On block-coded modulation with interblock memory

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 11, NOVEMBER 1997 1401

On Block-Coded Modulationwith Interblock Memory

Mao-Chao Lin,Member, IEEE, Jia-Yin Wang, and Shang-Chih Ma

Abstract—Block-coded modulation with interblock memory(BCMIM) is a variation of block-coded modulation (BCM) whichis designed for multilevel coding. By providing interblock mem-ory between adjacent blocks, the coding rate of a BCMIM schemecan be increased without decreasing the minimum squared Eu-clidean distance (MSED) as compared to the original BCM. Inan early version of BCMIM, interblock coding is provided onlybetween the first two coding levels of adjacent blocks. In thispaper, we design BCMIM with a more general form for whichinterblock coding can be introduced among many coding levels.In this way, we can further increase the coding rate of BCMIMwithout decreasing the MSED. We provide many examples toshow the advantages of BCMIM with the general form. Most ofthe examples are designed based on multidimensional signal sets,since a multidimensional signal set can provide more coding levelsthan a two-dimensional (2-D) signal set.

Index Terms— Block-coded modulation, coded modulation,coding, interblock coding.

I. INTRODUCTION

I N 1982, Ungerboeck [1] proposed the concept of codedmodulation which combines error-control coding and the

concept of set partitioning on the expanded two-dimensional(2-D) signal set (signal constellation). The concept of setpartitioning can be briefly described as follows: Letbe a 2-D signal set which consists of 2signal points.Each 2-D signal point in can be represented by abinary -tuple . Let , where

. The squaredEuclidean distance (SED) between is denoted by

. For , let be the setfor

. Let . Then,is partitioned into two subsets, which are and

its coset. Equivalently, is partitioned into 2 subsets

Paper approved by S. B. Wicker, the Editor for Coding Theory andTechniques of the IEEE Communications Society. Manuscript received May2, 1995; revised June 19, 1996 and April 20, 1997. This work was supportedby the Industrial Technology Research Institute of R.O.C. Contract AC-84103. This paper was presented in part at the International Symposiumon Information Theory and Its Applications, Sydney, Australia, November20–24, 1994.

M.-C. Lin is with the Department of Electrical Engineering, NationalTaiwan University, Taipei, 106 Taiwan, R.O.C. (e-mail: [email protected]).

J.-Y. Wang is with the Department of Electrical Engineering, NationalTaiwan University, Taipei, 106 Taiwan, R.O.C. (e-mail: [email protected]).

S.-C. Ma is with the Digital Video Broadcast System Department, Computerand Communication Research Laboratories, Industrial Technology ResearchInstitute, Hsinchu, 310 Taiwan, R.O.C. (e-mail: scma@cclvlab. ccl.itri.org.tw).

Publisher Item Identifier S 0090-6778(97)08202-0.

consisting of and its cosets. The intraset SED ofor its cosets is shown in (1), found at the bottom of the nextpage. Clearly, . With the partition chain

, we say that has an -level partitionstructure and a distance profile of .

There are many additional research results in the area ofcoded modulation. Some of them may be found in the refer-ences listed in [2]–[20]. Among the known coded modulationsystems, there are many which have the multilevel codingstructure [11]–[20]. Let be a sequence of signalpoints in where . Amultilevel coded modulation can be designed by considering

as the labelings of signal points at theth level,. For traditional multilevel coding [11]–[14],is designed to be a codeword of a binary code

. If each is a binary block code oflength , then the associated scheme is called block-codedmodulation (BCM), where each block consists of2-D signalpoints in . In 1994 [20], a variation of BCM called block-coded modulation with interblock memory (BCMIM) wasproposed. This is designed by modifying the BCM to allowan interdependency between the second coding level of thecurrent block and the first coding level of the next block.These two levels are encoded together in the first codingstage. The th coding level of the current block is encodedas a normal binary block code for . We saythat this BCMIM has an -level and -stage codingstructure. The introduction of interdependency between blocks(i.e., interblock coding) can increase the coding rate withoutdecreasing the minimum SED (MSED) as compared to theoriginal BCM.

