CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
A CONVOLUTIONAL CODING SYSTEM FOR
1\
COMMUNICATION SATELLITE APPLICATIONS
A graduate project submitted in partial satisfaction of the
requirements for the degree of t~aster of Science ·in
Electrical Engineering
by
Ralph Jiro Yamamoto
June, 1980
The Graduate Project of Ralph Jiro Yamamoto is approved:
California State University, Northridge
ii
DEDICATION
To my wife, Jean
iii
TABLE OF CONTENTS
Page
DEDICATION
. iii
ABSTRACT . . . • •
• vii
I.
II.
III.
IV.
Introduction •
. ..•.
1
Convolutional Coder Simulation .
9
Viterbi Decoder Simulation . .
27
Conclusions
36
.•..
REFERENCES
49
APPENDICES
A
Encoder Program Listing
• • • •
50
B
Code Test Program Listing
• • • •
63
c
Viterbi Decoder Program Listing
. • • • 73
iv
LIST OF FIGURES
Page
1.
Limiting Values of E(R) for very noisy channels •
2.
General form, Rate k/v Convolutional encoder
10
3.
Convolutional encoder K=3, v=2, k=l ••
12
4.
Tree diagram for encoder in Figure 3
13
5.
Trellis diagram for encoder in Figure 3 •.
14
6.
General form of Convolutional encoder,
Rate 1/2, constraint l~ngth 5 •
. •••
16
7.
Encoder Algorithm • • . . • • .
18
8.
Encoder program, typical output .
19
9.
Code Test Algorithm • . . . . • .
20
10.
Code Test program, typical output •
21
11.
Code test program, Catastrophic error condition
25
12.
Code test program, selected encoder •
26
13.
Viterbi Algorithm • • • . . • • . •
30
14.
Quantization intervals, demodulator
output normalized to noise level
32
15.
Noise distribution
33
16.
Metric table
35
17.
Viterbi decoder program, typical output •.
37
18.
Performance comparison of Viterbi decoding using
rate 1/2, K=5 code with 8-, 16-, and 32-bit path
lengths and 2- and 8-level quantization . • • • •
39
.••.
v
5
. Page
Bit error probability bound for
rate 1/2, K=5
• • • •
40
20 ..
Encoder output
41
21.
Decoder output, noiseless input
42
22.
Decoder output, noisy input
43
23.
Decoder output, loss of branch synchronization .
44
24.
Decoder output, loss of node synchronization
45
19.
vi
LIST OF TABLES
Page
1.
Basic coding gain for soft-decision
Viterbi decoding and hard-decision
sequential decoding • . . . •
7
2.
Maximum-minimum free distance convolutional codes •
23
3.
State-transition table
Convoiutional encoder, XSEL=37, YSEL=25
28
vii
ABSTRACT
A CONVOLUTIONAL CODING SYSTEM FOR
COMMUNICATION SATELLITE APPLICATIONS
by
Ralph Jiro Yamamoto
Master of Science in Electrical Engineering
Communication satellite channels are characterized by limited
transmission power and bandwidth.
The primary disturbance on these
channels can be quite accurately modeled as additive Gaussian noise
which~is
11
White" enough to be essentially independent from one bit
interval to the next.
Convolutional coding with soft-decision
Viterbi decoding has proven to be a very cost-effective system for
these channels.
A coding system, consisting of a convolutional
encoder and a Viterbi maximum likelihood decoder is developed
utilizing a constraint length of 5 and a half rate code with 8
level soft quantization.
The primary emphasis of this graduate project has been to develop
simulations of the convolutional encoder and the Viterbi decoder
that can be used in the coding system described above.
viii
The encoder
simulation is used to develop an optimum coding structure and to
generate test code sequences.
The decoder simulation is used to
test the performance of the Viterbi algorithm in the presence of
white Gaussian noise and with the loss of either branch or node
synchronization.
ix
I.
Introduction
The basic question that must be answered at the start of a
communication system design is how much transmission capability
is required to communicate the source output to the receiver.
The standard measure of the transmission capability (relative
efficiency) of a communication system is the average energy per
bit to noise power density ratio (Eb/No) required to obtain a
given level of
et~ror
rate performance.
The Eb/No can be expressed
in terms of the average modulating signal pmver by the
relationship
Eb/No
where
= (Ps/R) 1/No
(1)
Eb is the average energy per bit
No is the single sided noise power density
Ps is the average power at the receiver input
R is the source information rate in bits per
second
The noise power density, No, is in general a characteristic
of the particular communication channel used and is therefore a
fixed constraint for any system operating on that channel.
It
will be assumed that the channel noise is primarily due to thermal
activity at the receiver front-end and that the noise introduced
by the actual receiver hardware is negligible.
The average received power is dependent on the transmitted
power, the transmitting and the receiving antenna gains and the
channel losses.
It can be expressed as
1
2
KaKc
Ps = - T
where
s:
s2(t)dt
Ka
is the combined antenna gain
Kc
is the channel loss
(2)
s(t) is the signal time waveform
T
is the duration (period) of the signal
waveform
If equation (1) is rewritten as
Ps/No
R=--
(3)
EtJNo
it can be seen that by reducing the required Eb/No for a given
bit error rate (BER) probability, the performance of the system can
be improved proportionately.
That is, for a given data rate the
required Ps/No is reduced, or conversely, for a given Ps/No the
allowable data rate is increased.
In general there are two methods
to reduce the required Eb/N 0; 1) proper selection of the carrier
modulation and 2) encoding techniques.
The purpose of this graduate project has been to study one
particular coding system, viz, convolutional coding with Viterbi
decoding, and to evaluate its performance on a communication
satellite channel.
The coding schemes currently in use can be classified as two
basic types:
block coding and convolutional coding.
Block
coding is a process of mapping equal length messages of source
3
symbols into fixed sequences of code symbols.
