IEEE TRANSACTIONS ON INFORMATION THEORY. VOL 39. NO 2. MARCH 1993 337

Block-Coded M-PSK Modulation Over GF(M) Magnus Isaksson and Lars H. Zetterberg, Fellow, IEEE

Abstract4hannel codes where the redundancy is obtained not from parity symbols, but from expanding the channel signal-set, are dealt with. They were initially proposed by Ungerboeck using a convolutional code. Here, a block coding approach is given. Rate m/(m + 1) coded 2"'-ary PSK is considered. The expanded signal-set is given the structure of a finite field. The code is defined by a square nonsingular circulant generator matrix over the field. Binary data is mapped on a dataword, of the same length as the codewords, over an additive subgroup of the field. The codes using trellises are described, and then the Viterbi algorithm for decoding is applied. The asymptotic coding gain ranges from 1.8 to 6.0 dB for QPSK going from blocklength 3 to 12. For 8-PSK, the gain is from 0.7 to 3.0 dB with blocklength 4 to 8. With only four states in the trellis, codes of any length for QPSK and 8-PSK are constructed, each having an asymptotic coding gain of 3.0 dB. Simulation results are presented. It is found that the bit-error rate performance at moderate signal-to-noise ratios is sensitive to the number of nearest and next-nearest neighbors.

Index Terms- Block-coded modulation, expanded signal-sets, phase-shift keying, trellis.

I. INTRODUCTION

HANNEL CODING where the redundancy is obtained C not from parity symbols, but from expanding the chan- ne1 signal-set, represents a combined coding and modulation technique. Such a coding technique is suitable for digital transmission over band-limited channels because the data rate and bandwidth are unaffected. It was originally proposed by Ungerboeck [ 11 using a binary convolutional code, and is often called trellis-coded modulation (TCM). The objective for the code design is to make the minimum Euclidean distance larger than for uncoded modulation, and to minimize the number of nearest neighbors.

Block-coded modulation has been studied by several au- thors. Imai and Hirakawa [2] described a multilevel coding method where the signal-set is binary partitioned and a binary code is associated with each partition. Ginzburg [3] has generalized this to permit more elaborate partitions. Cusack (41 presented codes for quadrature amplitude modulation (QAM) using the idea of set partitioning. Sayegh [5] extended Cu- sack's work and presented block codes aimed at both QAM and phase shift keying (PSK). Kasami et al. [lo], [ 111 have

Manuscript received February 4, 1991; rcvised December 30, 1991. This work was presented in part at the 1989 URSl International Symposium o n Signals, Systems and Electronics, Erlangen, W. Germany, Sept., 1989, and at the 1990 IEEE International Symposium on Information Theory, San Diego, CA, January 14-19, 1990.

M. lsaksson was with the Department of Telecommunication Theory, Royal Institute of Technology. 100 44 Stockholm, Sweden. He is now with Ericsson Radio Systems AB, 164 80 Stockholm, Sweden.

L. H. Zettcrberg is with the Department of Telecommunication Thcory, Royal Institute of Technology, 100 44 Stockholm. Sweden.

IEEE Log Number 9203862.

further developed componentwise encoding. Forney et al. [6] dealt with a block coding approach for QAM. In the latter case, dense N-dimensional lattices were constructed using two- dimensional rectangular lattices, Ungerboeck's mapping by set partitioning idea [I], and binary block codes. More recently, Forney [7] has proposed a way of constructing dense lattices, suitable for QAM, by the use of two simple geometrical constructions called the squaring and cubing constructions, respectively. Kschischang et al. [8] adopted an algebraic approach in which the basic modulation signals are associated with the elements of a finite group, and binary block codes are used to give the required redundancy. A different scheme was proposed by Huber [9]. The idea is the contraction of an no bit-sequence to an 710 - 1 signal point sequence, giving a redundancy of I bits that can be used by a Reed-Solomon code.

In this paper, a class of codes with expanded signal-sets for PSK modulation for the additive white Gaussian noise (AWGN) channel is constructed using linear transformations [12]-[ 141. The expanded channel signal-set is given the struc- ture of a finite field. The codes are defined by a square nonsingular circulant generator matrix over the finite field. Binary data is mapped on a dataword, of the same length as the codewords, over an additive subgroup of the finite field. For soft maximum-likelihood (ML) decoding, the codes are described using trellises and the Viterbi algorithm is applied.

