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
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
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}.
Copyright G. Caire (Sample Lectures) 256
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
“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�`
Copyright G. Caire (Sample Lectures) 257
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
• 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.
Copyright G. Caire (Sample Lectures) 258
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
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
Copyright G. Caire (Sample Lectures) 259
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
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
]
Copyright G. Caire (Sample Lectures) 260
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
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.
Copyright G. Caire (Sample Lectures) 261
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
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
Copyright G. Caire (Sample Lectures) 262
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
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.
Copyright G. Caire (Sample Lectures) 263
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
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
Copyright G. Caire (Sample Lectures) 264
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
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.
Copyright G. Caire (Sample Lectures) 265
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
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
Copyright G. Caire (Sample Lectures) 266
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
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.
Copyright G. Caire (Sample Lectures) 267
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
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
Copyright G. Caire (Sample Lectures) 268
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
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
= siA + uiB, ci = siC + uiD
Copyright G. Caire (Sample Lectures) 269
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
Finding G(D) from the state equation
• Consider the ABCD minimal state space realization:
si+1
= siA + uiB, ci = siC + uiD
• 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
Copyright G. Caire (Sample Lectures) 270
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
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.
Copyright G. Caire (Sample Lectures) 271
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
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.
Copyright G. Caire (Sample Lectures) 272
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
• 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.
Copyright G. Caire (Sample Lectures) 273
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
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
Copyright G. Caire (Sample Lectures) 274
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
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
Copyright G. Caire (Sample Lectures) 275
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
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.
Copyright G. Caire (Sample Lectures) 276
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
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.
Copyright G. Caire (Sample Lectures) 277
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
• 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)}
Copyright G. Caire (Sample Lectures) 278
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
End of Lecture 8
Copyright G. Caire (Sample Lectures) 279
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