Convolutional coding
is a process of generating code using any constant linear sequential
circuit.
These coding schemes are fundamentally different in their
structures.
Block coding breaks the source data into individual
source words, then encodes each word independently of preceding or
succeeding words.
Convolutional coding encodes the source data in
a continuous fashion and the bits of encoded data are interdependent
over a finite interval called the constraint length of the code.
Convolutional codes have been shown to outperform block codes
of equivalent complexity in a majority of communication systems.
applications [4] • In particular, Viterbi has shown [1] that a
significant improvement is provided in the limiting case of "very
noisy•• channels (e.g. additive white Gaussian noise with
Eb/N~l).
The probabilities of error in this case can be
represented as
= 2-KcEc{R)/R
Ps = 2-KBEB(R)/R
Pc
where
Pc, PB are the probabilities of error for
convolutional and block coding
Kc
is the constraint length
K8
is the block length
Ec{R), Es(R) are the error bounds
The error bounds for convolutional codes
Ec(R) = C/2,
C-2,
05:R~C/2
C/2~R
<C
4
The error bound for block codes
Es(R)
= C/2- R,
(C - R)2
os;:R~C/4
, C/4~ R<C
The error bounds are shown in Figure 1 for comparison.
[8]
The curves
of Figure 1 make it clear that in the case of very noisy channels
convolutional codes have the potential of much better
pet~formance
than a corresponding block code of the same order of complexity.
In
particular~
using above inequalities, at R = C/2,
Ec(R)
=
C/2
5.83 Es(R)
=
indicating that to achieve the equivalent error performance the
block length must be over five times the constraint length of
the convolutional code.
The communicat'ion channel that we are considering can be
classified as a Very noisy channel
11
11
;
therefore, the choice
bet\'Jeen a convolutional code and a block code is obvious.
However,
there sti11 remains a problem of selecting between certain types
of convolutional coding schemes.
There are basically three types
of convolutional coding methods and they are classified by their
decoding structure:
1) threshold (feedback) decoding,
2) sequential (probabilistic} decoding, and 3) Viterbi decoding.
A quick sm·vey of coding literature [3, 4, 6]
allows one to
eliminate threshold decoding as a primary coding method on a
satellite channel.
The error correcting capability of a threshold
decoder is equivalent to very short constraint length codes, therefore
its efficiency is reduced as the number of errors increases.
The
5
f.(R)
C/2
~-:--"-----......
''
CONVOLUTIONAL
CODES
'
BLOCK ' - ,
CODES
....._
C/2
Figure 1
--
c
R
Limiting values of E(R) for very
noisy channels
6
major application for threshold decoding appears to be on fading
and bursty channels due to its burst correcting capabilities.
It has found application on satellite channels, but generally in
a back up role to the primary coding system.
Both sequential and Viterbi decoding methods have found
extensive use in satellite and space channel applications.
As
a means of comparison we can examine the basic coding ga·ins of
each method on a white Gaussian channel.
Given a particular
modulation scheme without any coding operating on a white Gaussian
channel, coding gain can be defined as the reduction, due to
coding, of the energy per bit error rate.
As a point of reference,
the lower bound on achievable Eb/No for an infinite bandwidth
white Gaussian channel is about -1.6dB [5] . For a bit error
rate of Io-5, uncoded data with BPSK (antipodal) modulation requires
an Eb/No of 9.6dB and with BFSK (orthogonal) modulation requires
an Eb/No of 12.6dB [5] .
The basic coding gains for various codes are given in
Table 1 for bit error probabilities of lo-3, lo-5, and lo-7 [6] •
All coding gains are relative to the value of Eb/No required by
uncoded BPSK.
The upper bound is obtained by observing that the pro-
bability of error for a code with rate R and minimum free distance is
Pe '2Q
(2EbRd/No)
where Q(x) is the integral of the tail beyond x of the Gaussian
density function.
For uncoded BPSK the probability of error is
Pe , uncoded
= Q (2Eb/N 0)
7
p '
Table 1
Basic coding gain (dB) for soft-decision Viterbi
decoding and hard-decision sequential decoding
.---·
Sequential
decoding
Viterbi decoding
Eb/No
Uncoded
(dB)
R=
1/3
1/2
2/3
~
7
5
6
7
6.8
w-3
4.2
3.3
3.5
3.8
9.6
w-5
5.7
4.3
4.6
11.3
w-7
6.2
4.9
---
Upper
Bound
7.0
5.4
6
1/2
41
47
2.9
1.5
3.0
5.1
4.2
3.8
4.2
5.3
5.8
4.7
4.8
6.5
6.0
7.0
5.2
7.4
7.4
8
Therefore, coding gainS:Rd
Two things are apparent from the table.
First, while both
Viterbi decoding and sequential decoding provide significant
coding gains, they both fall far short of the -l.6dB lower bound
for the white Gaussian noise channel.
Second, the advantage of
using a sequential decoder increases as the required probability
of error decreases.
A more detailed comparison of Viterbi decoding and sequential
decoding is deferred until after the development of the Viterbi
decoder in Section III.
For now let it suffice to say that there
is a speed-complexity versus error probability tradeoff in the
selection of a convolutional decoder.
For short constraint length
(K<10) codes and with moderate bit error rates (lo-5) the
Viterbi decoder turns out to be much less complex and more cost
effective than an equivalent sequential decoder.
On the basis of the previous discussion this writer has chosen
to develop simulations for a convolution coding system with
Viterbi decoding.
9
II.
Convolutional Encoder Simulation
The encoder and decoder simulations were designed to run on
a Teradyne M365CX minicomputer.
This particular processor is
optimized to control semiconductor test_equipment and therefore
does not have sufficient speed to be used for any real time code
processing.
The simulations were intended to test the function of
an equivalent hardware implementation.
