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1
Introduction to
Mobile Communications
Lecture 4 (Diversity)
Dong In Kim
College of Info/Comm Engineering
Sungkyunkwan University (SKKU)
2
Introduction to Diversity
Basic Idea
Send same bits over independent fading paths
Independent fading paths obtained by time, space,
frequency, or polarization diversity
Combine paths to mitigate fading effectsTb
tMultiple paths unlikely to fade simultaneously
3
Combining Techniques
Selection Combining
Fading path with highest gain used
Maximal Ratio Combining
All paths cophased and summed with optimal weighting to
maximize combiner output SNR
Equal Gain Combining
All paths cophased and summed with equal weighting
Array/Diversity gain
Array gain is from noise averaging (AWGN and fading)
Diversity gain is change in BER slope (fading)
5
Maximal Ratio Combining
1 1 2 2 1 1 2 2
total noise
2 2 2 2 2*
1 2
2 2 2
1 22 2 2
1 2
array gain
Combined output ( )
where / and .
M M M M
w
i i M i o
M
M
h h h x w w w
h h h h E w E w N B
h h hh h h SNR M SNR
diversity gain
2 2 2
1 2
2 2 2
1 2
div
1 1 Pr[ 1] Pr
!
where the is defined as the ersity order expone of inverse / .
Note: 1 (no fading) as
nt
out M M
s o
M
M
P h h hSNR M SNR
SNR E N
h h h
M
with probability one!M
7
Diversity Performance
Selection Combining (SC) Combiner SNR is the maximum of the branch SNRs.
CDF easy to obtain, pdf found by differentiating.
Diminishing returns with number of antennas.
Can get up to about 20 dB of gain.
Maximal Ratio Combining (MRC)
Optimal technique (maximizes output SNR)
Combiner SNR is the sum of the branch SNRs.
Distribution of SNR hard to obtain.
Can use MGF approach for simplified analysis.
Exhibits 20-40 dB gains in Rayleigh fading.
1
1M
i i
1
M
ii
at high SNRM
bP
8
MRC and its Performance
With MRC, =i for branch SNRs i
Optimal technique to maximize output SNR Yields 20-40 dB performance gains Distribution of hard to obtain
Standard average BER calculation
Hard to obtain in closed form Integral often diverges
MGF Approach
1 2 1 2( ) ( ) ( ) ( ) ( ) ( )b b b M MP P p d P p p p d d d
/ 2
210
;sin
M
b i i
i
gP d
M
/ 2
202 exp with
sinb b b
gP Q g d P E P
1 1 1, , 1 1, , ( ) ( )
M MM MP P P
9
EGC and Transmit Diversity
EGQ simpler than MRC Harder to analyze Performance about 1 dB worse than MRC
Transmit diversity With channel knowledge, similar to receiver
diversity, same array/diversity gain Without channel knowledge, can obtain
diversity gain through Alamouti scheme: works over 2 consecutive symbols
10
Main Points
Diversity typically entails some penalty in terms of rate, bandwidth, complexity, or size.
Techniques trade complexity for performance.
MRC yields 20-40 dB gain, SC around 20 dB.
Analysis of MRC simplified using MGF approach
EGC easier to implement than MRC: hard to analyze. Performance about 1 dB worse than MRC
Transmit diversity can obtain diversity gain even without channel information at transmitter.
11
Time Diversity
• Time diversity can be obtained by interleaving and coding over
symbols across different coherence time periods.
12
Example: GSM
• Amount of time diversity limited by delay constraint and how fast
channel varies.
• In GSM, delay constraint is 40ms (voice).
• To get full diversity of 8, needs v > 30 km/hr at fc = 900Mhz.
500kHz and sub-channel 200KHz
Total bandwidth of uplink/downlink 25MHz
cB
5ms, interleaving of 40ms diversity order 8!cT
13
Antenna Diversity
Receive Transmit Both
SIMO MISO MIMO
14
Receive Diversity
Same as repetition coding in time diversity,
except that there is a further power gain.
Optimal reception is via matched-filtering (MRC!)
(receive beamforming).
15
Transmit Diversity
If transmitter knows the channel, send:
maximizes the received SNR by in-phase addition of
signals at the receiver (transmit beamforming).
Reduce to scalar channel:
same as receive beamforming.
What happens if transmitter does not know the channel?
16
Space-Time Codes
• Transmitting the same symbol simultaneously at the
antennas doesn’t work.
• Using the antennas one at a time and sending the same
symbol over the different antennas is like repetition
coding.
• More generally, can use any time-diversity code by
turning on one antenna at a time.
• Space-time codes are designed specifically for the
transmit diversity scenario.
17
Alamouti Scheme
Over two symbol times:
Projecting onto the two columns of the H matrix yields:
• double the symbol rate of repetition coding.