In this paper, we generalize the method of providing in-terblock coding so that BCMIM has an -level and -stage

coding structure. In this way, we can provideadditional interblock coding. That implies a further increase ofcoding rate without sacrificing the MSED. Good examples ofthe generalized form of BCMIM based on 2-D signal setsare rare. However, we can easily construct good examplesof the generalized BCMIM based on multidimensional signalsets. We use the method proposed by Pietrobonet al. [9]–[10]to partition a 2 -D (2 -dimensional) signal set into an -level structure. Since the number of partition levels of a2 -D signal set is greater, there is more room fordesigning interblock coding in BCMIM. Several examples areconstructed based on this idea. We show that large codinggains can be achieved while the decoding complexities arerelatively low.

0090–6778/97$10.00 1997 IEEE

1402 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 45, NO. 11, NOVEMBER 1997

Fig. 1. Coding configuration of BCMIM used in [20].

II. BLOCK-CODED MODULATION WITH INTERBLOCK

MEMORY USING TWO-DIMENSIONAL SIGNAL SETS

Let be a 2 -D signal set which consists of 2 2 -D signal points, where each 2-D signal point is a sequenceof 2-D signal points in . Let

represent a 2-D signal point in andbe an binary linear block code for .The set

is a BCM scheme of length [12]–[14],which has an -level coding structure. For this BCM scheme,the coding rate is

(2)

information bits per 2-D signal point, and the MSED is

(3)

The BCMIM scheme proposed in [20] is constructed byproviding interblock coding between the first coding level ofone block and the second coding level of the following block.Let represent a blockand ) representthe next block. For the BCMIM scheme proposed in [20],

is a codeword of a binarylinear code of block length 2 , while and

are codewords of a binary linear code ofblock length for . The coding configurationis given in Fig. 1. We may say that such a BCMIM schemehas an -level and -stage coding structure.

We now extend the -level and -stage BCMIMscheme proposed in [20] to a more general form which hasan -level and -stage structure, where .The coding configuration is shown in Fig. 2. Every blockis a 2 -D signal point. The blackened parts are affectedby the encoding for the current time unit, the gray partshave already been determined for earlier time units, and theblank parts will be decided by the encoding for later timeunits. Let and . At the th coding stage

there is some form of interdependency amongthe th, th, , and th level of theconsecutive blocks represented by

, respectively. The

part of theseconsecutive blocks is a codeword of a binary linear code

of block length , which is constructed bycombining and some of their cosets.A proper design of for each can increase the codingrate of the BCMIM without reducing the MSED as comparedto the BCM.

We now show a very simple design of , which is anbinary linear code. The

generator matrix of has the following form:

(4)

where is the generator matrix of the code andis the generator matrix of an binary code . Theminimum Hamming distance of the binary code

is . At the th stage of encod-ing, a message is en-

coded into

+ where is an -tuple and isa -tuple for .

For this BCMIM scheme, the coding rate is

(5)

information bits per 2-D signal point and the MSED is

(6)

If ,

then this BCMIM scheme can achieve additional

and for(1)

LIN et al.: BLOCK-CODED MODULATION WITH INTERBLOCK MEMORY 1403

Fig. 2. Coding configuration of a generalized BCMIM.

information bits per 2-D signal point while the MSED isunchanged as compared to the BCM . Notethat the existence of provides theinterblock coding for theth coding stage and hence provides

additional information bits per block for this stage.It is possible to design , which is further generalized

from that given in (4), so that we can have BCMIM schemesfor which the coding rates are further increased while theMSED’s are not changed. We will show several generalizeddesigns of by specific examples appearing in Construc-tions 1–5.

In this paper, for , the (8, ) binary blockcode with generator matrix

(7)

is called the RM(8,) code, where implies that transposeof a matrix and

(8)

The minimum Hamming distances of the RM(8,1), RM(8,2),RM(8,3), RM(8,4), RM(8,5), RM(8,6), RM(8,7), and RM(8,8)codes are 8, 4, 4, 4, 2, 2, 2, and 1, respectively. Moreover, the(8,0, ) binary code is called the RM(8,0) code.

Construction 1: Let and be the 16-quadrature amplitude modulation (QAM) (16-QASK) signalset as given in [1]. Let and be the RM(8,1),RM(8,4), RM(8,7), and RM(8,8) codes, respectively. The set

based on is a BCM (BCM-1). It followsfrom (2) and (3) that BCM-1 has a coding rate of 2.5 bits per2-D signal point and an MSED of 3.2. The asymptotic codinggain (ACG) of BCM-1 over uncoded 8-AMPM [1] is 5.23 dB.

