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R EFLECTIONS ON THE G AUSSIAN B ROADCAST C HANNEL : P ROGRESS AND C HALLENGES Shlomo Shamai (Shitz) Department of Electrical Engineering TechnionIsrael Institute of Technology 2007 IEEE International Symposium on Information Theory June 26,

  1. Gaussian Scalar Broadcast Channel Capacity Region C APACITY R EGION Scalar Gaussian BC 0.35 0.3 Superposition coding + successive decoding � αP + σ 2 z 1 − αP 0.25 ✒ ✓ 1 + α P R y ≤ 1 1 + 2 log 0.2 σ 2 � y 2 log ✒ ✓ △ 0.15 ¯ 1 + ( 1 − α ) P R c + R z = 1 = R c + R z ≤ 1 P , y = 7 R z 2 log σ 2 α P + σ 2 z P TDMA z = 1 0.1 σ 2 0 ≤ α ≤ 1 . 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 � � R y = 1 1 + αP 2 log σ 2 y Achievability by superposition coding [Cover ’72]. X = X z + X y superposition coding, E ( X 2 z ) = ( 1 − α ) P , E ( X 2 y ) = α P . X z = X zc + X zz – carries the messages ( M c , M z ) , X y – carries the message ( M y ) . ⇒ noise level: α P + σ 2 @ receiver z = z = ⇒ decodes ( M c , M z ) . ⇒ decodes ( M c , M z ) and strips out X z = ⇒ noise level: @ receiver y = σ 2 y = ⇒ decodes ( M y ) . - superposition: interference removed @ receiver y . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 8 / 95

  2. Gaussian Scalar Broadcast Channel Capacity Region C APACITY R EGION Scalar Gaussian BC 0.35 0.3 Superposition coding + successive decoding � αP + σ 2 z 1 − αP 0.25 ✒ ✓ 1 + α P R y ≤ 1 1 + 2 log 0.2 σ 2 � y 2 log ✒ ✓ △ 0.15 ¯ 1 + ( 1 − α ) P R c + R z = 1 = R c + R z ≤ 1 P , y = 7 R z 2 log σ 2 α P + σ 2 z P TDMA z = 1 0.1 σ 2 0 ≤ α ≤ 1 . 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 � � R y = 1 1 + αP 2 log σ 2 y Achievability by superposition coding [Cover ’72]. X = X z + X y superposition coding, E ( X 2 z ) = ( 1 − α ) P , E ( X 2 y ) = α P . X z = X zc + X zz – carries the messages ( M c , M z ) , X y – carries the message ( M y ) . ⇒ noise level: α P + σ 2 @ receiver z = z = ⇒ decodes ( M c , M z ) . ⇒ decodes ( M c , M z ) and strips out X z = ⇒ noise level: @ receiver y = σ 2 y = ⇒ decodes ( M y ) . - superposition: interference removed @ receiver y . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 8 / 95

  3. Gaussian Scalar Broadcast Channel Capacity Region C APACITY R EGION Scalar Gaussian BC 0.35 0.3 Superposition coding + successive decoding � αP + σ 2 z 1 − αP 0.25 ✒ ✓ 1 + α P R y ≤ 1 1 + 2 log 0.2 σ 2 � y 2 log ✒ ✓ △ 0.15 ¯ 1 + ( 1 − α ) P R c + R z = 1 = R c + R z ≤ 1 P , y = 7 R z 2 log σ 2 α P + σ 2 z P TDMA z = 1 0.1 σ 2 0 ≤ α ≤ 1 . 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 � � R y = 1 1 + αP 2 log σ 2 y Achievability by superposition coding [Cover ’72]. X = X z + X y superposition coding, E ( X 2 z ) = ( 1 − α ) P , E ( X 2 y ) = α P . X z = X zc + X zz – carries the messages ( M c , M z ) , X y – carries the message ( M y ) . ⇒ noise level: α P + σ 2 @ receiver z = z = ⇒ decodes ( M c , M z ) . ⇒ decodes ( M c , M z ) and strips out X z = ⇒ noise level: @ receiver y = σ 2 y = ⇒ decodes ( M y ) . - superposition: interference removed @ receiver y . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 8 / 95

  4. Gaussian Scalar Broadcast Channel Capacity Region C APACITY R EGION Scalar Gaussian BC 0.35 0.3 Superposition coding + successive decoding � αP + σ 2 z 1 − αP 0.25 ✒ ✓ 1 + α P R y ≤ 1 1 + 2 log 0.2 σ 2 � y 2 log ✒ ✓ △ 0.15 ¯ 1 + ( 1 − α ) P R c + R z = 1 = R c + R z ≤ 1 P , y = 7 R z 2 log σ 2 α P + σ 2 z P TDMA z = 1 0.1 σ 2 0 ≤ α ≤ 1 . 0.05 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 � � R y = 1 1 + αP 2 log σ 2 y Achievability by superposition coding [Cover ’72]. X = X z + X y superposition coding, E ( X 2 z ) = ( 1 − α ) P , E ( X 2 y ) = α P . X z = X zc + X zz – carries the messages ( M c , M z ) , X y – carries the message ( M y ) . ⇒ noise level: α P + σ 2 @ receiver z = z = ⇒ decodes ( M c , M z ) . ⇒ decodes ( M c , M z ) and strips out X z = ⇒ noise level: @ receiver y = σ 2 y = ⇒ decodes ( M y ) . - superposition: interference removed @ receiver y . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 8 / 95

  5. Gaussian Scalar Broadcast Channel DPC D IRTY P APER C ODING (DPC) ENCODER DECODER ˆ M X Y M Σ Σ ( M, S n ) E ( X 2 ) < P Y n S ∼ N (0 , Q ) N ∼ N (0 , σ 2 ) - state { S n } available un-causally @ transmitter. [Gelfand-Pinsker, PCIT’80] – coding idea: binning. C = I ( U : Y ) − I ( U : S ) , P U , X , S , Y ; U − ( X , S ) − Y Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 9 / 95

  6. Gaussian Scalar Broadcast Channel DPC D IRTY P APER C ODING (DPC) Dirty Paper: [Costa, IT’83]: X � P U = X + α S , − S , α = P + N ⇒ C = 1 1 + P � � = 2 log σ 2 Extended to vectors ( X , S , N , Y ) [Yu-Sutivong-Julian-Cover-Chiang, ISIT’01]. Practical aspects of DP coding [Erez-Shamai-Zamir, IT’02], [Bennatan-Burstein-Caire-Shamai, IT’06], [Sun-Liveris-Stankovic-Xiong, ISIT’05]. ∗ Vector-perturbation [Peel-Hochwald-Swindlehurst, TCOM’05]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 9 / 95

  7. Gaussian Scalar Broadcast Channel DPC - Achievability A CHIEVABILITY BY “D IRTY -P APER C ODING ” (DPC) X = X z + X y X z = X zc + X zz – as in superposition coding conveys messages ( M c , M z ) E ( X 2 z ) = ( 1 − α ) P X y – conveys messages ( M y ) by DPC against the ‘interference’ X z accounting for additive noise σ 2 y . E ( X 2 � y ) = α P & X y − X z , Rates: ✒ 1 + ( 1 − α ) P ✓ R c + R z = 1 2 log α P + σ 2 z ✒ ✓ R y = 1 1 + α P 2 log σ 2 y ∗ DPC: interference for receiver – y removed @ transmitter. ∗ receiver y decodes also, in parallel, ( R c , R z ) . ∗ receiver z operates as in superposition coding. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 10 / 95