The paper is organized as follow. Definition of codes and trellis description are given in Section 11. Section 111 treats different construction of the codes. Decoding and simulation results are dealt with in Section IV. Finally, some comments are made in Section V.

11. FUNDAMENTALS

A. Definition of Codes

In order to transmit m information bits per signaling inter- val, we use rate m / ( m + 1)-coded 21n+1-ary PSK. Compari- son will be made with uncoded modulation using 2m-ary PSK. The objective for the code design is to make the minimum distance in Euclidean space d,,,,, larger than for uncoded modulation due. and to minimize the number of nearest neighbors Ng. Assuming AWGN and soft ML decoding this results in an improved noise immunity, referred to as coding gain. The coding gain is usually measured in dB's and corresponds to the energy savings that can be made at some specific bit-error rate (BER). The asymptotic coding gain, i.e., at large signal-to-noise ratios, is defined as

OOlX-Y448/93$03.00 0 1993 IEEE

338 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

*k -

1 = (01) f

Map Ui ci Map on zi 2"+'-PSK - c = u T on^ -

a = (10) --I- 0 = (00)

t a2 = (11)

Fig. 1 . GF(4) mapped on the signal-set for QPSK.

a = (010)

*1=(001)

as = (011)

a* = (100) + 0 = (ooo)

(16 = (111)

a' = (110) a6 = (101)

I

Fig. 2. GF(8) mapped on the signal-set for 8-PSK.

TABLE 1 FIELDS AND SUBSETS P

Modulation Field Subset P P(.)

To modulator Binary data

Fig. 3. Encoder structure.

The code is defined by an n x n nonsingular generator matrix

t,_l t o . . . t n - 2

t l t 2 .. . t o

The expanded signal-set is given the structure of the finite a a primitive field GF(2"+') = {0,1, a , a 2 , . . . , a

element. The mapping onto the signal-set is such that the

2 7 + - 2 } ,

by interpreting the vector representations of the elements as binary numbers. The signal-points are labeled consecutively with the field elements represented as binary numbers, see Figs. 1 and 2. For 8-PSK, see Fig. 2, the set of four elements rotated ~ / 2 , { 0, a , a2 , a4} , constitutes an additive group. The remaining set of four elements rotated r / 4 relative to the first set, { 1, a3, a6, a5}, is a coset of the additive group in GF(8). In the same way, we see that the set of two elements rotated T , {O,az}, constitutes an additive group, and the remaining pair of elements are cosets of this additive group in GF(8). The same discussion also applies to quaternary phase-shift keying (QPSK).

Let the data to be encoded, corresponding to mn bits, be represented by the vector

and n is the blocklength in PSK symbols. The subset 2) is defined as an additive group with 2" elements,

c = UT.

or equivalent as the cyclic convolution

~ ( z ) = u(z) t ( z ) (mod xn - l),

where t ( z ) = tn-1zn-l + . . . + t lz + t o is the generator polynomial, and c(z) and U(.) are the polynomials corresponding to c and U , respectively. As opposed to ordinary cyclic codes, the generator polynomial t ( z ) is not necessarily a divisor of zn - 1. Let C be the set of codewords. Each codeword is transmitted as a sequence of n PSK-symbols. The encoder structure is shown in Fig. 3.

Inverse encoding (or information retrieval), i.e., to deduce from an estimated codeword which dataword was sent, is necessary since the proposed decoder only gives the estimated codeword and the encoding is nonsystematic. This can be done by matrix operations if and only if T is nonsingular. We then get

V = { ~ = [ b m , b m ~ ~ , ~ ~ ~ , b ~ ] E G F ( ~ ~ + ' ) : b m = O } , U = CT-' for c E C.

where bi E GF(2) is the coefficient of in the polynomial

2) is formed from m data bits, {bm-ll bm-2, . . . , bo} , bi E GF(2), by padding a zero {0,bm-l,bm-2,...,b0} . This is interpreted as an element in GF(2"+l) represented as a vector over GF(2), and n such elements form a dataword U. Table I gives the subsets 2) and the primitive polynomials p ( z ) used to generate the finite fields in this paper.

Also the inverse generator matrix T-l is a circulant matrix.

representation of GF(2m+'). For encoding, a data symbol in Using polynomia1 notation this can be expressed as

U(.) = c(x)t-'(z) (mod xn - l),

assuming that t-'(x), the inverse of t ( z ) , exists. It satisfies

t(z)t-l(z) = 1 (mod zn - 1).