• 3dB loss of received SNR compared to transmit
beamforming (penalty because of no CSIT!).
18
Alamouti Scheme
19
Space-Time Code Design
A space-time code is a set of matrices
Full diversity is achieved if all pairwise differences
have full rank.
Coding (Array) gain determined by the determinants of
Time-diversity codes have diagonal matrices and the
determinant reduces to squared product distances.
4
det[( )( ) ]
L
A B L
A B A B
PSNR
X XX X X X
*( )( )A B A B X X X X
20
Cooperative Diversity
• Different users can form a distributed antenna array
to help each other in increasing diversity.
• Distributed versions of space-time codes may be
applicable.
• Interesting characteristics:
– Users have to exchange information and this
consumes bandwidth.
– Operation typically in half-duplex mode.
– Broadcast nature of the wireless medium can be
exploited.
21
Cooperative Diversity
22
Frequency Diversity
• Resolution of multipaths provides diversity.
• Full diversity is achieved by sending one symbol every
L symbol times.
• But this is inefficient (like repetition coding).
• Sending symbols more frequently may result in
intersymbol interference.
• Challenge is how to mitigate the ISI while extracting the
inherent diversity in the frequency-selective channel.
23
Approaches
• Time-domain equalization (eg. GSM)
• Direct-sequence spread spectrum (eg. IS-95 CDMA)
• Orthogonal frequency-division multiplexing OFDM
(eg. 802.11a, Flash-OFDM)
24
ISI Equalization
• Suppose a sequence of uncoded symbols are
transmitted.
• Maximum-likelihood sequence detection is performed
using the Viterbi algorithm.
• Can full diversity be achieved?
25
Reduction to Transmit Diversity
0 1 2. . 3 { , , }e g L h h h
26
MLSD Achieves Full Diversity
Space-time code matrix for input sequence
Difference matrix for two sequences first differing at
is full rank.
A( ) or ( )A B B x X x X
x
27
Direct-Sequence Spread Spectrum
• Information symbol rate is much lower than chip rate
(large processing gain).
• Signal-to-noise ratio per chip is low.
• ISI is not significant compared to interference from other
users and matched-filtering (Rake) is near-optimal.
28
Frequency Diversity via Rake
• Considered a simplified situation (uncoded).
• Each information bit is spread into two pseudorandom sequences xA and xB (xB= -xA).
• Each tap of the matched-filter is a finger of the Rake.
29
ISI vs Frequency Diversity
• In narrowband systems, ISI is mitigated using a complex
receiver.
• In asynchronous CDMA uplink, ISI is there but small
compared to interference from other users.
• But ISI is not intrinsic to achieve frequency diversity.
• The transmitter needs to do some work too!
30
OFDM: Basic Concept
• Most wireless channels are underspread
(delay spread << coherence time) .
• Can be approximated by a linear time invariant channel
over a long time scale.
• Complex sinusoids are the only eigenfunctions of linear
time-invariant channels.
• Should signal in the frequency domain and then
transform to the time domain.
31
OFDM
01 2
0
1
0
1[ ] ( )
( / , )
b b
N j nf t
nt mT n t mT
b
d m d t d eN
f W N W T
0 1
cyclic convolut[ [ ], , [ 1]] ( )
[ , , ,0, ,0] , [ [0], , [ 1]
ion
]
t
t t
L
y L y N L
h h d d N
y h d w
h d
0
DFT( ) DFT( ) DFT( ) DFT( )
Note: ( )!n
n n n n n
n n n
y N
h
h H nf
d w
h d w h d w
32
cont …
OFDM transforms the communication problem into the
frequency domain:
a bunch of non-interfering sub-channels, one for each
sub-carrier.
Can apply time-diversity techniques.
33
Cyclic Prefix Overhead
• OFDM overhead
= length of cyclic prefix / OFDM symbol time
• Cyclic prefix dictated by delay spread.
• OFDM symbol time limited by channel coherence time.
• Equivalently, the inter-carrier spacing should be much
larger than the Doppler spread.
• Since most channels are underspread, the overhead can
be made small.
34
Example: Flash OFDM (Flarion)
• Bandwidth = 1.25 Mz
• OFDM symbol = 128 samples = 100 m s
• Cyclic prefix = 16 samples = 11 m s delay spread
• 11 % overhead.
35
Channel Uncertainty
• In fast varying channels, tap gain measurement errors
may have an impact on diversity combining performance.
• The impact is particularly significant in channel with
many taps each containing a small fraction of the total
received energy. (eg. Ultra-wideband channels)
• The impact depends on the modulation scheme.
36
Summary
• Fading makes wireless channels unreliable.
• Diversity increases reliability and makes the channel
more consistent.
• Smart codes yields a coding gain in addition to the
diversity gain.
• This viewpoint of the adversity of fading will be
challenged and enriched in later parts of the course.