  8. Gaussian Scalar Broadcast Channel DPC - Achievability A CHIEVABILITY BY “D IRTY -P APER C ODING ” (DPC) X = X z + X y X z = X zc + X zz – as in superposition coding conveys messages ( M c , M z ) E ( X 2 z ) = ( 1 − α ) P X y – conveys messages ( M y ) by DPC against the ‘interference’ X z accounting for additive noise σ 2 y . E ( X 2 � y ) = α P & X y − X z , Rates: ✒ 1 + ( 1 − α ) P ✓ R c + R z = 1 2 log α P + σ 2 z ✒ ✓ R y = 1 1 + α P 2 log σ 2 y ∗ DPC: interference for receiver – y removed @ transmitter. ∗ receiver y decodes also, in parallel, ( R c , R z ) . ∗ receiver z operates as in superposition coding. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 10 / 95

  9. Gaussian Scalar Broadcast Channel EPI E NTROPY P OWER I NEQUALITY Converse by EPI [Bergmans, IT’74] EPI [Shannon, BSTJ’48], [Stam, IC’59],[Blachman, IT’65] Z n = X n + Y n ( X n , Y n ) independent n -component vectors given U (conditioned version). n h ( Z n | U ) ≥ e n h ( X n | U ) + e 2 2 2 n h ( Y n | U ) e Equality X n , Y n | U independent Gaussian with proportional covariance matrices ∗ . Proportionality always satisfied for n = 1 . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 11 / 95

  10. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  11. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  12. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  13. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  14. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  15. Gaussian Scalar Broadcast Channel Converse by EPI C ONVERSE BY EPI Converse by EPI (A. El Gamal, Lecture Notes, EE478) ∗ I ( U ; Z ) = h ( Z ) − h ( Z | U ) ∗ I ( X ; Y | U ) = h ( Y | U ) − h ( Y | U , X ) = h ( Y | U ) − h ( N y ) ∗ Z = X + N z = X + N y + N ∆ = Y + N ∆ . ❤ ✐ h ( Z ) ≤ 1 2 π e ( σ 2 z + P ) , equality X ∼ N ( 0 , P ) . 1 2 log 2 log [ 2 π e σ 2 1 z ] ≤ h ( Z | U ) ≤ h ( Z ) ≤ 1 2 log [ 2 π e ( σ 2 z + P )] 2 ⇒ h ( Z | U ) = 1 2 log [ 2 π e ( σ 2 = z + α P )] , 0 ≤ α ≤ 1 e 2 h ( Z | U ) ≥ e 2 h ( Y | U ) + e 2 h ( N ∆ ) EPI: 3 e 2 h ( Z | U ) − 2 π e ( σ 2 ✏ ✑ h ( Y | U ) ≤ 1 z − σ 2 = ⇒ y ) 2 log = 1 2 log [ 2 π e ( σ 2 y + α P )] ✏ ✑ 1 + ( 1 − α ) P ⇒ I ( U ; Z ) ≤ 1 = 2 log σ 2 z + α P ✏ ✑ ⇒ I ( X : Y | U ) ≤ 1 1 + α P = 2 log σ 2 y Classics of EPI (conditional version) applications: instrumental in the proof. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 12 / 95

  16. Gaussian Scalar Broadcast Channel I-MMSE I-MMSE The I-MMSE relation [Guo-Shamai-Verdú, IT’05]. Y = √ snr X + N − X Input signal. − Output signal. Y N − Gaussian noise ∼ N ( 0 , 1 ) . − Signal-to-Noise Ratio. snr d snr I ( X ; Y ) = 1 d 2mmse ( X : snr ) �� 2 � � mmse ( X : snr ) = E X − E X | Y . Generalization: Vectors, continuous time process [Guo-Shamai-Verdú, IT’05]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 13 / 95

  17. Gaussian Scalar Broadcast Channel I-MMSE: Examples I-MMSE - EXAMPLES I-MMSE: Gaussian Example: X ∼ N ( 0 , 1 ) . √ snr � 2 � 1 mmse ( X g : snr ) = E X − = 1 + snr Y 1 + snr , I ( X g ; Y ) = I g ( snr ) = 1 2 log ( 1 + snr ) . I-MMSE: Binary Example: X b = ± 1 , symmetric. ∞ 2 π tanh ( snr − √ snr y ) dy e − y 2 / 2 � mmse ( X b : snr ) = 1 − √ −∞ ∞ 2 π log cosh ( snr − √ snr y ) dy e − y 2 / 2 I ( X b : Y ) = I b ( snr ) = snr − � √ −∞ Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 14 / 95

  18. Gaussian Scalar Broadcast Channel I-MMSE: Examples I-MMSE - EXAMPLES I-MMSE: Gaussian Example: X ∼ N ( 0 , 1 ) . √ snr � 2 � 1 mmse ( X g : snr ) = E X − = 1 + snr Y 1 + snr , I ( X g ; Y ) = I g ( snr ) = 1 2 log ( 1 + snr ) . I-MMSE: Binary Example: X b = ± 1 , symmetric. ∞ 2 π tanh ( snr − √ snr y ) dy e − y 2 / 2 � mmse ( X b : snr ) = 1 − √ −∞ ∞ 2 π log cosh ( snr − √ snr y ) dy e − y 2 / 2 I ( X b : Y ) = I b ( snr ) = snr − � √ −∞ Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 14 / 95

  19. Gaussian Scalar Broadcast Channel I-MMSE d snr I ( X ; Y ) = 1 d 2mmse ( X : snr ) 1.2 Gaussian I g � snr � 1 0.8 Binary I b � snr � 0.6 0.4 0.2 Gaussian mmse � X g :snr � Binary mmse � X b :snr � 10 snr 2 4 6 8 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 15 / 95

  20. Gaussian Scalar Broadcast Channel mmse Properties mmse: Unique crossing point between MMSEs of a Gaussian and an arbitrary X variable. X be arbitrary zero mean: E ( X 2 ) = 1 . X g ∼ N ( 0 , 1 ) . = mmse ( √ ρ X g : snr ) − mmse ( X : snr ) △ ∆ mmse ( snr ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest number: ∆ mmse ( snr 0 ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Equality: X ∼ N ( 0 , 1 ) = ⇒ ∆ mmse ( snr ) ≡ 0 . Arbitrary: E ( X 2 ) → b 2 mmse ( X : b snr ) = mmse ( bX : snr ) . note: mmse ( √ ρ X g : snr ) = ρ 1 ∼ snr . 1 + ρ snr ρ snr ≫ 1 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 16 / 95

  21. Gaussian Scalar Broadcast Channel mmse Properties mmse: Unique crossing point between MMSEs of a Gaussian and an arbitrary X variable. X be arbitrary zero mean: E ( X 2 ) = 1 . X g ∼ N ( 0 , 1 ) . = mmse ( √ ρ X g : snr ) − mmse ( X : snr ) △ ∆ mmse ( snr ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest number: ∆ mmse ( snr 0 ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Equality: X ∼ N ( 0 , 1 ) = ⇒ ∆ mmse ( snr ) ≡ 0 . Arbitrary: E ( X 2 ) → b 2 mmse ( X : b snr ) = mmse ( bX : snr ) . note: mmse ( √ ρ X g : snr ) = ρ 1 ∼ snr . 1 + ρ snr ρ snr ≫ 1 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 16 / 95

