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 **for**cing, compute and **for**ward and scheduling)

Giuseppe Caire

EE Department, University of Southern Cali**for**nia, 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).

• In**for**mation 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 be**for**e **for**warding.

• 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 per**for**m 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 per**for**m approximate

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

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

**for**warding a deterministic function of the in**for**mation messages.

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

the in**for**mation 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 in**for**mation 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 in**for**mation 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 trans**for**mation

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 **for**ward

(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 Beam**for**ming (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 **for**ms a subnetwork, and CoF/RCoF need

not use the same code across sub**networks**.

• 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: un**for**tunately, 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 In**for**mation 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 per**for**mance 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 **for**ming 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 per**for**med.

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:

• Feed**for**ward 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 per**for**ms 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