ISAKSSON AND ZETTERBERG: BLOCK-CODED M-PSK MODULATION OVER GF(M 339

B. Algebraic Structure

We have the following properties of the codes. Theorem 1: Let the rate m/(m + 1) code C for 2""l-PSK of length n be generated by the generator polynomial t ( z ) .

a) C is an additive subgroup of GF(2"+l) [x]/(z" - l) , the ring of polynomials over GF(2m+1) with degree less than n.

b) If c ( x ) E C, then xc(z) (mod :xn - 1) E C, i.e., a cyclic shift of a codeword is also a codeword.

Proof: We introduce the notation u(z ) E D[x],,, where D [ z ] , denotes the set of polynomials over D with degree less than n. This set is an additive group, but in general not a ring.

Property a: Let q(z) , c2(x) E C,

c;(.T) = 74(z)t(x) (mod 2:'' - I ) . i = 1, 2

with U,(.) E D[x],. This gives

Q(Z) + c ~ ( . T ) [?L~(z) + Z L ~ ( X ) ] ~ ( X ) (mod T" - 1).

D is defined as an additive group, which implies u l ( z ) + 712(2) E D[z],. Hence, q ( z ) + ~ ( 2 ) E C and C is closed under addition. Since C is a finite subset of the ring, the closure under addition implies that C is a subgroup of the ring under addition.

Property a: Let c(z) E C.

c(z) = u ( z ) t ( x ) (mod xn - l), where U($) E D[x], , .

A cyclic shift of c ( x ) can be written

zc(2) (mod xn - 1) = z[u(z)t(z)(mod T" - I)]

= [zv(z)(mod zn - l ) ] t ( x ) (mod Z" - 1) (mod xn - 1).

xu(z)(mod xn - 1) is a cyclic shift of u ( : E ) , thus m ( x ) (mod xR - 1) E D [ x ] , and zc(z)(mod ,xn - 1) E C. U

It is easily shown by an example that C in general is not an ideal of the ring GF(2m+1)[x]/(z7L - I) , as opposed to ordinary cyclic codes. The code C could be called a nonideal cyclic code. C is not linear in the sense of being a vector space [15], however, the code is linear in the sense of being a group.

C. Code Description by Means of a Trellis

A trellis of length n is a set of nodes located in 71 + 1 planes. Nodes are interconnected from one plane to the next in order 0,1, . . . , n. For trellises considered here, the first and last plane contain only one node each, the zero node.

A trellis defines a code when each connection is labelled with a symbol c from the signal set GF(2'"+l). A code word is describes by a path through the trellis.

The construction of a trellis for a code C is similar to the method used by Wolf [16] for block codes. His con- struction is based on the parity relation z H T = 0, with H being the parity-check matrix and x a codeword. We will

introduce a function f~ to serve a similar purpose. With y = [yo. y l . . . . . yIL-l], y, E GF(2""'). let

f D ( Y ) = [ f D ( Y O ) . fD(Y1). . ' ' * fD(Vn-l)l,

where

With vector representation of a field element /3 = [b,. b,, . . . . bo] , we have

Recall that U = cT-', hence cT-' E V n . if and only if c is a codeword. This can be expressed as

f&+) = 0. (1 1 where 0 is the all-zero vector. This is the parity-check relation needed to construct a trellis.

The function f L, can easily be shown to be linear over GF(2), and we obtain

where tz-' is the zth row of T I . The first sum is taken componentwise over GF (2n'+1). while the second one is taken over GF(2).

Each node or state in the trellis is associated with a vector or label q J ( k ) , which is an n-tuple with elements in GF(2). To indicate node planes (depths) we use the variable k and enumerate nodes with j belonging to a set d k . The set d k may be different for different node planes.

For each ccleword c E C. a distinct path through the trellis will be defined by the sequence of transition labels c g , c1. . . . . r7,- 1. The first plane only contains the zero node, i.e., qo(0) = 0. For each codeword, e, the transition between node planes at depth k and k + 1 is defined by the relation

q , ( k + 1 ) = ~ ~ ( k ) + f p ( r . h t ~ ~ ) . f o r k = 0 , 1 :... n - I .

(3)

where the transition between the states q , ( k ) and q J ( k + 1) is labeled by the value of ch. From (1) and (2) , it follows that

7 1 - 1

(4) k=o

Thus, at the last plane, the trellis contains only qo(n,). the zero node. To reach the final node according to (4), the trellis must contain a tail as in the case with convolutional codes and the codes considered by Wolf [ 161. It requires an expurgation of possible transitions.

