On Two-user Fading Gaussian Broadcast Channels. with Perfect Channel State Information at the Receivers. Daniela Tuninetti

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1 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 1 On Two-user Fading Gaussian Broadcast Channels with Perfect Channel State Information at the Receivers Daniela Tuninetti Mobile Communication Laboratory EPFL, CH-1015 Lausanne, Switzerland daniela.tuninetti@epfl.ch Joint work with Prof. Shlomo Shamai (Technion - Israel)

2 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 2 Motivation Fading Gaussian Broadcast channels (BS) model the down-link of wireless cellular systems with delay constraints much larger than the fading coherence time and no feedback available. The BC problem was formalized by Thomas Cover in Most of the available results on general BCs date back to the 70s. For the subsequent two decades few relevant works appeared. The BC problem regained attention recently in the unfaded multi-antenna Gaussian case (Caire-Shamai 2000).

3 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 3 The broadcast problem (Cover 1972) A general discrete memoryless two-user Broadcast channel without feedback consists of an input X alphabet, two output Y alphabets and Z, and a transition P probability YZjX; A ((2 nr 0 ; 2 nr y ; 2 nr z );n;p (n) e )-code consists of - three message W sets = [2 nr u ] u u = for f0;y;zg, - one W encoder W y W z! X n 0 ; - two Y decoders! W n W y 0 Z and! W n W z 0, and probability of error, (w with ;w y ;w z ) 0 uniformly distributed, (n) e = Pr[ bw 0 6= w 0 or bw y 6= w y or bw z 6= w z ]; P The capacity region is the convex closure of rates 0 ;R y ;R z ) 2 R 3 + for which there exists a sequence of (R nr 0 ; 2 nr y ; 2 nr z );n;p (n) e )-codes with P (n) e! 0. ((2

4 P (n) e P Y jx = X n Y n Z n P YZjX and P ZjX = P YZjX DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 4 DECODER Y ( c W 0 ; c Wy) (W 0 ; Wy; Wz) ENCODER P Y ZjX DECODER Z ( c W 0 ; c Wz)! 0 depends only on the (conditional) marginals Z Z z y The capacity region of a general BC is NOT known. If channel P Y jx is stronger (classification by Korner-Marton 1975) than P channel ZjX then...

5 V ο PV P XjV P Y jx P ZjX Y ο PY Z ο PZ DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 5 X The channel is said to be: = P ZjX, degraded: 9 e PY jz : P y ep Y jzp Y jx less noisy: I(V ; Z)» I(V ; Y ) for all P VX, more capable: I(X; Z)» I(X; Y ) for all P X Clearly fdegradedg ρ fless noisyg ρ fmore capableg ρ fgeneral BCg. (deg.:bergmans 1973, Gallager 1974), (l.n.: Korner-Marton 1975) (m.c.: A.El Gamal 1979) (general:???).

6 C (mc) = DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 6 Capacity region for more capable BCs (A.El Gamal 1979) If I(X; Z)» I(X; Y ) for all P X then 8 < R 0 + R y + R z» minfi(x; Y ) ; I(V ; Z) + I(X; Y jv )g R 0 + R z» I(V ; Z) where V! X! (Y;Z) and jvj» jx j + 2. : NB. The capacity is also known in the following cases: the channel has one deterministic component (Marton 1979, Gelanfd-Pinsker 1980) and R y = 0 or R z = 0, the so called degraded message set case, (Korner-Marton 1977). sum and product of reversely degraded BCs (A.El Gamal 1980).

7 I = >< >: R 0 + R y» I(WU; Y ) R 0 + R z» I(WV; Z) minfi(w ; Y );I(W ; Z)g+ DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 7 The best inner bound (Marton 1979) For a general two-user BC, the best inner bound is: 8 R 0 + R y + R z» ; Y jw ) + I(V ; ZjW ) I(U ; V jw ) +I(U (W;U;V)! X! (Y;Z) where. No bounds on the cardinality W; U; V of...

