Name Iterative Source- and Channel Decoding Speaker: Inga Trusova Advisor: Joachim Hagenauer.

transcript

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

0,1Ik 0,1 NL

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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

conditional pdf of the received value at the channel output, given that code bit has been transmitted, is given by

211 2, ,22

y vk n k nne

p y vc k n k nn

0,1,vk n

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System ModelWhere,

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

show dependencies

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

,1I Ik k

/ 1P I Ik k

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Joint Source-Channel Decoding

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)

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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

0, 1,...,y y y yk k

ˆ ˆx E x I x I P I yk k k k kI yk kI Lk

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Joint Source-Channel DecodingBitvector a-posteriori probabilities (APPs), using the Bayes-

rule, are given by

is the bitvector a-priori probability

is a normalizing constant

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

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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

P I y P I I yk k k k kI Lk

P I I P I yk k k kI Lk

, 1p y I yk k k

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Iterative Source-Channel Decoding

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

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Bitvector probability densities are approximated by the product over the corresponding bit probability densities

With the bits

, ,, 11, 1

Np y I yk k n k

np y I yk k k Np I yk n k

0,1,Ik n

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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.

N P I yk n kP I y P I yk k k k P I yk n kn

,P I yk n k

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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

P I y P I yk n k k kI L Ik k n

0,1,Ik n

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Let interpret the fraction in (11) as the extrinsic information that we get from the channel decoder:

Superscript “(C)” is used to indicate that

is the extrinsic information produced by the channel decoder .

( )1 ,

NCP I y P I y P Ik k k k e k n

CP Ie k n

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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?

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

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Bit extrinsic information from the source decoder:

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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SUMMARY OF ISCD:

Step 1

At each time k, compute the initial bitvector a-priori probabilities by:

, 1 1 1

P I I P I yk k k kI Lk

,1 1 1

P I y P I I yk k k k kI Lk

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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

P I y P I yk n k k kI L Ik k n

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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 ,

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Step 6:

Use the bitvector APPs of step 4 in

to compute extrinsic bit information from the source redundancies

( ),( )

, ( ),

SP I yk n kSP Ie k n CP Ie k n

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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

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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

are fixed real numbers

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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

( )1 ,

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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 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

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

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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

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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!

Name Iterative Source- and Channel Decoding Speaker: Inga Trusova Advisor: Joachim Hagenauer.

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