Lecture 3 Capacity of Multiuser Gaussian Channels The Gaussian - - PDF document

Lecture 3

Capacity of Multiuser Gaussian Channels

• The Gaussian uplink: 6.1
• The fading Gaussian uplink: 6.3 (parts)
• The Gaussian downlink: 6.2
• The fading Gaussian downlink: 6.4 (parts)

Mikael Skoglund, Theoretical Foundations of Wireless 1/20

The Gaussian Uplink

α1 α2 β W1 W2 x(1)

m

x(2)

m

ym f(y|x(1), x(2)) ˆ W1 ˆ W2

• Study the case of two users, for simplicity.
• The information-theoretic multiple access channel, with transition

density f(y|x(1), x(2))

Mikael Skoglund, Theoretical Foundations of Wireless 2/20

• The Gaussian multiple access channel: ym = x(1)

m + x(2) m + wm,

where {wm} is i.i.d complex Gaussian CN(0, σ2)

• Coding:
• Data: W1 ∈ I1 = {1, . . . , M1} and W2 ∈ I2 = {1, . . . , M2},
• uniformly distributed and independent
• Encoders: α1 : I1 → Cn and α2 : I2 → Cn
• Power constraints:

1 n

n

X

m=1

|x(1)

m |2 ≤ P1,

1 n

n

X

m=1

|x(2)

m |2 ≤ P2

• Rates: R1 = log M1/n and R2 = log M2/n
• Decoder: β : Cn → I1 × I2, β(yn

1 ) = ( ˆ

W1, ˆ W2)

• Error probability:

P (n)

e

= Pr ` ( ˆ W1, ˆ W2) = (W1, W2) ´

Mikael Skoglund, Theoretical Foundations of Wireless 3/20

• Capacity: Two (or more) rates, R1 and R2 =

⇒ cannot consider

• ne maximum achievable rate =

⇒ study sets of achievable rate-pairs (R1, R2),

• achievable rate-pair: (R1, R2) is achievable if (α1, α2, β)n exist such

that P (n)

e

→ 0 as n → ∞

• capacity region:

C = the closure of the set of all achievable (R1, R2)

Mikael Skoglund, Theoretical Foundations of Wireless 4/20

• Capacity-region of the Gaussian uplink: (R1, R2) ∈ C if and only

if R1 ≤ log

• 1 + P1

σ2

• R2 ≤ log
• 1 + P2

σ2

• R1 + R2 ≤ log
• 1 + P1 + P2

σ2

• Mikael Skoglund,

Theoretical Foundations of Wireless 5/20

• The two-user Gaussian multiple access region (figure from the textbook).

Noise variance σ2 = N0.

Mikael Skoglund, Theoretical Foundations of Wireless 6/20

• Achieving capacity:
• Independent ’Gaussian codebooks’ C1 and C2, rates R1 and R2,

powers P1 and P2

• Users encode W1 and W2 independently using C1 and C2
• To achieve point ’B’ in the figure, use interference cancellation,
• decode user 1 treating the codeword of user 2 as noise
• subtract the codeword of user 1
• decode user 2
• Change order to achieve ’A’
• Points on the segment AB achieved by time sharing
• The points on AB are ’optimal’

Mikael Skoglund, Theoretical Foundations of Wireless 7/20

• Orthogonal multiple access; take TDMA for simplicity, let

α ∈ [0, 1]:

• user 1 uses the channel a fraction α of time
• user 2 uses the channel a fraction (1 − α) of time

gives the region R1 ≤ α log

• 1 + P1

ασ2

• R2 ≤ (1 − α) log
• 1 +

P2 (1 − α)σ2

• Any orthogonal scheme will give an equivalent expression

Mikael Skoglund, Theoretical Foundations of Wireless 8/20

R1 R2

T/F-DMA general Capacity region for P1 = P2. Note that T/F-DMA is only optimal when α/(1 − α) = P1/P2.

Mikael Skoglund, Theoretical Foundations of Wireless 9/20

• K-users,
• capacity region: straightforward generalization. . .
• sum capacity: some of the rates

K

X

k=1

Rk < Csum = log „ 1 + P

k Pk

σ2 « are always achievable ⇒ sum rates < Csum achievable

• symmetric capacity: Csym = largest R such that

R1 = R2 = · · · = RK = R are in the capacity region. With P1 = P2 = · · · = PK = P we get Csym = 1 K log „ 1 + KP σ2 «

• can be achieved with orthogonal access

Mikael Skoglund, Theoretical Foundations of Wireless 10/20

The Fading Gaussian Uplink

• Slow fading, perfect CSIR, no CSIT:
• Received signal

ym = h1x(1)

m + h2x(2) m + wm

{wm} is i.i.d complex Gaussian CN(0, σ2), and hi, i = 1, 2, are fixed channel gains, drawn according to f(h1, h2)

• Conditional capacity region: (R1, R2) ∈ C(h1, h2) iff

R1 ≤ log „ 1 + |h1|2P1 σ2 « R2 ≤ log „ 1 + |h2|2P2 σ2 « R1 + R2 ≤ log „ 1 + |h1|2P1 + |h2|2P2 σ2 «