The number of nodes or states at any plane is at most 2" but is in most cases much fewer. This will be shown in the following two examples. Both codes have a trellis with only

340 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

Fig. 4. Trellis for rate 1/2 coded QPSK of length 3 with an asymptotic coding gain of 1.76 dB over uncoded BPSK.

four states. Note however that each state may have different labels at different node planes.

Example 1: As a simple example, we construct the trellis for a rate 112 QPSK code of length 3 with asymptotic coding gain 1.76 dB over uncoded binary phase-shift keying (BPSK) given by

T = circ(a2 11).

The inverse generator matrix is

T-' = circ(a 1 1 ) .

Hence, we have

ti' = [a 111, t l ' = [l a 11, and tgl = [l 1 a].

The codewords are

[000] , [a211], [ 1 a 2 1 ] , [aaO], [ 1 1 a 2 ] , [aOa], [Oaa], [a2a2a2].

Following the procedure outlined above, we find the trellis illustrated in Fig. 4. The number at each node is the decimal notation j of the binary n-tuples q j ( k ) .

Example 2: As another example, the trellis for a rate 213 8-PSK code of length 7 with asymptotic coding gain 3.01 dB over uncoded QPSK, given by

T = circ(a4a4a2a2a2a2a2),

and the inverse generator matrix

T-' = circ(aa5 am5 aa5 a ) ,

is shown in Fig. 5. This trellis contains parallel transitions and will be discussed in detail subsequently.

The structure of these codes are similar to those considered by Forney [7]. We may identify the subgroup and cosets used to describe transitions. Note in Example 1 that ( 0 , ~ ) is an additive subgroup of GF(4) with {1,a2} being its coset. Similarly (0, a , a2, a4} is a subgroup of GF(8) and (1, a3, a5, a 6 } its coset. As a result, transitions taken from the subgroup will generate one set of node labels, while transitions from the coset will generate a different set of node labels. This result holds in general.

A = { O , a z } , B = {a,.'), C = {l,aa}, D = (a3,a6}

Fig. 5 Trellis for rate 2/3 coded 8-PSK of length 7 with an asymptotic coding gain of 3.01 dB over uncoded QPSK.

D. Parallel Transitions in a Trellis

Transitions between states are defined in (3) and parallel transitions implies that there are two elements /'?I and 102 in GF(2"+l) for which

fD(P1ti') = f D ( P 2 t 3 P1 # 102.

We find

fD[(Pl - P 2 ) t L 1 ] = 0.

Let p = ,& - ,& and observe that it belongs to GF(2"+'). It is easily seen that elements /3 for which fD (@ti ' ) = 0 form a subgroup GO of the additive group of G F (2mf'). For elements y belonging to cosets of GO it is true that f .~(rt i ' ) # 0 and furthermore all elements in a specific coset will give parallel transitions. To see this, let y1 and 72 belong to a coset of Go, then

YI = YO + 71 and 7 2 = YO + E ? ,

with 70 the coset leader and TI, 7 2 belonging to 00. We have

Hence, y1 and 7 2 give parallel transitions. So far we have considered a fixed index I C , but it is easy to

see that parallel transitions will either occur for all transitions or for none. This result follows because t i1 is a cyclic shift of t i l . Results will be summarized as follows.

Theorem 2: Parallel transitions occur for code elements 0 belonging either to the additive subgroup of GF(2"+') or to one of its cosets. GO is defined by fD(Ptil) = 0 for any

As an application consider the 8-PSK code in Example 2. is found to be {0 ,a2} with cosets 1 + Go = {l,a6}, a +

GO = { a , a4}, and a3 + GO = { a3, a s } . It can be seen from Fig. 5 that this holds at all depths of the trellis.

k = 0, l ; . . . , 71. - 1.