8 O y = 8 >< >: R y» I(X; Y ) DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 8 The best outer bound (Korner-Marton 1979) For a general two-user BC, the best outer bound is O = O y O z where (exchange the role of the users to obtain O z ) R z» I(V ; Z) y + R z» I(X; Y jv ) + I(V ; Z) R V! X! (Y;Z) where jvj» jx j + and 2.

9 C (deg) = C (mc) I y R y» I(X; Y jv ) = IjX=U; W=V DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 9 Remarks: If the channel is degraded, or less noisy, the capacity is 8 < : R 0 + R z» I(V ; Z) where V! X! (Y;Z) and jvj» jx j (Gallager 1974). In general C (deg) in neither an inner bound nor an outer bound! On the contrary, C (mc) is ALWAYS an inner bound, i.e., We shall refer to I y as the more-capable-like inner bound.

10 Y = AX + N y DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 10 AWGN BCs with RX-CSI only We consider complex-valued fading Gaussian BCs 8 < : Z = B X + N z - A and B: ergodic processes known at the receivers only, - N u ο N c (0;N): Gaussian noise at receiver u 2 fy; zg and - X: subject to the power constraint E[jXj 2 ]» P. The output of channel-y (resp. Z) is the pair (Y;A) (resp. (Z; B)). (W;U;V;X) are independent of A and B. In general, this channel is not more capable.

11 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 11 Example: jaj 2 2 f2; 16g and jbj 2 2 f4; 8g equiprobable C [bit/dim] Y: Gaussian input 0.2 Z: Gaussian input Y: Binary input Z: Binary input SNR=P/N0 [linear]

12 4 V X I y = 8 >< R z» C z (P ) C z (ffp ) DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 12 Remarks: Capacity for A and B constant (unfaded case) is achieved by 3 0 0;P Nc ο 5 1 ff p 3 1 ff p 4 ff 1 1 A for ff 2 [0; 1] The more-capable-like inner bound region (with R 0 = 0) computed for the above jointly Gaussian distribution reduces to [ R y + R z» C y (P ) ff2[0;1] >: R y + R z» C z (P ) + C y (ffp ) C z (ffp ) where C y (P ) = E log(1 + jaj 2 P=N ) Λ is the single user capacity which is achieved by X ο N c (0;P).

13 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 13 Certain authors claim (GLOBECOM 2000) that if C y (P ) C z (P ) then the degraded region, computed for jointly Gaussian input, is an inner bound for the general fading case! Now, with Gaussian input, stripping at the receiver with largest single user capacity is possible if y (P ) C z (P ) C y (ffp ) C z (ffp ) 8ff 2 [0; 1] C in other words, if d(ff) = C y (ffp ) C z (ffp ) as a function of ff 2 [0; 1] achieves its absolute maximum for ff = 1.

14 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 14 Example: jaj 2 2 f2; 16g and jbj 2 2 f4; 8g equiprobable Cy Cz [bit/dim] E[ A 2 ] = 9 and E[ B 2 ] = 6 E[log 2 A 2 ] = E[log 2 B 2 ] = 5/ SNR=P/N0 [linear]

15 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 15 Example: jaj 2 2 f2; 16g and jbj 2 2 f4; 8g equiprobable R z [bit/dim] R y +R z < C y (P/N) Degraded like region R y [bit/dim]; P/N=0.5

16 h = H(XjV )» H(X)» log(ßevar[x])» log(ßep ) = g(p ) DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 16 How far are we from capacity? The outer bound is of use if the union over ALL P VX is taken... IDEA (our converse!): [ [ [ = since PVX:E[jXj 2 ]»P h:h»g(p ) PVX:E[jXj 2 ]»P ; H(XjV )=h

