Transcript Capacity of Flat-Fading Channels

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Channel Capacity
Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
Capacity of Freq.-Selective Fading Channels
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Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
Capacity of Freq.-Selective Fading Channels
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Shannon’s coding theorem and its converse
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In AWGN, Shannon’s formula:
C  B log 2 (1   )
Where γ is the received signal to noise ratio given
by
P

N0 B
Shannon’s coding theorem and its converse
If the information transmission rate R<C, then codes
exist that guarantees arbitrarily small probability of
error.
Conversely if R>C, then has a non-vanishing
probability of error.
AWGN Channel Capacity
Also known as ergodic capacity.
Defined as the maximum MI of channel
Maximum error-free data rate a channel can
support.
Theoretical limit (not achievable)
Channel characteristic
Not dependent on design techniques
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Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
Capacity of Freq.-Selective Fading Channels
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Capacity of Flat-Fading Channels
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Capacity defines theoretical rate limit
Maximum error free rate a channel can support
Depends on what is known about channel
Channel distribution information
• Fading Statistics Known on the transmitter and receiver.
– Hard to find capacity
Receiver CSI
• Both transmitter and receiver know the distribution of g[i],
the value of g[i] is known to the receiver at time i.
Transmitter and receiver CSI
• Both transmitter and receiver know the distribution of g[i]
and the value of g[i] at time i.
Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
Capacity of Freq.-Selective Fading Channels
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CSI at Receiver Only
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Fading Known at Receiver Only: Assumes that
the receiver knows the channel gain, also
known as channel state information (CSI), and
both transmitter and receiver know the
distribution of the fading.
Shannon (Ergodic) Capacity
Capacity with Outage
Shannon (Ergodic) Capacity
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Shannon (Ergodic) Capacity

C  E  B log 2 (1   )    B log 2 1    p( )d 
0
How does this capacity compare to the AWGN
capacity ? Use Jensen’s inequality for a concave
function f(.) : E  f ( x)  f (E[ x])
E  B log 2 (1   )  B log 2 (1  E[ ])  B log 2 (1   )
Thus ergodic capacity of fading channel cannot be
better than AWGN channel capacity with same
average SNR.
Some technicalities
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 The ergodic capacity makes sense only for fast flat
fading channel where codeword length (each block)
spans many coherence time (Tc).
 In the case of slow flat fading channel, there is a nonzero probability that the entire codeword is in a deep
fade. Coding cannot average out the channel fade
which affects all coded symbols.
 Strictly speaking then, the ergodic capacity of a slow
(quasi-static) flat fading channel is zero! In other words,
it is practically impossible to get error-free
transmission no matter what the transmission rate.
 More useful to speak of outage capacity.
Outage Capacity
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 More relevant for slow-fading channels.
 Defined as the maximum rate of transmission with a specified
probability of error (outage probability).
 Allows coded data to be decoded correctly unless the channel
is in a deep fade.
 If ε is the outage probability,
then the outage capacity is
C  B log(1  F 1 ( ))  B log(1   min )
where   P(   min )  F ( min )
 The average transmission rate
without any errors is:
(1   )C  (1   ) B log(1  F 1 ( ))
Fade Margin and Outage
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 Consider the Rayleigh fading channel
 1 
F ( x)  P(  x)  1  e  x /  F 1 ( x)   ln 

 1 x 
 With outage prob of ε

 1
C  B log(1  F 1 ( ))  B log 1   ln 
 1 




 Compare this to the AWGN channel capacity CAWGN  B log 2 (1   )
 we require a fade margin of 10log10 1/ ln 1/ 1     in Rayleigh
fading channel to achieve the same rate as AWGN.
 Example: let ε =0.01, then we require a fade margin of 20 dB!
Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
• Optimal Rate and Power Adaptation
• Channel Inversion with Fixed Rate
Capacity of Freq.-Selective Fading Channels
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Fading Known at Transmitter and Receiver
For fixed transmit power, same as with only
receiver knowledge of fading
Transmit power S() can also be adapted
Leads to optimization problem
max
C
S ( ) : E[ S ( )]  S