  22. Gaussian Scalar Broadcast Channel mmse Properties mmse: Unique crossing point between MMSEs of a Gaussian and an arbitrary X variable. X be arbitrary zero mean: E ( X 2 ) = 1 . X g ∼ N ( 0 , 1 ) . = mmse ( √ ρ X g : snr ) − mmse ( X : snr ) △ ∆ mmse ( snr ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest number: ∆ mmse ( snr 0 ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Equality: X ∼ N ( 0 , 1 ) = ⇒ ∆ mmse ( snr ) ≡ 0 . Arbitrary: E ( X 2 ) → b 2 mmse ( X : b snr ) = mmse ( bX : snr ) . note: mmse ( √ ρ X g : snr ) = ρ 1 ∼ snr . 1 + ρ snr ρ snr ≫ 1 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 16 / 95

  23. Gaussian Scalar Broadcast Channel mmse Properties mmse: Unique crossing point between MMSEs of a Gaussian and an arbitrary X variable. X be arbitrary zero mean: E ( X 2 ) = 1 . X g ∼ N ( 0 , 1 ) . = mmse ( √ ρ X g : snr ) − mmse ( X : snr ) △ ∆ mmse ( snr ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest number: ∆ mmse ( snr 0 ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Equality: X ∼ N ( 0 , 1 ) = ⇒ ∆ mmse ( snr ) ≡ 0 . Arbitrary: E ( X 2 ) → b 2 mmse ( X : b snr ) = mmse ( bX : snr ) . note: mmse ( √ ρ X g : snr ) = ρ 1 ∼ snr . 1 + ρ snr ρ snr ≫ 1 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 16 / 95

  24. Gaussian Scalar Broadcast Channel mmse Properties mmse: Unique crossing point between MMSEs of a Gaussian and an arbitrary X variable. X be arbitrary zero mean: E ( X 2 ) = 1 . X g ∼ N ( 0 , 1 ) . = mmse ( √ ρ X g : snr ) − mmse ( X : snr ) △ ∆ mmse ( snr ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest number: ∆ mmse ( snr 0 ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Equality: X ∼ N ( 0 , 1 ) = ⇒ ∆ mmse ( snr ) ≡ 0 . Arbitrary: E ( X 2 ) → b 2 mmse ( X : b snr ) = mmse ( bX : snr ) . note: mmse ( √ ρ X g : snr ) = ρ 1 ∼ snr . 1 + ρ snr ρ snr ≫ 1 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 16 / 95

  25. Gaussian Scalar Broadcast Channel I-MMSE d snr I ( X ; Y ) = 1 d 2mmse ( X : snr ) Scaling: ρ = 0 . 8 1 0.8 Gaussian I g � Ρ snr � 0.6 Binary I b � snr � 0.4 mmse ����� � Ρ X g :snr � 0.2 mmse � X b :snr � snr 1 2 3 4 snr 0 snr z Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 17 / 95

  26. Gaussian Scalar Broadcast Channel mmse Properties U NIQUE C ROSSING P OINT : E XTENSION - X u be a zero mean RV dependent on U = u . - U – an arbitrary RV. - X g ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 . mmse ( √ ρ X g : snr ) − mmse ( X u : snr ) ∆ mmse ( snr , u ) = ∆ mmse ( snr ) = E U ∆ mmse ( snr , u ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest positive number: ∆ mmse ( snr ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Properties useful on their own: improved Gaussian based bounds on mmse ( X : snr ) , improved bounds on entropy via differential entropy. Proof Outline: [Guo-Shamai-Verdú’07] ✧ ✛ 2 ★ ✚✏ ✑ 2 d d snr mmse ( X : snr ) = − E E X − E ( X | Y ) | Y & Jensen. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 18 / 95

  27. Gaussian Scalar Broadcast Channel mmse Properties U NIQUE C ROSSING P OINT : E XTENSION - X u be a zero mean RV dependent on U = u . - U – an arbitrary RV. - X g ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 . mmse ( √ ρ X g : snr ) − mmse ( X u : snr ) ∆ mmse ( snr , u ) = ∆ mmse ( snr ) = E U ∆ mmse ( snr , u ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest positive number: ∆ mmse ( snr ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Properties useful on their own: improved Gaussian based bounds on mmse ( X : snr ) , improved bounds on entropy via differential entropy. Proof Outline: [Guo-Shamai-Verdú’07] ✧ ✛ 2 ★ ✚✏ ✑ 2 d d snr mmse ( X : snr ) = − E E X − E ( X | Y ) | Y & Jensen. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 18 / 95

  28. Gaussian Scalar Broadcast Channel mmse Properties U NIQUE C ROSSING P OINT : E XTENSION - X u be a zero mean RV dependent on U = u . - U – an arbitrary RV. - X g ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 . mmse ( √ ρ X g : snr ) − mmse ( X u : snr ) ∆ mmse ( snr , u ) = ∆ mmse ( snr ) = E U ∆ mmse ( snr , u ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest positive number: ∆ mmse ( snr ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Properties useful on their own: improved Gaussian based bounds on mmse ( X : snr ) , improved bounds on entropy via differential entropy. Proof Outline: [Guo-Shamai-Verdú’07] ✧ ✛ 2 ★ ✚✏ ✑ 2 d d snr mmse ( X : snr ) = − E E X − E ( X | Y ) | Y & Jensen. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 18 / 95

  29. Gaussian Scalar Broadcast Channel mmse Properties U NIQUE C ROSSING P OINT : E XTENSION - X u be a zero mean RV dependent on U = u . - U – an arbitrary RV. - X g ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 . mmse ( √ ρ X g : snr ) − mmse ( X u : snr ) ∆ mmse ( snr , u ) = ∆ mmse ( snr ) = E U ∆ mmse ( snr , u ) Given any snr 0 > 0 , let ρ ≤ 1 be the largest positive number: ∆ mmse ( snr ) = 0 . Then: d ∆ mmse ( snr ) ∆ mmse ( snr ) ≤ 0 , ≥ 0 , 0 ≤ snr < snr 0 d snr ∆ mmse ( snr ) ≥ 0 , snr 0 ≤ snr Properties useful on their own: improved Gaussian based bounds on mmse ( X : snr ) , improved bounds on entropy via differential entropy. Proof Outline: [Guo-Shamai-Verdú’07] ✧ ✛ 2 ★ ✚✏ ✑ 2 d d snr mmse ( X : snr ) = − E E X − E ( X | Y ) | Y & Jensen. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 18 / 95

  30. Gaussian Scalar Broadcast Channel Converse via I-MMSE P ROOF ON C ONVERSE – Gaussian Broadcast channel Z = √ snr z X + N z , Y = √ snr y X + N y . N y , N z ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 , snr y ≥ snr z capacity region: R y ≤ I ( X ; Y | U ) △ ¯ = R c + R z ≤ I ( U ; Z ) = I ( X , U ; Z ) − I ( X ; Z | U ) R z U − X − Y I ( X ; Z ) − I ( X ; Z | U ) = I ( X ; Z ) ≤ 1 2 log ( 1 + snr z ) . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 19 / 95

