Lecture 8: Convolutional Codes

TU Berlin | Sekr. HFT 6 | Einsteinufer 25 | 10587 Berlin

www.mk.tu-berlin.de

Faculty of Electrical Engineering and Computer Systems Department of Telecommunication Systems Information and Communication Theory Prof. Dr. Giuseppe Caire Einsteinufer 25 10587 Berlin Telefon +49 (0)30 314-29668 Telefax +49 (0)30 314-28320 [email protected] Sekretariat HFT6 Patrycja Chudzik Telefon +49 (0)30 314-28459 Telefax +49 (0)30 314-28320 [email protected]

Firma xy Herrn Mustermann Beispielstraße 11 12345 Musterstadt

Berlin, 1. Month 2014

Subject: Text…& Prof. Dr. Giuseppe Caire Lecture 8:

Convolutional Codes

Copyright G. Caire (Sample Lectures) 255


www.mk.tu-berlin.de




Subject: Text…& Prof. Dr. Giuseppe Caire

Introducing memory

• A binary linear block encoder is a linear transformation Fk2

! Fn2

.

• What about using a Linear time-invariant linear system for encoding?

• Convolutional codes consider small k and n, but introduce memory into theencoding process of a sequence of consecutive blocks.

• We may see this a the convolution of a sequence of information blocks {ui}with a matrix G of impulse responses, in order to generate a sequence ofcoded blocks {ci}.



www.mk.tu-berlin.de





“MA” k ⇥ n linear systems

• A k ⇥ n Moving Average (MA) system is defined by:

c(1)

i =

m1,1

X

`=0

g(1,1)` u(1)

i�` + · · · +

mk,1X

`=0

g(k,1)` u(k)

i�`

c(2)

i =

m1,2

X

`=0

g(1,2)` u(1)

i�` + · · · +

mk,2X

`=0

g(k,2)` u(k)

i�`

...

c(n)

i =

m1,n

X

`=0

g(1,n)

` u(1)

i�` + · · · +

mk,nX

`=0

g(k,n)

` u(k)

i�`



www.mk.tu-berlin.de





• Defining a vector output sequence c0

, c1

, c2

, . . . such that ci = (c(1)

i , . . . , c(n)

i )

and a vector input sequence u0

,u1

,u2

, . . . such that ui = (u(1)

i , . . . , u(k)

i ), wecan write

ci =

mX

`=0

ui�`G`

where we let m = max{m(i,j)}.

• We obtain a block-Toeplitz notation

(c0

, c1

, c2

, c3

, . . .) = (u0

,u1

,u2

,u3

, . . .)

2

6

6

6

6

4

G0

G1

· · · Gm 0 · · ·0 G

0

Gm�1

Gm... 0

. . . ... Gm�1

. . .G

0

...0 G

0

. . .

3

7

7

7

7

5

• The impulse responses g(i,j) are called the code generators.



www.mk.tu-berlin.de





D-transform

• D-transform domain

ui ! u(D) =

X

i

uiDi (Laurent series)

• Convolutional encoding in the D-transform domain:

c(D) = u(D)G(D)

or, equivalently,

(c1

(D), . . . , cn(D)) = (u1

(D), . . . , uk(D))

2

6

6

4

g1,1(D) g

1,2(D) · · · g1,n(D)

g2,1(D) g

2,2(D) · · · g2,n(D)

... ...gk,1(D) gk,2(D) · · · gk,n(D)

3

7

7

5



www.mk.tu-berlin.de





Example: a (2, 1) convolutional code

+

+

ui

c(2)

i = ui + ui�1

+ ui�2

c(1)

i = ui + ui�2

• Generator matrix:G(D) = [1 + D2, 1 + D + D2

]



www.mk.tu-berlin.de





Encoder canonical forms

• A code C is defined as the set of all output sequences (code sequences).

• As for block codes, a convolutional code C may have several input-ouputencoder implementations.

• We seek encoders in canonical form: a general problem in system theory isfor a given system, defined as the ensemble of all its output sequences, whatis the minimal canonical realization?