Fig. 3. Coding configuration of BCMIM-1.

We now combine levels 1, 2, and 3 into one coding stageto form a BCMIM (BCMIM-1A). The coding configurationis given in Fig. 3. The code is a combination of threecodes and where

a) is the RM(8,5) code andis the RM(8,4) code;

b) is the RM(8,8) code andis the RM(8,7) code;

c) is the RM(8,8) code.

The generator matrix of has the following form:

(9)

Note that and are binary codes used forlevels 1, 2, and 3 respectively. The code isthe RM(8,8) code. BCMIM-1A has a four-level and two-stage coding structure. Let repre-sent the signal block generated for theth time unit whichcontains eight 2-D signal points. The encoding for theth


(a)

(b)

Fig. 4. (a) Decoding trellis ofC(1) for BCMIM-1A and (b) decoding trellis ofC(1) for BCMIM-1B.

time unit involves three consecutive blocks, represented by, which is called a superblock. At

the first stage of encoding, message bits areencoded into

using the (24,16) code . At thesecond stage of encoding, message bits areencoded into using the (8,8,1) code .Now we consider the distance property. Letand represent two different message blocksand let andrepresent the corresponding superblocks. Suppose that themessages for previous time units have been correctly decoded.Consider the following conditions:

i) if , then the MSED betweenand is

;ii) if and , then the

MSED between andis ;

iii) if and , thenthe MSED between and is ;

iv) if and, then the MSED

between and is ;v) if and

, then the MSED between andis ;

vi) If and, then the MSED between and

is .

It follows from conditions i)–vi) that the MSED of BCMIM-1A is 3.2. For BCMIM-1A, the ACG over uncoded 8-AMPMis 6.02 dB and the coding rate is three bits per 2-D signal point.Both BCM-1 and BCMIM-1A can be decoded by using multi-stage decoding. BCM-1 can be decoded by using the trellisesof and for the first, second, third, and fourthdecoding stages, respectively. BCMIM-1A can be decodedby using the trellises of and for the firstand second decoding stages, respectively. The 16-state trellisfor as shown in Fig. 4(a) is composed of three partialtrellises for and , respectively. The threepartial trellises are used for levels 3, 2, and 1, respectively.


Fig. 5. Simulation results for BCM and BCMIM based on 16-QAM or 2�16-QAM.

Note that for BCMIM-1A, we have. We now modify the

generator matrix used in BCMIM-1A by setting. The

resultant BCMIM (BCMIM-1B) has the same coding rate andMSED as BCMIM-1A. The MSED of BCMIM-1B can beverified by a procedure similar to that for BCMIM-1A exceptfor the condition of i). For BCMIM-1B, the MSED between

and isif and

and is if and. This verifies that the MSED of BCMIM-1B is

3.2. For BCMIM-1B, the trellis for can be simplified tobe an eight-state trellis as shown in Fig. 4(b). Note that all 14bits on the right-hand side of of BCMIM-1B areall zero. Such an arrangement for BCMIM-1B can make thetrellises for the two cosets of the code with generator matrix

(10)

differ only in the first two bit positions. Thus, the trellis forof BCMIM-1B can be simplified. In either Fig. 4(a)

or (b), a branch in the partial trellis corresponding to(used in level 2) represents a coset of the (4,1,4) binary blockcode. Decoding complexity of a coded modulation schemecan be evaluated by ADD and COM, which are the averagenumbers of additions and comparisons for each decoded bit,respectively. We have ADD and COM for BCM-1; ADD and COM for BCMIM-1A; ADDand COM for BCMIM-1B.

Simulation results are given in Fig. 5. For a bit error rate(BER) of 10 , the coding gains of BCM-1, BCMIM-1A,and BCMIM-1B over uncoded 8-AMPM are 3.31 dB, 3.66dB, and 3.66 dB, respectively. Due to the large number

of nearest neighbors for the multilevel coded modulationsystem using multistage decoding, the coding gain of eachof BCM-1, BCMIM-1A, and BCMIM-1B for a BER of 10is appreciably lower than the associated ACG.