ISAKSSON AND ZETTERBERG: BLOCK-CODED M-PSK MODULATION OVER GF(M) 341

TABLE I1 RESULTS OF COMPUTER SEARCH FOR RATE 1 /2 CODED QPSK

3 1.76 4 3.01 5 3.01 6 3.01

7 3.01 8 3.98 9 4.77

10 4.77 11 5.44

12 6.02

Sayegh (%in -YE T = circ( . ) G[dBI

6.0 4 <,z 11 1.8 8.0 14 0 1 1 0 3.0 8.0 10 t r n l 0 0 3.0

8.0 6 n o ' 1 1 0 0 3.0

8.0 7 <> C l 1 0 0 0 0 3.0 10.0 24 o o 0 1 0 0 0 0 3.0

12.0 102 n r , ~ o 1 0 0 0 0 0 3.0 12.0 40 < l < I 1 1 0 1 0 0 0 0 3.0

14.0 176 ( l ( I O 2 1 0 1 0 0 0 0 0 4.0

16.0 759 ( I ( r 2 (1 n2 (1'0 1 0 0 0 0 0 4.8

TABLE I11 RESULTS OF COMPUTFR SEARCH FOR RATE 2 / 3 CODED 8-PSK

4 0.69 2.34 1.0 < , I 0 3 0 0

h 2.45 3.51 16.0 ,I ' 0 . 3 O r 1 0 2 1

5 1.66 2.93 16.0 o1 o1 (12 r)' or

7 3.01 4.00 91.0 0' o1 o 2 o2 o L o2 r?

8 3.01 4.00 24.0 1 1 (1 0 1 1 0 0

8* 3.01 4.00 120.0 0 . 3 11 11 11 1

Hamming Space Construction by Kschischang et al. [8] except for blocklengths 7 and 8. From Kasami et al. [IO], only two codes are directly comparable. The QPSK code of length 16 achieves an asymptotic coding gain of 6.02 dB, which is the same as for our code of length 12. The 8-PSK code of length 8 by Kasami et al. performs the same as our code of the same length, and both can be described by 4-state trellises. In fact, the codes are identical. In [ 111 Kasami et al. propose a method to reduce the number of nearest neighbors.

Tables IV and V give results on complexity of the codes in terms of number of states at different node planes and also the average number of transitions per symbol. The best QPSK code of length 3 can be implemented with the trellis

*Constructed in the following subsection.

111. CONSTRUCTION OF CODES in Example 1 and likewise, the best 8-PSK code of length 7 with the trellis in Example 2.

B. A Class of Codes with 4-State Trellises A. Codes Found by Computer Search

A n exhaustive search has been made to find good and rather short codes with 4- and 8-phase values. The criteria have been to first maximize the minimum distance dmir,, then within the class of codes having this dmin minimize the number of nearest neighbors N B .

8-PSK codes are in general not superlinear as defined by Benedetto et al. [17]. Consequently it is necessary to examine all pair of codewords to find &in. This can be overcome partially by using an effective lower bound on dmin [ 171. Fortunately, QPSK codes are superlinear which greatly simplified the search.

Table I1 contains results for QPSK codes of length up to 12 with coding gains over uncoded BPSK. Similarly, Table I11 contains results for 8-PSK where comparison is made with uncoded QPSK. The asymptotic coding gain is between 1.76 and 6.02 dB in the former case and between 0.69 and 3.01 dB in the latter case.

Comparisons can be made with some of the other previously mentioned approaches to block coded modulation. Regarding Sayegh [5], [IS] our QPSK codes perform better for 71, > 7, see Table 11, whereas the 8-PSK codes have identical d,,,;,, but our codes have generally a smaller number of nearest neighbors. Our QPSK codes are equally good as the superior

Examples 1 and 2 gave codes with 4-state trellises for 4- and 8-phase modulation. It is possible to generalize these codes to include general 2m+1-PSK and arbitrary length n of code- words, These codes are shown to give a modest 3-dB asymp- totic coding gain.

We start by partitioning the set of 2m-t1 phase value into four subsets So. SI, Sz and S3. Let 6 be the additive group of GF(2"+').

1) So is an additive subgroup of G with SI, S2 and s3 2) So U S2 is an additive subgroup of G with S1 U S3 being

Example 3: 8-PSK will give So = { 0 , a 2 } , SI = {1>a6}, S2 = {@.a4}. and S3 = {(u~~N'} with the representation in Fig. 2.

We next postulate the trellis structure of Fig. 6 with transi- tion sets as shown. The trellis contains two subtrellises each with a butterfly structure. As can be seen there are two ways of terminating the lower subtrellis. Note that if m > 1, parallel transitions must occur.

Let q z ( k ) . q 1 + 2 ( k ) . S,. and S,+2 represent the upper subtrellis for i = 0, and the lower subtrellis for z = 1. TO

being its cosets.

its coset.

