FROM ALMOST GAUSSIAN TO GAUSSIAN Max H. M. Costa and Olivier Rioul Unicamp and Télécom-ParisTech

22/09/2014

MaxEnt 2014 – Amboise, France

Summary  

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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

W1

a

b W2 Power P2

^ W1

^ W2

Z-Gaussian Interference Channel

The possibilities: Things that we can do with interference: 1. 2. 3. 4. 5.

Ignore (take interference as noise (IAN) Avoid (divide the signal space (TDM/FDM)) Partially decode both interfering signals Partially decode one, fully decode the other Fully decode both (only good for strong interference, a≥1)

Brief history 

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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) 

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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

(1-)P

N2

P P 1

Superposition coding

(1-)P

P N2 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

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Let Z1 + Z2 + Z3 be distributed as g Z1, Z2, Z3 are Gaussian variables

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Have:

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h(g) – h(f) ≤ 𝑛1 (the almost Gaussian hypothesis)

Key variables (cont.)   

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Y1 = X1 + Z1 Y2 = X1 + Z1 + Z2 + X2 Y3 = X1 + Z1 + Z2 + Z3

X1 ~ p Y2 ~ f•p Y3 ~ g•p

The missing inequality  

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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)     

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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) -g log f h(f) = -f log f h(g) = -g log g    D(f||g) D(g||f) -f log g

Smoothing by p:

h(Y3) = -g•p log g•p -g•p log f•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

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(f•p - g•p) log f•p does not change sign.

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This implies:

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h(Y3) - h(Y2) ≥ -21

Rational     

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

Remarks  

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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!

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Many thanks!