III. B LOCK-CODED MODULATION WITH INTERBLOCK

MEMORY USING MULTIDIMENSIONAL SIGNAL SETS

For a 2-D signal set with small , it is difficult to designBCMIM with much interblock coding. The reason is thatfor a 2-D signal set with small , such as 8-PSK, thereis no room for additional interblock coding design. Withthis observation, we resort to multidimensional signal setsto further upgrade the performance of BCMIM. In [9] and[10], Pietrobonet al. proposed a procedure to partition the2 -D signal set into a structure withpartition levels. From [9] and [10], we can have a partitionchain where is theunion of and its coset. In the thpartition level, there are 2 cosets of the subset in

. These 2 cosets of are denoted byor where

and . Each -D signalpoint orin can be generated from . For

is the labeling of for the th level.In this way, the distance profile of the -PSK signal set

, and the distance profile of the -QAM signal set can be derived, whereis the intraset MSED within each coset of . With thesesignal sets, we are able to construct very efficient BCMIMschemes. For a BCMIM scheme using a 2-D signal set, theparameter of in Section II is replaced by , the distanceprofile is replaced by ,and the denominator in (2) and (5) must be replaced by.


Fig. 6. Signal set of 2�8-PSK.

A. BCMIM Based on 28-PSK

Consider the set partitioning of the 28-PSK signal set,. As shown in Fig. 6, each four-dimensional (4-D) signal

point in is represented bywhere is a 2-D signal point

in the 8-PSK set for . The distance profile is[9].

Construction 2: Let and be theRM(8,0), RM(8,1), RM(8,4), RM(8,7), RM(8,7), and RM(8,8)codes, respectively. The set basedon is a BCM (BCM-2) which has a coding rate of

bits per 2-D signal point. For BCM-2, theMSED is 8 and the ACG over uncoded QPSK is 5.28 dB.

We can construct a BCMIM (BCMIM-2A) with a six-leveland three-stage coding structure. Let

represent the signal block generated for theth timeunit which contains eight 4-D signal points. The encodingfor the th time unit involves four consecutive blocks,represented by . Thecoding configuration is shown in Fig. 7. At the first stageof encoding, message bits are encoded into

using a (24,9) code .At the second stage, message bits areencoded intousing a (16,15) code . At the third stage, message bits

are encoded into us-ing the (8, 8, 1) code . The code bits

were already determined at earlier time units. Hence, for theth time unit,

are mapped onto by

for [9].The code is a combination of three codes

and , where

a) is the RM(8,1) code;b) is the RM(8,5) code and

is the RM(8,4) code;

Fig. 7. Coding configuration for BCMIM-2A and BCMIM-2B.

c) is the RM(8,8) code andis the RM(8,7) code.

The generator matrix of has the following form:

(11)

The code is a combination of two codes andwhere is the RM(8,8) code for and. The generator matrix of has the following form:

(12)

The code is the RM(8,8) code. With a proceduresimilar to that for BCMIM-1A, we can show that the MSEDof BCMIM-2A is 8.

We can slightly modify BCMIM-2A into anotherBCMIM (BCMIM-2B) by setting [

. With a procedure similarto that for BCMIM-1B, we can show that the MSED ofBCMIM-2B is eight. For either BCMIM-2A or BCMIM-2B,the coding rate is two bits per 2-D signal point and the ACGover the uncoded QPSK is 6.02 dB. The 16-state decodingtrellis of for BCMIM-2A is shown in Fig. 8(a), andthe eight-state decoding trellis of for BCMIM-2B isshown in Fig. 8(b). The comparison between BCMIM-2Aand BCMIM-2B is similar to that between BCMIM-1A andBCMIM-1B. For BCM-2, ADD and COM . ForBCMIM-2A, ADD and COM . For BCMIM-2B,ADD and COM .

Simulation results are given in Fig. 9. For a BER of 10,BCM-2, BCMIM-2A, and BCMIM-2B have coding gains of3.27 dB, 3.27 dB, and 3.21 dB over uncoded QPSK, respec-tively. For each of BCM-2, BCMIM-2A, and BCMIM-2B,the coding gain for a BER of 10 is much smaller than theassociated ACG due to the huge number of nearest neighbors.In Section IV, we will show a modification of BCMIM-2B forwhich the number of nearest neighbors is reduced.

For either BCMIM-2A or BCMIM-2B, with interblockcoding between levels 4 and 5, we can increase the codingrate by 1/16 bit per 2-D signal point as compared to BCM-


(a)

(b)

Fig. 8. (a) Decoding trellis ofC(1) for BCMIM-2A and (b) decoding trellis ofC(1) for BCMIM-2B.