• State-space representation (ABCD):

si+1

= siA + uiB, ci = siC + uiD

a minimal representation is a representation with the minimum number ofstate variables.



www.mk.tu-berlin.de





More on the (2, 1) example

• In the (2, 1) example of before, the state is defined as the content of thememory elements,

si = (ui�1

, ui�2

)

therefore we have

si+1

= si

0 1

0 0

�

| {z }

A

+ui

⇥

1 0

⇤

| {z }

B

andci = si

0 1

1 1

�

| {z }

C

+ui

⇥

1 1

⇤

| {z }

D



www.mk.tu-berlin.de





Generalization ...

• A convolutional code can be seen as a block code defined on the field Fq(D)

of rational functions over Fq.

• Roughly speaking: rational functions are to polynomials as rationals Q to theintegers Z.

• Generalizing what seen before, we can consider G(D) with rational elementsgi,j(D).

• In system theory, this corresponds to AR-MA linear systems.

• The code is preserved by elementary row operations.

• It follows that for any G(D), we can find a systematic generator matrix in theform

G(D) = [I|P(D)]

where I is the k ⇥ k identity, and P(D) is a k ⇥ (n � k) matrix of rationalfunctions.



www.mk.tu-berlin.de





Example: systematic convolutional encoders

• For the (2, 1) example with memory m = 2 and generator matrix:

G(D) = [1 + D2, 1 + D + D2

]

we have:G0

(D) =

1,1 + D + D2

1 + D2

�

+

ui+

c(1)

i = ui

ai

c(2)

i = ai + ai�1

+ ai�2



www.mk.tu-berlin.de





State diagram

• A state-space realization with m binary state variables is a finite-statemachine (FSM) with a state space ⌃ = Fm

2

.

• In general, a FSM is described by its state transition diagram, i.e., by agraph with |⌃| vertices, corresponding to all possible state configurations,and edges connecting those states for which a transition is possible.

• Each edge (s, s0) 2 ⌃ ⇥ ⌃ is labeled by input and output vectors b 2 Fk

2

andc 2 Fn

2

, corresponding to the state transition between s and s0.



www.mk.tu-berlin.de





Example: the (2, 1) 4-state code

• The input-output labels of the state diagram depend on the encoding function:this example corresponds to the feedforward (non-systematic) encoder forthe (2, 1) example seen before.

(0, 0)

(1, 0)

(1, 1)

(0, 1)

0/00

1/11

1/10

0/01

0/11

1/00

1/01

0/10



www.mk.tu-berlin.de





Trellis diagram

• An alternative representation consists of the trellis section, i.e., by a bipartitegraph with |⌃| state vertices on the left and |⌃| state vertices on the right.

• Left vertices represent the possible states at time i, and right verticesrepresent the possible states at time i + 1. Edges represent the possiblestate transitions corresponding to input ui and output ci.

• The trellis representation follows from the state transition diagram byintroducing the time axis.

• A trellis diagram for a convolutional code consists of the concatenation of aninfinite number of trellis sections.

• Given an initial state at time i = 0, an input sequence u(D) determines anoutput sequence c(D) and a state sequence s(D) that correspond to a pathin the trellis.



www.mk.tu-berlin.de





Example: the (2, 1) 4-state code

(0, 0) (0, 0)

(1, 0) (1, 0)

(1, 1) (1, 1)

(0, 1) (0, 1)

0/00

1/11

1/10

0/01

1/01

0/10

1/00

0/11



www.mk.tu-berlin.de





The Factor Graph for a convolutional code

• Three types of variable nodes: information bitnodes ui, coded bitnodes ci

and states nodes si.

• The function nodes correspond to the state and output mappings

si+1




www.mk.tu-berlin.de





Finding G(D) from the state equation

• Consider the ABCD minimal state space realization:

si+1


• By applying D-transform we obtain

D�1s(D) = s(D)A + u(D)B, c(D) = s(D)C + u(D)D

• Eliminating the state s(D), we arrive at c(D) = u(D)G(D) with

G(D) = B�

D�1I + A��1

C + D



www.mk.tu-berlin.de





MAP decoding of convolutional codes

• We consider the transmission of the block code CN obtained by trellistermination of a convolutional code, over a memoryless channel with inputci 2 Fn

2

and output yi 2 Y.

