BANDWIDTH EFFICIENT CODING 

FOR 

ERROR CONTROL 

Technical Report 

to 

NASA 

Goddard Space Flight Center 

Greenbelt, Ma.ryland 20771 

Grant Number NAG 5-931 

Report Number NASA 89-001 

Shu Lin 

Principal Investigator 

Department of Electrical Engineering 

University of Hawaii at Manoa 

Honolulu, Hawaii 96822 

August 1,1989 

(NASA-a-185836) BBIDWIDTH EPPICIBbiT CODIIG 

FOR ERROR COITElOL Final Technical Report 

(Hauaii Univ,) 38 p .CSCL 178 

889-27062 ~- 

Unclas 

63/32 0224582 
L. 

1 1 

BANDWIDTH EFFICIENT CODING FOR ERROR CONTROL 

Shu Lin 

Department of Electrical Engineering 

University of Hawaii at Manoa 

Honolulu, Hawaii 96822 

ABSTRACT 

One of the dramatic developments in bandwidth-efficient communications over the past few years is the introduction and rapid applications of 

combined coding and bandwidth efficient modulation, known as coded modulation, for error control. Using coded modulation, reliable data transmission 

can be attained without compromising bandwidth efficiency. In this report, 

we present the basic concepts of coded modulation. We use two simple examples to demonstrate how significant coding gains can be achieved without 

bandwidth expansion. This report serves as an introduction to subsequent 

technical reports on coded modulation. 

1 
1. Introduction 

Conventionally in digital communications, channel coding and modulation are designed and performed separately. Error control is achieved by 

adding structured redundant bits (or parity check bits) to each transmitted 

message. Transmission of these redundant bits results in either a bandwidth 

expansion or reduction of data rate. This type of channel coding is suitable 

for power limited channels without bandwidth constraint, such as deep space 

and wideband satellite channels. Coding gain is achieved at the expense of 

bandwidth expansion. However, for many channels, such as voiceband telephone, terrestrial microwave, mobile and some satellite channels, bandwidth 

is a very precious resource, and hence bandwidth expansion is not desirable 

or even not possible. For this reason, conventional channel coding is rarely 

used for error control in these bandlimited channels. 

An obvious approach to conserve bandwidth is to enlarge the signal set of 

the modulation. However, if coding and modulation are still designed and performed independently, this approach gives very disappointing results. For example, we compare the performance of an uncoded QPSK modulation system 

with that of an 8-PSK modulation system using a rate-2/3 convolutional code 

of constraint length v = 6 and free distance dfiee = 7 which is decoded with a 

Viterbi decoder. Suppose the uncoded QPSK system has an output bit-errorrate (BER) of at some signal-to-noise ratio (SNR). At the same SNR, 

the raw BER of an 8-PSK modulation system is because the Euclidean 

distance between signal points becomes smaller. When we use the rate-2/3 

convolutional code with Viterbi decoder in conjunction with this 8-PSK modulation, the resultant output BER of the overall coded system is After 

all the effort expended in coding and in building a rather complex 64-state 

Viterbi decoder, the error performance of 8-PSK modulation only breaks even 

with that of an uncoded QPSK system. This example simply demonstrates 

2 
that, if we want to achieve coding gain while preserving bandwidth, coding 

and modulation should not be performed independently. This simple fact motivated communication system designers to combine coding and modulation 

as an integral part to achieve coding gain without sacrificing bandwidth. In 

1982, Ungerboek [l] showed that it is indeed possible to design a system by 

combining coding and modulation to attain signifficant coding gain without 

bandwidth expansion. His work has stirred a rapid research and development 

in this area of combining coding and modulation which is now known as coded 

modulation. 

The basic concept of coded modulation is to encode information symbols 

onto an expanded channel signal set (relative to that needed for uncoded 

modulation). The channel signal set expansion provides the needed redundancy for error control without increasing the bandwidth requirements, while 

coding is used to produce a certain interdependency between succesive 

channel signals, such that only certain sequences of channel signals are permitted. Using properly designed coded modulation, reliable data transmission 

can be achieved without compromising the bandwith efficiency. Coded modulation can be classified based on either the structure of the code being used or 

the type of modulation constellation being used. Based on the code structure, 

coded modulation can be classified into two categories: 

(1) trellis coded modulation (TCM) in which convolutional (or trellis) codes are used, and 

(2) block coded modulation (BCM) in which block codes are used. 