2. With interblock coding between levels 2 and 3, the codingrate is further increased by 3/16 bit per 2-D signal point. Westill need 1/16 bit per 2-D signal point to have the codingrate equal to uncoded QPSK. This comes from the interblockcoding among levels 1, 2, and 3.

In fact, we can modify either BCMIM-2A or BCMIM-2Bto further increase the coding rate to be 33/16 = 2.0625 bitsper 2-D signal point. In the following, we will only considerthe further modification of BCMIM-2B. This is obtained bycombining levels 1, 2, 3, 4, and 5 as one coding stage. Thegenerator matrix of code for this stage is

(13)

where. is the (8,8,1) code.

It can be shown that the MSED of this two-stage BCMIM(BCMIM-2C) is also eight with a technique similar to thatfor BCMIM-2B. The ACG of BCMIM-2C over uncodedQPSK is 6.15 dB. Simulation results are shown in Fig. 9. Thecomplexity of decoding BCMIM-2C is almost the same asthat of BCMIM-2B.

In the following construction, a BCMIM scheme with blocklength 4 is shown.

Construction 3: Let and beand

binary codes respectively. The set is aBCM (BCM-3) with block length 4. For BCM-3, the MSEDis 8 and the coding rate is bits per 2-D signalpoint.

We have ADD and COM , if six-stage decodingis used.

A BCMIM scheme (BCMIM-3) which has a six-level andtwo-stage coding structure can be designed as follows.


Fig. 9. Simulation results for BCM and BCMIM based on 2�8-PSK.

i) is a (16,10) code with generator matrix

(14)

whereand

.ii) is the (4,4,1) code.

BCMIM-3 has an MSED of 8 and a coding rate of 14/8 =1.75 bits per 2-D signal point. Using a two-stage decoding,we have ADD and COM for BCMIM-3. ForBCM-3 and BCMIM-3, the ACG over uncoded QPSK are 4.39dB and 5.44 dB, respectively. Simulation results are shownin Fig. 9. The coding gains of BCM-3 and BCMIM-3 overuncoded QPSK are 2.90 dB and 3.52 dB, respectively, for aBER of 10 . It is difficult to achieve a high coding rate fora BCMIM with a small block length. However, the decodingcomplexity of a BCMIM with a small block length can bevery low.

B. BCMIM Based on 216-QAM

According to [10], the distance profile for the eight-levelpartition structure of 2 16-QAM is

. With this, we can con-struct efficient BCMIM schemes.

Construction 4: Let andbe RM(8,1), RM(8,4), RM(8,4), RM(8,7), RM(8,7),

RM(8,8), RM(8,8), and RM(8,8) codes, respectively. The

BCM of block length 8 (BCM-4) basedon the 2 16-QAM signal set has an MSED of 3.2 and thecoding rate is bits per 2-D signal point. Wehave ADD and COM , if an eight-stage decodingis used.

We now combine levels 1, 2, 3, 4, and 5 to form a BCMIM(BCMIM-4) which has an eight-level and four-stage codingstructure. The design is as follows:

i) The code has the generator matrix in the form of(13), where:

a) is the RM(8,3) code;

b) is the RM(8,7) code forand ;

c) ;

d) is the RM(8,8) code forand ;

e) and

.

ii) and are all RM(8,8)codes.

BCMIM-4 has an MSED of 3.2 and a coding rate ofbits per 2-D signal point. Using four-stage

decoding for BCMIM-4, we have ADD and COM. The decoding trellis of is shown in Fig. 10. In

Fig. 10, each branch in the partial trellises corresponding tolevels 2 and 3 represents a coset of a (4,1,4) binary code.For BCM-4 and BCMIM-4, the ACG over uncoded 8-AMPMare 5.93 dB and 6.53 dB, respectively. Simulation results areshown in Fig. 5. The coding gains of BCM-4 and BCMIM-4over uncoded 8-AMPM are 3.07 dB and 3.36 dB, respectively,for a BER of 10 .


Fig. 10. Decoding trellis forC(1) of BCMIM-4.

BCMIM-4 is a BCMIM scheme with a high coding rate.In the following construction, a BCMIM scheme with a veryhigh coding gain is given.

