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Advantage of Diversity Increase the system reliability (decrease the bit or packet error rate) Increase the achievable data rate and hence system capacity Increase the coverage area Decrease the required transmit power
Diversity Techniques Instructor: Prof. Rajarshi Mahapatra Dept. of ECE, GEU Email:
[email protected] Cabin: ECE Dept.
R. Mahapatra, GEU
Why Diversity? Average BER in fading channel P P p d The average BER varies approximately as b
Ps Ps p d
and
0
SNR with 20 dB, BER 5×10-3 For BER=10-6, we need SNR 57 dB
Ps
y hx n
0
1 P r SNR
– BER in wired system varies exponentially with SNR
• which is clearly unpractical,
to make sure that the SNR has a smaller probability of being low.
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1/ SNR
1/ SNR
0
0
1
p(r )dr 2rdr SNR
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It is the technique used to compensate for fading channel impairments Diversity is usually employed to reduce the depth & duration of the fades experienced by a receiver Diversity is achieved by creating several independent paths between the transmitter and receiver
J k0v 2 1 2 S2 f 2 f1 2 0 2
Temporal separation can be easily converted into spatial separation Applied to spatial, temporal, and frequency diversity Assumptions:
Correlation between the received signals need to be minimized
Receiver combines the received signal for the several paths using some method Diversity-combining uses the fact that independent signal paths have a low probability of experiencing deep fades simultaneously
(i) validity of the Wide Sense Stationary Uncorrelated Scatterer (WSSUS) model, (ii) no existence of Line Of Sight (LOS), (iii) exponential shape of the Power Delay Profile (PDP), (iv) isotropic distribution of incident power, and (v) use of omnidirectional antennas. Wireless Communication
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What is Diversity
Correlation coefficient of two signals that have a temporal separation τ and a frequency separation f1 − f2.
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1 SNR
System performance will improve, if we decrease the probability of feed fade. Diversity is a way to achieve this
Correlation Coefficient
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r
Deep Fade
BER Probability of deep fade
to improve the BER
xy
2
In Rayleigh fading, the probability of deep fade
• Need more transmit power, more than 1000 times than wired system with achieve same BER
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h P r2P
1 r2 P SNR
r 2 P n2
1 2 s
n2
n2
Performance is poor, when
in Rayleigh fading the BER decreases only linearly with the SNR
For example:
2
Need of Diversity
b
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What is Diversity
Types of Diversity
In diversity, the same data has been send over independent fading paths. These independent paths are combined in some way such that the fading of the resultant signal is reduced Two types: Microdiversity & Macrodiversity Microdiversity (simply Diversity): to mitigate the effect of multipath fading
Spatial diversity: several antenna elements separated in space. Temporal diversity: transmission of the transmit signal at different times. Frequency diversity: transmission of the signal on different frequencies. Angular diversity: multiple antennas (with or without spatial separation) with different antenna patterns. Polarization diversity: multiple antennas with different polarizations (e.g., vertical and horizontal).
by combining signals received by several antennas
Macrodiversity (Cooperative Diversity): to mitigate the effects of shadowing from buildings and objects by combining signals received by several base stations or access points R. Mahapatra, GEU
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Spatial Diversity
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Temporal Diversity Tx
Rx
SISO
Use multiple transmit and multiple SIMO Tx Rx receive antennas MISO Rx Tx A large correlation between signals at antenna elements is undesirable, as it MIMO Tx Rx decreases the effectiveness of diversity Thus, it is important to to establish a relationship between antenna spacing and the correlation coefficient Multiple Tx: split power over several Tx antennas. More antennas = more power split Multiple Rx: collect signal by several Rx antennas. More antennas = more collected power Rule of thumb: Antenna separation about l/2 is required Larger separation is required for directional antennas R. Mahapatra, GEU
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As the wireless propagation channel is time variant, signals that are received at different times are uncorrelated. For “sufficient” decorrelation, the temporal distance must be > coherence distance Temporal diversity can be realized in different ways Repetition coding: signal is repeated several times Automatic Repeat reQuest (ARQ) Combination of interleaving and coding
Reduces overall transmission data rate R. Mahapatra, GEU
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Frequency diversity
Angular diversity
Same signal is transmitted at two (or more) different frequencies These frequencies are spaced apart by more than the coherence bandwidth of the channel, then
A fading dip is created when multipath components, come from different directions, interfere destructively. Angular diversity is usually used in conjunction with spatial diversity; it enhances the decorrelation of signals at closely spaced antenna Form a directional beam towards the receiver or group of receiver Smart antennas are antenna arrays with adjustable phase at each antenna element: such arrays form directional antennas that can be steered to the incoming angle of the strongest multipath component
Their fading is approximately independent, and The probability is low that the signal is in a deep fade at both frequencies simultaneously.