Based on the modulation signaling, coded modulations are divided into two 

types: 

3 
(1) constant-envepole-type coded modulation in which the modulation signals have constant envelope, e.g., MPSK, and 

(2) lattice-type coded modulation in which the signal constellation has 

lattice structure, e.g., QASK. 

In this report, we present the basic concepts of coded modulation. We 

use two simple examples to illustrate these concepts and show how coding 

gain can be achieved without bandwidth expansion. This report serves as an 

introduction to subsequent technical reports on coded modulation. 

4 
I I 

2. Modulation Codes 

In a coded modulation system, information sequences are coded into 

signal sequences over a certain modulation signal set (e.g., the 8-PSK signal 

set). These signal sequences form a modulation code. In order to achieve 

good error performance, modulation codes are generally decoded with a softdecision decoding algorithm. For an additive white Gaussian noise (AWGN) 

channel, the error performance (or coding gain) of a modulation code is largely 

determined by its minimum squared Euclidean distance. -_ 

Let (X(s), Y(s)) denote a point s in a two dimensional Euclidean space 

R2, where X(s) and Y(s) are the z- and y- coordinates of s. The squared 

Euclidean distance between two points, s and s', in R2, denoted d(s, s'), is 

defined as follows: 

d(s, s') (X(s) - x(s'))2 + (Y(s) - Y(s'))2. (2.1) 

Let V= (s1,s2,..-,sn) andg = (s{,sk,*--,sI) be twon-tuplesover R2. The 

squared Euclidean distance between 6 and v', denoted IV - #I2, is defined as 

follows: 

n 

j=1 

Consider a modulation code C with signals from a two-dimensional modulation signal set s. The minimum squared Euclidean distance of C, 

denoted D[C], is defined as follows: 

(2.3) 

A D[C] = min { IV - V' I2 : ;ii, V' E C and 6 # a}. 

For a trellis modulation code, D[C] becomes the squared free Euclidean distance, denoted dLee. Suppose IC information bits are encoded into a sequence 

of n signals based on C. The effective rate of C, denoted R[C], is defined as 

5 
which is simply the average number of information bits transmitted by C 

per dimension. For R[C] 2 1, coding is achieved without compromising the 

bandwidth efficiency or reducing the data rate. With a properly designed 

modulation code (block or trellis), it is possible to attain significant coding 

gain without bandwidth expansion. The effectiveness of a modulation code is 

measured by its effective rate, minimum squared Euclidean distance and the 

ease of its decoding implementation. Just like in the design of conventional 

coding, for a given effective rate R[C] and code length n (or constraint length 

for TCM), we want to design a modulation code with a minimum squared 

Euclidean distance as large as possible and a structure which makes maximum 

likelihood decoding practical. 

Suppose a modulation code C is used for an AWGN channel and d 

the code sequences (or codewords) in C axe equally likely to be transmitted. 

Let v = (z1,y1,x2,y2,..-) be the output sequence of the demodulator at 

the receiving end, where xi and yi are the two ouputs of the demodulator 

corresponding to the i-th received signal. The sequence V is called the received 

sequence. Note that xi and ?/; are two real numbers. The squared Euclidean 

distance between 'i; and a code sequence V = ($1, s2, - . , sj, .) in C is given 

by, 

For maximum likelihood decoding (MLD), the received sequence 'i; is decoded 

into the code sequence ijl for which the following condition holds, 

for i # e. To perform a soft-decision MLD, it is desirable for a modulation 

code to have a trellis structure so that the Viterbi decoding algorithm can be 

applied. 

6 
3. Signal Set Partition, Labeling and Distance Structure 

The design of a coded modulation system depends on the choice of the 

modulation signal set, signal set partition, signal labeling, choice of coding 

and code bits-t+signal mapping. In this section, we present the concepts 

of signal set partitioning, signal labeling, and distance properties among the 

signal points in a set of a partition and between the sets of a partition. 

