SYSC 4607 – Slide Set 15 Outline

Review of Previous Lecture

Diversity Systems
- Diversity Combining Techniques
- Performance of Diversity in Fading Channels
- Transmitter Diversity
- CSI at Tx
- No CSI at Tx (Alamouti Scheme)
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Review of Previous Lecture

Diversity overcomes the effects of flat fading by
combining multiple independent fading paths

Diversity typically entails some penalty in terms
of rate, bandwidth, complexity, or size.

Different combining techniques offer different
levels of complexity and performance.
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Diversity Combining Techniques

Selection Combining (SC)
- Strongest signal is selected. Cophasing not required.

Threshold (Switching) Combining
- Signal above a given threshold is used. Switching to a different branch if it
drops below the threshold.

Maximal Ratio Combining (MRC)
- Signals are cophased and summed after optimal weighting proportional to
individual SNR’s. Goal is to maximize SNR at the combiner output.

Equal Gain Combining (EGC)
- Branch signals are cophased and added (Maximal Ratio with equal
weights).
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Linear Diversity Combining

Individual branches are
weighed by αi and summed

Selection and Threshold
Combining: all αi = 0, except
one. Cophasing not required

Maximal Ratio Combining: αi
function of γi. Co-phasing
required

Equal Gain: αi = 1. Cophasing required
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Linear Diversity Combining


  is a random variable with PDF p

( ) and CDF P  ( ) which
depends on the type of fading and the choice of combining
Most often PDF is obtained by differentiating CDF
Pout  p(    0 ) 

0
 p

( )d  P  ( 0 )
0
Ps   Ps ( ) p  ( )d
0

Ps ( ) is the random probability of error for AWGN non-fading

channel
Most often closed form solution for CDF, Pout and Ps unavailable.
Results based on computer simulation.
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Array and Diversity Gains

Array Gain
- Gain in SNR from coherent addition of signals and non-coherent
addition (averaging) of noise over multiple antennas
- Gain in both fading and non-fading channels

Diversity Gain
- Gain in SNR due to elimination of weak signals (deep fades).
Changes slope of probability of error.
- Gains in fading channels
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Selection Combining

Combiner outputs the signal with the
2
r
highest SNR i / N i

The chance that all the branches are in
deep fade simultaneously is very low.

Since at each instant only one signal is
used co-phasing is not required.
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Selection Combining
P  ( )  p(    )  p(max[ 1 ,,  M ]   )
M
 p( 1       M   )   p( i   )
i 1
(Assuming independent branches)
For iid Rayleigh fading (ri Rayleigh, γi exponential):
M
P  ( )   (1  e  /  i )  (1  e  /  ) M
i 1
  E( i ), same for all branches
p  ( ) 
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dP  ( )
d

M

(1  e  /  ) M 1 e  / 
81
Selection Combining

M
1
    p  ( )d   
i 1 i
0
Pout ( 0 )  p(    0 )  P  ( 0 )  (1  e  0 /  ) M

The average SNR gain (array gain)
increases with M, but not linearly.
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Threshold (Switching) Combining

Branches are scanned sequentially. First
one above a given threshold is selected.
The signal is used as long as its SNR is
above threshold.

Since at each instant only one signal is
used, co-phasing is not required.
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Threshold (Switching) Combining

For two-branch diversity with iid branch statistics:
P 1 ( T ) P 2 ( )
 T

P  ( )  
 p( T   2   )  P 1 ( T ) P 2 ( )    T

For iid Rayleigh fading with   E( i )
1  e  T /   e  /   e  ( T  ) / 
P  ( )  
 (  T  ) / 
 / 
1

2
e

e

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 T
 T
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Maximal Ratio Combining

In the general model set  i  ai e  j
M

M
Then, r   i ri e j   ai ri
i
i 1

i
i 1
Assuming the same noise psd at all branches:
M
r2
1
 

N tot N 0
( ai ri ) 2
i 1
M
2
a
 i
i 1

Maximizing   by Cauchy-Schwartz inequality:
2
2
a

r
i / N0
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Maximal Ratio Combining

Assuming iid Rayleigh fading in each branch
with equal average branch SNR,  , resulting  
has chi-squared distribution with 2M degrees of
freedom:
1 M 2 M
 
ri    i

N 0 i 1
i 1
 M 1e  / 
p ( )  M
,  0
 (M  1)!