 S ( ) 
0 B log 2 1  S  p( )d
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Optimal Adaptive Scheme
Power Adaptation
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1
S ( )   0  

S
 0
  0
Waterfilling
1

else
0
Capacity

 
R
  log 2   p( )d  .
B 0
 0 
1

0

Channel Inversion
Fading inverted to maintain constant SNR
Simplifies design (fixed rate)
Greatly reduces capacity
Capacity is zero in Rayleigh fading
Truncated inversion
Invert channel above cutoff fade depth
Constant SNR (fixed rate) above cutoff
Cutoff greatly increases capacity
• Close to optimal
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Capacity in Flat-Fading
Rayleigh
Log-Normal
Outline
Shannon Capacity
Capacity of Flat-Fading Channels
 Fading Statistics Known
 Fading Known at RX
 Fading Known at TX and RX
Capacity of Freq.-Selective Fading Channels
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Frequency Selective Fading Channels
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For time-invariant channels, capacity
achieved by water-filling in frequency
Capacity of time-varying channel unknown
Approximate by dividing into subbands
Each subband has width Bc (like MCM).
Independent fading in each subband
Capacity is the sum of subband capacities
1/|H(f)|2
P
Bc
f
Capacity: frequency selective channels
Split freq selective channel into
Nc parallel channels, each with
constant channel gain. Each “flat”
channel behaves like AWGN
channel with gain Hi.
Channel capacity (assuming TX &
RX knows Hi)
2

H j Pj 

C  max  B log 1 
{ P1 , P2 , PNc }

N0 B 
j 1


Nc
Nc
subject to  Pj  P
j 1
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The Water-Filling Solution
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 Solution: Optimal power allocation
 P(1/  0  1/  j )
Pj  
 0
 j  0
 j  0
 Where  j  H j P /  N0 B  is the
SNR of jth channel with power P
and γ0 is the Lagrange multiplier
chosen such that
2
Nc
P
j 1
j
Nc
 P   1/  0  1/  j   1
Allocate more tx power to
the better channel!!
j 1
 With above optimal power allocation, capacity becomes:
C
Nc

 
j: j 
B log( j /  0 )
0
Continuous Freq.-Selective Fading Channels
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 If H(f) is continuous (or cannot be split into parallel channels), a
similar water filling solution can be derived.
 The channel capacity is given by:
2

H ( f ) P( f ) 
 df
C
max
 log2 1 

P ( f ): P ( f ) df  P
N0


 The power allocation over frequency, via the Lagrangian technique,
the resulting optimal allocation is water-filling over frequency:
P( f ) 1/  0  1/  ( f )  ( f )   0

 ( f )  0
P
0
where  ( f )  H ( f ) P / N 0
2
 This results in channel capacity:
(f )
C
log 2 
 df
f :  f  0
 0 
Time-varying Frequency-Selective fading channel
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 When frequency selective channel is time-varying and fading, then split
the channel into Nc sub-channels each with coherence bandwidth Bc.
 The channel within each sub-channel is approximately flat and they are
independent.
 Capacity is approximately the sum of all the capacities of these
subchannels subject to total transmit power constraint.
 The optimal two-dimensional water-filling and the corresponding Shannon
capacity:
Nc 
C    Bc log( j /  0 ) p( j )d  j
j 1  0
where  0 is found from
1 1
   p ( j )d  j  1


j 
j 1  0   0
Nc 
Main Points
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Fundamental capacity of flat-fading channels depends
on what is known at TX and RX.
 Capacity when only RX knows fading same as when TX and
RX know fading but power fixed.
 Capacity with TX/RX knowledge: variable-rate variable-power
transmission (water filling) optimal
 Almost same capacity as with RX knowledge only
 Channel inversion practical, but should truncate
Capacity of wideband channel obtained by
breaking up channel into subbands
Similar to multicarrier modulation