Construction 5: Let and beRM(8,0), RM(8,1), RM(8,1), RM(8,4), RM(8,4), RM(8,7),RM(8,7), and RM(8,8) binary codes, respectively. The BCM

of block length 8 (BCM-5) based on the2 16-QAM signal set has an MSED of 6.4 and a coding rateof two bits per 2-D signal point. We can combine levels 2,3, 4, 5, 6, and 7 to form a BCMIM (BCMIM-5) which hasan eight-level and two-stage coding structure. The design isas follows:

i) is a (48,35) binary code with the generator matrix

(15)

where

a) is the RM(8,4) code forand ;

b) is the RM(8,7) code forand ;

c) is the RM(8,8) code forand ;

d)

e)and

.

ii) is the RM(8,8) code.

BCMIM-5 has an MSED of 6.4 and a coding rate ofbits per 2-D signal point. Using a two-stage

decoding for BCMIM-5, we have ADD and COM, where is decoded by using a 16-state trellis.

Simulation results are shown in Fig. 5. The coding gain ofBCMIM-5 over uncoded 8-AMPM is 4.98 dB for a BER of10 .

IV. PERFORMANCE ANALYSIS

The error performance of a coded modulation system ismainly determined by its MSED and the number of nearestneighbors for each code path. Remember that in each of theBCM or BCMIM described in Section II or III, the coding gainfor a BER of 10 is much smaller than the asymptotic codinggain. This is due to the large number of nearest neighborsfor each code path. This phenomenon is a part of coded


modulation systems using multidimensional signal sets andmultistage decoding.

We now consider the signal set of 28-PSK. In the mappingfrom toas given in Section III, the numbers of nearest neighbors to

at the six levels arelisted below.

i) At level 1, the number of neighbors in the form ofat an MSED of 0.586 is

.ii) At level 2, the number of neighbors in the form of

at an MSED of1.172 is .

iii) At level 3, the number of neighbors in the form ofat an

MSED of 2 is .iv) At level 4, the number of neighbors in the form of

atan MSED of 4 is .

v) At level 5, the number of neighbors in the form of

at an MSED of 4 is .vi) At level 6, the number of neighbors in the form of

at an MSED of 8 is .

In general, at level , suppose that the number of binarycode paths at a Hamming distance of from a referencebinary code path is . Then, using multistage decoding, thenumber of code paths at an MSED of from the referencecode path is . For small , usually is small andlarge is required. Hence, in designing a BCM or BCMIMusing a multidimensional signal set, the usage of levelwhen

is small will usually result in degradation of the coding gainat low to moderate signal to noise ratio. Hence, in a design,the usage of level for small should be avoided unless theassociated MSED is very large.

Consider the case of BCMIM-2B. At the first stageof coding, we encode message bits into

using a (24, 9) binary code . Amongthe nine bits of only the encoding ofwill involve the first level of the partition structure of 28-PSK. The MSED between any two code paths with differentvalues of is 8.688. Using the multistage decoding, thenumber of neighbors at an MSED of 8.688 is about 2.Such an enormous number of neighbors will appreciablydeteriorate the error performance. In the following, we showa modification of BCMIM-2B which has fewer neighbors.

In Section III, there is a BCMIM (BCMIM-2C) that ismodified from BCMIM-2B and has a coding rate of 33/16 bitsper 2-D signal point for which the generator matrix for isgiven in (13), where and are zero matrices. We nowset and of (13) to be zero matrices, and let

and be the same as those for BCMIM-2C. In this way, wehave a new BCMIM scheme (BCMIM-2D), for which level 1is not used while the MSED remains eight and the coding rate

is two bits per 2-D signal point. Using two-stage decoding,simulation results are given in Fig. 9. The decoding trellis for

is similar to Fig. 4(b). In fact, the only modification is toreplace the two-state partial trellis of length eight on the left-hand side of Fig. 4(b) by a two-state partial trellis of length16. For BCMIM-2D, we have ADD , COM . Asexpected, the error performance of BCMIM-2D is better thanthat of BCMIM-2B. For a BER of 10 , BCMIM-2D has acoding gain of 3.49 dB over uncoded QPSK.

In Sections II and III, we compared the performances ofeach BCMIM scheme with the associated BCM scheme. Wenow list two trellis-coded modulation (TCM) schemes and aBCM scheme for comparison.

a) TCM-6: This is the 16-state TCM based on 8-PSKproposed by Ungerboeck [1].

b) TCM-7: This is the eight-state TCM based on 16-QAMproposed by Ungerboeck [1].

c) BCM-8: This is a BCM based on 8-PSK,where and are RM(8,1) code, RM(8,7) code,and RM(8,8) code, respectively.

