Lecture 2 Capacity of Fading Gaussian Channels Slow fading - - PDF document

Lecture 2

Capacity of Fading Gaussian Channels

• Slow fading channels: Ch. 5.4.1–4
• Fast fading channels: Ch. 5.4.5–6

Mikael Skoglund, Theoretical Foundations of Wireless 1/16

The Flat Fading Gaussian Channel

decoder encoder CSIT CSIR

hm α β xm ym wm ω ˆ ω

• Discrete-time, complex baseband: xm ∈ X = C, ym ∈ Y = C,

channel gain hm ∈ C, noise wm ∈ C

• Complex-valued channel gains {hm},
• channel-state information at the transmitter (CSIT):

hm known at the transmitter

• channel-state information at the receiver (CSIR):

hm known at the receiver

• The noise {wm} is i.i.d. complex Gaussian CN(0, σ2)

Mikael Skoglund, Theoretical Foundations of Wireless 2/16

Slow fading, perfect CSIR, no CSIT

• Slow fading (non-ergodic, quasi-static, block fading):

hm = h, m = 1, . . . , n, with h drawn according to a pdf fh

• Coding:
• Equally likely information symbols ω ∈ IM = {1, . . . , M}
• An (M, n) code with power constraint P

1 Codebook C =

n xn

1 (1), . . . , xn 1 (M)

• , with

n−1

n

X

m=1

|xm(i)|2 ≤ P, i ∈ IM

2 Encoding:

ω = i ⇒ xn

1 = α(i) = xn 1 (i)

3 Decoding:

yn

1 received, hm = h known ⇒ ˆ

ω = β(yn

1 ; h)

Mikael Skoglund, Theoretical Foundations of Wireless 3/16

• Conditional capacity, conditioned on a value hm = h

C(h) = log

• 1 + |h|2 P

σ2

• Outage probability,

pout(R) = Pr

• C(h) < R
• probability that a code of rate R will not work
• Definition of ε-capacity:
• The rate R is ε-achievable if there exists a sequence of (⌈2nR⌉, n)

codes such that lim

n→∞ P (n) e

≤ ε where P (n)

e

= Pr(ˆ ω = ω)

• The ε-capacity Cε is the supremum of the ε-achievable rates

Mikael Skoglund, Theoretical Foundations of Wireless 4/16

• Let γ = |h|2, Fγ(x) = Pr(γ ≤ x)
• ε-capacity (’ε-outage capacity’) of the slow fading channel,

Cε = log

• 1 + F −1

γ (ε) P

σ2

• Since, when trying a ’Gaussian codebook’ of rate R,

( C(h) > R ⇒ P (n)

e

→ 0 C(h) < R ⇒ P (n)

e

→ 1 ⇒ P (n)

e

→ pout(R)

• Fγ such that Fγ(x) > 0 for any x > 0 ⇒

C = lim

ε→0+ Cε = 0

• The ’ordinary’ (Shannon) capacity C is zero!
• true e.g. for exponential γ (Rayleigh fading)

Mikael Skoglund, Theoretical Foundations of Wireless 5/16

Parallel slow fading channels, perfect CSIR, no CSIT

• General block fading model:

Assume n = LTc and hm = gℓ, m = t + (ℓ − 1)Tc, ℓ = 1, . . . , L, t = 1, . . . Tc and {gℓ} i.i.d,

• the channel is constant for Tc channel uses = the “coherence

interval,” i.i.d realizations in different intervals

• Coding:
• Block length n = LTc — coding over L coherence intervals,
• Codebook:

C = n xn

1 (1), . . . , xn 1 (M)

• , power constraint P
• Encoding:

ω = i ⇒ xn

1 = α(i) = xn 1 (i)

• Decoding:

yn

1 received, hn 1 known ⇒ ˆ

ω = β(yn

1 ; hn 1 )

Mikael Skoglund, Theoretical Foundations of Wireless 6/16

• ε-capacity, the general block fading model:
• R = n−1 log M is ε-achievable if there exists a sequence of

