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Capacity Bounds for Amplitude-Constrained AWGN MIMO Channels with Fading

• A. Favano1,2, M. Ferrari2, M. Magarini1, and L. Barletta1

1Politecnico di Milano, Milano, Italy, 2CNR-IEIIT, Milano, Italy

2

Motivation

Information capacity in two cases of practical interest

PER-ANTENNA

S/P

. . .

PA PA PA X1 X2 XN

TOTAL AMPLITUDE

S/P

. . .

PA

X1 X2 XN

3

System Model

The Input-Output relationship is

• Y =

H · X + Z (1) where Z ∼ CN(0, IN) and H is the channel matrix

• H =

  

• H1,1

. . .

• H1,N

. . . ... . . .

• HN,1

. . .

• HN,N

   .

4

System Model

Using SVD the MIMO model can be simplified Y = UH Y = UH

• U
• Λ
• λmax
• λmaxVH ·

X + Z

• = Λ · X + Z,

(2) where λmax is the largest element of Λ. The matrix Λ is Λ =    λ1 . . . . . . ... . . . . . . λN    , with λ1 = 1 and λ1 ≥ λ2 ≥ · · · ≥ λN.

Per-Antenna Amplitude Constraint

S/P

. . .

PA PA PA X1 X2 XN

6

Per-Antenna Constraint

Per-antenna amplitude constraint on the complex input vector X

• X ∈

X = Box( a) { x : |˜ xi| ≤ ai , i = 1, . . . , N}, (3) where a = ( a1, . . . , aN) ∈ RN

+ is the set of amplitude constraints.

• X1
• X2
• X3

2 a1 2 a2 2 a3

• X

7

Per-Antenna Constraint

We focus on channel matrices H = U ΛVH with VH = IN. After the SVD, the original per-antenna constraint is equivalent to X ∈ X = Box (a) , (4) with a = a λmax and ak = ak λmax = a, ∀k.

X1 X2 X3

2a 2a 2a

X

8

Per-Antenna Constraint

Constraint for complex-valued vector models = = = for vectorized real-valued models

X = X1 × X2 × · · · × XN

with Xk = B2 (a) {xk : |xk| ≤ a} for k = 1, . . . , N.

9

Per-Antenna Constraint

|xk| ≤ a not equivalent to Re{xk} ≤ a, Im{xk} ≤ a

Re{Xk} Im{Xk} a a > a

Xk

Per-Antenna Amplitude Constraint

Upper Bound

11

Per-Antenna Upper Bound

Assumptions The complex noise Z ∼ CN(0, IN) The channel fading matrix Λ is diagonal VH = IN An upper bound on the MIMO channel capacity is

CMIMO = max

FX: supp(FX)⊆X I (X ; Y)

(5) (full CSI) = max

FΛX: supp(FΛX)⊆ΛX I (ΛX ; Y)

(6) (Upper bound on MI) ≤ max

FΛX: supp(FΛX)⊆ΛX N

• k=1

I (λkXk ; Yk) (7) (Swap max and sum) ≤

N

• k=1

max

FλkXk : supp(FλkXk )⊆λkXk

I (λkXk ; Yk) =

N

• k=1

Ck. (8)

12

Per-Antenna Upper Bound

For the kth subchannel, the McKellips-Type upper bound from [1] is Ck ≤ Ck,PA = log

• 1 + λka
• π/2 + λ2

ka2

2e

• ,

(9) and the total MIMO upper bound is CMIMO ≤

N

• k=1

Ck ≤

N

• k=1

Ck,PA. (10)

[1] Thangaraj, Kramer, and B¨

• cherer, “Capacity Bounds for Discrete-Time,

Amplitude-Constrained, Additive White Gaussian Noise Channels,” TIT, 2017

Per-Antenna Amplitude Constraint

Lower Bound

14

Per-Antenna Lower Bound

Under the constraint X ∈ X the EPI lower bound is given by CEPI(N) = N log

• 1 + Vol (ΛX)

1 N

2πe

• ,

(11) and for the per-antenna we have

ΛX = λ1X1 × λ2X2 × · · · × λNXN with λkXk = B2 (λka)

Vol (ΛX) = Vol (λ1X1 × · · · × λNXN) =

N

• k=1

Vol (B2 (λka)) . (12)

