Capacity Bounds for Amplitude-Constrained AWGN MIMO Channels with Fading A. Favano 1 , 2 , M. Ferrari 2 , M. Magarini 1 , and L. Barletta 1 1 Politecnico di Milano, Milano, Italy, 2 CNR-IEIIT, Milano, Italy
Motivation Information capacity in two cases of practical interest PER-ANTENNA TOTAL AMPLITUDE X 1 X 1 PA X 2 X 2 PA S/P PA S/P . . . . . . X N X N PA 2
System Model The Input-Output relationship is � Y = � H · � X + � Z (1) where � Z ∼ CN ( 0 , I N ) and � H is the channel matrix � � . . . H 1 , 1 H 1 , N � . . H = ... . . . . . � � . . . H N , 1 H N , N 3
System Model Using SVD the MIMO model can be simplified � � � λ max V H · � Y = U H � Λ � X + � Y = U H = Λ · X + Z , U Z (2) � λ max where � λ max is the largest element of � Λ. The matrix Λ is λ 1 . . . 0 . . Λ = ... , . . . . . . . λ N 0 with λ 1 = 1 and λ 1 ≥ λ 2 ≥ · · · ≥ λ N . 4
Per-Antenna Amplitude Constraint X 1 PA X 2 PA S/P . . . X N PA
Per-Antenna Constraint Per-antenna amplitude constraint on the complex input vector � X � X ∈ � X = Box ( � a ) � { � x : | ˜ x i | ≤ � a i , i = 1 , . . . , N } , (3) where � a = ( � a 1 , . . . , � a N ) ∈ R N + is the set of amplitude constraints . � X 3 � X 2 � a 3 � X 2 2 � a 1 2 � a 2 � X 1 6
Per-Antenna Constraint ΛV H with V H = I N . We focus on channel matrices � H = U � After the SVD, the original per-antenna constraint is equivalent to X ∈ X = Box ( a ) , (4) a � a k � with a = � λ max and a k = � λ max = a , ∀ k . X 3 X 2 a X 2 2 a X 1 2 a 7
Per-Antenna Constraint Constraint for complex-valued vector models � = � = � = for vectorized real-valued models X 1 × X 2 × · · · × X N X = with X k = B 2 ( a ) � { x k : | x k | ≤ a } for k = 1 , . . . , N . 8
Per-Antenna Constraint | x k | ≤ a not equivalent to Re { x k } ≤ a , Im { x k } ≤ a Im { X k } a > a X k a Re { X k } 9
Per-Antenna Amplitude Constraint Upper Bound
Per-Antenna Upper Bound Assumptions � The complex noise Z ∼ CN ( 0 , I N ) � The channel fading matrix Λ is diagonal � V H = I N An upper bound on the MIMO channel capacity is C MIMO = F X : supp ( F X ) ⊆X I ( X ; Y ) max (5) (full CSI) = F Λ X : supp ( F Λ X ) ⊆ Λ X I ( Λ X ; Y ) max (6) � N (Upper bound on MI) ≤ max I ( λ k X k ; Y k ) (7) F Λ X : supp ( F Λ X ) ⊆ Λ X k = 1 � � N N (Swap max and sum) ≤ max I ( λ k X k ; Y k ) = C k . (8) F λ kXk : supp ( F λ kXk ) ⊆ λ k X k k = 1 k = 1 11
Per-Antenna Upper Bound For the k th subchannel, the McKellips-Type upper bound from [1] is � � � π/ 2 + λ 2 k a 2 C k ≤ C k , PA = log 1 + λ k a , (9) 2 e and the total MIMO upper bound is � � N N C MIMO ≤ C k ≤ C k , PA . (10) k = 1 k = 1 [1] Thangaraj, Kramer, and B¨ ocherer, “Capacity Bounds for Discrete-Time, 12 Amplitude-Constrained, Additive White Gaussian Noise Channels,” TIT, 2017
Per-Antenna Amplitude Constraint Lower Bound
Per-Antenna Lower Bound Under the constraint X ∈ X the EPI lower bound is given by � � 1 1 + Vol ( Λ X ) N C EPI ( N ) = N log , (11) 2 π e and for the per-antenna we have × × · · · × with λ k X k = B 2 ( λ k a ) Λ X = λ 1 X 1 λ 2 X 2 λ N X N � N Vol ( Λ X ) = Vol ( λ 1 X 1 × · · · × λ N X N ) = Vol ( B 2 ( λ k a )) . (12) k = 1 14
Per-Antenna Lower Bound From the two previous equations the total MIMO lower bound is � N 2 i = 1 λ N 1 + . C MIMO ≥ C PA ( N , a ) = N log i a 2 (13) 2 e 15
Per-Antenna Amplitude Constraint refined Lower Bound
Per-Antenna refined Lower Bound × × · · · × Λ X = λ 1 X 1 λ 2 X 2 λ N X N For a limited SNR, if λ k → 0 we have λ k → 0 Vol ( λ k X k ) = 0 lim (14) 17
Per-Antenna refined Lower Bound If a singular value approaches zero (e.g. λ N → 0) the EPI lower bound becomes loose at low SNR Vol ( Λ X ) � N lim Vol ( λ k X k ) = 0 λ N → 0 C PA ( N , a ) = 0 lim λ N → 0 k = 1 because the EPI lower bound is a volume-based bound × × · · · × Λ X = λ 1 X 1 λ 2 X 2 λ N X N 18
Per-Antenna refined Lower Bound 1 � ( X 1 , X 2 , . . . , X k ) T and similarly for Y k Given that X k 1 , we have � � C MIMO = max 1 ; Y N X N I (15) 1 F X N 1 � � (data processing inequality) ≥ max 1 ; Y k ≥ C EPI ( k ) . X k I (16) 1 F X k 1 We define a new and refined bound , and we call it piecewise-EPI lower bound C p-EPI � max ( C EPI ( k )) = max ( C PA ( k , a )) . (17) k k 19
