Draft. On Limiting Expressions for the Capacity Region of Gaussian Interference Channels. Mojtaba Vaezi and H. Vincent Poor

Transcription

1 Preliminaries Counterexample Better Use On Limiting Expressions for the Capacity Region of Gaussian Interference Channels Mojtaba Vaezi and H. Vincent Poor Department of Electrical Engineering Princeton University Asilomar, Pacific Grove, CA November 10, 2015 M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

2 Preliminaries Counterexample Better Use Outline 1 Preliminaries Single-Letter vs Multiletter Problem Setup 2 Counterexample Channel model Inner bound Outer bound Comparison 3 Better Use of the Multiletter Expression Find a Single-Letter Expression Find its Optimal Inputs M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

3 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Single-letter/multiletter capacity expressions Tx1 Rx1 A Single-User Channel Capacity expressions for the single-user channel [Shannon 1948] 1 Discrete memoryless channel 1 Multiletter: n I (X n ; Y n ), maximization over all p(x n ), n Single-letter: I (X ; Y ), maximization over all p(x) A single-letter capacity expression includes the channel input and output random variables involved in one use of the channel. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

4 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Single-letter/multiletter capacity expressions Tx1 Rx1 A Single-User Channel Capacity expressions for the single-user channel [Shannon 1948] 1 Discrete memoryless channel 1 Multiletter: n I (X n ; Y n ), maximization over all p(x n ), n Single-letter: I (X ; Y ), maximization over all p(x) A single-letter capacity expression includes the channel input and output random variables involved in one use of the channel. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

5 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Single-letter/multiletter capacity expressions Tx1 Rx1 A Single-User Channel Capacity expressions for the single-user channel [Shannon 1948] 1 Discrete memoryless channel 1 Multiletter: n I (X n ; Y n ), maximization over all p(x n ), n Single-letter: I (X ; Y ), maximization over all p(x) A single-letter capacity expression includes the channel input and output random variables involved in one use of the channel. 2 Gaussian channel with E( X 2 ) P Optimal input p(x) N (0, P) C = 1 2 log(1 + P) γ(p). M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

6 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Single-user channel Discrete memoryless channel R 1 n I (X n ; Y n ) + ɛ n (a) = 1 n I (X i ; Y i ) + ɛ n n (b) i=1 = I (X ; Y ) + ɛ n, (a) follows by the memoryless property of the channel (b) follows by the definition of the information capacity Gaussian channel: optimal inputs are Gaussian p(x n ) N (0, PI n ) p(x) N (0, P) Single-letter and multiletter capacity expression are equal Gaussian inputs are optimal for the Gaussian channel M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

7 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Introduction capacity region Two-User Interference Channel (IC) multiletter: is known [Ahlswede 1973] single-letter: is not known, in general M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

8 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Channel model Discrete memoryless interference channel (DM-IC) Source 1 Source 2 M 1 Encoder 1 M 2 Encoder 2 X1 n X2 n p(y 1, y 2 x 1, x 2 ) Limiting/multiletter capacity expression C IC = lim n co p(x n 1 )p(xn 2 ) Y1 n Decoder 1 Y2 n Decoder 2 ˆM 2 { R1 1 n I (X 1 n; Y 1 n) } R 2 1 n I (X 2 n; Y 2 n), where co( ) denotes the convex hull. [Ahlswede 1973] ˆM1 M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

9 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Limiting capacity expression The above limiting capacity expression is computationally excessively complex does not provide any insight into how to best code does not give a hint about optimal input for the Gaussian channel M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

10 Preliminaries Counterexample Better Use Single-Letter vs Multiletter Problem Setup Limiting capacity expression The above limiting capacity expression is computationally excessively complex does not provide any insight into how to best code does not give a hint about optimal input for the Gaussian channel Q Can we limit the inputs of the limiting capacity to be Gaussian to compute the capacity region of the Gaussian interference channel M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

11 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Gaussian One-Sided IC X 1 X Y 1 a 1 Y 2 + Y 1 = X 1 + ax 2 + Z 1, Y 2 = X 2 + Z 2, Z 1 N (0, 1) Z 2 N (0, 1) and input i is subject to an average power constraint P i E( X i 2 ) P i, i = 1, 2. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

12 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Inner bound Time-sharing: (R 11, R 21 ) and (R 12, R 22 ) are achievable τ 1 (R 11, R 21 ) + τ 2 (R 12, R 22 ) is achievable, for any τ 1 + τ 2 = 1. Achievability τ 1 τ 2 (R 11, R 21 ) (R 12, R 22 ) (R 1, R 2 ) = τ 1 (R 11, R 21 ) + τ 2 (R 12, R 22 ) Time-sharing in two dimensions Han-Kobayashi (HK) during τ 1 fraction of channel uses Single-user transmission during τ 2 (R 21 = 0) M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

13 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Inner bound Theorem 1 The set of non-negative (R 1, R 2 ) satisfying in which R 1 τ 1 R 11, R 2 τ 1 R 21 + τ 2 R 22, ( P 1 ) τ R 11 γ 1, 1 + aβ 1 P 21 ( a β 1 P ) 21 R 21 γ 1 + P + γ(β 1 1 P 21 ), τ 1 + aβ 1 P 21 R 22 γ(p 22 ), is achievable for the one-sided Gaussian interference channel where τ 1 + τ 2 = 1, τ 1 P 21 + τ 2 P 22 = P 2, 0 β 1 1, and β 1 = 1 β 1. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

14 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Inner bound Weighted sum-rate µr 1 + R 2 = τ 1 (µr 11 + R 21 ) + τ 2 R 22 ( P 1 ) ( τ τ 1 [µγ 1 a β 1 P ) 21 + γ 1 + aβ 1 P P 1 τ 1 + aβ 1 P 21 ] ( P2 τ 1 P ) 21 + γ(β 1 P 21 ) + τ 1 γ, τ 1 in which 0 τ 1 1, 0 β 1 1, and 0 P 21 P 2 τ 1. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