15

Per-Antenna Lower Bound

From the two previous equations the total MIMO lower bound is CMIMO ≥ CPA(N, a) = N log  1 + N

i=1 λ

2 N

i

2e a2   . (13)

Per-Antenna Amplitude Constraint

refined Lower Bound

17

Per-Antenna refined Lower Bound

ΛX = λ1X1 × λ2X2 × · · · × λNXN

For a limited SNR, if λk → 0 we have lim

λk→0 Vol (λkXk) = 0

(14)

18

Per-Antenna refined Lower Bound

If a singular value approaches zero (e.g. λN → 0) the EPI lower bound becomes loose at low SNR

Vol(ΛX)

lim

λN→0 N

• k=1

Vol (λkXk) = 0 lim

λN→0 CPA(N, a) = 0

because the EPI lower bound is a volume-based bound

ΛX = λ1X1 × λ2X2 × · · · × λNXN

19

Per-Antenna refined Lower Bound

Given that Xk

1 (X1, X2, . . . , Xk)T and similarly for Yk 1, we have

CMIMO = max

FXN

1

I

• XN

1 ; YN 1

• (15)

(data processing inequality) ≥ max

FXk

1

I

• Xk

1 ; Yk 1

• ≥ CEPI(k).

(16) We define a new and refined bound, and we call it piecewise-EPI lower bound Cp-EPI max

k

(CEPI(k)) = max

k

(CPA(k, a)) . (17)

20

Per-Antenna refined Lower Bound

piecewise-EPI vs EPI lower bounds for a 2 × 2 real MIMO system with λ = (λ1, λ2)T = (1, 0.05)T 10 20 30 40 50 5 10 15 20 25 a2/(2N) [dB] Rate [bpcu] Cp-EPI CPA(2, a) CPA(1, a)

Per-Antenna Amplitude Constraint

Asymptotic Gap

22

Per-Antenna Asymptotic Gap

We compute the asymptotic gap between the upper and lower bounds to evaluate their tightness lim

a→∞ g(a) = lim a→∞ CPA(a) − CPA(N, a),

(18) and the result is lim

a→∞ g(a) = lim a→∞ N

• k=1

log

• λ2

ka2

2e

• − N log
• |Λ|

2 N a2

2e

• = 0.

(19)

Total Amplitude Constraint

S/P

. . .

PA

X1 X2 XN

24

Total Amplitude Constraint

For the total amplitude constraint we have

• X ∈

X = B2N

• a
• {

x : x ≤ a}, (20) where B2N

• a
• is a 2N-dimensional ball in R2N centered in 02N and with

radius a

• X2
• X3
• X1
• a
• X

25

Total Amplitude Constraint

Since VH X = X, the total amplitude constraint after the SVD is X ∈ X = B2N (a) , (21) with a = a λmax.

X2 X3 X1

a

X

26

Total Amplitude Constraint

Equivalent capacity with full CSI CMIMO = max

FX: supp(FX)⊆X I (X ; Y)

(22) (full CSI) = max

FΛX: supp(FΛX)⊆ΛX I (ΛX ; Y)

(23) Resulting constraint region ΛX

λ2X2 λ3X3 λ1X1

λ1a λ3a λ2a

ΛX

Total Amplitude Constraint

Upper Bound

28

Total Amplitude Upper Bound

Using an enlarged region S ⊇ ΛX provides CMIMO ≤ max

FΛX: supp(FΛX)⊆S I (ΛX ; Y) .

(24) We want a region S similar to ΛX simpler than ΛX

29

Total Amplitude Upper Bound

With S = B2N (λ1a) we can use McKellips-Type bound [1].

λ1x1 λ2x2 ΛX S λ1a λ2a λ1x1 λ2x2 ΛX S λ1a λ2a

but very spread singular values → very loose bound.

[1] Thangaraj, Kramer, and B¨

• cherer, “Capacity Bounds for Discrete-Time,

Amplitude-Constrained, Additive White Gaussian Noise Channels,” TIT, 2017

30

Total Amplitude Upper Bound

Given λ = (1, 1, 1/5)T, a better choice than Sb is Sc Sb = B3 (a) is Sc = B2 (a) × B1 (a/5)

ΛX Sb ΛX Sc

Is it still simple to evaluate the upper bound?