In the following, we use an example to illustrate the signal set partitioning process. Consider the 8-PSK signal set (or constellation) as shown 

in Figure 1, where each signal point is labeled with an integer i from the 

set S = {0,1,2,3,4,5,6,7}. Each integer i E S can be represented in the 

following binary form: 

i = bo + bl 2 + b2 22 

with bj = 0 or 1 for 0 5 j 5 2. The binary sequence (3- tuple), (bo, bl, b), is 

the binary representation of the integer i, where bo is the least significant bit 

and b2 is the most significant bit. The binary representations for the eight 

integers in S are: 

0 e (000) 

2 e (010) 

4 e (001) 

1 +=3 (100) 

3 e (110) 

5 e (101) 

6 e (011) 7 a (111) 

The squared Euclidean distance between two ignal points, s and st, in 

the &PSK signal set S is given by: 

From (3.1)) we find that 

7 
d(1,O) = d(2,l) = 0.586 

d(2,O) = d(5,3) = 2 

d(4,O) = d(5,l) = 4. 

The minimum and maximum squared Euclidean distances between signal 

points in the 8-PSK signal set S are 0.586 and 4 respectively. 

Let X be a subset of the signal set S. Define the minimum squared 

Euclidean distance of X as follows: 

where d(z, xt) is given by (3.1). This distance is called the intra-set distance 

of S. The minimum squared Euclidean distance between X and Y, denoted 

d(X, Y), is defined as follows: 

d[X,Y] fi min{d(z,y) : z E X and y E Y}. 

This distance is a measure of the separation of two sets of signals. 

Now we form a chain of partitions of the signal set S with increasing number of subsets, increasing intra-set distances and set separations as 

shown in Figure 2. First we partition S into two subsets, So = {0,2,4,6} 

and Sl = {1,3,5,7} where, for k = 0 or 1, Sk consists of those integers in 

S with bo = k in their binary representations. The intra-set distance of S is 

d[S] = 0.586. The first partition of S produces two subsets SO and SI, both 

subsets having intra-set distances equal to 2, i.e., d[So] = d[Sl] = 2. We see 

that there is an increase in intra-set distance from the signal set S to the first 

partition {So, SI}. The separation between SO and SI is d[So, SI] = 0.586. 

From Figures 1 and 3,we see that So and SI are actually two QPSK signal 

sets. 

8 
At the second step of partitioning, each subset Sh in the first partition 

with k = 0 or 1 is partitioned into two subsets, Sko and Sh1, where for j = 0 

or 1, Shj consists of those integers in S with bo = k and = j in their binary 

representations. The partitioning process results in the following subsets: 

The intra-set-distance of each of the above 4 subsets is 4, i.e., d[Skj] = 4 for 

0 5 k 5 1 and 0 5 j 5 1. The separations between these subsets are: 

I 

d( so0 , sol) = d(S10, Sll) = 2 

d( so0 ) SlO) = d( so0 , Sll) = d( SlO , sol) 

= d(So1, S11) = 0.586 

We see that the second partitioning of the signal set results in an increase 

of the intra-set distance from 2 to 4. Furthermore, we see from Figure 3 that 

each subset Skj in the second partition is a BPSK signal set. Two BPSK 

signal sets which are contained in the same QPSK signal set are separated by 

a squared Euclidean distance of 2. Two BPSK signal sets from two different 

QPSK signal sets are separated by a squared Euclidean distance at least 0.586. 

At the last step of partitioning, we partition each subset Skj in the second 

partition into two subsets, Skjo and Skjl, where Skji consists of ody one 

integer whose binary representation is (bo, bl, b) = (IC, j, i). Hence, 

so00 = {o}, so01 = {4}, so10 = (2}, so11 = (6)) 

SlOO = {I}, SlOl = {5}, SllO = {3}, s111 = (7). 

Since each subset in this last partition contains only one signal point, its 

intra-set distance is regarded as infinity. The set separations are: 

I 

I 

9 
d(skj0, skjl) = 4, 

d(SkOi, Skli) = 2, 

d( Soji, S1 ji) = 0.586. 

The above partitioning process is called binary partition which results in 3 

partitions with increasing intr&set distances, as shown in Figure 4, 

d[S] = 0.586, d[Sk] = 2, d[Skj] = 4, d[Skji] = 00. 

At each partitioning stage, a subset is labeled by a binary sequence, and 

at the end of the partitioning process, each signal point is labeled by a binary 

sequence (in this case, the labeling sequence is also the binary representation 

of the integer which represents the signal point). Two signal points with labels 

different at the first position (i.e., bo position) are at a squared Euclidean 

distance at least 0.586 apart. Two signal points with labels identical at the 

first position but different at the second position are separated by a squared 

Euclidean distance at least 2 apart. Two signal points with labels identical 

at the first two positions but different at the last position are at a squared 

Euclidean distance 4 apart. For convenience, we denote the partition chain 

with S/SQ/SOO/SOOO. 