15 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Outer bound From limiting capacity expression we have µr 1 + R 2 1 n [ µi (X n 1 ; Y n 1 ) + I (X n 2 ; Y n 2 ) ], for some p(x1 n)p(x 2 n ), when n. We let p(x n 1 ) N (0, K 1 ), p(x n 2 ) N (0, K 2 ), tr(k X n 1 ) np 1, tr(k X n 1 ) np 1. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

16 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Outer bound From limiting capacity expression we have µr 1 + R 2 1 n [ µi (X n 1 ; Y n 1 ) + I (X n 2 ; Y n 2 ) ], for some p(x1 n)p(x 2 n ), when n. We let p(x n 1 ) N (0, K 1 ), p(x n 2 ) N (0, K 2 ), tr(k X n 1 ) np 1, tr(k X n 1 ) np 1. [ ( P 1 ) ] ξ µr 1 + R 2 ξ µγ 1 + ap 2 + γ(p 2) + ξγ( P2 ξp 2 ), ξ in which 0 ξ 1, ξ = 1 ξ, and 0 P 2 P 2. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

17 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Comparison R Inner bound Outer bound R 1 Figure: The inner and the outer bounds for a = 0.6, P 1 = 7 and P 2 = 7. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

18 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Remarks In the multiletter capacity of the IC, we cannot limit the inputs to Gaussian for Gaussian IC This does not imply that Gaussian inputs are not capacity-achieving for the Gaussian IC We may still find single-letter or a different multiletter expression the Gaussian IC in which Gaussian inputs are optimal (see [Cheng and Verdú 1993] for the MAC). M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

19 Preliminaries Counterexample Better Use Channel model Inner bound Outer bound Comparison Remarks In the multiletter capacity of the IC, we cannot limit the inputs to Gaussian for Gaussian IC This does not imply that Gaussian inputs are not capacity-achieving for the Gaussian IC We may still find single-letter or a different multiletter expression the Gaussian IC in which Gaussian inputs are optimal (see [Cheng and Verdú 1993] for the MAC). Q Can multiletter expression still be useful? Yes! if we can find a single letter expression or a multiletter expression for which Gaussian inputs are optimal. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

20 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs How to better use the multiletter expression? 1 Manipulate it to get a single-letter expression (e.g., IC in the strong interference [Costa and A. El Gamal 1987]) R 1 + R 2 1 n I (X n 1 ; Y n 1 ) + 1 n I (X n 2 ; Y n 2 ) (a) 1 n I (X n 1 ; Y n 1 X n 2 ) + 1 n I (X n 2 ; Y n 2 ) (b) 1 n I (X n 1 ; Y n 2 X n 2 ) + 1 n I (X n 2 ; Y n 2 ) (c) = 1 n n I (X 1i, X 2i ; Y 2i ) i=1 (a) is due to the independence of X n 1 and X n 2 (b) from the strong IC condition I (X 1 ; Y 1 X 2 ) I (X 1 ; Y 2 X 2 ) (c) by the memoryless property of the channel, and M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

21 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs How to better use the multiletter expression 2 We can write the multiletter expression as µr 1 + R 2 lim or equivalently, µr 1 + R 2 1 n 1 n n p(x1 n)p(xn 2 ) [ µi (X n 1 ; Y n 1 ) + I (X n 2 ; Y n 2 ) ] [ µi (X n 1 ; Y n 1 ) + I (X n 2 ; Y n 2 ) ], = 1 n [ µh(x n 1 + ax n 2 + Z n 1 ) µh( ax n 2 + Z n 1 ) + h(x n 2 + Z n 2 ) h(z n 2 ) ] 1 n U o. and find the optimal input for all µ. M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

22 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs Improving the Outer Bounds Finding the optimal inputs for the µ-sum rate is not easy! Instead, we may rewrite the terms inside the bracket as W = U o + h(z n 2 ) = µh(x n 1 + ax n 2 + Z n 1 ) µh( ax n 2 + Z n 1 ) + h(x n 2 + Z n 2 ) and try to outer bound it. 1 First Outer bound (trivial!): µh(x1 n + ax2 n + Z1 n ) µh( ax n }{{} 2 + Z n n 1 ) +h(x2 + Z2 n ) }{{}}{{} W 1 W 2 W 3 M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

23 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs Improving the Outer Bounds 2 Second Outer bound: µh(x1 n + ax2 n + Z1 n ) }{{} W 1 3 Third Outer bound (Conjecture!): µh( ax n 2 + Z1 n ) + h(x2 n + Z2 n ) } {{ } W 2 M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

24 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs Improving the Outer Bounds 2 Second Outer bound: µh(x1 n + ax2 n + Z1 n ) }{{} W 1 3 Third Outer bound (Conjecture!): h(x1 n + ax2 n + Z1 n ) + }{{} W 1 µh( ax n 2 + Z1 n ) + h(x2 n + Z2 n ) } {{ } W 2 (µ 1)h(X n 1 + ax n 2 + Z n 1 ) µh( ax n 2 + Z n 1 ) + h(x n 2 + Z n 2 ) }{{} W 2 M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

25 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs Summary Gaussian inputs are not sufficient to achieve the border of the multiletter capacity of the IC This does not imply that Gaussian inputs are not necessarily capacity-achieving for the Gaussian IC The multiletter capacity region still can be useful to improve outer bound M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar

26 Preliminaries Counterexample Better Use Simplify It Find its Optimal Inputs Thank you! M. Vaezi & H. V. Poor (Princeton) On Limiting Capacity Expressions Asilomar