The speed of the hardware
would be determined by the particular implementation (TTL, ECL, etc.)
The programs are written in an assembly level language and are
run using a master operating system, both of which are unique to
Teradyne equipment.
The language itself, however, is very similar
to other assemb1y languages and these simulations could easily be
adapted to other minicomputers.
All data is in octal format
unless followed by a decimal point in which case it is a decimal
number.
A convolution encoder can be described as any constant linear
sequential circuit.
It can be represented by a kK stage shift
register and v modulo 2 adders as shown in Figure 2.
Each of
the adders is connected to certain stages of the shift register.
The pattern of these connections specifies the code output.
source
infot~ation
time.
After each k bit shift the outputs of the adders are
The
bits are shifted into the encoder k bits at a
sampled sequentially to yield the code symbols.
There are v code
symbols generated for each set of k source bits that are input so
the code rate RN
= k/v, where k<v for any practical code. The
10
kK stage shift register
v MOD 2 adders
----+------binary code
symbols
Figure 2
General form, Rate k/v
Convolutional encoder
11
constraint length of code is K which is the number of k bit shifts
over which a single source bit can influence the encoder output.
Since the encoder is a linear finite-state machine it is
apparent that after some fixed interval the code output patterns
would tend to repeat.
tree diagram.
v
= 2,
This can be easily shown by means of a
A common example of an encoder with k
is shown in Figure 3.
= 1, K = 3,
The tree diagram associated with
this encoder is shown in Figure 4.
The tree diagram is used to
illustrate the code output that is associated for all the
various combinations of source inputs.
The convention used in
this particular tree is that if the source bit is a zero, the
code symbols are shown on the upper branch5 and if it is a one
the code
s~nbols
are on the lower branch.
The tree is shown for
the first five source bits (initial state of the shift register
is a11 zeroes).
It can be seen from the tree that after the first three
branches the structure becomes repetitive.
This is caused by
the fact that any given source bit can influence the code output
only while it is in the shift register.
The tree structure
repeats after K branches where K is the constraint length of the
code (the number of stages in the shift register).
The repetitive nature of the tree allows it to be redrawn as
a trellis which is a tree-like structure with remerging branches.
The trellis diagram for the encoder example is shown in Figure 5.
From the treliis diagram it can be seen that the encoder can be
12
Figure 3
Convolutional encoder
K=3 ~ v=2, k=l
13
00
~11
00
10
1
01
11
10
00
11
01
01
10
00
11
L...--11
10
01
11
01
0
10
0
1
....r-oo
0
-en
00
01
10
00
11
11
00
10
01
11
10
00
OLC01
10
00
01
11
11
10
11
00
01
01
01
10
10
Figure 4
10
c11
00
01
10
Tree diagram for encoder
in Figure 3
14
Dashed lines indicate a "1 transition
Solid lines indicate a "0" transition
11
STATE
00
00
""
" 11
" "-
"
~
00
00
11
""
"
Ol
\
\
]0
\ 01
\
\
\
10
01
\
Figure 5
11
Trellis diagram for encoder
in Figure 3
15
represented by a fixed number of states or nodes and the transitions
between these states.
The states correspond to the oldest two bits
in the shift register and the transitions correspond to the most
current bit entered into the register.
Therefore, an encoder will,
in general, have 2K-l states with 2k transitions between states.
The states ar'e represented by k(K-1) bits and the transitions by
k bits.
The convolutional encoder being simulated is a binary, constraint
length 5, rate 1/2 encoder.
In this case k
= 1, K = 5, and v = 2.
The encoder has 16 states with two transitions in each state.
The general form of the encoder is shown in Figure 6.
There are
two modulo 2 adders which are designated as the X and Y adders.
The pattern of connections to the adders is designated by the
X select and Y select words where each bit set indicates a
connection.
The data flow in the encoder is from right to left
and the X adder output precedes the Y adder output.
There were bJO objectives in the encoder simulation:
first,
to generate a general purpose encoder structure which would encode
any fixed pattern of source bits, and second, to develop a code
test which would determine an optimum connection pattern to the
two adders (optimum code).
programs were written:
To accomplish these objectives, two
Encoder and Code Test.
The object listings
of these programs are in Appendix A and Appendix B.
The Encoder program is designed to encode any number of source
bits up to a maximum of 90 bits (18 constraint lengths).
The
16
X adder
'II \
/1 I \ '
/I
/
I
code bits ---1
\
I
I
I
I
I \ /I I
I
\
X select
\ \
\ \
\
\
- - d a t a bits
J
I
\ I 1 I
\ ' 1 I 1
\.\ill
fc;-/E
I
Y se 1ect
,,
Y adder
Figure 6
General form of Convolutional encoder
Rate 1/2, constraint length 5
17
interconnection pattern to the adders is user selectable.
The
flowchart for the program is shown in Figure 7.
The initial state of the shift register is zero.
data is shifted in one bit at a time.
The source
After each shift, each bit
in the shift register is checked to see if it is set, i.e. if
it has a logical value of one.
If the bit is set and there is a
corresponding connection to an adder, the adder is incremented.
After a 11 5 bits are checked, the adder outputs are combined to
form the code symbol and the next source bit is shifted into the
register.
Figure 8 shows a typical output from this program.
The Code Test program generates code for all possible
combinations of source inputs over one constraint length for any
particular pattern of adder connections.
The flowchart for
the program is shown in Figure 9.
First, the user defines the pattern of adder connections.
The program then generates code for all 32 combinations of source
bits over one constraint length (5 bits) of input.
state'of the shift register is zero.
The initial
After each segment of
code is generated the program computes the free distance from
that code word to the all zero code word.
After all combinations
have been run, the program computes the average free distance to
the all zero code word for this particular code.
Figure 10 shows
a typical output from this program.
