddah Ali and Tse, 2010) ...

• Various flavors of IA have been proposed:

algebraic (lattices, rational “dimensions”).

linear spaces, signal levels,

• Practical applications of IA has been so-far limited, due to two main issues:

1) need of CSIT knowledge; 2) poor performance at finite SNRs.

26


K = 3 M × M Gaussian interference channel

• Perhaps the most relevant configuration in actual wireless networks:

y 1 = H 11 x 1 + H 12 x 2 + H 13 x 3 + z 1

y 2 = H 21 x 1 + H 22 x 2 + H 23 x 3 + z 2

y 3 = H 31 x 1 + H 32 x 2 + H 33 x 3 + z 3 ,

• Linear transmit and receive beam forming matrices {V k } and {U k }, of size

M × d k .

• Transmitted signal x k = V k s k , where s k ∈ X d k, where X is some discrete

constellation.

27


• The perfect alignment condition

rank ( )

U H kH k,k V k = d k , ∀ k = 1, . . . , 3

U H kH k,j V j = 0, ∀ j ≠ k

• Explicit alignment solution given y Cadambe and Jafar.

• Alternating Optimization: optimize alternatively the receivers for fixed

transmitters, and exchange the role of transmitters and receivers (reminiscent

of “multiuser time-reversal”).

28


Blind Interference Alignment (BIA)

• 3-user interference channel, each transmitter has 2 antennas, each receiver

has one reconfigurable antenna with two modes.

• Precoding over 4 time slots: At time 1), all Tx’s send their 2 × 1 symbol vector

x j : j = 1, 2, 3; At times 2,3,4) the Tx’s take turns and repeat their symbol

vector alone.

• User 1 switches as:

mode } {{ h}

, mode

} {{

g

}

, mode } {{ h}

, mode } {{ h}

time1 time2 time3 time4

29


• Stacking the received signal into a 4 × 1 vector along time, we have




y 1 [1]

y 1 [2]

y 1 [3]

y 1 [4]

⎤ ⎡


⎦ = ⎢


h H 11 h H 21 h H 31

g11 H 0 0

0 h H 21 0

0 0 h H 31

} {{ }

4×6



⎥ ⎣ x ⎤

1


x 2

x 3

⎦ +




z 1 [1]

z 1 [2]

z 1 [3]

z 1 [4]




30


Delayed CSIT (MAT)

• K = 2 MISO 2 × 1 interference channel, with outdated CSIT.

• Precoding over 3 slots: At time 1), Tx 1 sends its 2 × 1 vector x 1 ; at time 2)

Tx 2 sends its 2 × 1 vector x 2 ; at time 2) Tx 1 sends the overheard (scalar)

signal by Rx 2 during time 1 and Tx 2 sends the overheard (scalar) signal by

Rx 1 during time 2.

• Stacking the received signal into a 3 × 1 vector along time, we have


⎣ y 1[1]

y 1 [2]

y 1 [3]


⎦ =



h H 11[1] 0

0

g 11 h H 21[1]

h H 12[2]

g 12 h H 12[2]



[

x1

x 2

]

+


⎣ z 1[1]

z 1 [2]

z 1 [3]



31


Detectors

• From the receiver perspective:

y = Ms + Jv + z,

where s ∈ X d denotes the vector of desired symbols, v ∈ X l denotes the

vector of interfering symbols, M ∈ C m×d has rank d < m, and J ∈ C m×l has

rank m − d.

• The alignment condition yields:

Span{M} ∩ Span{J} = {0},

and rank([M, J]) = m

32


Linear detection

• y lin = U H y.

• Zero-Forcing of interference: U zf given by the unitary term in the SVD

UΛU H = I m − J(J H J) −1 J H .

• Linear MMSE receiver: U mmse given by

U H mmse = M H [ ρ −1 I m + JJ H + MM H] −1

.

• After interference (approximate) removal, any classical detection of the

desired user symbols can be performed.

33


Linear Integer Forcing detection

• The objective is to remove interference while providing a rank-d integer linear

combination of the desired symbols.

y if = U H y = A H s + w,

where w contains the residual noise and interference and non-integer error.

• This can be combined with lattice coding and CoF at each receiver.

• Exact IF:

U H ex = A H (M H M) −1 M H .

• “Optimal” IF:

U H opt = A H U H mmse.

34


Joint Decoding of signal and interference

• Approximate MAP detector for the useful signal only, that takes into account

the discrete nature of the interference.

• Extended data vector: ̂x = [ŝ T , ̂v T ] T .

• Minimum distance decoding: let H = [M, J] has dimensions m × (d + l),

min ‖y − Hx‖2 .

x∈X d+l

• Since d + l is generally larger than m, this is an underdetermined integer LS

problem.

35


• MMSE-GDFE regularized LD:

• Feedforward filter F given by

F = BH H [ ρ −1 I m + HH H] −1

.

• Feedback filter B given by upper triangular Cholesky factor such that

BB H = [ ρ −1 I m+l + H H H ] .

• Then, the MMSE-GDFE LD performs minimum distance lattice decoding on

ỹ = Fy = Bx + w,

where w is non-Gaussian with covariance I (by construction).

36


Numerical experiments: 3-user 4 × 4, d k = 2, 4QAM

3−user 4x4−MIMO Leakage Minimization

10 0 SNR

10 −1

10 −2

SER

10 −3

10 −4

10 −5

ZF

ZF−LD

MMSE−IF

GDFE−LD

14 16 18 20 22 24 26 28 30 32 34

37


Numerical experiments: 3-user 4 × 4, d k = 2, 4QAM

3−user 4x4−MIMO Max SINR

10 −1 SNR

10 −2

SER

10 −3

10 −4

ZF

ZF−LD

MMSE−IF

GDFE−LD

14 16 18 20 22 24 26 28 30 32 34

38


Numerical experiments: 3-user BIA, 4QAM

Blind IA

10 0 SNR

10 −1

SER

10 −2

10 −3

10 −4

ZF

ZF−LD

MMSE−IF

GDFE−LD

5 10 15 20 25 30

39


Numerical experiments: 2-user MAT, 4QAM

delayed CSIT

10 0 SNR

10 −1

SER

10 −2

10 −3

ZF

ZF−LD

MMSE−IF

GDFE−LD

0 5 10 15 20 25 30

40


Challenges

• In AO or other numerical optimization of the IA transmitters: how can we

incorporate the presence of non-linear/IF detectors?

• Efficient solution of underdetermined integer LS problems.

• Why MMSE-GDFE regularized LD works so well?

41


Conclusions

• New wireless network paradigm: focus on signal interaction.

• Role of algebraic structures.

• Role of non-linear decoding.

• Can we build robust PHY layers from this novel approach?

42


Thank You!

43

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