Also wideband signals achieve frequency diversity, like OFDM techniques over wideband (WLAN, WiMax, LTE) R. Mahapatra, GEU
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Polarization diversity
Benefits of Spatial Diversity
Transmit and/or receive with both vertical and horizontal polarization Scattering is independent for each polarization, giving independent paths Limited to 2 transmit and 2 receive diversity Tx polarization diversity: half power for each polarization
Main advantage of spatial diversity relative to time and frequency diversity is that no additional bandwidth or power is needed However, the cost of each additional antenna, its RF chain, and the associated signal processing required to modulate or demodulate multiple spatial streams may not be negligible, A trade-off is often very attractive for a small number of antennas Follows two techniques
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Receive diversity: multiple receive antennas Transmit diversity: multiple transmit antenna
Diversity Gain
Achieves its performance enhancement by coherently combining the energy received by each of the antennas Does not rely on statistical diversity between the channels Achieved even if there is no fading In case of LOS system, the received SNR increases linearly with the number of receive antennas Because the channels are all correlated in this case, there is no diversity gain Array gain is equal to number of Receive antennas Helps to improve system capacity
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Diversity Order
If antennas are sufficiently spaced, system achieve full diversity order N Ps c1 the average BEP, If the antennas are not sufficiently spaced or the channel is LOS c2 , c2 are the constant the average BEP, Ps c2 N d 1 d
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Several methods to combine the receiver branches:
Diversity order Nd Nt Nr
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Combining Scheme
It is defined as number of uncorrelated channel paths between the transmitter and the receiver If Nt is the number of Tx antennas and Nr is the number of Rx antennas
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Array Gain
Results from the creation of multiple independent channels between the transmitter and the receiver and is a product of the statistical richness of those channels Due to the better (than exponential) pdf p() of the combined SNR Better BER results of the integration Achieve in Receive & Transmit diversity R. Mahapatra, GEU
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Selection combining (SC) Threshold combining Maximal ratio combining (MRC) Equal gain combining (EGC) Mix of the above
Tradeoff between performance and complexity R. Mahapatra, GEU
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Selection combining (SC)
Selection combining (SC)
simplest type of combiner Estimates the instantaneous Transmitter strengths of each of the streams Select the highest one Channel path gain to each branch ri exp( j ) Noise PSD at each branch is Ni instantaneous SNR ri2 / Ni If noise power N0 is same on all branches, this is equivalent to select largest ri2 N 0 Combiner output SNR = best branch SNR
Select Best Antenna
Rx Filter
If the BER is determined by noise, then RSSI-driven diversity is the best of all the selection diversity methods, as maximization of the RSSI also maximizes the SNR
RSSI Comparison RSSI
Rx Filter
If the BER is determined by co- Rx Filter channel interference, then RSSI is no longer a good Tx Sequence selection criterion. The selection-diversity in BERdriven
Demodulator
Correlation Comparison Correlation
Rx Filter
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For M branch diversity, the CDF of the combined SNR Probability of best SNR < = Probability all SNR <
P
p
p i exp i i i 1
pmax 1 , 2 ,.., M p i M
i 1
The outage probability for a target o on the ith branch in Rayleigh fading is M M Pout 0 p i 0 1 exp 0 i 1
i 1
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Outage Probability of Selection Combining in Rayleigh Fading
Selection combining (SC) The SNR distribution in Rayleigh fading CDF of M branches is given by
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Demodulator
Most benefit from M=1 (no diversity) to M=2 Benefit reduces as M increases