By using the group property of convolutional codes we can
now proceed to select an optimum code.
The code will be optimum
18
INPUT
SOURCE
SEQUENCE
SHIFT
lBIT
No
No
INCREMENT
X ADDER
FOR EACH BIT
No
INCREMENT
Y ADDER
FOR EACH BIT
OUTPUT CODE
Figure 7
Encoder Algorithm
19
CONVOLUTIONAL ENCODER
CONSTRAINT LENGTH 5, RATE 1/2
SOURCE INPUT
72365
14253
66241
55342
65732
13527
1230
26657
56023
67245
76756
54605
66302
10756
70743
64273
CODE OUTPUT
3521
16212
N:
XSEL:
YSEL:
90
11
15
Figure 8
Encoder program, typ·i ca 1 output
20
START
INITIALIZE
VARIABLES
I LOOP 32 I
INPUT
SOURCE
WORD
LOOP ENCODER
ALGORITHM
10 BITS
OUTPUT
CODE
COMPUTE &OUTPUT
FREE DISTANCE
COMPUTE &OUTPUT
AVERAGE FREE
DISTANCE
I END I
Figure 9
Code Test Algorithm
21
CONVOLUTIONAL ENCODER TEST
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
XSELECT
YSELECT
=
=
11
15
SOURCE WORD
CODE HORD MSH
0
0
0
0
0
0
0
0
1
0
40
100
140
200
240
300
34-0
400
440
500
540
600
640
700
740
1000
1040
1100
1140
1200
1240
1300
1340
1400
1440
1500
1540
1600
1640
1700
1740
1
1
1
1
1
1
1
6
6
6
6
6
6
6
6
7
7
7
7
7
7
7
7
AVERAGE FREE DISTANCE
=
Figure 10
CODE WORD LSH
0
1514
6460
7174
32300
33614
34760
35274
51400
50114
57060
56574
63700
62214
65360
64674
46000
47514
40460
41174
74300
75614
72760
73274
17400
16114
11060
10574
25700
24214
23360
22674
FREE DISTANCE
0.
5.
5.
8.
5.
8.
8.
9.
5.
6.
8.
11.
8.
7.
9.
10.
5.
10.
6.
9.
8.
11.
11.
12.
8.
9.
7.
10.
9.
8.
10.
11.
8.
Code test program, typical output
22
in the sense that it maximizes the minimum free distance between
any two code words and that it does not allow catastrophic error
propagation.
The group property requires that the set of
distances between the all zero code word and all other code words is
the same as the set of distances between any specific code word and
all others.
The free distance between any two code words is simply the
Hamming distance, that is, the number of bit positions in
the code words differ.
~tJhich
Maximizing the m·inimum free distance
minimizes the error probability since it allows the most errors
to occur in the channel before an error will be made by the
decoder.
In general if the free distance is d, the number of
errors that can be tolerated by a maximt.rm likelihood decoder is
d-1 for d odd and (d/2)-1 for d even.
2
The previous discussion assumes that all code words are
equiprobable in occurence.
If this is not the case, the optimum
code would be one which maximized the minimum free distances
between the more probable code words.
Maximum minimum free distances have been previously determined
[4] for both systematic and nonsystematic codes.
the distances for various constraint lengths.
Table 2 shows
Systematic codes do
not perform as well as the nonsystematic codes because a systematic
code requires that one of the adders must be eliminated so that
the input bi.t can be passed through to the output.
An optimum
code for a constrai.nt length 5 encoder,. therefore, would be
23
Table 2
Maximum-Minimum Free Distance
Convolutional Codes
K
Systematic
Nonsystematic
2
3
3
3
4
5
4
4
6
5
5
7
24
nonsystematic with a minimum free distance of 7.
Catastrophic error propagation occurs when a finite number of
errors introduced by the channel causes an infinite number of
decoding errors.
A necessary and sufficient condition for
catastrophic error propagation [4] is that any nonzero data path
in the trellis produces K-1 consecutive branches with all zero
code symbols.
{R
Also~
it can be shown that for binary tree codes
= 1/v) catastrophic propagation will occur if each adder has
an even number of connections (see Figure 11).
The output of the Code Test program provides both minimum
free distance and catastrophic error information.
Using this
program an encoder was selected to generate test code for the
Viterbi decoder.
The encoder output is shown in Figure 12.
The code has an average free distance of 9.
the
average~
Therefore~
on
over one constraint length of code (10 bits) the
decoder could tolerate 4 incorrect bits.
The code is not optimum
in the sense that there are 3 code words that have a free distance
of only 6 from the all zeroes code word.
Table 2 indicates that
for a constraint length 5 code the maximum minimum free distance
should be 7.
Storage limitations of the minicomputer did not
permit an exhaustive search to find an optimum code, but this code
is sufficiently "good" to test the various aspects of the Viterbi
decoder operation.
Due to the large number of states and transitions, a trellis or
state diagram becomes unwieldy so this information is presented in
25
CONVOLUTIONAL ENCODER TEST
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
XSELECT
YSELECT
= 22
= 22
SOURCE WORD
CODE WORD MSH
CODE WORD LSH
FREE DISTANCE
0
40
100
140
200
240
300
340
400
440
500
540
600
640
700
740
1000
1040
1100
1140
1200
1240
1300
1340
1400
1440
1500
1540
1600
1640
1700
1740
0
0
0
0
1
1
1
1
6
6
6
6
7
7
7
7
30
30
30
30
31
31
31
31
36
36
36
36
37
37
37
37
0
6060
30300
36360
41400
47460
71700
77760
6000
60
36300
30360
47400
41460
77700
71760
30000
36060
300
6360
71400
77460
41700
47760
36000
30060
6300
360
77400
71460
47700
41760
4.
4.
8.
4.
8.
8.
12.
4.
4.
8.
8.