  31. Gaussian Scalar Broadcast Channel Converse via I-MMSE P ROOF ON C ONVERSE – Gaussian Broadcast channel Z = √ snr z X + N z , Y = √ snr y X + N y . N y , N z ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 , snr y ≥ snr z capacity region: R y ≤ I ( X ; Y | U ) △ ¯ = R c + R z ≤ I ( U ; Z ) = I ( X , U ; Z ) − I ( X ; Z | U ) R z U − X − Y I ( X ; Z ) − I ( X ; Z | U ) = I ( X ; Z ) ≤ 1 2 log ( 1 + snr z ) . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 19 / 95

  32. Gaussian Scalar Broadcast Channel Converse via I-MMSE P ROOF ON C ONVERSE – Gaussian Broadcast channel Z = √ snr z X + N z , Y = √ snr y X + N y . N y , N z ∼ N ( 0 , 1 ) , E ( X 2 ) = 1 , snr y ≥ snr z capacity region: R y ≤ I ( X ; Y | U ) △ ¯ = R c + R z ≤ I ( U ; Z ) = I ( X , U ; Z ) − I ( X ; Z | U ) R z U − X − Y I ( X ; Z ) − I ( X ; Z | U ) = I ( X ; Z ) ≤ 1 2 log ( 1 + snr z ) . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 19 / 95

  33. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS I ( X ; Z | U ) = E U I ( X ; Z | U = u ) snr z ❩ = 1 E U mmse ( X u : ν ) d ν 2 0 I ( X ; Y | U ) E U I ( X ; Y | U = u ) = snr y ❩ 1 = E U mmse ( X u : ν ) d ν 2 0 snr y ❩ = I ( X ; Z | U ) + E U mmse ( X u : ν ) d ν snr z Now, there is 0 ≤ α ≤ 1 snr z ❩ 1 2 log ( 1 + α snr z ) = 1 I ( X ; Z | U ) = E U mmse ( X u : u ) d ν 2 0 snr z ❩ α 1 = 1 + αν d ν 2 0 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 20 / 95

  34. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS I ( X ; Z | U ) = E U I ( X ; Z | U = u ) snr z ❩ = 1 E U mmse ( X u : ν ) d ν 2 0 I ( X ; Y | U ) E U I ( X ; Y | U = u ) = snr y ❩ 1 = E U mmse ( X u : ν ) d ν 2 0 snr y ❩ = I ( X ; Z | U ) + E U mmse ( X u : ν ) d ν snr z Now, there is 0 ≤ α ≤ 1 snr z ❩ 1 2 log ( 1 + α snr z ) = 1 I ( X ; Z | U ) = E U mmse ( X u : u ) d ν 2 0 snr z ❩ α 1 = 1 + αν d ν 2 0 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 20 / 95

  35. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS I ( X ; Z | U ) = E U I ( X ; Z | U = u ) snr z ❩ = 1 E U mmse ( X u : ν ) d ν 2 0 I ( X ; Y | U ) E U I ( X ; Y | U = u ) = snr y ❩ 1 = E U mmse ( X u : ν ) d ν 2 0 snr y ❩ = I ( X ; Z | U ) + E U mmse ( X u : ν ) d ν snr z Now, there is 0 ≤ α ≤ 1 snr z ❩ 1 2 log ( 1 + α snr z ) = 1 I ( X ; Z | U ) = E U mmse ( X u : u ) d ν 2 0 snr z ❩ α 1 = 1 + αν d ν 2 0 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 20 / 95

  36. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS I ( X ; Z | U ) = E U I ( X ; Z | U = u ) snr z ❩ = 1 E U mmse ( X u : ν ) d ν 2 0 I ( X ; Y | U ) E U I ( X ; Y | U = u ) = snr y ❩ 1 = E U mmse ( X u : ν ) d ν 2 0 snr y ❩ = I ( X ; Z | U ) + E U mmse ( X u : ν ) d ν snr z Now, there is 0 ≤ α ≤ 1 snr z ❩ 1 2 log ( 1 + α snr z ) = 1 I ( X ; Z | U ) = E U mmse ( X u : u ) d ν 2 0 snr z ❩ α 1 = 1 + αν d ν 2 0 Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 20 / 95

  37. Gaussian Scalar Broadcast Channel Converse via I-MMSE d snr I ( X ; Y ) = 1 d 2mmse ( X : snr ) Scaling: α = ρ = 0 . 8 1 0.8 Gaussian I g � Ρ snr � 0.6 Binary I b � snr � 0.4 mmse ����� � Ρ X g :snr � 0.2 mmse � X b :snr � snr 1 2 3 4 snr 0 snr z Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 21 / 95

  38. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS This implies that: α 0 ≤ snr ≤ snr 0 ≤ snr z E U mmse ( X u ; snr ) > 1 + α snr , α E U mmse ( X u ; snr ) < 1 + α snr , snr ≥ snr 0 α E U mmse ( X u ; snr 0 ) = 1 + α snr 0 Thus: α E U mmse ( X u ; snr ) < 1 + α snr , snr z < snr ≤ snr y snr y snr y 1 � E U mmse ( X u ; ν ) d ν ≤ 1 � α 1 + αν d ν 2 2 snr z snr z 1 2 log ( 1 + α snr y ) − 1 = 2 log ( 1 + α snr z ) Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 22 / 95

  39. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS This implies that: α 0 ≤ snr ≤ snr 0 ≤ snr z E U mmse ( X u ; snr ) > 1 + α snr , α E U mmse ( X u ; snr ) < 1 + α snr , snr ≥ snr 0 α E U mmse ( X u ; snr 0 ) = 1 + α snr 0 Thus: α E U mmse ( X u ; snr ) < 1 + α snr , snr z < snr ≤ snr y snr y snr y 1 � E U mmse ( X u ; ν ) d ν ≤ 1 � α 1 + αν d ν 2 2 snr z snr z 1 2 log ( 1 + α snr y ) − 1 = 2 log ( 1 + α snr z ) Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 22 / 95

  40. Gaussian Scalar Broadcast Channel Converse via I-MMSE I-MMSE E XPRESSIONS This implies that: α 0 ≤ snr ≤ snr 0 ≤ snr z E U mmse ( X u ; snr ) > 1 + α snr , α E U mmse ( X u ; snr ) < 1 + α snr , snr ≥ snr 0 α E U mmse ( X u ; snr 0 ) = 1 + α snr 0 Thus: α E U mmse ( X u ; snr ) < 1 + α snr , snr z < snr ≤ snr y snr y snr y 1 � E U mmse ( X u ; ν ) d ν ≤ 1 � α 1 + αν d ν 2 2 snr z snr z 1 2 log ( 1 + α snr y ) − 1 = 2 log ( 1 + α snr z ) Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 22 / 95

  41. Gaussian Scalar Broadcast Channel Converse via I-MMSE ⇒ I ( U ; Z ) = I ( X ; Z ) − I ( X ; Z | U ) � � 1 + ( 1 − α ) snr z 2 log ( 1 + snr z ) − 1 1 2 log ( 1 + α snr z ) = 1 ≤ 2 log 1 + α snr z 1 2 log ( 1 + α snr y ) − 1 I ( X : Y | U ) ≤ I ( X : Z | U ) + 2 log ( 1 + α snr z ) 1 = 2 log ( 1 + α snr y ) But MMSE is related to entropy [Guo-Shamai-Verdú, IT’05] ∞ � � � h ( X ) = 1 2 log ( 2 π e ) − 1 1 1 + ν − mmse ( X : ν ) d ν 2 0 and can be used elegantly to prove the EPI [Verdú-Guo, IT’06]. I-MMSE and EPI are related to de Bruijn’s identity ∂ h ( x + √ tN ) √ = 1 2 J ( X + tN ) ∂ t Yet the proof here is based on first principles, addressing only mutual information in a natural way. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 23 / 95