342 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

TABLE IV COMPLEXITY OF TRELLISES FOR U T E 1/2 CODED QPSK

GQPSK fBPSK Transitions n [dBl N B per Symbol States

3 4 5 6 7 8 9 10 11 12

1.76 3.01 3.01 3.01 3.01 3.98 4.71 4.17 5.44 6.02

4 14 10 6 7

24 102 40

116 159

5.33 10.00 14.40 28.00 19.43 53.00 61.33

272.80 434.18 910.00

1-4-4-1 1-4-16-4-1 1-4-16-1 6-4-1 1-4-16-64-16-4-1 1-4-16-16-16-16-4-1 1-4- 16-64-64-64-16-4-1 1-4-16-64-64-64-64-1 6-4-1 1-4-16-64-256-1024-256-64-16-4-1 1-4- 16-64-256- 1024-1024-256-64-16-4-1 1-4- 16-64-256- 1024-4096- 1024-256-64-16-4-1

TABLE V COMPLEXITY OF TRELLISES FOR RATE 2/3 CODED 8-PSK

GBPSK fQPSK Transitions n [dBl Ns per symbol States

4 0.69 1 .o 12.00 1-4-4-4-1 SOP) Sa

5 1.66 16.0 12.80 1-4-4-4-4-1 6 2.45 16.0 34.67 1-4-16-16-16-4-1 I 3.01 91.0 13.71 1-4-4-4-4-4-4-1 8 3.01 24.0 106.00 1-4-16-64-64-64-16-4-1 Sl

8* 3.01 120.0 14.00 1-4-4-4-4-4-4-4-1

... S3

sa ss

sdl) sdz) qdn-2) qdn- 1)

Fig. 6. Trellis structure for a code of arbitrary length with four states. *Constructed in the following subsection.

is a coset in GF(2)n. The elements in the group are given by ( 5 ) and (6).

As the last condition it is required that the states at depths 1 through n - 1 are distinct, i.e.,

qo( k ) , q1 ( k ) , q 2 ( k ) , and q3( k ) all unequal

obtain the trellis structure in Fig. 6, two conditions have to be satisfied:

1) each subtrellis shall have the butterfly structure; 2) at depths 1 through n- 1, the four states must be distinct.

Lemma 3: The butterfly is obtained if The first condition on T-l is given in the following lemma.

for k = 1, 2 , . . . , n - 1. (7)

i = 0, 1, It can be shown by an example that there are matrices T-' with Pi E S; and E Si+2.

resulting in nondistinct states. We have the following result

with k = 1, 2 , , . . . , n - 1, if Lemma 4: The trellis will have distinct states at depth k, Proofi See Appendix A.

It is easily deduced that SO is identical to the subgroup SO defined in Theorem 2, hence. k-1

From this relation it follows that for m > 1 only a subset of GF(2"+') can be used in T - l .

A' property of the code construction and the subset parti- proof: See Appendix A. 0 We also give a lemma that states how the lower subtrellis

tioning {S i } is that the set of state-changes in the upper trellis between states at depth k and k + 1

is terminated. {fD(Ptkl) : P E so U sa}

is an additive group, and the corresponding state-changes in the lower subtrellis

Lemma 5: Let 5 be the sum of the components of t i l . For the lower subtrellis ql(n - 1) is followed by a transition from set s1 if fD(pl5) = 0, p1 E sl, while it is followed by a transition from S3 if f~(P15) = 1.

0 {fD(Ptkl) : P E s1 us,} Proof: See Appendix A.

ISAKSSON AND ZETTERRERG: BLOCK-CODED M-PSK MODULATION OVER GF(M) 343

C. Existence of Codes with State Trellises and Their Minimum Distances

To prove the existence of 4-state codes we must find generator matrices T for which the conditions in Lemmas 3 and 4 are satisfied. Already some codes are known from Tables I1 and 111.

QPSK: The transition sets are So = (0) . SI = {I}. s~ = { a } , and 5 3 { a 2 } . Hence, there will be no parallel transitions. We propose

T-’ = circ.(a I 1.. . I )

It is easy to verify that the required conditions are satisfied for any length 7~. According to Lemma 5 , the state q l ( n - 1) is followed by a transition from set $3. There are many other matrices that also satisfy the conditions.

8-PSK: The transition sets have already been defined in Example 3. We propose

It is easy to verify also now that the two conditions are satisfied. For even lengths 7 i , q1(71 - I ) is followed by a transition from set S3, and for odd lengths, SI. Also here, there are many other matrices that satisfy the conditions.