As another example, we consider the 16-QASK signal constellation S as 

shown in Figure 4, where each signal point is represented by an integer and 

labeled by the binary representation of the integer. Let dl denote the intra-set 

distance of S, i.e., dl = d[S]. The partition chain is shown in Figure 5. We 

see that the intr&set distance increases form dl to 8d1. 

Signal set Partitioning is used to map encoded bits onto signal set so that 

the channel signal sequences have largest possible minimum squared Euclidean 

distance. 

10 
4. Trellis Coded Modulation (TCM) 

In this section, we use a simple example to illustrate the concepts and 

process of TCM. The code to be used is a rate-1/2 convolutional code with a 

4-state trellis diagram, and the modulation signal set is the 8-PSK as shown 

in Figure 1. The schematic diagram of the coded modulation system is shown 

in Figure 6 [2]. 

At each unit of time, two information bits are mapped into an &PSK 

signal through convolutional encoding. At time I, the input information bits 

are ci2) and cI1), and the corresponding output coded bits are vi2), vi1) and 

vio). The information bit c12) is uncoded and vi2) = ci2). The information bit 

cil) is encoded into two bits, vi1) and vio), based on a rate-1/2 convolutional 

code of memory m = 2. The parity equations are: 

The code has a 4-state trellis diagram as shown in Figure 7. The free (Hamming) distance of the code is 3. When two paths diverge at a certain node, it 

takes at least three branches for them to remerge. A branch coming out from 

an I-th order node is labeled by an output pair (vio),vil)) corresponding to 

an input bit cl (1) . 

Now we need to include the uncoded information bit ci2) in the trellis diagram to make the overall code a rate-2/3 trellis code. This can be 

achieved by splitting each branch in the trellis into two parallel branches, one 

corresponding to c!~) = 0 and the other corresponding to ci2) = 1. The resultant trellis is shown in Figure 8. Now each branch is labeled with a triplet, 

(VI (O) )Vl (l) ,211 (2)), with vi2) = ci2) and vi1) = Two parallel branches are 

11 
v(') 0) and (vl (0) , vl (1) , 1). The new trellis has the following labeled with (vl , 

structure: 

(1) There are two parallel branches connecting two adjacent nodes (or 

states); 

(2) There are two groups of two parallel branches diverging from each 

node; 

-- 

(3) There are two groups of two pa.r*allel branches merging into a node. 

Every path in the trellis corresponds to a binary code sequence. 

Now we need to map each triplet (V~~),V/~),V~~)) on a branch into an 

8-PSK signal. The mapping should be done in such a way that the resultant signal sequences in the trellis have largest possible minimum squared 

Euclidean distance (or squared free Euclidean distance). The mapping is 

done based on the set partitioning. Obviously, the two triplets on two parallel branches should be mapped into two signal points in a set Skj with the 

largest intra-set distance. For this case, d[skj] = 4. For example, so0 = {0,4}. 

The two groups of two parallel branches diverging from or merging into a 

code should be assigned to signal points from two subsets, S~O and ski, with 

largest set separation, d[Sko, Ski]. In this case, d[Sko, Ski] = 2. For example, 

So0 = {0,4} and Sol = {2,6}. 

Using the above mapping rule, we obtain a 4-state trellis with signal 

symbols from the 8-PSK signal set as shown in Figure 9. Note that mapping 

is done simply by using the binary representation of each signal point, i.e., the 

triplet (vl (O) ,vl (l) , v(~)) issimply the binary representation of a signal point. 

For two paths in the trellis which diverge from the same node through two 

non-parallel branches, it will take at least 3 branches before they can remerge 

into a later node in the trellis. Since the free Hamming distance of the rate12 
1/2 convolutional code being used is 3, it guarantees that two diverging and 

remerging paths must be separated by a squared Euclidean distance at least 

d(Sk0, sk1) + d(S) + d(Sk0, Ski) = 2 + 0.586 + 2 = 4.586. 

Two paths diverging and remerging through two parallel branches are separated by a squared Euclidean distance of 4. As a result, two signal sequences 

in the trellis axe separated by a squared Euclidean distance at least 4. Hence ~ 

the squared free distance d&ee is 4. 