  42. Gaussian Scalar Broadcast Channel Converse via I-MMSE ⇒ I ( U ; Z ) = I ( X ; Z ) − I ( X ; Z | U ) � � 1 + ( 1 − α ) snr z 2 log ( 1 + snr z ) − 1 1 2 log ( 1 + α snr z ) = 1 ≤ 2 log 1 + α snr z 1 2 log ( 1 + α snr y ) − 1 I ( X : Y | U ) ≤ I ( X : Z | U ) + 2 log ( 1 + α snr z ) 1 = 2 log ( 1 + α snr y ) But MMSE is related to entropy [Guo-Shamai-Verdú, IT’05] ∞ � � � h ( X ) = 1 2 log ( 2 π e ) − 1 1 1 + ν − mmse ( X : ν ) d ν 2 0 and can be used elegantly to prove the EPI [Verdú-Guo, IT’06]. I-MMSE and EPI are related to de Bruijn’s identity ∂ h ( x + √ tN ) √ = 1 2 J ( X + tN ) ∂ t Yet the proof here is based on first principles, addressing only mutual information in a natural way. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 23 / 95

  43. Gaussian Scalar Broadcast Channel Fading Scalar BC F ADING S CALAR B ROADCAST C HANNEL Z i = H z , i X i + N z , i H y , i X i + N y , i , i -time index Y i = - { X i } – power limited input, E ( X 2 ) = P . - { N z , i } , { N y , i } – AWGN, E ( N 2 z ) = σ 2 z ≥ E ( N 2 y ) = σ 2 y . - { H z , i } , { H y , i } – ergodic fading processes known @ respective receivers. [Biglieri-Proakis-Shamai, IT’98], [Tuninetti-Shamai, DIMACS’04]. Symmetric fading H z ∼ H y ∼ H = ⇒ degraded BC. ⇒ Gaussian superposition codes = � 1 + | H | 2 ( 1 − α ) snr z � 1 snr z = P /σ 2 R c + R z ≤ E H , z , 2 log 1 + | H | 2 α snr z 1 � � 1 + | H | 2 α snr y snr y = P /σ 2 R y ≤ E H , y . 2 log Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 24 / 95

  44. Gaussian Scalar Broadcast Channel Fading Scalar BC F ADING S CALAR B ROADCAST C HANNEL Z i = H z , i X i + N z , i H y , i X i + N y , i , i -time index Y i = - { X i } – power limited input, E ( X 2 ) = P . - { N z , i } , { N y , i } – AWGN, E ( N 2 z ) = σ 2 z ≥ E ( N 2 y ) = σ 2 y . - { H z , i } , { H y , i } – ergodic fading processes known @ respective receivers. [Biglieri-Proakis-Shamai, IT’98], [Tuninetti-Shamai, DIMACS’04]. Symmetric fading H z ∼ H y ∼ H = ⇒ degraded BC. ⇒ Gaussian superposition codes = � 1 + | H | 2 ( 1 − α ) snr z � 1 snr z = P /σ 2 R c + R z ≤ E H , z , 2 log 1 + | H | 2 α snr z 1 � � 1 + | H | 2 α snr y snr y = P /σ 2 R y ≤ E H , y . 2 log Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 24 / 95

  45. Gaussian Scalar Broadcast Channel Fading Scalar BC F ADING S CALAR B ROADCAST C HANNEL Z i = H z , i X i + N z , i H y , i X i + N y , i , i -time index Y i = - { X i } – power limited input, E ( X 2 ) = P . - { N z , i } , { N y , i } – AWGN, E ( N 2 z ) = σ 2 z ≥ E ( N 2 y ) = σ 2 y . - { H z , i } , { H y , i } – ergodic fading processes known @ respective receivers. [Biglieri-Proakis-Shamai, IT’98], [Tuninetti-Shamai, DIMACS’04]. Symmetric fading H z ∼ H y ∼ H = ⇒ degraded BC. ⇒ Gaussian superposition codes = � 1 + | H | 2 ( 1 − α ) snr z � 1 snr z = P /σ 2 R c + R z ≤ E H , z , 2 log 1 + | H | 2 α snr z 1 � � 1 + | H | 2 α snr y snr y = P /σ 2 R y ≤ E H , y . 2 log Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 24 / 95

  46. Gaussian Scalar Broadcast Channel Fading Scalar BC: Challenges F ADING B ROADCAST C HANNEL Challenges: Fading BC: Degraded Capacity Region: R c + R z ≤ I ( U ; Z | H ) U − X − ( Y , H ) ≤ I ( X ; Y | U , H ) R y Is Gaussian ( U , X ) optimal as conjectured [Tuninetti-Shamai-Caire, ITA’07] ?? Problem: Jensen’s Penalty in EPI ✏ e E ( U ) + 1 ✑ ✏ e U + 1 ✑ log ≤ E log . Partial results: - On-Off ( 0 , 1 ) fading. - Finite state fading (uniformly degraded region), = ⇒ I-MMSE methodology. Other cases: - known transmitter CSI [Li-Goldsmith, IT’01] - more capable settings [Tuninetti-Shamai, DIMACS’04] - cases with one sided CSI [Agrawal-Cioffi, ALLERTON’06] - and others. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 25 / 95

  47. Gaussian Scalar Broadcast Channel Fading Scalar BC: Challenges F ADING B ROADCAST C HANNEL Challenges: Fading BC: Degraded Capacity Region: R c + R z ≤ I ( U ; Z | H ) U − X − ( Y , H ) ≤ I ( X ; Y | U , H ) R y Is Gaussian ( U , X ) optimal as conjectured [Tuninetti-Shamai-Caire, ITA’07] ?? Problem: Jensen’s Penalty in EPI ✏ e E ( U ) + 1 ✑ ✏ e U + 1 ✑ log ≤ E log . Partial results: - On-Off ( 0 , 1 ) fading. - Finite state fading (uniformly degraded region), = ⇒ I-MMSE methodology. Other cases: - known transmitter CSI [Li-Goldsmith, IT’01] - more capable settings [Tuninetti-Shamai, DIMACS’04] - cases with one sided CSI [Agrawal-Cioffi, ALLERTON’06] - and others. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 25 / 95

  48. Gaussian Scalar Broadcast Channel Fading Scalar BC: Challenges F ADING B ROADCAST C HANNEL Challenges: Fading BC: Degraded Capacity Region: R c + R z ≤ I ( U ; Z | H ) U − X − ( Y , H ) ≤ I ( X ; Y | U , H ) R y Is Gaussian ( U , X ) optimal as conjectured [Tuninetti-Shamai-Caire, ITA’07] ?? Problem: Jensen’s Penalty in EPI ✏ e E ( U ) + 1 ✑ ✏ e U + 1 ✑ log ≤ E log . Partial results: - On-Off ( 0 , 1 ) fading. - Finite state fading (uniformly degraded region), = ⇒ I-MMSE methodology. Other cases: - known transmitter CSI [Li-Goldsmith, IT’01] - more capable settings [Tuninetti-Shamai, DIMACS’04] - cases with one sided CSI [Agrawal-Cioffi, ALLERTON’06] - and others. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 25 / 95