• Block MAP decoding rule (aka, Maximum Likelihood (ML) decoding rule):

bc = arg max

c2CN

N�1

X

i=0

log PY |X(yi|ci)

• Since |CN | = 2

k(N�⌫max

) and N � ⌫max

is generally large, the brute-forceevaluation of the MAP decoding rule is intractable.



www.mk.tu-berlin.de





The Viterbi Algorithm

• The ML decision metric is additive and that the codewords are representedby paths in the code trellis.

• In this case, the ML decoding rule can be computed efficiently by the ViterbiAlgorithm (VA).

• For a state transition (s0, s), let c(s0, s) = (c(1)

(s0, s), . . . , c(n)

(s0, s)) denote thecorresponding block of coded symbols.

• We define the branch metric at trellis section i as

Mi(c) = ↵ log PY |X(yi|c) � �, c 2 Fn2

where ↵ > 0 and � are suitable constant.



www.mk.tu-berlin.de





• The VA maintains one path metric for each surviving path terminating at eachstate, and recursively updates the path metrics as follows:

1. Initialization: M0

(0) = 0, M(s) = �1 for all s 6= 0.2. Add Compare and Select (ACS) recursion: for i = 1, 2, . . . , N , and for all

states s 2 ⌃, let

Mi(s) = max

s02P(s){Mi�1

(s0) + Mi�1

(c(s0, s))}

where P(s) is the set of “parent” states of s, i.e., the set of states s0 2 ⌃

for which there exists a transition (s0, s) in the code state diagram.3. The n-tuple of symbols c(s0, s) achieving the maximum in the ACS step is

added to the survivor path terminating in state s at time i.4. Final decision (trace-back): the maximum of the total metric is obtained

as MN(0). The MAP decoded codeword bc is the corresponding survivorpath.



www.mk.tu-berlin.de





Example: the (2, 1) 4-state code over the BSC

00 00 00 00 00 00 0011

11

0110

001001

01 00 10 00 00 01 00

0 11

1322

2223

2233

2333

3

3

3



www.mk.tu-berlin.de





Example: the (2, 1) 4-state code over the BSC

00 00 00 00 00 00 0011

11

0110

001001

01 00 10 00 00 11 00

0 11

1322

2223

2233

2333

3

4



www.mk.tu-berlin.de





Practical implementation issues

• Path metric re-normalization: the output of the VA does not change if at eachstep the maximum path metric is subtracted from all path metrics.

• Minimizing instead of maximizing: the output of the VA does not change if wechoose ↵ < 0 and min instead of max in the ACS step.

• Register exchange vs. traceback: .... better explained through an example ...

• Handling puncturing: the contribution to the branch metric of a punctured bitis zero.

• Handling ties: if two competing candidate paths terminating in the same statehave the same metric, the VA takes an arbitrary, or random, decision, with noimpact on the average error probability.



www.mk.tu-berlin.de





Optimality of the VA

Theorem 13. The final surviving path in the VA is the ML-decoded codeword.⇤

Proof:M�(a)

M�(b) M+

(b)

M+

(a)

• Claim: if M�(b) < M�(a) then the semi-path b� cannot be part of the MAPcodeword and can be dropped from the metric count.



www.mk.tu-berlin.de





• Compare the 4 path metrics

M�(b) + M+

(b), M�(b) + M+

(a), M�(a) + M+

(b), M�(a) + M+

(a)

• If M�(b) < M�(a) then

max {M�(b) + M+

(b), M�(b) + M+

(a)}<

max {M�(a) + M+

(b), M�(a) + M+

(a)}



www.mk.tu-berlin.de





End of Lecture 8


Date post:	14-Feb-2017
Category:	Documents
Upload:	ngoxuyen
View:	294 times
Download:	6 times

Lecture 8: Convolutional Codes

Documents