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BER Performance of MRC

Average Pb for Maximal Ratio Combining with iid Rayleigh Fading
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Equal Gain Combining

In maximal ratio combining, set ai  1, i.e., i  e -j
1 

  ri 
N 0 M  i 1 
M
2

Then,

In general, no closed-form solution for P ( ) .
For iid two-branch Rayleigh channel with same 
CDF in terms of Q function:
 

P  ( )  1  e 2 /    /  e  /  [1  2Q( 2 /  )] ,   0
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i
Diversity Improvements with M

For Selection Combining, average SNR increases with
M. The increase, however, is not linear. Maximum
benefit gained when M is increased form 1 to 2.

For Maximal Ratio and Equal Gain Combining diversity
gain and array gain both contribute to the performance
improvement. For large M, array gain dominates
performance improvements.
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Diversity Improvements with M



For Selection Combining SNR
saturates with modest increase in
M. Maximal Ratio and Equal
Gain do not exhibit saturation,
but slope changes
Maximal Ratio and Equal Gain
outperform Selection; however,
their relative performance is close
(within 1 dB). Equal Gain simpler
to implement
Improvements in average SNR
(10log10 (  /  )) for M-Branch
compared to one branch. (a)
Maximal Ratio, (b) Equal Gain,
(c) Selection
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Transmitter Diversity



Multiple transmit antennas, with power divided among
them
Suitable for systems with greater capabilities at the
transmit site (example: cellular, downlink)
Implementation depends on channel knowledge
- With transmitter channel knowledge (CSIT), performance is
similar to receiver diversity (same array/diversity gain)
- Without channel knowledge, can obtain diversity gain (but not
array gain) through Alamouti scheme: transmission over space and
time (2 consecutive symbols)
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Channel Known at Tx

We assume path gain ri e j is known at the
transmitter.
i
s(t), transmitted signal with energy per
symbol Es , is multiplied by constant
complex gain αi :  i  ai e  j , 0  ai  1
M
2
a
 Total energy constraint Es requires  i  1

i
i 1

Signals transmitted from M antennas are
M
combined in the air: r (t )   ai ri s(t )
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i 1
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Channel Known at Tx

Similar to MRC at Rx coefficients are found to maximize SNR:
ai 
ri
M
r
i 1

2
i
Es


Resulting SNR: 
N0
M
r
i 1
Ps   M Q(  M   )   M e
M
1
Ps   M 
i 1 1   M  i / 2
i
M
2
  i
i 1
M   / 2
 M 
M


  M exp  M  i / 2
i 1


M

For large SNR: Ps   M 

Similar to MRC full diversity order of M is achieved

 2 
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Channel Unknown at Tx
Alamouti Scheme

Transmission in space and time:
First symbol period; Second symbol period
Antenna 1: s1
Antenna 1: -s2*
Antenna 2: s2
Antenna 2: s1*
Each transmission has energy Es/2
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The Alamouti Scheme
Channel gains: h1  r1e j , h2  r2 e j
1
Received signals:
y=
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 y1   h1
 *    *
 y 2   h2
2
 y1  h1 s1  h2 s 2  n1

*
*
y


h
s

h
s
1 2
2 1  n2
 2
h2  s1   n1 
    * 
* 
 h1  s 2   n2 
= HAs + n
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The Alamouti Scheme
Define z = HAH y, where HAH is the conjugate
transpose of HA.
We have HAH HA =(|h1|2+|h2|2)I2
Thus z =
(z1 z2)T =(|h1|2+|h2|2)I2
~
n
s+ ,
where n~  H AH n is zero-mean complex Gaussian
2
2
H
~
~
with E (n n )  ( h1  h2 ) N 0 I 2 .
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The Alamouti Scheme
( h1  h2 ) E s
2
So,
z i  ( h1  h2 ) si  n~i , i  1, 2, and
2
2
i 
2
2N 0
(Factor 2 represents transmission of half energy Es/2 per
symbol per antenna)
Observations:
1.
Alamouti scheme achieves a diversity order of
2 although channel knowledge not available at
transmitter
2.
Alamouti scheme achieves an array gain of 1
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Performance of the Alamouti Scheme
BER performance of Alamouti Scheme vs. SNR γb: comparison of BPSK with
Maximal Ratio and two-branch transmit diversity in Rayleigh fading
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Main Points


MRC provides best diversity performance
EGC easier to implement compared to MRC
- Performance about 1 dB worse than MRC

Performance of Transmit Diversity depends on
channel knowledge
- With channel knowledge at transmitter (CSIT) same
performance as receiver diversity
- Without CSIT, the Alamouti scheme provides same
diversity gain, but array gain is 3 dB lower.
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