  49. Gaussian Scalar Broadcast Channel Fading Scalar BC: Challenges F ADING B ROADCAST C HANNEL Challenges: Fading BC: Degraded Capacity Region: R c + R z ≤ I ( U ; Z | H ) U − X − ( Y , H ) ≤ I ( X ; Y | U , H ) R y Is Gaussian ( U , X ) optimal as conjectured [Tuninetti-Shamai-Caire, ITA’07] ?? Problem: Jensen’s Penalty in EPI ✏ e E ( U ) + 1 ✑ ✏ e U + 1 ✑ log ≤ E log . Partial results: - On-Off ( 0 , 1 ) fading. - Finite state fading (uniformly degraded region), = ⇒ I-MMSE methodology. Other cases: - known transmitter CSI [Li-Goldsmith, IT’01] - more capable settings [Tuninetti-Shamai, DIMACS’04] - cases with one sided CSI [Agrawal-Cioffi, ALLERTON’06] - and others. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 25 / 95

  50. Gaussian Scalar Broadcast Channel Fading Scalar BC: Challenges F ADING B ROADCAST C HANNEL Challenges: Fading BC: Degraded Capacity Region: R c + R z ≤ I ( U ; Z | H ) U − X − ( Y , H ) ≤ I ( X ; Y | U , H ) R y Is Gaussian ( U , X ) optimal as conjectured [Tuninetti-Shamai-Caire, ITA’07] ?? Problem: Jensen’s Penalty in EPI ✏ e E ( U ) + 1 ✑ ✏ e U + 1 ✑ log ≤ E log . Partial results: - On-Off ( 0 , 1 ) fading. - Finite state fading (uniformly degraded region), = ⇒ I-MMSE methodology. Other cases: - known transmitter CSI [Li-Goldsmith, IT’01] - more capable settings [Tuninetti-Shamai, DIMACS’04] - cases with one sided CSI [Agrawal-Cioffi, ALLERTON’06] - and others. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 25 / 95

  51. MIMO Gaussian Broadcast Channel The Model D OWNLINK C HANNEL OF A M ULTI -A NTENNA M OBILE S YSTEM y k = H k x k + n k , k = 1 ... K H k - Channel fading, n k ∼ CN ( 0 , N k ) - additive noise, y k - Received signals. Each user receives a different message! ( R c = 0 ) ! Possible average power constraint: E ( x † x ) ≤ P . Can we obtain an M-fold increase in throughput? In general not a degraded channel! Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 26 / 95

  52. MIMO Gaussian Broadcast Channel TDMA T IME D IVISION M ULTIPLE A CCESS TDMA: 2-User example 4 3.5 3 2.5 R 2 2 1.5 TDMA 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 R 1 No multiplicative increase in throughput compared to the single antenna transmitter. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 27 / 95

  53. MIMO Gaussian Broadcast Channel Beamforming B EAM -F ORMING AND Z ERO -F ORCING Beam-forming: 2-User example 4 3.5 3 2.5 R 2 2 Beam-forming 1.5 TDMA 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 R 1 A 2–fold increase in throughput (maximum sum-rate). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 28 / 95

  54. MIMO Gaussian Broadcast Channel Beamforming - Zero-Forcing B EAM -F ORMING AND Z ERO -F ORCING Sum-Rate: 2-User example 25 Zero-Forcing Region 20 Maximum Throughput 15 Optimal Beam-Forming for 2 users 10 5 Beam-Forming to best user 0 0 5 10 15 20 25 30 35 40 Transmit Power [dB] A 2–fold increase in throughput (maximum sum-rate). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 29 / 95

  55. MIMO Gaussian Broadcast Channel Dirty-Paper-Coding B EAM -F ORMING AND Z ERO -F ORCING Dirty Paper Coding: 2-User example 4 3.5 DPC 3 2.5 R 2 2 Beam-forming 1.5 TDMA 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 R 1 DPC [Caire-Shamai, IT’03]. DPC a must not an alternative to superposition! Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 30 / 95

  56. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  57. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  58. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  59. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  60. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  61. MIMO Gaussian Broadcast Channel Overview - Historical Perspective H ISTORICAL P ERSPECTIVE Non degraded = ⇒ open in general. The 2–User case ( K = 2 ) : [Caire-Shamai, IT’03] sum rate. - Costa DPC (achieves Marton’s region), [Marton, IT’79]. - Sato’s cooperated bound, [Sato, IT’78]. General M -antennas, K -Users sum-rate: [Tse-Viswanath, IT’03], [Vishwanath-Jindal-Goldsmith, IT’03] - MAC-Broadcast duality concepts. - An MMSE-DFE approach [Yu-Cioffi, IT’04]. Optimality of DPC under a Gaussian assumption. - Degraded Same Marginal Bound. [Tse-Viswanath, DIMACS’03], [Vishwanath-Kramer-Shamai-Jafar-Goldsmith, DIMACS’03] Capacity region: [Weingarten-Steinberg-Shamai, IT’06] - Optimality of DPC via the notion of an Enhanced Channel. Capacity region via extremal entropy inequalities [Liu-Viswanath, IT’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 31 / 95

  62. MIMO Gaussian Broadcast Channel Duality Concepts MIMO MAC C HANNEL M ODEL : D UALITY C ONCEPTS “Reciprocal” MIMO Gaussian MAC: � H † y = k x k + n k n ∼ CN ( 0 , N ) . Input constraints: individual transmit power, E [ x † k x k ] ≤ P k , k E [ x † total transmit power � k x k ] ≤ P . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 32 / 95

  63. MIMO Gaussian Broadcast Channel MAC MIMO MAC: C LASSICAL R ESULTS Capacity region (known from Cover-Wyner): ✭❳ ✥ ✦ ✮ I + N − 1 ❳ H † C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) = R k ≤ log det , ∀ A k P k H k k ∈A k ∈A Capacity region under sum-power constraint: - achieved by Gaussian codes, ❬ C mac ( P ; H 1 ,..., K , N ) = c . h . C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) P k P k ≤ P Polymatroid structure (Wyner-Cover pentagon): vertices π ✏ ✑ i ≤ k H † N + P det π i P π i H π i R π k = log ✏ ✑ i < k H † N + P det π i P π i H π i Vertices achieved by successive decoding and cancellation: MMSE-DFE interpretation (“Multiuser Detection”). Successive decoding order: π K , π K − 1 , . . . , π 1 . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 33 / 95

  64. MIMO Gaussian Broadcast Channel MAC MIMO MAC: C LASSICAL R ESULTS Capacity region (known from Cover-Wyner): ✭❳ ✥ ✦ ✮ I + N − 1 ❳ H † C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) = R k ≤ log det , ∀ A k P k H k k ∈A k ∈A Capacity region under sum-power constraint: - achieved by Gaussian codes, ❬ C mac ( P ; H 1 ,..., K , N ) = c . h . C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) P k P k ≤ P Polymatroid structure (Wyner-Cover pentagon): vertices π ✏ ✑ i ≤ k H † N + P det π i P π i H π i R π k = log ✏ ✑ i < k H † N + P det π i P π i H π i Vertices achieved by successive decoding and cancellation: MMSE-DFE interpretation (“Multiuser Detection”). Successive decoding order: π K , π K − 1 , . . . , π 1 . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 33 / 95

