name iterative source- and channel decoding speaker: inga trusova advisor: joachim hagenauer
TRANSCRIPT
Name
Iterative Source- and Channel Decoding
Speaker: Inga Trusova
Advisor: Joachim Hagenauer
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1. Introduction
2. System model
3. Joint Source-Channel Decoding(JSCD)
4. Iterative Source-Channel Decoding(ISCD)
5. Simulation Results
6. Conclusions
Content
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Introduction
PROBLEMS EXIST• Limited block length for source and channel coding
• Data-bits issued by a source encoder contain residual redundancies
• Infinite block-length for achieving “perfect” channel codes
• Output bits of a practical channel decoder are not error free
Application of the separation theorem of information theory is not justified in practice!
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IntroductionGOAL:
To improve the performance of communication systems without sacrificing resources
SOLUTION:
Joint source-channel coding & decoding (JSCCD)• Several auto correlated source signals are considered• Source samples are
1. quantized
2. their indexes appropriately mapped into bit vectors
3. bits are interleaved & channel-encoded
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Introduction
AREA OF INTEREST:
Joint source-channel decoding (JSCD)
Key idea of JSCD:
To exploit the residual redundancies in the data bits in order
To improve the overall quality of the transmission
The turbo principle (iterative decoding between components) is a general scheme, which we apply to JSCD
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System Model
Initial data• AWGN channel is assumed for transmission• A set of input source signals has to be transmitted at each
time index k• Only one of the inputs, the samples , is considered
• are quantized by the bit vector
with and , denoting the set of all possible N-bit vectors
,1,..., , ,..., ,I I I I Lk k k n k N
XkXk
0,1Ik 0,1 NL
(1)
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System Model
Figure 1: System Model
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System ModelAs
coherently detected binary modulation (phase shift keying)
is assumed
Than
conditional pdf of the received value at the channel output, given that code bit has been transmitted, is given by
,yk n
211 2, ,22
, ,2
y vk n k nne
p y vc k n k nn
0,1,vk n
(3)
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System ModelWhere,
2 0
2
Nn Es
Es
0N
the variance
energy that is used to transmit each channel-code bit
One-sided power spectral density of the channel noise
Note: The joint conditional pdf for a channel word Nvy IRk
to be received , given that codeword is transmitted, is the product of (3) over all code-bits, since the channel noise is statistically independent
0,1,vk n
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System ModelIF
are autocorrelated
THAN
show dependencies
AND
are modeled by first-oder stationary Markov-process, which
is described by transition probabilities
ASSUMPTIONS• Transition probabilities and probability-distributions of the
bitvectors are known• Bitvectors are independent of all other data, which is
transmitted in parallel by bitvector
Xk
,1I Ik k
/ 1P I Ik k
IkUk
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Joint Source-Channel Decoding
GOAL:
Distortion of the decoder output signal min
JSCD for a fixed transmitter
Optimization criterion is given by the conditional expectation of the mean square error:
2ˆ 2D E x x I yk k kI yk k (4)
xk
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Joint Source-Channel DecodingIn (4)
is the quantizer reproduction value corresponding to the bitvector , which is used by the source encoder to quantize
is a set of channel output words which were received up to the current time k
D min results in the minimum mean – square estimator
x̂ IkIk
Xk
0, 1,...,y y y yk k
ˆ ˆx E x I x I P I yk k k k kI yk kI Lk
(5)
(6)
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Joint Source-Channel DecodingBitvector a-posteriori probabilities (APPs), using the Bayes-
rule, are given by
Where
is the bitvector a-priori probability
is a normalizing constant
Since
1 , 1P I y B P I y p y I yk k k k k k k k
1P I yk k
/1B p y p yk k k
1, 1 1P I I y P I Ik k k k k
(7)
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Joint Source-Channel DecodingA-priori probabilities are given by
At k=0 the unconditional probability distribution is used in stead of the “old” APPs
Drawback :
From (7) the term is very hard to compute analytically
,1 1 1
1
P I y P I I yk k k k kI Lk
1 1 1
1
P I I P I yk k k kI Lk
, 1p y I yk k k
(8)
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Iterative Source-Channel Decoding
Goal:
To find more feasible, less complex way to compute at least a good approximation
Solution:
Iterative Source-Channel Decoding (ISCD)
We write:
, , 1, 1, 1
p y I yk k kp y I yk k k p I yk k
(9)
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Iterative Source-Channel Decoding
Now,
Bitvector probability densities are approximated by the product over the corresponding bit probability densities
With the bits
, ,, 11, 1
,, 11
Np y I yk k n k
np y I yk k k Np I yk n k
n
0,1,Ik n
(10)
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Iterative Source-Channel Decoding
If we insert (10) into the formula (7) which defines bitvector a-posteriori probabilities we obtain:
The bit a-posteriori probabilities can be efficiently computed by the symbol-by-symbol APP algorithm for a binary convolution channel code with a small number of states.
