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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

  1. 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

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. 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

  24. 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

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

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