  65. MIMO Gaussian Broadcast Channel MAC MIMO MAC: C LASSICAL R ESULTS Capacity region (known from Cover-Wyner): ✭❳ ✥ ✦ ✮ I + N − 1 ❳ H † C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) = R k ≤ log det , ∀ A k P k H k k ∈A k ∈A Capacity region under sum-power constraint: - achieved by Gaussian codes, ❬ C mac ( P ; H 1 ,..., K , N ) = c . h . C mac ( P 1 , . . . , P K ; H 1 ,..., K , N ) P k P k ≤ P Polymatroid structure (Wyner-Cover pentagon): vertices π ✏ ✑ i ≤ k H † N + P det π i P π i H π i R π k = log ✏ ✑ i < k H † N + P det π i P π i H π i Vertices achieved by successive decoding and cancellation: MMSE-DFE interpretation (“Multiuser Detection”). Successive decoding order: π K , π K − 1 , . . . , π 1 . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 33 / 95

  66. MIMO Gaussian Broadcast Channel DPC Achievable Region DPC A CHIEVABLE R EGION OF THE MIMO BC Let S ∈ S + be an input covariance constraint. The region R dpc ( S ; H 1 ,..., K , N 1 ,..., K ) � �� � �   H † det N π k + H π k i ≤ k B π i π k   � � = c . h .  R : R π k ≤ log � � H † �� � det N π k + H π k i < k B π i π π k  P k B k ≤ S is achievable by DPC. Achieved by individual Gaussian coding with input covariance matrices B k . While coding for user π k , invoke Costa precoding to account all users π i with i > k . - Successive precoding order: π K , π K − 1 , . . . , π 1 . R dpc ( P ; H 1 ,..., K , N 1 ,..., K ) = � tr ( S ) ≤ P R dpc ( S ; H 1 ,..., K , N 1 ,..., K ) . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 34 / 95

  67. MIMO Gaussian Broadcast Channel Duality Concepts D UALITY C ONCEPTS Two−user MIMO−BC capacity region 3.5 User 1 encoded last 3 Dominant face (maximum sum rate) 2.5 2 User 1 encoded first R 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 R dpc ( P ; H 1 ,..., K ) = C mac ( P ; H † 1 ,..., K ) BC region via convex-hull of MAC regions. Power allocation and optimal receivers (MMSE-DFE) for the reciprocal MAC are easy to compute. General method: solve the dual MAC and map back the solution to the MIMO BC. [Yu, IT’06]. Duality: min over noise covariance under diagonal based constraints – accounts for linear input constraints, i.e. individual powers per antenna. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 35 / 95

  68. MIMO Gaussian Broadcast Channel Duality Concepts D UALITY C ONCEPTS Two−user MIMO−BC capacity region 3.5 User 1 encoded last 3 Dominant face (maximum sum rate) 2.5 2 User 1 encoded first R 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 R dpc ( P ; H 1 ,..., K ) = C mac ( P ; H † 1 ,..., K ) BC region via convex-hull of MAC regions. Power allocation and optimal receivers (MMSE-DFE) for the reciprocal MAC are easy to compute. General method: solve the dual MAC and map back the solution to the MIMO BC. [Yu, IT’06]. Duality: min over noise covariance under diagonal based constraints – accounts for linear input constraints, i.e. individual powers per antenna. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 35 / 95

  69. MIMO Gaussian Broadcast Channel Capacity Region [W EINGARTEN -S TEINBERG -S HAMAI , IT’06] n h ( X + Y ) ≥ e n h ( X ) + e n h ( Y ) tight only for ( X , Y ) Gaussians, 2 2 2 Vector EPI e with proportional covariances! Why not EPI a la Bergmans? � � E ( XX † ) � S Optimality for given covariance constraint . Optimality for square invertible H k . Aligned MIMO BC – canonic form: y k = x + n k , n k ∼ CN ( 0 , N k ) , k = 1 , 2 . . . K . Enhanced Channel: y ′ k = x + n ′ k , k = 1 , . . . , K . The y ′ k channel is an enhanced version of the y k channel if N ′ k � N k ∀ k . Clearly, the capacity of the { y ′ k } channel is larger than that of the { y k } channel. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 36 / 95

  70. MIMO Gaussian Broadcast Channel Capacity Region [W EINGARTEN -S TEINBERG -S HAMAI , IT’06] n h ( X + Y ) ≥ e n h ( X ) + e n h ( Y ) tight only for ( X , Y ) Gaussians, 2 2 2 Vector EPI e with proportional covariances! Why not EPI a la Bergmans? � � E ( XX † ) � S Optimality for given covariance constraint . Optimality for square invertible H k . Aligned MIMO BC – canonic form: y k = x + n k , n k ∼ CN ( 0 , N k ) , k = 1 , 2 . . . K . Enhanced Channel: y ′ k = x + n ′ k , k = 1 , . . . , K . The y ′ k channel is an enhanced version of the y k channel if N ′ k � N k ∀ k . Clearly, the capacity of the { y ′ k } channel is larger than that of the { y k } channel. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 36 / 95

  71. MIMO Gaussian Broadcast Channel Capacity Region P ROOF I DEA FOR THE N ON -D EGRADED G AUSSIAN V ECTOR C HANNEL DPC rate region of a two user 4 × 4 AMBC 3 2.5 Supporting Hyperplane Rate region of an 2 Enhanced and Degraded channel R 2 1.5 1 Dirty Paper Coding Region External point 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 Step 1: for every point R / ∈ R dpc ( S ; N 1 ,..., K ) , there exists an Enhanced aligned degraded MIMO BC whose DPC region outer bounds the original capacity region and does not contain R . Step 2: the capacity region of an Aligned degraded MIMO BC coincides with its DPC region, covariances at the tangential point satisfy equality in vector EPI. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 37 / 95

  72. MIMO Gaussian Broadcast Channel Capacity Region P ROOF I DEA FOR THE N ON -D EGRADED G AUSSIAN V ECTOR C HANNEL DPC rate region of a two user 4 × 4 AMBC 3 2.5 Supporting Hyperplane Rate region of an 2 Enhanced and Degraded channel R 2 1.5 1 Dirty Paper Coding Region External point 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 Step 1: for every point R / ∈ R dpc ( S ; N 1 ,..., K ) , there exists an Enhanced aligned degraded MIMO BC whose DPC region outer bounds the original capacity region and does not contain R . Step 2: the capacity region of an Aligned degraded MIMO BC coincides with its DPC region, covariances at the tangential point satisfy equality in vector EPI. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 37 / 95

  73. MIMO Gaussian Broadcast Channel Capacity Region P ROOF I DEA FOR THE N ON -D EGRADED G AUSSIAN V ECTOR C HANNEL DPC rate region of a two user 4 × 4 AMBC 3 2.5 Supporting Hyperplane Rate region of an 2 Enhanced and Degraded channel R 2 1.5 1 Dirty Paper Coding Region External point 0.5 0 0 0.5 1 1.5 2 2.5 3 R 1 Step 1: for every point R / ∈ R dpc ( S ; N 1 ,..., K ) , there exists an Enhanced aligned degraded MIMO BC whose DPC region outer bounds the original capacity region and does not contain R . Step 2: the capacity region of an Aligned degraded MIMO BC coincides with its DPC region, covariances at the tangential point satisfy equality in vector EPI. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 37 / 95

