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

FROM ALMOST GAUSSIAN TO GAUSSIAN Max H. M. Costa and Olivier Rioul Unicamp and Télécom-ParisTech 22/09/2014 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 ^ W 1 W 1 a b ^ W 2 W 2 Power P2

Z-Gaussian Interference Channel

The possibilities: Things that we can do with interference: Ignore (take interference as noise (IAN) 1. Avoid (divide the signal space (TDM/FDM)) 2. Partially decode both interfering signals 3. Partially decode one, fully decode the other 4. Fully decode both (only good for strong inter- 5. ference , a≥1)

Brief history  Carleial (1975): Very strong interference does not reduce capacity (a 2 ≥ 1+P)  Sato (1981), Han and Kobayashi (1981): Strong interference (a 2 ≥ 1) : IFC behaves like 2 MACs  Motahari, Khandani (2009), Shang, Kramer and Chen (2009), Annapureddy, Veeravalli (2009): Very weak interference (2a(1+a 2 P) ≤ 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 α P=1/ a 2

Degraded Gaussian Interference Channel

Differential capacity Discrete time channel as a band limited channel

Gaussian Broadcast Channel

Superposition coding (1-  )P N 2 P  P 1

Superposition coding (1-  )P P N 2  P 1

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(g) = -  g log g -  g log f h(f) = -  f log f    D(f||g) D(g||f) -  f log g Smoothing by p: -  g•p log f•p h(Y3) = -  g •p log g •p -  f•p log g•p    h(Y2) = -  f•p log f• p By DPI: 0 ≤ D(f•p || g•p) ≤ D(f ||g ) ≤ n  1 0 ≤ D( g•p || f•p) ≤ D(g||f) ≤ n  1

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 = g -  f = 2g – f is also a valid density, then  If 𝑔 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!