1. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Brief Overview of Information TheoryBrief Overview of Information Theory
and Channel Codingand Channel Coding
Steven D. Gray
1
2. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
OutlineOutline
• Information theory
– Gaussian channel
– Rayleigh fading channels
• Two approaches for achieving the same rate
• Convolutional encoding
• Convolutional decoding
• Hardware implementation of a Viterbi
• Conclusions
2
3. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Brief Introduction to InformationBrief Introduction to Information
TheoryTheory
);(max
)(
YXIC
xp
=
Encoder
Channel
)|( xyp
Decoder
Message
Estimate
of Message
W n
X n
Y Wˆ
n
X Is a codeword from an alphbet of size n (ex. A point in an 8 PSK consellation)
Channel capacity is the highest rate in bits per channel use at which information
can be sent with arbitrary low probability of error.
3
4. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
A Little Information TheoryA Little Information Theory
Capacity for the Gaussian ChannelCapacity for the Gaussian Channel
( )
( )YXI
PXxp
C ;
E:
max
2
≤
=
X Y
Z
+
For a Gaussian Channel with Bandwidth, W
+=
W
SNR
WC 1log
0N
P
SNR =:
bits per second
4
5. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
A Little Information TheoryA Little Information Theory
Capacity for the Flat Rayleigh ChannelCapacity for the Flat Rayleigh Channel
−⋅⋅⋅= Γ
P
EeeWC i
1
log
1
2
Average Capacity
where
∑
∞
= ⋅
−
++=−
1 !
)(
)ln()(
k
k
i
kk
x
xExE
P is the average power and E is Euler's constant
Source: W.C.Y. Lee, "Estimate of Channel Capacity in Rayleigh Fading Environment,"
IEEE Transactions on Vehicular Technology, Vol. 39, No 3, August 1990.
5
6. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
A Little Information TheoryA Little Information Theory
Capacity Region ComparisonCapacity Region Comparison
5 6 7 8 9 10 11 12 13 14 15
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
bits/sec/Hz
SNR or Average Power (dB)
Shannon - Gaussian Channel
Shannon - Flat Rayleigh Fading
• For channels of interest (heuristically speaking)
- Gaussian capacity is an upper bound
- Flat Rayleigh capacity is a lower bound
6
7. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
A Little Information TheoryA Little Information Theory
Gaussian Channel CapacityGaussian Channel Capacity
Shannon Capacity vs. Existing 2.4 GHz Wireless LAN at 10-6
BER
0 1 2 3 4 5 6 7 8 9 10
0
0.5
1
1.5
2
2.5
3
3.5
4bits/sec/Hz
SNR (dB)
Shannon
Barker
CCK
PBCC
7
8. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
A Little Information TheoryA Little Information Theory
ConclusionsConclusions
• Shannon tell us that there is room for exploitation
• Approaches should be pursued to exploit cases when
the SNR is good
– With a good code, 20 Mbps is possible in the
Gaussian channel when the SNR is 10 dB or less
– Good codes are available with reasonable
complexity
8
9. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Two Approaches for AchievingTwo Approaches for Achieving
Same RateSame Rate
• Approach 1
– Uncoded BPSK modulation
+ IEEE802.11a without convolutional coding
+ Perfect synchronization and channel estimation
+ Rate = 12 Mbps
– Additive White Gaussian Noise (AWGN)
• Approach 2
– Coded QPSK modulation
+ IEEE802.11a PHY with convolutional coding
+ Rate 1/2, 64 state convolutional code
+ Perfect synchronization and channel estimation
+ Rate = 12 Mbps
– AWGN
9
10. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Two Approaches for AchievingTwo Approaches for Achieving
Same RateSame Rate
-4 -2 0 2 4 6 8
10
-7
10
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR
BER
Bit Error Rate, IEEE802.11a 12 Mbits in AWGN, uncoded BPSK and Rate 1/2 QPSK
Uncoded BPSK
Rate 1/2 QPSK
10
11. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Two Approaches for AchievingTwo Approaches for Achieving
Same RateSame Rate
-4 -2 0 2 4 6 8
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
SNR
PER
64 byte Packet Error Rate, IEEE802.11a 12 Mbits in AWGN, uncoded BPSK and Rate 1/2 QPSK
Uncoded BPSK
Rate 1/2 QPSK
Conclusion: Channel Coding can Improve Spectrum Efficiency Bandwidth Reduction
11
12. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Convolutional EncodingConvolutional Encoding
Data Source
+
+
[ ]1,0][ ∈nb
30][ ≤≤∈ psmw p
Storage Element
Generic Rate 1/2 Encoder
00
11
01
10
S 0
S 1
S 2
S 3
11 11 11
11
10
01
10
01
00
01
00 00 00 00
11
00
11 00
10 10
01 01 01
10
11
10
Trellis Diagram
• R=1/2
• 4 state
• Start from all
zero state
12
13. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Convolutional DecodingConvolutional Decoding
• Optimal, bit error rate, decoding is achieved by maximizing
the likelihood function for a given codeword
– Compare the received codeword to all possible
codewords and pick output with smallest distance
• Viterbi in 1967 published a dynamic programming algorithm
for decoding
• Complexity in decoding is proportional to the number of
states and the number of branches into each state
– Example: 64 state code used in PBCC or IEEE802.11a
+ 128 metric calculations per transition in the trellis
13
14. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Hardware Implementation of ViterbiHardware Implementation of Viterbi
• 64 state code from PBCC and IEEE802.11a
• 32 Add Compare and Select (ACS) units (32 butterflies)
• Trace back length is 32 (should be 4 - 5 times constraint
length)
• Input is <3,2,t> and path metrics are <10,9,t>
Branch
Metric
Computation
Add Compare
Select
Trace Back
Unit
Set Initial
State
Store
Path
Metric
Branch
History
Bit StreamSoft
Inputs
14
15. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
Hardware Implementation of ViterbiHardware Implementation of Viterbi
• Register Transfer Logic (RTL) synthesis for Viterbi VHDL
is done using Synopsys Design Compiler
• Target for RTL is Xilinx Virtex 1000e Field Programmable
Gate Array (FPGA)
• Design complexity
– 55.7K logic gates
– 8Kbytes of Xilinx RAM (4 RAM blocks) for convience
– Actual required RAM is 500 bytes
15
16. Submission
May, 2000 Doc: IEEE802.11-00/086
Steven Gray, NokiaSlide
ConclusionsConclusions
• Channel coding is a means to improve spectrum
efficiency over an uncoded system
• Particularly for achieving rates above 20 Mbps, channel
coding will make required SNR's reasonable
• Hardware complexity is absorbed in the digital ASIC
– Impact on IC costs are small
– Engineering design costs are always a factor for a
more complex design
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