8.
8.
12.
12.
4.
8.
4.
8.
8.
12.
8.
12.
8.
8.
8.
8.
12.
12.
12.
12.
AVERAGE FREE DISTANCE
=
Figure 11
8.
Code test program,
Catastrophic error condition
o.
26
CONVOLUTIONAL ENCODER TEST
BINARY~ CONSTRAINT LENGTH 5~ RATE 1/2
XSELECT = 37
YSELECT = 25
SOURCE HORD
CODE WORD MSH
CODE WORD LSH
FREE DISTANCE
0
0
7354
35660
32534
67'300
60054
0.
8.
8.
8.
8.
6.
8.
10.
8.
12.
6.
10.
8.
10.
10.
8.
8.
12.
12.
12.
6.
8.
10.
0
40
0
100
140
200
240
300
340
400
440
500
540
600
640
0
0
1
1
52560
1
55634
35400
32754
260
7134
52700
55454
67160
60234
66000
61354
53660
54534
1300
6054
34560
33634
53400
54754
66260
61134
34700
33454
1160
6234
1
7
7
7
7
6
6
6
6
35
35
35
35
34
34
34
34
32
32
32
32
33
33
700
740
1000
1040
1100
1140
1200
1240
1300
1340
1400
1440
1500
1540
1600
1640
1700
1740
33
33
AVERAGE FREE DISTANCE
=
Figure 12
12.
8.
12.
10.
10.
10.
12.
8.
10.
9.
Code test
program~
selected encoder
27
tabular form in Table 3.
This table is used later to generate the
transition table for the Viterbi decoder.
III.
Viterbi Decoder Simuiation
The Viterbi decoder simulation program can be functionally
divided into tvm parts:
second~
first~
there is the decoder itself and
there is a test routine which takes a code sequence generated
by the Encoder program and produces a quantized code word input fm"
the decoder.
The quantized code word input can be generated in a
noiseless fashion or \'lith pseudo random Gaussian noise added
and/or unsynchronized by node (1 code bit) or branch (2 code bits).
The Viterbi decoder being simulated is a binary, constraint
length 5, rate 1/2 decoder.
It has 8 level soft quantization,
integer metrics, and a path memory of 30 bits (six constraint lengths).
It was designed to handle 540 bits of quantized code word
(equivalent to 90 source bits) and provide 60 usable bits of
decoded output (the last 30 bits remain in the path memory).
The Viterbi decoding algorithm [1] is a maximum likelihood
algorithm derived by utilizing the recursive structure of the
convolutional code trellis.
sequence.
Thus consider an L bit information
A brute-force maximum likelihood decoder would calculate
the likelihood of all possible paths through the trellis for the
L bit sequence and would select the path with the largest likelihood.
The information bits corresponding to that path would then be the
decoder output.
However, because there are 2L paths through the
28
Table 3
State-transition Table
Convolutional encoder, XSEL = 37, YSEL
NEXT
STATE
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
INPUT
PREVIOUS
- -STATE
---
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0000
1000
0000
1000
0001
1001
0001
1001
0010
1010
0010
1010
0011
1011
0011
1011
0100
1100
0100
1100
0101
1101
0101
1101
0110
1110
0110
1110
0111
1111
0111
1111
= 25
CODE
SYMBOL
00
11
11
00
10
01
01
10
11
00
00
11
01
10
10
01
10
01
01
10
00
11
11
00
01
10
10
01
11
00
00
11
"
'
29
trellis for a given sequence this method quickly becomes impractical
as the length of the information sequence increases.
The Viterbi algorithm on the other hand, has a computation
complexity which is independent of L due to its recursive nature.
Recall from the encoder development that the convolutional code
trellis forms a repetitive structure after a depth corresponding
to the code constraint length has been reached,
The trellis
could be represented by a set of 2k{K-1) states and 2k paths entering
each state,
The Viterbi decoder examines all the paths entering
each state and retains only the maximum likelihood path.
Thus,
as code corresponding to each information bit is processed the
decoder determines a maximum likelihood path for each of the 2k(K-l)
states.
The information bits corresponding to each of the paths
is stored in the path memory.
After a fixed number of bits is
stored, corresponding to the length of the path memory, the decoder
will output the oldest bit on the maximum likelihood path as
each new information bit is processed.
The complexity of the Viterbi decoder can be seen to depend
exponentially on the constraint length K rather than on the length
L of the information sequence.
Therefore, the Viterbi decoder
tends to be a good choice for relatively short constraint
length codes.
The flowchart of the Viterb"i decoder program is shown in
Figure 13.
parts:
The Viterbi algorithm can be divided into three basic
1) input processing and metric computation, 2) add-
30
I START 1
J
I
INPUT 3 BITS OF
QUANTIZED CODEWORD
I
CALCULATE ALL
BRANCH
J
I
~1ETRICS
-
l
I LOOP 16 I
I
DETERMINE WHICH
TWO TRANSITIONS
ARE POSSIBLE
em
l
EACH TO
S PREVIOUS
ATE NETRIC
l
--
SELECT LARGER
METRIC & STORE
CORRESPONDING BIT
I
PATH
MEMORY
FULL?
No
Yes
H LARGEST
[~ECT1:]
METRIC
-,
OUTPUT THE
OLDEST BIT
I
Figure 13
Viterbi Algorithm
31
compare-store routine, and 3) path memory and output processing.
The object listing of this program is in Appendix C,
The input processing and metric computation routine processes
the quantized code word three bits at a time.
The code word received
from the channel is quantized into one of eight levels as shown
in Figure 14.
The quantization process can occur in a noiseless
fashion or there can be noise added.
A noiseless input is geneiated
by replacing a Zero" code symbol with a zero in the quantized
11
code word and replacing a 11 one" code symbol with a seven in the
quantized code word.