In the above coded modulation system, every two information bits are 

encoded into a signal point in a two-dimensional 8-PSK signal set. Hence the 

information rate per dimension is 1 and the effective rate R[C] is 1. 

Now we evaluate the coding gain of the above trellis coded 8-PSK system 

over an uncoded reference modulation system. Consider a QPSK system 

without coding. Every two information bits axe encoded into a signal point 

in a 2-dimensional QPSK signal set & shown in Figure 10. The effective rate 

of this system is also 1 bit per dimension. Two QPSK signal sequences may 

only differ in one place with two symbols separated by the smallest squared 

Euclidean distance, 2. Therefore, the minimum squared Euclidean distance 

of an uncoded QPSK system is 2. The coded 8-PSK system described above 

also has effective rate 1 but minimum squared Euclidean distance 4. Both 

systems have the same signaling rate. Hence, for the same SNR, the coded 

8-PSK modulation system provides smaller output BER than the uncoded 

QPSK without bandwidth expansion or reducing data rate. For the same 

output BER, the coded 8-PSK modulation system requires less SNR than the 

uncoded QPSK system. This reduction, usually expressed in decibels, in the 

required SNR to achieve a specified BER of the coded system over an uncoded 

system is called the coding gain. 

The coding gain is related to the ratio of the minimum squared Euclidean I 

13 
distance of a coded system to the minimum squared Euclidean distance of a 

reference system. The asymptotic coding gain of a coded system over an ’ 

uncoded reference system is given by [1,2] 

(MSED)coded system 

(MSED) reference system 

7 = lolog,, 

where MSED stands for minimum squared Euclidean distance. 

The asymptotic gain of the above coded 8-PSK system over the uncoded 

QPSK system is 

The real coding gain at BER is 2.6 dB (see Figure 11). Note that this 

coding gain is achieved without bandwidth expansion. We see that simple 

coded 8-PSK modulation system achieves quite significant coding gain with a 

little additional decoding complexity. Due to the trellis structure, the simple 

coded 8-PSK modulation code can be decoded with a 4-state Viterbi decoder 

using a soft-decision MLD algorithm. 

Coded modulation is based on the signal constellation, signal set partitioning and labeling, code selection and bits-to-signal mapping. The TCM 

illustrated in the above example can be applied to other types of modulation 

(e.g., 16-PSK, 16-QASK) and convolutional codes of higher rates and larger 

number of states. A general TCM system is shown in Figure 12. Larger 

coding gains can be achieved with larger number of trellis states. Roughly 

speaking, it is possible to gain 3 dB with 4 states, 4 dB with 8 states, nearly 

5 dB with 16 states and up to 6 dB with 128 or more states. 

14 
. 

5. Block Coded Modulation 

Again we use an example to illustrate the design process of BCM. The 

example given here is a block coded 8-PSK modulation system. The twodimensional 8-PSK signal set is shown in Figure 1, where each signal point is 

represented by an integer in S = {O,l, 2,3,4,5,6,7}. 

Like the design of a TCM system, the design of a BCM system is also 

based on code selection, set partitioning and bits-to-signal mapping. In our 

example, we choose three binary block codes of length n = 8 as follows: 

(1) cbl is the (8,l) repetition code which consists of the d-zero and done vectors,and hence its minimum Ha.mming distance is 61 = 8; (2) cb2 is 

the (8,7) code with d the even-weight vectors and has minimum Hamming 

distance 62 = 2; and (3) CM is the (8,8) code which consists of d the binary 

8-tuples and has minimum Hnmming distance 63 = 1. 

Consider the following three subsets of S: 

B1 = (0, l}, B2 = {0,2} and B3 = {0,4}. 

These subsets have increasing intmset distances, 

d[B1] 0.586, d[&] = 2, d[&] = 4. 

From the binary code Cfi we construct a block code Ci with symbols from 

Bi as follows: every 1-component in a codeword 5 in Cfi is repleaced by the 

symbol bi. Denote Ci with biC&. Then the minimum squared Euclidean 

distance of Ci is D[Ci] = Sid[Bi] for 1 5 i 5 3. The above transformation 

results in three codes, 

15 
with minimum squared Euclidean distances, 

respectively. Now take the direct-sum of C1, C2 and C3 as follows: 

C is then a block code of length n = 8 with symbols from S = {0,1,2,3,4,5,6,7}. 