17 power entropy ineq: e H(BXjV B) + e H(N zjv B) log PV PXjV PVXB=PB g = E H(Nz) e B[H(b XjV;B=b)] + e e E B[log(jBj 2 )]H(XjV )» jbj 2 eh(xjv ) g + N DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 17 A lower bound to the (conditional) entropy Similarly to Bergman s proof (1974): H(ZjV B) = log ψ! eg(n) + ße ße ineq: Jensen EB ße

18 H(ZjV B) c ( ; ) max: H N EB;V» PV PXjV PVXB=PB EB;V = h N + jbj 2 E[ff 2XjV (v)] i g DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 18 An upper bound to the (conditional) entropy h ff 2ZjV B (v; b) i g h N + jbj 2 ff 2XjV (v) i g ineq: Jensen EB» where ff 2 XjV (v) is the variance of the random variable XjV = v, i.e., Z fi fi 2 fi fi fi ff 2 XjV (v) = Z P XjV (xjv)jxj 2 fi fi P XjV (xjv)x fi The above chain holds with = iif XjV = v ο N c ( ;g 1 H(XjV ) ).

19 » EB» EA PVX:E[jXj 2 ]»P ;H(XjV )=h g N + jbj 2 P Λ EB» N + jaj 2 eh(xjv ) g» N + jbj 2 eh(xjv ) g g(n ) DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 19 Everything can be expressed in terms of H(XjV ): I(Z; V jb) = H(ZjB) H(ZjV B) ße I(X; Y jv A) = H(Y jv A) H(Y jxa) Hence, for fading AWGN BCs with RX-CSI only ße [ [ = PVX:XοN c (0;P ) ;XjV οn c ( ;g 1 (h)) [ [ = h:h»g(p ) ff2[0;1]:h=g(ffp )

20 O y = I y = 8 >< 8 >< R z» C z (P ) C z (ffp ) R z» C z (P ) C z (ffp ) DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 20 We get [ R y» C y (P ) ff2[0;1] >: and recall that R y + R z» C z (P ) + C y (ffp ) C z (ffp ) [ R y + R z» C y (P ) ff2[0;1] >: it follows that... R y + R z» C z (P ) + C y (ffp ) C z (ffp )

21 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 21 Main results Theorem 1. The degraded region computed for Gaussian input is the capacity region for fading AWGN BCs with RX-CSI for which the condition for successive cancellation is verified, i.e., = C y (ffp) C z (ffp) has absolute maximum for ff = 1. d(ff) Theorem 2. The more-capable region computed for Gaussian input is the capacity region for fading AWGN BCs with RX-CSI for d(ff) = C which (ffp) C z (ffp) 0 y for ff 2 [0; all 1]. Theorem 3. If the fading AWGN BCs with RX-CSI BC is more-capable then Gaussian input is optimal.

22 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 22 For fading AWGN BCs with RX-CSI, Theorem 2 comprises channels that are NOT more-capable, i.e., for which no coding theorem exists! 1.5 A 2 {2,16} equiprobable B 2 {4, 8} equiprobable P/N = 0.5 R z [bit/dim] R y [bit/dim] We are working the case d(ff) is not non-negative in ff 2 [0; 1] and has not its absolute maximum for ff = 1 (62 Th:1 [ Th:2)

23 DIMACS Workshop on Network Information Theory - March 2003 Daniela Tuninetti 23 Conclusions We consider a general fading AWGN BC with RX-CSI: 1. We first derive an inner bound region which is valid for any fading distribution and has the flavor of the more capable capacity region. 2. From the inner bound with Gaussian inputs: the best receiver (largest single user capacity) must decode jointly the two messages and successive cancellation might incur inherent degradation. 3. Then, by using the entropy power inequality, we show that Gaussian inputs exhaust Korner-Marton s outer bound region. 4. We conclude by stating conditions under which inner and outer bound meet, yielding thus the capacity region. 5. Our capacity result holds for channels that neither degraded NOR more capable.