The infinite family of 4-state codes can also be obtained from a multilevel code construction as in [lo] with the component codes (n , 1, n) repetition code, (71,rr - 1.2) parity check code, and (n, n. 1) uncoded.

Minimum Distances: In the trellis three different types of distances must be compared to find the minimum distance, namely,

a) parallel transitions, d:, b) within subtrellis, d i , c) between subtrellises, d:.

We have

Let M = 2m+’ be the number of phase values; then with signal amplitude normalized to one,

d: = n . 2 ( 1 -cos;).

For QPSK ( M = 4) there are no parallel transitions, thus d: = CO. The squared minimum distances for QPSK, 8-PSK, and 16-PSK are shown in Table VI. The asymptotic coding gain G has been computed according to Section I1 with

d i , 2 1 - C O S - ( 3 The coding gain in all cases is 3.01 dB if the code length is greater than or equal to a certain minimum length, which is also indicated in the table.

TABLE VI M I N I M U M D l S l A N C t 5 t O K CODES WITH FOLJR STATES

4 8 3.01 4

8 4 3.01 7

1 0 2 [ 2 - Jz) 3.01 X

With a 4-state trellis i t is not possible to obtain more than a 3.01 dB gain. With more states we may achieve better results as shown in Table 11.

D. Relation between QPSK Codes and Binary Block Codes

It is instructive to compare our QPSK codes with binary block codes, see also [ 8 ] . With Gray coding of phase values there will be a simple relation between the Euclidean distance d ~ ( c ~ . c l ) between QPSK codewords c1 and c2 of length 71,

and codewords bl and 6 2 from a binary block code of length 277, namely,

t d ; ( c l , ~ 2 ) = 2dH(b l .b2 ) , (9)

where d~ is the Hamming distance, and QPSK signals have amplitude one. Thus, the corresponding minimum distances are related as

(1;. I l l i l l = 2dH. r r i i r i ’ (1 0)

If our codes are compared with the best known linear block codes of equal length it is found that they have the same performance in terms of &+irl 1211.

Example 4: It is known that the extended Golay (24,12) code is the best linear block code of length 24 with d H , m i n = 8. The QPSK code of length 12 in Table I1 has d$,min = 16 which is consistent with (10). Indeed it has been found that the distance distribution of the extended Golay code equals that for the QPSK code when they are compared properly.

For coded QPSK, binary block codes and Gray mapping out- performs multilevel block modulation codes, see [8, Fig. 111 and Table I1 in this paper. However, the opposite seems to be the case for convolutional codes according to Woerz and Hagenauer [22].

Iv . DECODING AND SIMULATION RESULTS

A. Channel Model

We assume that the codewords are transmitted on a channel where the only impairment is AWGN with single-sided power spectral density No. Intersymbol interference-free signaling and perfect carrier phase tracking at the receiver are also assumed. The output of the channel becomes

T, = T , + 111,.

where rz is the transmitted complex-valued discrete channel signal transmitted at modulation time iT, and wz is an in- dependent normally distributed noise sample with zero-mean

344 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

BER and variance o2 = N0/2 along each dimension. The symbol interval is 2'.

In the simulation results, BER is plotted versus the param- eter &/No, where Ea is the average energy per information bit. We have

lo-l

Eb - 1 Es 1 Es - - - NO m No m 202'

where m is the number of information bits per transmitted symbol, and E, the average signal energy.

B. Maximum-Likelihood Decoding The trellis description of our codes makes it possible to

use the Viterbi algorithm for decoding [16], [19]. The squared

10-4

+ : n = 8, G = 3.98 dB, NB = 24 : n = 9, G = 4.77 dB, NB = 102

1 0 - 6 Euclidean distance is used as the metric. Let 0 2 4 6 8 10

Eb/NO [dBl

be the complex sequence of transmitted PSK signals with amplitude one, and let r be the received complex sequence.

We want to use soft maximum-likelihood decoding, which implies that r should be the sequence of unquantized demod- ulator outputs. The likelihood function equals the conditional probability density p ( r I c) . For the AWGN channel, maximiz- ing p(r I c) is equivalent to minimizing the squared Euclidean distance (r - ~ ( c ) 1 2 . The path metric can hence be defined as

n-1 2

~ ( c ) = Ir; - .(.;)I . i=O

Since this is an additive measure it is straightforward to use the Viterbi algorithm with (ri - z(ci)I2 as branch metric. The Viterbi algorithm finds the most likely codeword c, and the corresponding dataword U is found from

U = cT-'.

