5: Capacity of Wireless Channels
5. Capacity of Wireless Channels
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5: Capacity of Wireless Channels
Information Theory
• So far we have only looked at specific communication
schemes.
• Information theory provides a fundamental limit to
(coded) performance.
• It succinctly identifies the impact of channel resources on
performance as well as suggests new and cool ways to
communicate over the wireless channel.
• It provides the basis for the modern development of
wireless communication.
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5: Capacity of Wireless Channels
Historical Perspective
Gugliemo Marconi
Claude Shannon
1901
• Wireless communication
has been around since
1900’s.
• Ingenious but somewhat
adhoc design techniques
1948
•Information theory says every
channel has a capacity.
• Many recent advances based
on understanding wireless
channel capacity.
New points of views arise.
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5: Capacity of Wireless Channels
Multipath Fading: A Modern View
16dB
Classical view: fading channels are unreliable
Modern view: multipath fading can be exploited to
increase spectral efficiency.
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5: Capacity of Wireless Channels
Capacity of AWGN Channel
Capacity of AWGN channel
If average transmit power constraint is
noise psd is
watts/Hz,
watts and
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5: Capacity of Wireless Channels
Power and Bandwidth Limited Regimes
Bandwidth limited regime
capacity logarithmic
in power, approximately linear in bandwidth.
Power limited regime
insensitive to bandwidth.
capacity linear in power,
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5: Capacity of Wireless Channels
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5: Capacity of Wireless Channels
Frequency-selective Channel
's are time-invariant.
OFDM converts it into a parallel channel:
where
is the waterfilling allocation:
with  chosen to meet the power constraint.
Can be achieved with separate coding for each sub-carrier.
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5: Capacity of Wireless Channels
Waterfilling in Frequency Domain
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5: Capacity of Wireless Channels
Fading Channels
Different notions of capacity, depending on scenarios:
• Slow fading channel: outage capacity
• Fast fading channel: “ergodic” capacity under infinitely
deep time interleaving
• Capacity also depends on whether channel knowledge is
known at the transmitter and/or receiver.
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5: Capacity of Wireless Channels
Slow Fading Channel
h random.
There is no definite capacity.
Outage probability:
-outage capacity:
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5: Capacity of Wireless Channels
Outage for Rayleigh Channel
Pdf of log(1+|h|2SNR)
Outage cap. as fraction of AWGN cap.
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5: Capacity of Wireless Channels
Receive Diversity
Diversity plus power gain.
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5: Capacity of Wireless Channels
Time Diversity (I)
Coding done over L coherence blocks, each of many
symbols.
This is a parallel channel. If transmitter knows the
channel, can do waterfilling.
Can achieve:
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5: Capacity of Wireless Channels
Time Diversity (II)
Without channel knowledge,
Rate allocation cannot be done.
Coding across sub-channels becomes now necessary.
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5: Capacity of Wireless Channels
Fast Fading Channel
Channel with L-fold time diversity:
As
Fast fading channel has a definite capacity:
Tolerable delay >> coherence time.
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5: Capacity of Wireless Channels
Capacity with Full CSI
Suppose now transmitter has full channel knowledge.
What is the capacity of the channel?
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5: Capacity of Wireless Channels
Fading Channel with Full CSI
This is a parallel channel, with a sub-channel for each
fading state.
where
is the waterfilling power allocation as a function of
the fading state, and  is chosen to satisfy the
average power constraint.
Can be achieved with separate coding for each fading
state.
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5: Capacity of Wireless Channels
Transmit More when Channel is Good
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5: Capacity of Wireless Channels
Performance
At high SNR, waterfilling does not provide any gain.
But transmitter knowledge allows rate adaptation and
simplifies coding.
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5: Capacity of Wireless Channels
Performance: Low SNR
Waterfilling povides a significant power gain at low SNR.
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5: Capacity of Wireless Channels
Waterfilling vs Channel Inversion
• Waterfilling and rate adaptation maximize long-term
throughput but incur significant delay.
• Channel inversion (“perfect” power control in CDMA
jargon) is power-inefficient but maintains the same data
rate at all channel states.
• Channel inversion achieves a delay-limited capacity.
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5: Capacity of Wireless Channels
Summary
• A slow fading channel is a source of unreliability: very
poor outage capacity. Diversity is needed.
• A fast fading channel with only receiver CSI has a
capacity close to that of the AWGN channel. Delay is
long compared to channel coherence time.
• A fast fading channel with full CSI can have a capacity
greater than that of the AWGN channel: fading now
provides more opportunities for performance boost.
• The idea of opportunistic communication is even more
powerful in multiuser situations, as we will see.
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