GAUSSIAN Max H. M. Costa and Olivier Rioul Unicamp and - - PowerPoint PPT Presentation

FROM ALMOST GAUSSIAN TO GAUSSIAN

22/09/2014 Max H. M. Costa and Olivier Rioul

Unicamp and Télécom-ParisTech

MaxEnt 2014 – Amboise, France

Summary

 Gaussian Interference Channel - standard form  Brief history  Z-Interference channel  Degraded Interference channel  Corner points of capacity region  Upper Bound  Lower bound  Discussion

Standard Gaussian Interference Channel

Power P1 Power P2

a b

W1 W2 W1 W2

^ ^

Z-Gaussian Interference Channel

The possibilities:

Things that we can do with interference:

1.

Ignore (take interference as noise (IAN)

2.

Avoid (divide the signal space (TDM/FDM))

3.

Partially decode both interfering signals

4.

Partially decode one, fully decode the other

5.

Fully decode both (only good for strong inter- ference, a≥1)

Brief history

 Carleial (1975): Very strong interference does not

reduce capacity (a2 ≥ 1+P)

 Sato (1981), Han and Kobayashi (1981): Strong

interference (a2 ≥ 1) : IFC behaves like 2 MACs

 Motahari, Khandani (2009), Shang, Kramer and

Chen (2009), Annapureddy, Veeravalli (2009): Very weak interference (2a(1+a2P) ≤ 1) :  Treat interference as noise (IAN)

History (continued)

 Sason (2004): Symmetrical superposition to beat

TDM – found part of optimal choice for α

 Etkin, Tse, Wang (2008): capacity to within 1 bit,

good heuristical choice of αP=1/a2

Degraded Gaussian Interference Channel

Differential capacity

Discrete time channel as a band limited channel

Gaussian Broadcast Channel

Superposition coding

N2 (1-)P P 1 P

Superposition coding

N2 (1-)P P 1 P

Multiple Access Channel

Degraded Interference Channel

• One Extreme Point

Degraded Interference Channel

• Another Extreme Point

Degraded Gaussian Interference Channel

Key variables

 Let Z1 + Z2 + X2 be distributed as f  Note: X2 is a codebook  Let Z1 + Z2 + Z3 be distributed as g  Z1, Z2, Z3 are Gaussian variables  Have: h(g) – h(f) ≤ 𝑜1  (the almost Gaussian hypothesis)

Key variables (cont.)

 Y1 = X1 + Z1  Y2 = X1 + Z1 + Z2 + X2  Y3 = X1 + Z1 + Z2 + Z3  X1 ~ p  Y2 ~ f•p  Y3 ~ g•p

The missing inequality

 Need a Fano type inequality based on  non-disturbance criterion:  -n ≤ h(Y3) – h(Y2) ≤ n  (with diminishing  )

Upper bound on h(Y3) – h(Y2)

 I(X1;Y2) = I(X1;Y2|X2) – I(X1;X2|Y2)  ≥ I(X1;Y2|X2) – n2  ≥ H(X1) – H(X1| X1+Z1+Z2) – n2  = I(X1;X1+Z1+Z2) – n2  ≥ I(X1;Y3) – n2  By the data processing inequality (DPI).  Therefore  h(Y3)-h(Y2) ≤ h(Y3|X1)-h(Y2|X1) + n2  = h(g) – h(f) + n2 ≤ n1 + n2

Lower Bound on h(Y3) – h(Y2)

h(f) = -f log f  h(g) = -g log g

• f log g

 

• g log f

By DPI: 0 ≤ D(f•p||g•p) ≤ D(f||g) ≤ n1 0 ≤ D(g•p||f•p) ≤ D(g||f) ≤ n1 h(Y3) = -g•p log g•p 

• f•p log g•p

 

• g•p log f•p

h(Y2) = -f•p log f• p D(f||g) D(g||f) Smoothing by p:

Lower Bound (cont.)

 Conjecture: We argue by continuity that  (f•p - g•p) log f•p does not change sign.  This implies:  h(Y3) - h(Y2) ≥ -21

Rational

 0 ≤ D(g•p||f•p) = ( g•p log g•p - ( g•p log f•p  + ( f•p log f•p - ( f•p log f•p  = h(f•p) – h(g•p) +(f•p – g•p) log f•p  ≤ D(g||f) ≤ (f – g) log f ≤ 2n1  Equivalently  h(Y3) - h(Y2) ≥ (f•p - g•p) log f•p +(g - f) log f  ≥ -2n1

Special case

 Let f = g + f.

Then expand

 0 ≤ D(f•p||g•p) ≤ ( f•p log f•p - ( f•p log g•p  + ( g•p log g•p - ( g•p log g•p  ≤ h(g•p) – h(f•p) +(g•p – f•p) log g•p  ≤ h(Y3) – h(Y2) +f•p log g•p  If 𝑔

= g -f = 2g – f is also a valid density, then can prove the lower bound by symmetry and upper bound.

Remarks

 Somewhat surprisingly,  h(Y2) can be greater then h(Y3).  Close to establish the corner points of the capacity

region of the standard interference channel.

 To whisper or to shout: Not to cause inconvenience, X1

needs to be decoded at Y2. Better to shout!

 Many thanks!