,1
, 11
N P I yk n kP I y P I yk k k k P I yk n kn
,P I yk n k
(11)
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Iterative Source-Channel Decoding
Note:
ALL the received channel words up to the current time are used for the computation of the bit APPs, because the bit-based a-priori information
For a specific bit
, 1 1
,
P I y P I yk n k k kI L Ik k n
0,1,Ik n
(12)
yk k
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Iterative Source-Channel Decoding
Let interpret the fraction in (11) as the extrinsic information that we get from the channel decoder:
Note:
Superscript “(C)” is used to indicate that
is the extrinsic information produced by the channel decoder .
( )1 ,
1
NCP I y P I y P Ik k k k e k n
n
(13)
( ),
CP Ie k n
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Iterative Source-Channel Decoding
As a result we have:
A modified channel-term (btw brackets ) that includes the reliabilities of the received bits and, additionally, the information derived by the APP-algorithm from the channel-code.
Drawback:
Bitvector APPs are only approximations of the optimal values, since the bit a-priori information didn’t contain the mutual dependencies of the bits within bitvectors
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Iterative Source-Channel DecodingHow to improve the accuracy of the bitvector APPs?
Idea:
Iterative decoding of turbo codes:
From the intermediate results for the bitvector APPs (13),
new bit APPs are computed by
( ),
,
SP I y P I yk n k k kI L Ik k n
(14)
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Iterative Source-Channel Decoding
Bit extrinsic information from the source decoder:
Note:
Computed extrinsic information is used as the new a-priori information for the second and further runs of the channel decoder.
( ),( )
, ( ),
SP I yk n kSP Ie k n CP Ie k n (15)
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Iterative Source-Channel Decoding
SUMMARY OF ISCD:
Step 1
At each time k, compute the initial bitvector a-priori probabilities by:
, 1 1 1
1
P I I P I yk k k kI Lk
,1 1 1
1
P I y P I I yk k k k kI Lk
(8)
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Iterative Source-Channel Decoding
Step 2:
Use the results from step 1 in to compute the initial bit a-priori information for the APP channel decoder.