If the average SNR for all of the branches are the same
1 exp
Pout 0 p R. Mahapatra, GEU
M
i
0
0
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Selection combining (SC) M 1 exp
M 1
exp
Probability of error is calculated by averaging the AWGN formula over the PDF of the SNR
Pb Pb p d
Now we use
The average SNR of the combiner output in i.i.d. Rayleigh fading is
p 0
d M 1 exp 0
M 1
exp
M 1 d i 1 i
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and
Ps Ps p d
0
p
0
M
1 e
M 1
e
For BPSK there is no closed form, needs numerical integration (or simulation) For DPSK we get
The average SNR gain increases with M, but not linearly R. Mahapatra, GEU
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Selection combining (SC)
The PDF of the combined SNR is given by differentiating Pout w.r.t p
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1 M Pb e p d 2 2 0 R. Mahapatra, GEU
M 1
1
m
M 1 m
1 m
m 0
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Average BEP of BPSK under SC with i.i.d. Rayleigh Fading
Maximal Ratio Combining (MRC) MRC compensates for the phases, and weights the signals from the different antenna branches according to their SNR. Instead of selecting one branch, all branches are added with weight ai exp(-ji)
Similarly, more gain when moving from M=1 to M=2 Gains reduces as M increases
where i is the phase of the incoming signal on the i-th branch
If a transmitted symbol is s with unity power |s|2=1 The received symbol at branch i is: s ri exp(ji) + ni The combined symbol is: S s M r a M n a comb
i 1
i
i
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Maximal Ratio Combining (MRC)
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Phase Correction
M ri ai i 1 M N o ai2
We need to select ai to maximize SNR Using Swartz inequality we find the ai ri optimum weights
Phase Correction
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Maximal Ratio Combining (MRC)
Measurement
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i
Noise Part
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Assume all ni have equal variance No, the SNR is Measurement
i
i 1
Signal Part
The resulting SNR is Combined SNR is the sum of all branches SNR Each i is exponential with mean i 27
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1 No
2
i 1
No M
r i 1
M
2
i
i i 1
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Maximal Ratio Combining (MRC)
Average BER of BPSK
Assuming all branch SNR have equal mean, is chi-square with 2M degrees of freedom (see M Proakis, chapter 2) and mean The PDF is given by e
The average BER is calculated by averaging the AWGN BER over the random SNR For BPSK
p
M 1
Pb Q
M M 1! o
p d
1e R. Mahapatra, GEU
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k 1 o k 1 k 1 ! M
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M 1 m 1 m m 0 2
M M 1
0
o
2 p d
1 2
The outage probability is found from PDF Pout o Prob o
0
m
1
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Performance of MRC
Equal Gain Combining (EGC)
Most used diversity method Phase Correction
Measurement
Measurement
Phase Correction
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Equal Gain Combining (EGC)
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Equal Gain Combining (EGC)
MRC required knowledge of the SNR on each branch Simpler approach is equal weight for all branches (all ai=1) 2 M The combined SNR is 1 ri
From the CDF we find the outage probability
There is no closed form solution for the CDF or PDF except for M=2
Also the BER for BPSK is
M N o i 1
CDF is P 1 e PDF is p
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1
2
e
Pout o 1 e 2 o
1 2Q
2
Pb Q
1 1 e 2 e 1 2Q 4
1 2Q
2 o
2 p d
0
2
33
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Space Time Block Codes
A special application of MIMO system with multiple antennas - Intelligent coding of symbols across space and time Signal processing wherein time (the natural dimension of digital communication) is complemented with spatial dimension inherent in the multiple spatially-distributed antennas Redundancy is increased by decreasing the independence between the signals transmitted by different antenna elements Time diversity is also introduced on the data stream from each antenna element Two major types of space time coding Space time block coding (STBC) Space time trellis coding (STTC)