Mikael Skoglund, Theoretical Foundations of Wireless 11/20

• Outage probability: The probability that coding at rates (R1, R2)

fails, pul

• ut = Pr
• (R1, R2) /

∈ C(h1, h2)

• ε-outage capacity region: The closure of the set

{(R1, R2) : pul

• ut(R1, R2) ≤ ε}

Mikael Skoglund, Theoretical Foundations of Wireless 12/20

• Fast fading, perfect CSIR, no CSIT:
• Received signal

ym = h(1)

m x(1) m + h(2) m x(2) m + wm

{wm} is i.i.d complex Gaussian CN(0, σ2), and h(i)

m , i = 1, 2, are

jointly stationary and ergodic ⇒ (R1, R2) ∈ C iff R1 ≤ E " log 1 + |h(1)

m |2P1

σ2 !# R2 ≤ E " log 1 + |h(2)

m |2P2

σ2 !# R1 + R2 ≤ E " log 1 + |h(1)

m |2P1 + |h(2) m |2P2

σ2 !#

• Fast fading, perfect CSIR, perfect CSIT: Next lecture

Mikael Skoglund, Theoretical Foundations of Wireless 13/20

The Gaussian Downlink Channel

(W1, W2) ˆ W1 ˆ W2 xm y(1)

m

y(2)

m

α β1 β2 f(y(1), y(2)|x)

• The information-theoretic broadcast channel, with transition density

f(y(1), y(2)|x)

Mikael Skoglund, Theoretical Foundations of Wireless 14/20

• Degraded broadcast channel,

f(y1, y2|x) f(y1|x) f(y2|y1) x x y1 y1 y2 y2

⇔

• A (general) broadcast channel is degraded if it can be split as in the
• figure. That is, y2 is a “noisier” version of x, and

f(y1, y2|x) = f(y2|y1)f(y1|x)

Mikael Skoglund, Theoretical Foundations of Wireless 15/20

• The Gaussian broadcast (downlink) channel,

y(i)

m = xm + w(i) m , i = 1, 2

• {w(i)

m } i.i.d zero-mean Gaussian, E[|w(i) m |2] = σ2 i

• the channel is degraded (why?)
• Coding:
• Data: W1 ∈ I1 = {1, . . . , M1} and W2 ∈ I2 = {1, . . . , M2}
• Encoder: α : I1 × I2 → Cn,

codewords xn

1 (w1, w2)

• Power constraint:

1 n

n

X

m=1

|xm|2 ≤ P

• Rates: R1 = log M1/n and R2 = log M2/n
• Decoders: β1 : Cn → I1, β2 : Cn → I2
• Error probability: P (n)

e

= Pr ` ( ˆ W1, ˆ W2) = (W1, W2) ´

Mikael Skoglund, Theoretical Foundations of Wireless 16/20

• Capacity,
• achievable rate-pair: (R1, R2) is achievable if (α, β1, β2)n exist such

that P (n)

e

→ 0 as n → ∞

• capacity region:

C = the closure of the set of all achievable (R1, R2)

• Capacity region for the Gaussian downlink,
• assume σ1 < σ2 ⇒ the pair

R1 < log „ 1 + αP σ2

1

« R2 < log „ 1 + (1 − α)P αP + σ2

2

« can be achieved for any α ∈ [0, 1]

Mikael Skoglund, Theoretical Foundations of Wireless 17/20

• Superposition coding achieves capacity:
• Assume σ1 < σ2 (user 1 is the ’good’ user)
• Let P1 = αP and P2 = (1 − α)P
• Generate two independent ’Gaussian codebooks’ C1 and C2 with

powers P1 and P2 and rates R1 and R2

• Code w1 into x(1)

m using C1 and w2 into x(2) m using C2,

transmit xm = x(1)

m + x(2) m — superposition coding

• β2 assumes {x(1)

m } is noise, and decodes only w2 using C2

• β1 first decodes w2 based on y(1)

m and subtracts the correct x(2) m to

form ¯ y(1)

m = y(1) m − x(2) m = x(1) m + w(1) m , then β1 decodes w1 based on

¯ y(1)

m

• interference cancellation
• works since user 1 has a better channel ⇒ must be able to order

users according to their ’goodness’

Mikael Skoglund, Theoretical Foundations of Wireless 18/20

• The two-user Gaussian broadcast region (figure from the textbook).

Mikael Skoglund, Theoretical Foundations of Wireless 19/20

The Fading Gaussian Downlink

• Fast fading, perfect CSIR, no CSIT:
• Received signal

y(i)

m = h(i) m xm + w(i) m

with the h(i)

m ’s, i = 1, . . . , M, jointly stationary and ergodic,

• general case unsolved!, the channel is non-degraded. . .
• the symmetric case, when the the h(i)

m ’s are identically distributed,

i = 1, . . . , K, the capacity region is

K

X

k=1

Rk ≤ E " log 1 + |h(i)

m |2P

σ2 !#

• Fast fading, perfect CSIR, perfect CSIT: Next lecture

Mikael Skoglund, Theoretical Foundations of Wireless 20/20