If each code symbol is mapped to its corresponding signal point in the 8-PSK 

signal set as shown in Figure 1, we then obtain a block 8-PSK modulation 

code of length 8. Note that each codeword in this 8-PSK modulation code is 

a sequence of eight 8-PSK signals. 

From the construction process, we see that a total of 16 = 1 + 7 + 8 information bits are encoded into a sequence of eight 8-PSK signals. Hence, 

on average, each signal carries two information bits and the effecitve rate is 

R[D] = 1. The overall coded modulation system is shown in Figure 13. The 

minimum squared Euclidean distance of this block 8-PSK modulation code is 

I 

I 

Now we consider the coding gain of the above block 8-PSK modulation 

code over an uncoded reference modulation system. Suppose an uncoded 

QPSK system is used. Then 16 information bits are encoded into a sequence of 

8 QPSK signals. For this uncoded system, the effective rate is R[CJref = 1 and 

the minimum squared Euclidean distance is D[C],,f= 2. Hence the simple 

block coded 8-PSK modulation system achieves a 

'I 

16 
li ---1 

. 

asymptotic coding gain over the uncoded QPSK system without bandwidth 

expansion. The real coding gain is shown in Figure 14. We see that, at 

decoded block-error-rate, the block coded 8-PSK system achieves a 2.42 dB 

coding gain over the uncoded QPSK. 

Next we consider the decoding of our example code. Note that each 

binary component code has a trellis structure as shown in Figure 15. Then 

the block 8-PSK modulation code has a 4-state 8-section trellis diagram as 

shown in Figures 16 and 17 which is the direct product of three trellis diagrams 

of the component codes shown in Figure 15. Hence the code can be decoded 

with a 4-state Viterbi decoder using a soft-decision MLD algorithm. From 

Figure 17, we see that the trellis consists of two identical parallel 2-state 

trellis sub-diagrams without cross connections between them. This structure 

suggests that the decoding can be done with two 2-state Viterbi decoders 

to process the two trellis sub-diagrams in parallel. This not only speeds up 

the decoding process but also simplifies the implementation. Furthermore, 

this block 8-PSK modulation code is phase invariant under multiples of 45" 

rotation. That is, if there is a 45" phase rotation, a codeword becomes another 

codeword in the code. This phase invariant property is useful in resolving 

carrier-phase ambiguity and ensuring rapid carrier-phase resynchronization 

after temporary loss of synchronization. 

-- 

The block 8-PSK modulation code given above is an equivalent to the 

trellis 8-PSK modulation code given in Section 4. Except the code structure, they are similar in many ways. The block 8-PSK modulation code has 

advantages in decoding complexity and decoding speed. The trellis 8-PSK 

modulation code is phase-invariant only under 180" rotation, however it has 

slightly larger coding gain (less than 0.2 dB) than the block 8-PSK modulation 

code. The method used in the construction of the example block 8-PSK modulation code can be used to construct any type of modulation codes (MPSK 

17 
I 

or QASK modulation codes). This method is known as the multi-level construction [3-61. The resultant code has trellis structure if the component codes 

have trellis structure. 

18 
6. Conclusion 

In this report, we have introduced the basic concepts of coded modulation. We used two simple coded modulation systems to demonstrate that 

coding gain can be attained without compromising the bandwidth efficiency 

(or bandwidth expansion). 

In our next two reports, we will present the multi-level construction of 

modulation codes. We will show that using the multi-level method, modulation codes with arbitrary large minimum squared Euclidean distance can be 

constructed. 

-_ 

19 
References 

[l] G. Ungerboek, ”Channel Coding with Multilevel/Phase Signals,” IEEE 

Trans. on Information Theory, Vol.IT-28, No.1, pp.55-67, January 1982. 

[2] G. Ungerboek, ”Trellis-Coded Modulation with Redundant Signal Sets, 

Part I: Introduction,” IEEE Communications Magazine, Vo1.25, No.2, 

pp.5-11, February 1987. 

[3] H. Imai and S. Hirakawa, ”A New Multilevel Coding Method Using Error Correcting Codes,” IEEE Trans. on Information Theory, Vol.IT-23, 

No.3, pp.371-376, May 1977. 

[4] V.V. Ginzburg, ”Multidimensional Signals for a Continuous Channel,” 

Problem Peredachi Informatsii, V01.20, No.1, pp.28-46, 1984. 