  74. MIMO Gaussian Broadcast Channel Cellular Downlink A PPLICATION : C ELLULAR D OWNLINK – THE W YNER MODEL [S OMEKH -Z AIDEL -S HAMAI , SPWC’05, AR X IV ’07] Cell-Site� Cell-Site� 2� 3� Cell-Site� Cell-Site� 1� 0� a� b� 3,r� b� 0,r� a� 1,k� 0,k� Cell 1� Cell 3� User r� User k� Cell 0� A “Wyner-type” multi-cell model with M cells ordered on a circle . Motivation: symmetry properties, more amenable to analytical analysis, equivalent to linear models for M ≫ 1 . A fully synchronous, optimally coded system is assumed, with cell-sites located at the cells’ boundaries. There are K users in each cell, and a single receive/transmit antenna at each cell-site. Each user “sees” only the two nearest cell-sites. Models a practical “soft-handoff” scenario at the cells’ boundaries. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 38 / 95

  75. MIMO Gaussian Broadcast Channel Cellular Downlink D OWNLINK S YSTEM M ODEL The received MK × 1 signal vector, is given by y dl = H † M x dl + n dl . H M [ M × KM ] - Channel transfer matrix. x dl [ M × 1 ] - The vector of signals transmitted by the M cell-sites. An equal individual per-cell-site power constraint is assumed: � � �� x dl x † ( m , m ) ≤ ¯ ∀ m . E P dl n dl [ MK × 1 ] ∼ N c ( 0 , I MK ) - Circularly symmetric AWGN vector. Full CSI is available to the joint multi-cell transmitter only. The mobile receivers are assumed to be cognisant of their own CSI, and of the employed transmission strategy. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 39 / 95

  76. MIMO Gaussian Broadcast Channel Cellular Downlink D OWNLINK A VERAGE P ER -C ELL S UM -R ATE C APACITY Using MIMO-Broadcast-MAC (minmax) duality [Yu, IT’06] the average per-cell sum-rate capacity is: � �  H M D M H †  det M + Λ M 1   C dl (¯ P ) = E H M M min Λ M max D M log  . det ( Λ M )  The optimization is over all nonnegative diagonal matrices: D M [ MK × MK ] , s.t. Tr ( D M ) ≤ 1 , Λ M [ M × M ] , s.t. Tr ( Λ M ) ≤ 1 / ¯ P . Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 40 / 95

  77. MIMO Gaussian Broadcast Channel Cellular Downlink D OWNLINK - N O -F ADING For non-fading channels a m , k = b m , k = 1 , ∀ m , k . - The channel transfer matrix becomes “block-circulant”. Average per-cell downlink sum-rate capacity ( M → ∞ ) is: √ � � 1 + 2 ¯ 1 + 4 ¯ P + P C dl-nf (¯ P ) = log . 2 - with either average or per cell power constraint and ∀ k . Other subsequent results: [Foschini-Huang-Karakayali-Valenzuela-Venkatesan, CISS’05], [Liang-Goldsmith, GLOBECOM’06], [Jing-Tse-Hou-Soriaga-Smee-Padovani, ITA’07]. Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 41 / 95

  78. MIMO Gaussian Broadcast Channel Cellular Downlink C ELLULAR B ROADCAST C HANNEL M ODELS : C HALLENGES Fading Models: Bounds in [Somekh-Zaidel-Shamai, arXiv’07]. ∗ Limiting eigenvalue distribution of finite diagonal HH † . Planar and general Wyner-like fading models. Limited multi-cell processing: cognition, back-haul rate limitations. Partial results in [Somekh-Zaidel-Shamai, arXiv’07], [Lapidoth-Shamai-Wigger, ISIT’07], [Marsch-Fettweis, EW’07], [Sanderovich-Somekh-Shamai, ISIT’07]. Feedback and impact of inaccurate CSI (to be discussed next). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 42 / 95

  79. MIMO Gaussian Broadcast Channel Cellular Downlink C ELLULAR B ROADCAST C HANNEL M ODELS : C HALLENGES Fading Models: Bounds in [Somekh-Zaidel-Shamai, arXiv’07]. ∗ Limiting eigenvalue distribution of finite diagonal HH † . Planar and general Wyner-like fading models. Limited multi-cell processing: cognition, back-haul rate limitations. Partial results in [Somekh-Zaidel-Shamai, arXiv’07], [Lapidoth-Shamai-Wigger, ISIT’07], [Marsch-Fettweis, EW’07], [Sanderovich-Somekh-Shamai, ISIT’07]. Feedback and impact of inaccurate CSI (to be discussed next). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 42 / 95

  80. MIMO Gaussian Broadcast Channel Cellular Downlink C ELLULAR B ROADCAST C HANNEL M ODELS : C HALLENGES Fading Models: Bounds in [Somekh-Zaidel-Shamai, arXiv’07]. ∗ Limiting eigenvalue distribution of finite diagonal HH † . Planar and general Wyner-like fading models. Limited multi-cell processing: cognition, back-haul rate limitations. Partial results in [Somekh-Zaidel-Shamai, arXiv’07], [Lapidoth-Shamai-Wigger, ISIT’07], [Marsch-Fettweis, EW’07], [Sanderovich-Somekh-Shamai, ISIT’07]. Feedback and impact of inaccurate CSI (to be discussed next). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 42 / 95

  81. MIMO Gaussian Broadcast Channel Cellular Downlink C ELLULAR B ROADCAST C HANNEL M ODELS : C HALLENGES Fading Models: Bounds in [Somekh-Zaidel-Shamai, arXiv’07]. ∗ Limiting eigenvalue distribution of finite diagonal HH † . Planar and general Wyner-like fading models. Limited multi-cell processing: cognition, back-haul rate limitations. Partial results in [Somekh-Zaidel-Shamai, arXiv’07], [Lapidoth-Shamai-Wigger, ISIT’07], [Marsch-Fettweis, EW’07], [Sanderovich-Somekh-Shamai, ISIT’07]. Feedback and impact of inaccurate CSI (to be discussed next). Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 42 / 95

  82. MIMO Gaussian Broadcast Channel Common Rate C HALLENGES – C OMMON R ATE y 1 M 1 , ˆ ˆ H 1 + Decoder 1 M c ( t × r 1 ) Encoder n 1 x M 1 , M 2 , M c ( R 1 , R 2 , R c ) y 2 + M 2 , ˆ ˆ H 2 Decoder 2 M c ( t × r 2 ) n 2 y k = H k x + n k , k = 1 , 2 . . . K ( K = 2) E xx † ≤ S n k ∼ N (0 , I ) , What is the capacity region ( C C ( S ) ) ? Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 43 / 95

  83. MIMO Gaussian Broadcast Channel Common Rate A CHIEVABLE R ATES – [J INDAL -G OLDSMITH , ISIT’04] Allocate powers Q 1 + Q 2 + Q c � S ( K = 2 ) . R 12 ( Q 1 , Q 2 , Q c ) = the set of all ( R 1 , R 2 , R c ) s.t. Common Message - Gaussian coding: log | H i Q c H T i + ( H i ( Q 1 + Q 2 ) H T � i + I ) | � R c ≤ min | H i ( Q 1 + Q 2 ) H T i + I | i = 1 , 2 Private Message #2 - Gaussian coding and successive cancellation decoding: R 2 ≤ log | H 2 Q 2 H T 2 + ( H 2 Q 1 H T 2 + I ) | | H 2 Q 1 H T 2 + I | Private Message #1 - Dirty paper coding: R 1 ≤ log | H 1 Q 1 H T 1 + I | Shlomo Shamai (Technion) Gaussian Broadcast Channel ISIT 2007 44 / 95

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