The coding rates of TCM-6, TCM-7, and BCM-8 are two,three, and two bits per 2-D signal point, respectively. TheMSED’s of TCM-6, TCM-7, and BCM-8 are 5.172, 3.2, and4, respectively. For TCM-6, ADD and COM .For TCM-7, ADD and COM . For BCM-8, ADD and COM if three-stage decoding isused; ADD and COM if the optimum (one-stage)decoding is used. Simulation results are given in Figs. 5 and9, respectively. We find that BCMIM-2D has lower BERthan TCM-6 for greater than 6.7 dB, and BCMIM-1B has lower BER than TCM-7 for greater than 8.8dB. BCMIM-2D also requires fewer additions and compar-isons as compared to TCM-6. However, TCM-6 has betterregularity. The comparison between BCMIM-1B and TCM-7 is similar. BCMIM-2D has significantly lower BER andrequires a somewhat higher decoding complexity as comparedto BCM-8.

V. CONCLUDING REMARKS

In this paper, the block length of the BCMIM are allconfined to four or eight 2-dimensional signal points, where

is only one or two. There are two ways to achieve alarger coding rate for BCMIM without sacrificing the MSED.One is to increase the block length. For , thedecoding complexity may still be acceptable. For ,the decoding complexities will usually be much greater. Theother way of increasing MSED is to increase. One problemassociated with the designing of BCMIM with large isthat the number of nearest neighbors will usually be verylarge. At low to moderate signal to noise ratio, this maygreatly reduce the coding gain obtained by increasing MSED.Moreover, for large , the effort of finding the bit metric ineach coding level will sharply increase. At the first codinglevel, the bit metric for is the MSED betweenthe received 2-dimensional signal point and all the 2-dimensional signal points in . The bit metric for

is the MSED between the received 2-dimensional signal


point and all the 2-dimensional signal points in . Atthe th coding level havebeen determined. The bit metric for is the MSEDbetween the received 2-dimensional signal point and all the2 -dimensional signal points in . Thebit metric for is the MSED between the received 2-dimensional signal point and all the 2-dimensional signalpoints in . In this paper, the additions(or comparisons) required for finding the bit metric of eachlevel are taken into account in calculating the number andADD (or COM) for decoding each coded modulation scheme.Since or in this paper, the number of additions(or comparisons) required for finding the bit metrics of eachlevel is not great. However, for large, calculating the bitmetrics will be difficult. The solutions of these problems willbe essential in constructing more efficient BCMIM.

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Mao-Chao Lin (S’86–M’86) was born in Taipei,Taiwan, R.O.C., on December 24, 1954. He receivedthe B.S. and M.S. degrees in electrical engineeringfrom National Taiwan University, Taipei, Taiwan,R.O.C., in 1977 and 1979, respectively. He receivedthe Ph.D. degree in electrical engineering from theUniversity of Hawaii, Manoa, in 1986.

From 1979 to 1982, he was an Assistant Scientistat Chun-Shan Institute of Science and Technology,Lung-Tan, Taiwan, R.O.C. He is currently withthe Department of Electrical Engineering, National

Taiwan University, Taipei, Taiwan, R.O.C., as a Professor. His researchinterests include coding theory and cryptography.

Jia-Yin Wang was born in Tainan, Taiwan, R.O.C.,on December 31, 1968. He received the B.S. degreein electrical engineering from National Taiwan Uni-versity, Taipei, Taiwan, R.O.C., in 1991. He beganhis graduate study at National Taiwan University inelectrical engineering in 1992 and is now a Ph.D.candidate.

His research interests are in the area of codingtheory.

Shang-Chih Ma was born in Keelung, Taiwan,R.O.C., on December 16, 1966. He received the B.S.and Ph.D. degrees in electrical engineering fromNational Taiwan University, Taipei, Taiwan, R.O.C.,in 1988 and 1994, respectively.

He is currently with the Digital Video BroadcastSystem Department, Computer and Communica-tion Research Laboratories, Industrial TechnologyResearch Institute, Hinschu, Taiwan, R.O.C., asa System Engineer. His research interests includecoding theory and digital video technology.

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On block-coded modulation with interblock memory