The Approximate Sum Capacity of the Symmetric Gaussian K -User - PowerPoint PPT Presentation

The Approximate Sum Capacity of the Symmetric Gaussian K -User Interference Channel Or Ordentlich Joint work with Uri Erez and Bobak Nazer July 5th, ISIT 2012 MIT, Cambridge, Massachusetts Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian 2-user IC : channel model z 1 y 1 x 1 1 ˆ w 1 E 1 D 1 w 1 g z 2 g x 2 1 y 2 ˆ w 2 w 2 E 2 D 2 y k = x k + g x ¯ k + z k Channel is static and real valued. Gaussian noises z k are of zero mean and variance 1. All users are subject to the power constraint � x k � 2 ≤ n SNR. Define INR � g 2 SNR and α � log(INR) log(SNR) . Channel is symmetric: sum capacity = 2 × symmetric capacity Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

GDoF of symmetric Gaussian 2-user IC Symmetric capacity is known to within 1 / 2 bit (Etkin et al. 08). DoF for each user is 1 / 2. GDoF gives more refined view d ( α ) 1 2 3 1 2 α 1 2 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

Symmetric Gaussian 2-user IC Noisy interference regime Treat interference as noise d ( α ) 1 2 3 1 2 α 1 2 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

Symmetric Gaussian 2-user IC Weak interference regime Jointly decode intended message and part of interference (Han-Kobayashi). d ( α ) 1 2 3 1 2 α 1 2 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

Symmetric Gaussian 2-user IC Strong interference regime Jointly decode intended message and interference d ( α ) 1 2 3 1 2 α 1 2 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

Symmetric Gaussian 2-user IC Very strong interference regime Decode interference and then successively decode intended message d ( α ) 1 2 3 1 2 α 1 2 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC : channel model z 1 y 1 x 1 1 w 1 ˆ E 1 D 1 w 1 g g z 2 g x 2 1 y 2 ˆ w 2 E 2 D 2 w 2 g . . . . . . g z K g x K 1 y K ˆ w K w K E K D K � y k = x k + g x m + z k m � = k INR � g 2 SNR and α � log(INR) log(SNR) . Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? DoF is discontinuous at the rationals (Etkin and E. Ordentlich 09, Wu et al. 11). GDoF of the symmetric K -user IC is independent of K , except for discontinuity at α = 1 (Jafar and Vishwanath 10). d ( α ) 1 2 3 1 2 1 K 1 2 α 1 2 2 3 Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? What about finite SNR? Adding interference cannot increase capacity → Outer bounds for K = 2 remain valid for K > 2. 3−user IC @ SNR=35dB 6 symmetric rate[bits/channel use] 5 4 3 2 1 0 −2 0 2 10 10 10 g Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? What about finite SNR? Can always use time-sharing 1 → C SYM > 2 K log(1 + K SNR). 3−user IC @ SNR=35dB 6 symmetric rate[bits/channel use] 5 4 3 2 1 0 −2 0 2 10 10 10 g Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? What about finite SNR? Can treat interference as noise → achieves the approximate capacity for noisy interference regime 3−user IC @ SNR=35dB 6 symmetric rate[bits/channel use] 5 4 3 2 1 0 −2 0 2 10 10 10 g Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Each receiver sees a K -user MAC � y k = x k + g x m + z k , m � = k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Assume x 1 , . . . , x K ∈ Λ. = ⇒ Effective 2-user MAC at each receiver y k = x k + g x int , k + z k , � where x int , k = x m ∈ Λ. m � = k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For the other regimes lattice codes are useful. Closed under addition = ⇒ K − 1 interferers folded to one effective interferer. Assume x 1 , . . . , x K ∈ Λ. = ⇒ Effective 2-user MAC at each receiver y k = x k + g x int , k + z k , � where x int , k = x m ∈ Λ. m � = k How to decode x k ? Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y g x int , k x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y g x int , k x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y g x int , k x int , k Decode x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y g x int , k x int , k Decode x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Cancel x int , k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Decode x k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Decode x k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Decode x k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Decode x k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? For large g , can decode sum of interferences, subtract and decode desired codeword (Sridharan et al. 08) x k x k z 1 y 0 x int , k x int , k Decode x k Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: what do we know? What about finite SNR? Successive decoding is optimal in the very strong interference regime. 3−user IC @ SNR=35dB 6 symmetric rate[bits/channel use] 5 4 3 2 1 0 −2 0 2 10 10 10 g Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC

The symmetric Gaussian K -user IC: strong interference y k = x k + g x int , k + z k , x k , x int , k ∈ Λ Assume strong interference: g > 1 but not ≫ 1. For 2-user IC jointly decoding intended message and interference is optimal. For K -user IC jointly decoding x k , x int , k seems like a good idea. Question What rates are achievable? Ordentlich, Erez, Nazer Approx. Sum Capacity of the Symmetric Gaussian K -User IC