(⌈2nR⌉, n) codes such that lim P (n)

e

≤ ε when Tc → ∞ for a fixed and finite L, with n = LTc

• Cε is the supremum of the ε-achievable rates

Mikael Skoglund, Theoretical Foundations of Wireless 7/16

• With

C(gL

1 ) = 1

L

L

X

ℓ=1

log „ 1 + |gℓ|2 P σ2 « and pout(R) = Pr ` C(gL

1 ) < R

´ it can be shown that Cε = p−1

• ut(ε)
• pout decays as (P/σ2)−L ⇒ L-fold diversity!
• the transmitter does not need to know {gℓ}
• coding needs to span L different coherence intervals

⇒ long delays

Mikael Skoglund, Theoretical Foundations of Wireless 8/16

Fast fading, perfect CSIR, no CSIT

• Fast fading (ergodic fading):

Assume {hm} is an i.i.d process (or more generally, stationary and ergodic),

• each time-instant m gives a new value for hm
• Coding:
• Codebook:

C = n xn

1 (1), . . . , xn 1 (M)

• , power constraint P
• Encoding:

ω = i ⇒ xn

1 = α(i) = xn 1 (i)

• Decoding:

yn

1 received, hn 1 known ⇒ ˆ

ω = β(yn

1 ; hn 1 )

Mikael Skoglund, Theoretical Foundations of Wireless 9/16

• Capacity (ergodic capacity),

C = E

• log
• 1 + |hm|2 P

σ2

• {hm} stationary ⇒ C does not depend on m
• no CSIT required
• This is the same value as obtained for Cε, any ε ∈ (0, 1), in the

block fading channel model when letting L → ∞,

• coding accross (infinitely) many coherence intervals necessary

⇒ long delays

Mikael Skoglund, Theoretical Foundations of Wireless 10/16

Fast fading, perfect CSIR, perfect CSIT

• {hm} known causally at the transmitter
• Coding:
• Encoding:

ω = i and hm

1 known ⇒ xm = α(i; hm 1 ), with power

constraint n−1

n

X

m=1

E|xm(i; hm

1 )|2 ≤ P, i ∈ IM

• Decoding:

yn

1 received, hn 1 known ⇒ ˆ

ω = β(yn

1 ; hn 1 )

Mikael Skoglund, Theoretical Foundations of Wireless 11/16

• Capacity (ergodic capacity),

C = E

• log
• 1 + |hm|2 P(hm)

σ2

• where

P(x) = 1 λ − σ2 |x|2 + and where λ is chosen such that E[P(hm)] = P

• Waterfilling over time. . .

Mikael Skoglund, Theoretical Foundations of Wireless 12/16

• Separate power control is optimal

‘Gaussian codebook’

πm π(·) hm zm xm ω

• Fixed (rate and power) ’Gaussian codebook’ {zn

1 (i)}, with

1 n

n

X

m=1

|zm|2 ≤ 1

• Based on the CSIT hm, multiply with πm = √Pm where

Pm = » 1 λ − σ2 |hm|2 –+ and transmit xm = πmzm — achieves capacity

Mikael Skoglund, Theoretical Foundations of Wireless 13/16

• Optimal power control,
• Notable gains only at low SNR’s. . .

Mikael Skoglund, Theoretical Foundations of Wireless 14/16

• Channel inversion:
• Assume real-valued transmission: ym = hm(πmzm) + wm
• Use

πm = λ hm with λ chosen such that E[π2

m] = P ⇒ ym = λzm + wm ⇒

C = 1 2 log ` 1 + λ2 σ2 ´

• Simple, but suboptimal in general
• Can give huge power peaks (when hm ≈ 0)
• For e.g. hm Rayleigh distributed,

E[h−2

m ] = ∞ ⇒ inversion does not work

Mikael Skoglund, Theoretical Foundations of Wireless 15/16

• Capacities, fast fading; plot from
• Goldsmith and Varaiya, “Capacity of fading channels with channel

side information,” IEEE Trans. on Inform. Theory, Nov. 1997

Mikael Skoglund, Theoretical Foundations of Wireless 16/16