There are few major types Transmit diversity: main goal is diversity gain Spatial multiplexing: main goal is increase data rate Eigen steering: main goal is both. Requires knowledge of the channel at the transmitter side Mix of the above: Lots of research
Transmit diversity, spatial multiplexing and simplified version of Eigen steering are used in 3G and 4G standards
STBC is simpler by STTC can provide better performance STBC is used in mobile communications. STTC is not used in any systems yet
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o
2 1 1 1 1 2 1
Space Time Coding
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Alamouti Transmit Diversity
2 Tx-1 Rx - s* 2
Simple yet effective Space/time encoding of data sequence at the transmitter Orthogonal block codes used Combining/processing scheme at the receiver Decision rule: maximum likelihood detection
Time
s1
s2
-s2*
s1 *
time t time t+T
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s 1*
y1
h4 s 1*
s2
n3 n4
s1
s2
-s2*
s1 *
2
2 1
h2
h2
h1
Combiner h4
h3
1 2
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2
Received SNR
1 1
12 22 2N0
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No Diversity Downlink with 2x1 ATD Uplink with 1x2 MRC -1
10
ML Detector
h3
-2
h4
4
3
3
10
-3
10
-4
10
4
0
5
10
15
20
25
30
Average Received CNR (dB)
1
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1×2 MRC in uplink has 3 dB better performance than 2×1 ATD in downlink. In ATD scheme, each transmitting antenna radiates half the energy in order to ensure the same total radiated power as with one transmit antenna
2 3 4 2 Wireless Communication
2 1
1 2
4 3
3 4
39
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Interesting Facts If the system is radiation power limited, in order to have the same total radiated power from two transmit antennas the energy allocated to each symbol should be halved. This results in a 3-dB penalty in the error performance A 3-dB reduction in amplifiers power handling is very significant and may be desirable in some cases
Sensitivity to Channel Estimation Errors
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Antenna Configurations: to provide sufficient decorrelation at the base station receiver, they must be on the order of ten wavelengths apart at the mobile station by about three wavelengths
Soft Failure the diversity gain is lost but the signal may still be detected
The scheme is very sensitive to CSI. it is assumed that the receiver has perfect knowledge of the channel
Impact on Interference
The Delay Effects
Although half the power is transmitted from each antenna, it appears that the number of potential interferers is doubled, i.e., we have twice the number of interferers, each with half the interference power
With N branch transmit diversity, if the transformed copies of the signals are transmitted at N distinct intervals from all the antennas, the decoding delay is N symbol periods
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Interesting Facts
Power Requirements
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1
=(12+22)s1+h1*n1+h2n2*
0
~ s1 12 22 32 42 s1 h2*n1 h2 n2* h3*n3 h4 n4* ~ s 2 2 2 2 s h* n h n* h* n h n*
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Where, s~1=h1*y1+h2y2*
y2=-h1s2*+h2s1*+n2
d2(y1,h1sl1+h2sl2)+d2(y2,-h1sl2*+h2sl1*) d2(y1,h1sk1+h2sk2)+d2(y2,-h1sk2*+h2sk1*) ~ ~ (12+22-1)|sl|2+d2(s,sl) (12+22-1)|sk|2+d2(s,sk) ~ ~ PSK – Signals: d2(s,sl) d2(s,sk)
y4 h3 s1* h4 s1* n4
2
ML Detector
~ s1=h1*y1+h2y2* ~ s =h *y -h y *
~ +n
-s2* s1*
y3 h3 s1 h4 s2 n3
s2
h2 ~ s1 ~ s2
y1=h1s1+h2s2+n1
~ s1 h1* y1 h2 y2* h3* y3 h4 y4* ~ s h* y h y * h* y h y *
y2 h1s2* h2 s1* n2
h2
10
h2 ~ s1 ~ s2
Channel Estimator
y1 h1s1 h2 s2 n1
s1
h1
h1
h3
Ant 2
=
Antenna 2
Bit Error Rate
Ant 1
h1
Comparison of BER performance between MRC and ATD
Channel Estimator
h2
s2
Combiner
37
n1 n2
h1
Channel Estimator
Ant 2
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s1
h1
n1 n2
h2
2 Tx-2 Rx - s* 2
h1 Ant 1
y2 Antenna 1
Space
s1
41
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MIMOh System
Matrix Representation
11
y1 h11 y h 2 21 . y N R hN R 1
h12
h21 Tx 1
Rx 1
h22 Rx 2
Tx 2
h1NT
hN R 1
h2 NT
h12 h22 . hN R 2
hN R NT
Tx NT
y1 h11s1 h12s2 . . h1 NT s NT n1
With simplest Space time coding
Channel matrix NR NT
Output NR 1 column matrix
Rx NR
h1 NT s1 n1 h2 NT s2 n1 . . . . hN R NT s NT nN R . .