[5] R.M. Tanner, ”Algebraic Construction of Large Euclidean Distance Combined Coding Modulation Systems,” Abstract of Papers, 1986 EEE International Symposium on Information Theory, Ann Harbor, October 

6-9, 1986, also IEEE Trans. on Information Theory to be published. 

[6] T. Kasami, T. Takata, T. Fujiwara and S. Lin, ”A Concatenated Coded 

Modulation Scheme for Error Control,” LEEE Trans. OR Communications, to be published. 

20 
8 

I d(2,0)=2 

(a) 8-PSK Signal Set 

3 

(1 10) 

e 

5 

(101) 

a 

1 

(1 001 

7 

(111) 

e 
S 

0 (000) l(100) 

4 (001 1 2 (010) 3 (1 10) S(l01) 

0 (000) 2 (010) 

4(001) 6 (01 1) 

0 cobo, 2 (01 0) 

4 (001 1 'lo d(011) 

0 42 6 

(000) (001) (010) (011) 

so00 so01 so10 so11 

51 l(100) 3(110) 

S(101) 7(111) 

l(100) 3 (1'10) 

sols(lol) 7(111) 

(100) (101) (110) (111) 

s 100 SlOl SllO Slll 

Figure 2. Partition chain of the 8-PSK signal set 
S 

0 so0 0 0 

(BPSK) 0 

0 0 

0 

0 0 

0 0 

0 

000 

so00 

- 1- 0.586 a 

0 

0 

a 

0 

a 

0 

lo- 2 

0 0 

0 0 O 0 so1 0 O 0 SlO 0 <*4 

0 0 

0 

./ \' 

0 0 

0 

ob 

0 

0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 0 0 00 0 

0 00 00 00 00 00 00 0 

0 0 0 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 

00 1 010 01 1 100 101 110 111 

so01 so10 so11 SlOO SlOl SllO Slll 

Figure 3. 8-PSK/QPSK/BPSK signal constellation chain 
. I 

0 0 

8 (0001) 13(1011) 

e e 

15(1111? lO(0101) 

0 e 

4 (0010) l(1000) 

e 

12 (001 1) 

e 

11 (1101) 

0 

0 (0000) 

0 

7(1 110) 

0 

9 ( 1001 1 

0 

14 (01 11 1 

0 

S(l010) 

0 

2 (0100) 

Figure 4. A 16-QASK signal set 
e... 

0. OP 1 \ 00 

0. 

O\ 

-0 0 0’0 0 0.0. 0000 

m 0000 0.0. 0000 

e0.0 0 e 0 e BOO sol 0 0 0 0 

0000 0.0. 0000 modo 

1 .000 /\ 00.0 0000 /\ 0000 000. / loo /’ \ 0000 0 0’0 0 

00.000 qz 8d1 

e000 00 0 

0000 0.00 000. 

e000 0000 0000 

so00 0000 

00.0 

0000 0000 

0.00 000. 

/\ /\ 0000 0000 0000 0000 0000 00.0 0000 .000 0000 0000 0000 0000 e000 0000 00.0 0000 

0600 oode 0000 0.00 0000 0000 0000 0000 0000 om00 0000 ooom 0000 0000 0000 0000 

00.0 0000 e000 0000 0000 0000 0000 0000 0.00 0000 ooom 0000 0000 0000 0000 0000 000. 0000 Ob00 0000 0000 0000 0000 0000 

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 

Figure 5. Partition chain of the 16-$ASK signal set 
i 

- 

n 

-uE 

Q) 

5 

+ 

I' 

I 

c 

0 

c 

0 

0 

0 
0 
3 

10 

11 

Figure 10. A QPSK signal constellation 
5 6 7 8 9 10 11 12 d8 

SNR per symbol (de) 

Figure 11. Coding gain over an uncoded QPSK 
C 

I 

a 

-u- 

-J 

6 

7 

U 

%J 

0 

0 

M 

+ 

n 

-u- 

- 

9 L 

t' t' 
m c IL 

ab B 

0) 

+, 

rn 

d 

0 .C( 

Y 

a 

0 

0 
1 

1 0-1 

lo-* 

1 0-3 

1 o4 

1 o5 

1 o4 

6.0 7.0 8.0 9.0 10.0 11.0 120 13.0 14.0 15.0 

SNR per symbol (dB) 

Figure 14. Error performance of the 4-state 8-PSK block code 
0 

0 

0 

0 

0 

c 

c 0 

c 0 

rc 0 

c 0 

c 0 

c 0