31

Total Amplitude Upper Bound

We want S to be a Cartesian Product S = S1 × S1 × S3 . . . ... . . . S3 S1 S2           Y1 Y2 Y3 Y4 Y5 Y6 Y7           =           λ1 . . . . . . λ7                     X1 X2 X3 X4 X5 X6 X7           +           Z1 Z2 Z3 Z4 Z5 Z6 Z7           We decompose the MIMO system in subsystems with independent constraints. CMIMO ≤

NS

• k=1

Ck

32

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

2D 3D 2D

S(p) = S(p)

1

× S(p)

2

× S(p)

3

⊇ ΛX

33

Total Amplitude Upper Bound

The optimal partition is such that

• = arg min

p Vol

• S(p)

. (25) The number of possible partitions grows rapidly with N.

34

Total Amplitude Upper Bound

We drastically reduce this number by noticing two properties of the optimal partition its subsets are always composed of consecutive singular values

λ1 λ2 λ3 λ4 λ5 λ1 λ2 λ3 λ4 λ5

if there are identical singular values they are always grouped together

λ1 . . . λk λk+1 . . . with λk = λk+1 λ1 . . . λk λk+1 . . .

35

Total Amplitude Upper Bound

Given the optimal partition o and the corresponding enlarged region S(o), the upper bound is

CMIMO = max

FΛX: supp(FΛX)⊆ΛX I (ΛX ; Y)

(26) (Enlarged region S(o)) ≤ max

FΛX: supp(FΛX)⊆S(o) I (ΛX ; Y)

(27) (Upper bound on MI) ≤ max

FΛX: supp(FΛX)⊆S(o) N(o)

• k=1

I

• Λ(o)

k X(o) k

; Y(o)

k

• (28)

(Swap max and sum) ≤

N(o)

• k=1

max I

• Λ(o)

k X(o) k

; Y(o)

k

• =

N(o)

• k=1

Ck, (29)

where Ck is the capacity of the kth subsystem.

36

Total Amplitude Upper Bound

Applying the McKellips-Type bound [1] CMcK(BN (a)) = log

• kN (a) + Vol (BN (a))

(2πe)N/2

• ,

(30) to each subsystem with constraint subregion S(o)

k

= λ(o)

k X (o) k

, induced by the optimal partition, we get CMIMO ≤ CTA =

N(o)

• k=1

Ck,TA =

N(o)

• k=1

CMcK

• λ(o)

k X (o) k

• .

(31)

[1] Thangaraj, Kramer, and B¨

• cherer, “Capacity Bounds for Discrete-Time,

Amplitude-Constrained, Additive White Gaussian Noise Channels,” TIT, 2017

Total Amplitude Constraint

Lower Bound

38

Total Amplitude Lower Bound

Under a total amplitude constraint the EPI lower bound is CTA(N, a) = N log    1 + N

i=1 λ

2 N

i πN Γ(N+1)a2N 1

N

2πe     , (32) and it is again possible to obtain a refined version via the piecewise-EPI lower bound Cp-EPI = max

k

(CEPI(k)) = max

k

(CTA(k, a)) . (33)

Total Amplitude Constraint

Asymptotic Gap

40

Total Amplitude Asymptotic Gap

The asymptotic gap for the optimal partition is go lim

a→∞ g(a) = f(λ, o).

(34) It is not possible to formulate it in a compact expression.

41

Total Amplitude Asymptotic Gap

Scatter plot of go versus N, with H entries drawn as Hi,j ∼ CN(0, 2), ∀i, j.

2 3 4 5 6 7 8 9 10 5 10 15 20 25 N go [bpcu] 95th percentile mean 5th percentile Channel realization

42

Total Amplitude Asymptotic Gap

We provide a simple upper bound to the optimal gap go ≤ g = min

• log

N

k=1 (λ1/λk)2

• , log (N!)
• .

(35) The bound g is already tighter than the best gaps found in the literature [2] gM − g ≥ 1 2 log (2πN) , (36) gD − g ≥ 0. (37)

[2] Dytso, Goldenbaum, Shamai, and Poor, “Upper and Lower Bounds on the Capacity of Amplitude-Constrained MIMO Channels,” GLOBECOM, 2017

Contributions

44

Contributions

We evaluated the asymptotic gap between upper and lower bounds For the per-antenna constraint the gap approaches zero For the total amplitude constraint we obtained a tighter gap compared to the best bounds in the literature.