Input NT 1 column matrix
Y HS N
y2 h21s1 h22s2 . . h2 NT s NT n2 . . . . . . . . . . . . . y N R hN R 1s1 hN R 2 s2 . . hN R NT s NT nN R
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Matrix Representation
44
Code Rate In the space-time block code, the number of symbols the encoder takes as its input in each encoding operation is k The number of transmission periods required to transmit the space-time coded symbols through the multiple transmit antennas is p.
The received signal covariance matrix E YY H RYY HRSS H R NN
noise NR 1 column matrix
Total received signal power can be expressed as trRYY NR
Rate
tr(A) denotes the trace of matrix A, obtained as the sum of the diagonal elements of A
where rb and rs are the bit and symbol rate, respectively, and B is the bandwidth
T
T
I NT is the NT NT identity matrix Wireless Communication
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Full Rate=1
4 transmit antenna
s1 s S4 2 s3 s4
8 transmit antenna s s s s
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S8
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s2 s1
s3 s4
s4 s3
s1 s2
s3 s2 s1
s5 s6 s7 s8
s6 s5 s8 s7
s7 s8 s5 s6
s7 s8 s5
s8 s7 s6
s1 s2 s3
s2 s1 s4
s3 s4 s1
s6
s5
s4
s3
s2
1
2
3
s2 s3 s4
s1 s4 s3
s3 s1 s2
s5 s6 s7
s6 s5 s8
s8
s7
4
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Transmit Antenna in Full Rate
s S2 1 s2
With 2 transmit antennas
k p
The spectral efficiency of the space-time block code is given by r r mR km b s bits/sec/Hz B rs p
The covariance matrix of the transmitted P signal RSS N I N R. Mahapatra, GEU
R
s8 s7 s6 s5 s4 s3 s2 s1
s2 s1
It is desirable to construct the full code rate R = 1 transmission schemes for any number of transmit antennas,
s4 s3 s2 s1
since full rate codes are bandwidth efficient
For NT transmit antennas, the minimum value of transmission periods p to achieve the full rate min( 24cd ), where 0 c,0 d 4 and 8c 2d NT
For NT ≤ 8, the minimum value of p is given by NT 2 p 2 NT 3 p 4 NT 4 p 4 NT 5 p 8 47
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and
NT 6 p 8 NT 7 p 8 NT 8 p 8
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Full Rate=1 With 3 transmit antennas
8 transmit antenna s1 s 2 S5 s3 s4 s5 R. Mahapatra, GEU
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s2 s1 s4 s3 s6
s2 s1
s3 s4
s4
s1
s4 s3 s2
s5 s6 s7
s6 s5 s8
s7 s8 s5
s1 s8
s8 s1
s7 s2
s6 s3
s1 S3 s2 s3
s3 s4 s1 s2 s7
Example
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Code Rate 1/2 s1 S s2 s3
s4 s3 s2
c 3
s8 s7 s6 s5 s4
s2 s1 s4
s3 s4 s1
s4 s3 s2
s1* s2* s3*
s2* s1* s4*
s3* s4* s1*
s4* s3* s2*
Code rate 3/4 s1 S s2 s3 h' 3
49
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s2* s1* 0
s3* 0 s1*
0 s3* s2*
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