When noise is added, the noise distribution
will be as shown in Figure 15.
The noise is zero mean,
variance one, and has an amplitude equal to the signal amplitude.
An error will occur when a "zero" code symbol is translated into
a value greater than three or a 0ne code symbol is translated
11
11
into a value less than four in the quantized code word.
In an actual decoder imp.lementation the quantization process
\•lOuld operate on the code symbols one at a time as they are input
into the decoder.
To ease the computational load in the
simulation, however, the entire string of 180 code bits is
quantized prior to the first input to the decoder.
The branch metric is a measure of the likelihood that a
particular branch is correct given the quantized code symbol
pair.
From Table 3 in the encoder development we can see that
each code symbol pair corresponds to a single transtion between
states in the code trellis.
For each code symbol pair there are
32
Code
Symbol
Demodulator
Outpu t
Quantization
level
7
1 .5
1
6
1 .0
5
.5
4
- No Signal
.5
0
-1 .0
-1 . 5 - 1---
3
2
1
0
Figure 14
Quantization intervals,
demodulator output normalized
to noise level
33
Relative value distribution
over intervals
36
32
21
12
I
-
-4
-3
-2
-1
0
+1
+2
+3
+4
Quantization
levels
··Figure 15
Noise distribution
34
four possible branches (00, 01, 10, 11).
Each of these branches
is compal"ed with each quantized code symbol pair that is input
and a metric is generated indicating the goodness of the match
(see Figure 16).
In order to avoid multiplication in the
algm"ithm it is a common practice to use the logarithm of the
likelihood function as the branch metric.
Simulations I5J have
shown that use of integer metrics rather than log likelihoods causes
negligible performance degradation with 2, 4, or 8 level quantization.
Figure 16 shows a set of integer metrics for an 8-level
quantized channel.
The add-compare-store (ACS) routine is used to determine the
maximum likelihood path into each state of the decoder.
Referring again to Table 3 we see that the decoder has sixteen
states \'lith two transitions (branches) into each state.
The
ACS routine, therefore, must select the more likely of two
paths into each state and store the corresponding information
bi.t in the path memory.
For each of the two paths into a current
state, the t'outine takes the branch metric and adds it to the path
metric corresponding to the previous state.
The routine then
compares the two path metrics and selects the path with the larger
metric as the maximum likelihood path into that state. The information
bit corresponding to that path is stored in the path memory
and metric is stored as the path metric for that state.
The
ACS operation repeats for each of the sixteen states, so there
are actually sixteen paths (information sequences) being stored
35
Code
Symbols
Quantization
level
0
3
6
2
5
1
2
0
1
7
0
5
6
4
4
3
2
1
7
0
3
4
5
6
7
1
Figure 16
Metric table
36
in the decoder.
The path memory and output processing routine determines
what the actual decoder output will be.
The path memory stores
information bits sequentially on each of the sixteen paths until
it becomes full (30 bits long).
At this time it compares the path
metrics of all sixteen states and outputs the oldest information
bit on the path with the largest metric.
The process will now
continue each time a new set of information bits is passed to the
path memory.
A typical output from the Viterbi decoder simulation is shown
in Figure 17.
The quantized code word was generated without noise
in this case, so that the branch metrics all had the maximum
value of fourteen.
For ninety information bits (90 transitions)
the end metric is 90 x 14
= 1260 (2354 octal). The end metric
(sum of all branch metrics on the maximum likelihood path) is
used in this simulation as a measure of the noise introduced in
the channel.
The smaller the end metric, the larger the number
of errors that have occurred.
IV.
Conclusions
The goals at the start of this graduate project were to develop
simulations of a convolutional coding system in order to analyze
and test the Viterbi decoding structure.
The simulations have been
developed to the point where they provide good qualitative
information about the decoder and provide a basis to measure its
..t.l,·
-
----
----
~
--·
·--· -~~-.~h."
37
VITERBI DECODER
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
8 LEVEL SOFT QUANTIZATION
INTEGER METRICS
30 BIT PATH MEMORY
CODE INPUT
15007
76744
60312
43537
44063
50112
64044
62106
62013
73717
QUANTIZED CODE WORD
770
77000
77070
77070
70007
77007
INPUT
77077
770
7
77000
77070
70
7770
7070
700
70707
77777
770
770
70070
77007
77777
70700
77707
70000
7
0
7777
70
77770
777
70077
7077
700
7070
7777
END STATE
13
DECODER OUTPUT
72365
14253
66241
55342
END NETRIC
2354
15556
65425
Fi_gure 17 .
Viterbi decoder program,
typical output
·
~.~------
38
performance.
Data storage limitations of the minicomputer precluded
processing the large volume of input and output data that would have
been required to obtain a true measure of the decoder error rate
performance.
As a point of reference, we can look at data from simulations
by Heller and Jacobs [5] . Figure 18 shows a comparison of
Viterbi decoding using rate 1/2, K=5 code for different path memory
lengths and different levels of quantization.
Figu1·e 19 shows
the bit err·or probabi 1i ty bound for this code.
The operation of encoder and decoder simulations is demonstrated
by
the example shown in Figures 20-24.
Figure 20 shows the encoder
output generated for an arbitrary input sequence.
Figure 21
shows the decoder operation with a noiseless quantized code word
input.
For this case the end metric is maximum (2354) as
expected and the original information sequence is recovered without
error.
Figure 22 shows decoder operation with noise added.
In
this particular instance, the noise has produced 25 actual errors in
the 180 code bit sequence (14 percent).
reflects the amount of noise added.
The end metric (2037)
It is about 26 percent lower
than maximum (some of the noise was not of sufficient magnitude
to produce errors).
Because the errors do not exhibit tight
grouping the original information sequence is still recovered
\'Ji thout error.
F·i gure 23 shows decoder operation with 1oss of
branch synchronization.