Per-Antenna refined Lower Bound 25 C p-EPI C PA ( 2 , a ) 20 C PA ( 1 , a ) Rate [bpcu] 15 piecewise-EPI vs EPI lower bounds for a 2 × 2 real MIMO system with 10 λ = ( λ 1 , λ 2 ) T = ( 1 , 0 . 05 ) T 5 0 0 10 20 30 40 50 a 2 / ( 2 N ) [dB] 20
Per-Antenna Amplitude Constraint Asymptotic Gap
Per-Antenna Asymptotic Gap We compute the asymptotic gap between the upper and lower bounds to evaluate their tightness a →∞ g ( a ) = lim lim a →∞ C PA ( a ) − C PA ( N , a ) , (18) and the result is � � � � � 2 N λ 2 | Λ | k a 2 N a 2 a →∞ g ( a ) = lim lim log − N log = 0 . (19) a →∞ 2 e 2 e k = 1 22
Total Amplitude Constraint X 1 X 2 PA S/P . . . X N
Total Amplitude Constraint For the total amplitude constraint we have �� � � X ∈ � X = B 2 N � { � x : � � x � ≤ � a } , a (20) �� � where B 2 N is a 2N-dimensional ball in R 2 N centered in 0 2 N and with a radius � a � X 3 � X � � X 2 a � X 1 24
Total Amplitude Constraint Since V H � X = � X , the total amplitude constraint after the SVD is X ∈ X = B 2 N ( a ) , (21) a � with a = � λ max . X 3 X X 2 a X 1 25
Total Amplitude Constraint Equivalent capacity with full CSI C MIMO = F X : supp ( F X ) ⊆X I ( X ; Y ) max (22) (full CSI) = F Λ X : supp ( F Λ X ) ⊆ Λ X I ( Λ X ; Y ) max (23) Resulting constraint region Λ X λ 3 X 3 Λ X λ 3 a λ 2 X 2 λ 1 a λ 2 a λ 1 X 1 26
Total Amplitude Constraint Upper Bound
Total Amplitude Upper Bound Using an enlarged region S ⊇ Λ X provides C MIMO ≤ F Λ X : supp ( F Λ X ) ⊆S I ( Λ X ; Y ) . max (24) We want a region S � similar to Λ X � simpler than Λ X 28
Total Amplitude Upper Bound With S = B 2 N ( λ 1 a ) we can use McKellips-Type bound [1]. λ 2 x 2 λ 2 x 2 S S λ 2 a λ 2 a Λ X λ 1 a λ 1 a Λ X λ 1 x 1 λ 1 x 1 but very spread singular values → very loose bound. [1] Thangaraj, Kramer, and B¨ ocherer, “Capacity Bounds for Discrete-Time, 29 Amplitude-Constrained, Additive White Gaussian Noise Channels,” TIT, 2017
Total Amplitude Upper Bound Given λ = ( 1 , 1 , 1 / 5 ) T , a better choice than S b is S c S b = B 3 ( a ) S c = B 2 ( a ) × B 1 ( a / 5 ) is S b S c Λ X Λ X Is it still simple to evaluate the upper bound? 30
Total Amplitude Upper Bound We want S to be a Cartesian Product S = S 1 × S 1 × S 3 λ 1 . . . Y 1 0 X 1 Z 1 S 1 X 2 Z 2 Y 2 Y 3 X 3 Z 3 . . ... = + S 2 . . X 4 Z 4 Y 4 . . Y 5 X 5 Z 5 X 6 Z 6 Y 6 S 3 . . . λ 7 Y 7 0 X 7 Z 7 We decompose the MIMO system in subsystems with independent constraints . � N S C MIMO ≤ C k k = 1 31
Total Amplitude Upper Bound We partition the MIMO dimensions in subspaces with similar singular values λ = ( 1 , 0 . 88 , 0 . 6 , 0 . 59 , 0 . 51 , 0 . 3 , 0 . 18 ) T 2 D 3 D 2 D S ( p ) = S ( p ) × S ( p ) × S ( p ) ⊇ Λ X 1 2 3 32
Total Amplitude Upper Bound The optimal partition is such that � S ( p ) � o = arg min p Vol . (25) The number of possible partitions grows rapidly with N . 33
Total Amplitude Upper Bound We drastically reduce this number by noticing two properties of the optimal partition � its subsets are always λ 1 λ 2 λ 3 λ 4 λ 5 composed of consecutive singular values λ 1 λ 2 λ 3 λ 4 λ 5 � if there are identical singular . . . . . . λ 1 λ k λ k + 1 values they are always with λ k = λ k + 1 grouped together . . . . . . λ 1 λ k λ k + 1 34
Total Amplitude Upper Bound Given the optimal partition o and the corresponding enlarged region S ( o ) , the upper bound is C MIMO = F Λ X : supp ( F Λ X ) ⊆ Λ X I ( Λ X ; Y ) max (26) (Enlarged region S ( o ) ) ≤ F Λ X : supp ( F Λ X ) ⊆S ( o ) I ( Λ X ; Y ) max (27) N ( o ) � � � Λ ( o ) k X ( o ) ; Y ( o ) (Upper bound on MI) ≤ max I (28) F Λ X : supp ( F Λ X ) ⊆S ( o ) k k k = 1 N ( o ) � � N ( o ) � � Λ ( o ) k X ( o ) ; Y ( o ) (Swap max and sum) ≤ max I = C k , (29) k k k = 1 k = 1 where C k is the capacity of the k th subsystem . 35
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