Part V. AWGN Channel Capacity AWGN Capacity Formula; Sphere - - PowerPoint PPT Presentation

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Part V. AWGN Channel Capacity

AWGN Capacity Formula; Sphere Packing; Resources in AWGN Channel

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Channel coding theorem

• For every memoryless channel, there is a definite number C (computable) such that:
• If the data rate R < C, then there exists a coding scheme that can deliver data at rate R
• ver the channel with vanishing error probability as the block length
• Conversely, if the data rate R > C, then no matter what coding scheme is used, the error

probability will converge to 1 as

• C is called the capacity of the channel, and it has a computable formula for all

kinds of channels (depending on the channel statistics)

• We focus on the additive white Gaussian noise (AWGN) channel in this course,

and give a heuristic argument to derive the AWGN channel capacity

n → ∞ n → ∞

AWGN channel model (discrete-time, real-valued)

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Codebook: Vm = um + Zm, m = 1, . . . , n, Zm

• ∼ N(0, σ2)

Zm um C comprises 2nR length-n codewords u ∈ Rn Power constraint: 1 n

n

X

m=1

|um|2 ⌘ 1 n kuk2  P, 8 u 2 C

unit: joule per channel use

Code design problem is equivalent to placing 2nR n-dimensional vectors within a sphere of radius , so that the error probability is minimized √ nP

Sphere packing interpretation

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Rn

p n(P + σ2)

V = u + Z

√ nσ2

• By LLN, as , the received V will lie

at the surface of the n-dimensional sphere centered at u with radius with probability 1

• Also by LLN, as , the received V

will lie within the n-dimensional sphere with radius with probability 1

• Asymptotically, vanishing error probability

is equivalent to non-overlapping spheres

• How many non-overlapping spheres can

be packed into the large sphere? n → ∞ √ nσ2 n → ∞ p n(P + σ2)

Necessary condition: capacity upper bound

• maximum # of non-overlapping spheres =

maximum # of codewords that can be reliably delivered

• A necessary condition is
• The channel capacity is hence upper

bounded by

• How to achieve it?

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Rn

p n(P + σ2)

V = u + Z

√ nσ2

2nR ≤ √

n(P +σ2)

n

√ nσ2n

⇐ ⇒ R ≤ 1

n log

√

n(P +σ2)

n

√ nσ2n

• =

1 2 log

• 1 + P

σ2

• 1

2 log

• 1 + P

σ2

Achieving capacity (1)

• Prove the existence of good codebook

by random coding, as we did before for linear block codes:

• Randomly generate 2nR length-n codewords

uniformly inside the “u-sphere” of radius

• Goal: ensure the average-over-random-code

average probability of error vanishes as

• Decoding:

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C u-sphere

√ nP u1 u2

√ nP n → ∞

α ,

P P +σ2 : the MMSE coefficient

V − → MMSE − → αV − → Nearest Neighbor − → ˆ u

Achieving capacity (2)

• Due to symmetry, we can assume WLOG

the true codeword sent by Tx is

• By LLN, the distance between and :
• As long as lies inside the sphere centered

at with radius , decoding will be correct w.h.p.

• Pairwise probability of error
• The probability that a random falls inside

the sphere!

• Ratio of the volume of the two spheres.

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u-sphere

√ nP

• n P σ2

P +σ2

u1 u2 αV

u1

u1 αV ∥αV − u1∥2 = ∥αZ + (α − 1)u1∥2 ≈ α2nσ2 + (α − 1)2nP = n P σ2

P +σ2

u1 αV

• n P σ2

P +σ2

P {Eu1→u2}

u2

Achieving capacity (3)

• Pairwise probability of error
• Ratio of the volume of the two spheres:
• Union bound:
• Sufficient condition for vanishing :

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u-sphere

√ nP

• n P σ2

P +σ2

u1 u2 αV

P {Eu1→u2}

P {Eu1→u2} = √

nP σ2/(P +σ2)

n

√ nP

n

=

• σ2

P +σ2

n/2 P(n)

e

≤ (2nR − 1)P {Eu1→u2} ≤ 2nR

σ2 P +σ2

n/2

P(n)

e

= 2n(R− 1

2 log(1+ P σ2 ))

R < 1 2 log ✓ 1 + P σ2 ◆

Continuous-time AWGN channel capacity

• For the continuous-time (waveform) channel model considered in Lecture 03:
• Power constraint P watts
• White Gaussian noise PSD N0/2 joules per second per hertz
• Total bandwidth W hertz (symbol duration T = 1/W)
• Recall we can convert the waveform channel to an equivalent discrete-time

complex-valued AWGN channel:

• Power constraint is PT = P/W joules per channel use
• Variance of the circular symmetric complex Gaussian noise is N0 joules per channel use
• For each real dimension, its capacity is bits per channel use
• The channel capacity is bits per channel use

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1 2 log

• 1 + P/2W

N0/2

• 2 × 1

2 log

• 1 + P/2W

N0/2

• = log
• 1 +

P W N0

• = W log

✓ 1 + P WN0 ◆ bits per second

AWGN channel capacity

• The capacity formula provides a high-level way of thinking about how the

performance fundamentally depends on the basic resources in the channel

• No need to go into details of specific coding and modulation schemes
• Basic resources: power P and bandwidth W

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CAWGN(P, W) = W log ✓ 1 + P N0W ◆

• = log (1 + SNR)

“spectral efficiency”

SNR

P N0W

Resources in AWGN channel

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30 5 Bandwidth W (MHz) Capacity Limit for W → ∞ Power limited region 0.2 1 Bandwidth limited region (Mbps) C(W ) 0.4 25 20 15 10 1.6 1.4 1.2 0.8 0.6 P N0log2 e

C(W) = W log ✓ 1 + P N0W ◆ ≈ W P N0W log2 e = P N0 log2 e

Fix P

Bandwidth-limited vs. power-limited

• When : power-limited regime
• Linear in power; Insensitive to bandwidth
• When : bandwidth-limited regime
• Logarithmic in power; Approximately linear in bandwidth

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CAWGN(P, W) = W log ✓ 1 + P N0W ◆

• = log (1 + SNR)

“spectral efficiency”

SNR

P N0W

SNR ⌧ 1 CAWGN(P, W) ≈ W

• P

N0W

• log2 e =

P N0 log2 e

SNR 1 CAWGN(P, W) ≈ W log

• P

N0W