In this noiseless case, the end metric
is still maximum and the original sequence is recovered correctly
~
~---
-----
.. o;-,·
39
lo-2
JJ
......
.........
..0
co
10-3
..0
0
s..
a.
s...
0
s...
s...
L!..l
lo-4
8
~
•r-
ca
lo-5
2-LEVEL
8-LEVEL
3
Figure 18
4
5
6
7
8
Performance comparison of Viterbi decoding
using rate 1/2, K=5 code with 8-, 16-,
and 32-bit path lengths and 2- and
8-level quantization [5]
~,
"----
--
-
--··----~-
-
40
10- 5
~
•r-
..,..
.Q
ro
1o-6
.0
0
s-
0..
s...
0
s...
s-
LJJ
lo-7
of-)
•r-
co
1o·-8
Figure 19
Bit error probability bound for
rate 1/2, K=5 [5]
--------------
--------
.,~~---
---------
--------
-----·-·"'-
41
f'
CONVOLUTIONAL ENCODER
CONSTRAINT LENGTH 5, RATE 1/2
SOURCE INPUT
62543
71066
66652
52507
55122
22200
CODE OUTPUT
. 15262
14631
7624
46041
30624
40225
74130
42401
50505
35353
5363
53660
N:
XSEL:
YSEL:
90
37
25
Figure 20
Encoder output
.
,~.·
------~------
42
f
VITERBI DECODER
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
8 LEVEL SOFT QUANTIZATION
INTEGER METRICS
30 BIT PATH MEMORY
CODE INPUT
15262
14631
7624
46041
30624
40225
74130
42401
50505
35353
5363
53660
QUANTIZED CODE WORD INPUT
70707
770
7700
7700
70700
7070
770
7700
70000
700
7770
70777
70070
70700
707
77007
70707
7077
77
77770
70
70077
70007
70707
77700
70
70777
7
7000
77707
70700
77000
70077
7
7
70000
71066
66652
52507
END STATE
0
DECODER OUTPUT
62543
END METRIC
2354
Figure 21
Decoder output, noiseless input
.
43
VITERBI DECODER
BINARY, CONSTRAINT LENGTH 5, RATE l/2
8 LEVEL SOFT QUANTIZATION
INTEGER METRICS
30 BIT PATH MEMORY
CODE INPUT
15262
14631
7624
46041
30624
40225
74130
42401
50505
35353
5363
53660
QUANTIZED CODE WORD
20671
5604
50700
20671
40004
6730
INPUT
40613
7501
4060
5610
2501
30567
30050
70501
40706
33006
70507
47256
4075
33340
11050
74075
30005
71406
67700
472
70334
1105
7402
73304
61300
56110
50077
1003
114
50040
END STATE
0
DECODER OUTPUT
62543
71066
66652
52507
END METRIC
2037
Figure 22
Decoder output, noisy input
~,-
------
--
44
VITERBI DECODER
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
8 LEVEL SOFT QUANTIZATION
INTEGER ~1ETRICS
30 BIT PATH MEMORY
CODE INPUT
15262
14631
7624
46041
30624
40225
74130
42401
50505
35353
5363
53660
7000
70077
70700
770
70770
7770
7777
77000
7070
7700
707
70777
70070
7077
77770
700
0
70770
70007
70
7700
770
707
0
62155
55525
25217
QUANTIZED CODE WORD INPUT
77070
70007
70007
77007
0
77070
70770
70070
7000
70077
70070
77707
END STATE
0
DECODER OUTPUT
45307
END METRIC
2354
Figure 23
Decoder output. loss of
branch synchronization
45
VITERBI DECODER
BINARY, CONSTRAINT LENGTH 5, RATE 1/2
8 LEVEL SOFT QUANTIZATION
INTEGER METRICS
30 BIT PATH MEMORY
CODE INPUT
15262
14631
QUANTIZED CODE WORD
7707
77000
7000
7700
0
77707
7624
46041
30624
40225
74130
42401
50505
35353
5363
53660
INPUT
7077
77007
70700
77007
7007
7770
700
7007
7070
70077
7077
70777
777
77700
707
770
70
7077
77007
707
7777
70
70000
77077
7000
70007
770
77
70
0
30200
30647
11504
END STATE
16
DECODER OUTPUT
50776
END METRIC
2104
Figure 24
Decoder output, loss of node·
synchronization
46
with a one bit offset.
Figure 24 shows the decoder operation with
loss of node synchronization.
Even in the noiseless case the end
metric is low and the output recovered is totally distorted.
The example above demonstrates that loss of node
synchronization is a very serious problem for the Viterbi decoder.
The most obvious solution to this problem would be to test the
path metrics periodically and attempt to resynchronize the decoder
if the metrics are too small.
This solution does not work in the
presence of large amounts of noise, however.
Referring back to
Figures 22 and 24 we see that the end metric for the noisy case
is actually lower than that for the unsynchronized case.
It
seems that for this case it would be necessary to put synchronization
data within the code sequence.
Further study could be done to
develop an efficient synchronization algorithm.
At this point, it is appropriate to make a more in-depth
comparison between the Viterbi and sequential decoding techniques.
As we have seen, the probability of error for a convolutional
code is directly related to its constraint length.
To reduce
the proba.bility of error we must increase the constraint length.
The complexity of the Viterbi decoder increases exponentially
with constraint length.
For K>lO the Viterbi algorithm becomes
impractical to implement and sequential decoding becomes a viable
alternative.
The reason for this is the complexity of a sequential
decoder is relatively independent of constraint length.
The decoding
algorithm sequentially searches the code tree to find a path whose
---
-·
-
---
--
-
--
~··
--
47
metric increases faster than some predetermined threshold.
As
each set of code symbols is input, the decoder selects the most
likely branch.
