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Wireless CommunicationLecture 4

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Channel Capacity

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Shannon CapacityDefined as the maximum mutual information of

channel

Maximum error-free data rate a channel can support.

Theoretical limit (usually don’t know how to achieve)

Depends on the channel characteristics

We focus on AWGN channel with fading

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AWGN Channel Capacity

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Power and Bandwidth Limited Regimes

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Band limited regime SNR>>1

0 0.5 1 1.5 20

20

40

60

80

100

120

140

B in Hz

C in

bits

P = 3 dB

P = 10 dB

P = 16 dB

P = 20 dB

0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

12

B in Hz

C in

bits

P = 4

P = 16

P = 256

P = 1024

N0=1 assumed

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Power limited regime SNR<<1

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

B (Hz)

C (

bits

)

P = -10dB

P = -6dBP = -3dB

P = 0

N0=1 assumed

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Capacity Curve

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Shannon Limit in AWGN channelWhat is the minimum SNR per bit (Eb/N0) for reliable communications?

)2ln(

)2ln(0NP

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Capacity of Flat-Fading ChannelsCapacity defines theoretical rate limit

Maximum error free rate a channel can support

Depends on what is known about channel CSI: channel state information CDI: channel distribution information

Unknown fading: Worst-case channel capacity

Fading Known at Receiver Only )1(log)(1log 2

0

2

BdpBC

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Capacity of Fading Channels

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Capacity of fading channel

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Fading channel, only Rx knows CSI

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Fading Known at both Transmitter and Receiver

For fixed transmit power, same as only receiver knowledge of fading

Transmit power P(g) can also be adapted

Leads to optimization problem:

dpP

PB

PPEPC rob )(

)(1log

)]([:)(

max

0

2

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Optimal Adaptive Scheme

Power Adaptation

Capacity

else0

)( 011

0

P

P

1

0

1g

g0 g

.)(log0

2

0

dpB

Rrob

Waterfilling

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An equivalent approach:power allocation over time

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Optimal SolutionThe water-filling solution is given by

To define the water level, solve:

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Asymptotic results

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Performance Comparison

At high SNR, water-filling does not provide any gain. Transmitter knowledge allows rate adaptation and simplifies coding.

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Channel InversionFading inverted to maintain constant SNRSimplifies design (fixed rate)Greatly reduces capacity

Capacity is zero in Rayleigh fadingTruncated inversion

Invert channel above cutoff fade depthConstant SNR (fixed rate) above cutoffCutoff greatly increases capacity

Close to optimal

]1[log]1[log ]/1[1

22 EBBC

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Frequency Selective Fading ChannelsFor time-invariant channels, capacity

achieved by water-filling in frequencyCapacity of time-varying channel unknownApproximate by dividing into subbands

Each subband has width Bc (like MCM).Independent fading in each subbandCapacity is the sum of subband capacities

Bcf

P

1/|H(f)|2