Step 3:
Perform APP channel decoding
, 1 1
,
P I y P I yk n k k kI L Ik k n
(12)
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Iterative Source-Channel Decoding
Step 4:
Perform source decoding by inserting the extrinsic bit information from APP channel decoding into
to compute new (temporary) bitvector APPs
Step 5:
If this is the last iteration proceed with step 8, otherwise continue with step 6
( )1 ,
1
NCP I y P I y P Ik k k k e k n
n
(13)
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Iterative Source-Channel Decoding
Step 6:
Use the bitvector APPs of step 4 in
to compute extrinsic bit information from the source redundancies
( ),
,
SP I y P I yk n k k kI L Ik k n
( ),( )
, ( ),
SP I yk n kSP Ie k n CP Ie k n
(14)
(15)
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Iterative Source-Channel Decoding
Step 7:
Set the extrinsic bit information from Step 6 equal to the new bit a-priori information for the APP channel decoder in the next iteration ; proceed with Step 3
Step 8:
Estimate the receiver output signals by
using the bitvector APPs from Step 4
ˆ ˆx E x I x I P I yk k k k kI yk kI Lk
(6)
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Iterative source channel decoding
Figure 2: Iterative Source-Channel Decoding according to the Turbo Principle
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Iterative source channel decoding Computation of the bitvector APPs by (13) requires bit
probabilities which can be computed from from the output L-values:
With inversion:
( ),( ) log, ( ) 1,
CP I oe k nCL Ie k n CP Ie k n
( )( ), ( )( ) , ,
, ( ),1
CL I Ce k n L I IeC e k n k nP I ee k n CL Ie k ne
( ),
CL Ie k n
(16)
(17)
are fixed real numbers
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Iterative source channel decoding
SIMPLIFICATION 1:
Reminder: formula (13) bitvector APPs computation:
Let’s insert (17) into (13) and turn the product over the exponential functions into summations in the exponents:
( )exp1 , ,1
NCP I y A P I y L I Ik k k k k e k n k n
n
(18)
( )1 ,
1
NCP I y P I y P Ik k k k e k n
n
(13)
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Iterative source channel decoding
Benefits of using (18) instead of (13):
• Normalizing constant Ak doesn’t depend on the variable Ik,n
• L-values from the APP channel decoder can be integrated into the Optimal-Estimation algorithm for APP source decoding without converting the individual L-values back to bit probabilities
• Strong numerical advantages
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Iterative source channel decodingSIMPLIFICATION 2: The computation of new bit APPs within the iteration is still
carried out by (14)Reminder:
But, instead of (15) for the new bit extrinsic informationReminder:
( ),
,
SP I y P I yk n k k kI L Ik k n
(14)
( ),( )
, ( ),
SP I yk n kSP Ie k n CP Ie k n (15)
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Iterative source channel decodingExtrinsic L-values are used:
Benefits of using (19) in stead of (15):• Division is turned into a simple subtraction in the L-value
domain
THUS,
In ISCD the L-values from the APP channel
decoder are used and the probabilities
are not required
( ) ( ), , ,
S S CL I L I L Ie k n k n e k n (19)
( ),
CL Ie k n ,CP Ie k n
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Quantizer Bit MappingAssumption:Input is a low-pass correlationThe value of the sample xk will be close to xk-1If: The channel code is strong enough L-values at
the APP channel decoder output have large magnitudes a-priori information for the source decoder is perfect ISCD: APP source decoder tries to generate extrinsic
information for a particular data bit , while it exactly knows all other bits
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Quantizer Bit Mapping
Figure 3: Bit Mappings for a 3-bit Quantizer to be used in ISCD
natural
optimized
Gray
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Simulation ResultsSimulation process:
• Correlation of independent Gaussian random samples by a first-order recursive filter (coefficient )
• Source encoders: 5-bit Lloyd Max scalar quantizers
• 50 mutually independent bitvectors were generated, all transmitted at time index k.
• The bits were scrambled by a random-interleaver
0.9
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Simulation Results
Simulation process (contd.):
• The bits were channel-encoded by a rate- ½ recursive systematic convolution code (RSC-code with memory 4, which were terminated after each block of 50 bitvectors (250 bits)
• AWGN-channel was used for transmission
• ISCD was performed at the decoder
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Simulation Results
Figure 4: Performance of ISCD for various 5-Bit Mappings
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Conclusions
Strong quality gains are achievable by:
• Application of the turbo principle in joint source-channel decoding
• Bitmapping of the quantizers is important for the performance
• Optimized bit mapping of the quantizers in ISCD allows to obtain strong quality improvements
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Thank you for your attention!