Unlike the Viterbi decoder, the sequential decoder
only tests the states on the one path that it has selected, hence
the independence of constraint length.
If at any time the decoder
selects the wrong branch, all the following bits on the path will
be in error.
When this occurs the calculated path metric wtll
diverge from the anticipated metric and the decoder has to back
track and try again.
Since the differences between the correct path
metric and any incorrect path metric increases with constraint length,
K for sequential decoders is usually made very large.
One drawback
of the sequential decoder is the number of incorrect path branches,
and consequently the computational complexity varies with the amount
of channel noise.
If there is enough noise to cause the computational
storage buffers to overflow, then all information currently in the
decoder is lost.
To prevent storage overflow it is necessary to make
the decoding speed larger than the bit rate, thus limiting the
maximum bit rate capacity.
So, in general, there is a tradeoff
between data rate and error probability in selection of a convolutional
decoder.
The crossover points occur at error probabilities some\'Jhere
between 10-3 and 10- 5 and data rates around 10 M bits/second [4] •
There are many areas of study concerning coding techniques
for satellite channels that were outside the scope of this paper.
The Viterbi technique has been used to reduce intersymbol
interference on bandwidth limited channels and is finding application
48
on nonlinear channels,
The basic concepts developed here have
application beyond the Viterbi algorithm itself.
As was pointed
out earlier, the performance achieved by available codes falls
far short of the Shannon bound for Gaussian channels.
Perhaps there
are other as yet undefined coding structures that will allow us to
approach this bound more closely.
i'
.
".
References
1) A.J. Viterbi, Error bounds for convolutional codes and an
asymptotically optimum decoding algorithm IEEE Trans.
Information Theory, val. IT-13, pp. 260-269, Apr. 1967
11
11
2)
,
J.K. Omura, 0n the Viterbi decoding algorithm IEEE Trans.
Information Theory, val. IT-15, pp. 177-179, Jan. 1969
11
11
,
3) G.D. Forney, Jr., Coding and its application in space
communications .. , IEEE Spectrum, val. 7, pp. 47-58, June 1970
11
4)
A.J. Viterbi, .. Convolutional codes and their performance in
communication systems IEEE Trans. Communications Technology,
val. COM-19, pp. 751-772, Oct. 1971
11
,
5)
J.A. Heller and I.M. Jacobs, Viterbi decoding for satellite
and space communication .. , IEEE Trans. Comnunications Technology,
val. COM-19, pp. 835-848, Oct. 1971
6)
I.M. Jacobs, "Practical applications of coding IEEE Trans.
Information Theory, val. IT-20, pp. 305-310, May 1974
7)
N. Abramson, Information Theory and Coding, New York:
McGraw-Hill, 1963
8)
J.M. Wozencraft and I.M. Jacobs, Principles of Communication
Engineering. New York: Wiley, 1965
9)
11
11
,
H. Taub and D.L. Schilling, Principles of Communication
New York: McGraw-Hill, 1971
Syste~s.
49
APPENDIX A
Encoder Program Listing
50
51
PAGE
l
MAY
••0003U
•
GENERAL PURPOSE CO'<VOLUTIO~AL ENCODER
CONSTRAINT LlNGTH 5, RAH 1/2 ~
5
5
7
8
9"
g
10
11
tl
S~AI
se 11
,,
5221
16
17
s~"
50-ll
SC~I
t9
23
21
TSM'I
lS0t I
22
23
24
25
TSr~l
T50(1
TSHI
TS~51
Et•l'lll
Ec~n'
t.C ldl21
29
EC>?-31
EC"'r41
3~
ECw~~l
>1AXI11UH INPUT
SOURCE INPUT loiORO Ill
I"'PUT WORD lf2
SOU>~!':E l'<PIJT kO'ID lfl
Sl'U~CE I'<PIJT wo~o
SOUQCE INPUT looi)R!) ~~~
S.OURCE INPUT ~<ORO ~0
TEMP 500 (LSW)
TE"'P 51'1
souqcf
Tf'1P
Tf.HP
~1?12
Tf.01P
Tf>IP
504
(MSW)
5~5
COOEIIORO Ill
(LSW)
r.oDE>~OIIO Ill
CIJ11E>~ORD
II!
COOEwoqO 114
lf5
115
EC•('III
EC•r!SI
I'C•~71
e:c .. ~a•
"'
118
119
IIU
35
35
31
3&
3D
EC•ll'll
EC ~Ill
CI)O(WQRO lilt
CODEWORD lfl2
CTRI
Nl
li5UHI
COU~TER
4A
YSU~I
•2
•J
XSELl
VSELI
SWPI
ECWI'I
.,
.,
u
(11SW)
SAJ
CODE~ORO
31
(LSN)
n•
CODE >lORD
C'JOE"ORO
CODEWORD
COOEWOQO
CODEWORD
32
33
:u
en
RESERVED LOCATIONS
1.
25
27
2!
ENCODER
Sli1ULATIIlN
~
18
198~
C01111ENT
2
u
••
(HSII)
tiLJMBEII OF SOURCE OIGtU +
X AOOEQ OUTPUT
y AOOEfl OUTPUT
X ADDER INPUT SELECT
y AOOER INPUT SELECT
SOURCE WORD POINTER
COOE~ORD
POINTE~
'
52
MA't 4,
1 U1!00l 0<'0132
:!
3 00~~~3 00~1~2 tNITe
4
:l~I'V04
~f':ll32
'
<10(11"'10
020132
JHP J8EG"
STl YSEI.
8
9
0iHI~I4
01H!132
If'
II
20PII'I&
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COPY
0~1'1117
~000~&
3
ZEIIO
l l 2~0~2~ ~111'20
13 00t'P21 01H3'3
..
S09
15
r0~r22
e~P~J•
15
11
~~Pii'23
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Code Test Program Listing
63
64
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