C. Simulation Results

The asymptotic coding gain G is only achieveable at high &/No. In this section we investigate the performance of coded QPSK and 8-PSK at low and moderate &/No. Simulation results will be presented for a few codes.

Fig. 7 shows BER versus Eb/No for coded QPSK for lengths 4, 8, and 9 and with the asymptotic coding gains 3.01, 3.98, and 4.77 dB, respectively. The codes are specified in Table 11. Also shown is uncoded BPSK. The coding gains achieved at moderated &/NO are smaller than the asymptotic values. At a BER of lo-', the gains are 2.2, 3.0, and 3.3 dB, respectively.

Fig. 8 shows BER versus &/No for coded 8-PSK for lengths 7 and 8 and with an asymptotic coding gain of 3.01 dB. The codes are specified in Table 111. Also shown is uncoded QPSK. At a BER of lo-', the gains are only 1.4-1.7 dB. This is caused by the large number of nearest and next-nearest neighbors. The trellises for the two codes of length 8 differ a great deal in complexity. The code with 91 nearest neighbors performs worse than the code with 120 nearest neighbors due

Fig. 7. Bit-error rate of coded QPSK of lengths 4, 8, and 9 and un- coded BPSK.

10-1

10-2

10-6

Fig. 8

BER

A : n = 7, G = 3.01 dB, NS = 91.0 + : n = 8, G = 3.01 dB, N B = 120.0

: n = 8, G = 3.01 dB, NB = 24.0

i t \% '

I I I I 1 0 2 4 6 8 10

Bit-error rate of coded 8-PSK of lengths 7 and 8 with 3.01 dB

Eb/No IdBJ

asymptotic coding gain and uncoded QPSK.

to the next-nearest neighbors being very close to the nearest neighbors.

In [20], the following rule of thumb was given for trellis coded modulation. In the first approximation for error rates around the effective coding gain is reduced by 0.2 dB for every increase in the number of nearest neighbors by a factor of 2. This rule gives estimates of the effective coding gain which are 0-0.3 dB larger than the simulated values. For our codes, we find the corresponding reduction factor to be 0.20-0.28 dB.

Performance at low Eb/No is strongly influenced by the number of nearest neighbors. According to Fomey et al. [6], the number of nearest neighbors is generally significantly larger for block codes than for trellis codes with comparable asymptotic coding gain. Hence at low &/No, Ungerboeck's trellis coded modulation is expected in general to perform better than the proposed codes. As an example, from [l, Fig. 161, we get that trellis coded 8-PSK with four states

ISAKSSON AND ZETTFRRFRG BLOCK-CODED M PSK MODULATION OVER GF(M1 345

and an asymptotic coding gain of 3.01 dB achieves a coding gain of 1.4 dB at a BER of IO-'. The coding gain of the corresponding code in Fig. 8 of length 8 with 4 states and N B = 120.0 is only 0.6 dB.

V. CONCLUSION

All codes considered in this paper are defined by circulant matrices T of order 71 by over GF2("'+'). Encoding is done by multiplying T by a data vector of length 71. These codes are mapped onto 2"'+' phase modulated signals. The trellis description will ensure ML-decoding by means of the Viterbi algorithm.

Performance has been evaluated for QPSK and 8-PSK. The best QPSK codes found by computer search require a large number of states in the trellis. By construction, classes of codes are found with only four trellis states and with an asymptotic coding gain of 3.0 dB.

This coding gain is the same as that of the best codes for modest blocklengths but 4-state codes have more nearest neighbors than the best codes. It is likely that constructive classes of codes with better performance for longer block- lengths can be found by increasing the number of states.

For 8-PSK and blocklengths up to 8 no code has been found with a coding gain better than 3.0 dB. This is shown to be the case also for the constructive class of codes with 4- state trellises. In this case, parallel transitions will occur that limits the coding gain to 3.0 dB. To avoid this and improve performance, we must increase the number of states to 16 or more.

A comparison has been made with the multilevel block modulation codes constructed by Sayegh (51 and others. The idea is to use two o r more binary block codes to separately encode different components in the binary representation of phase values. As mentioned, our codes perform better or at least as well as those. A reasonable explanation is that in the multilevel codes components are encoded independently while in our case the encoding is highly dependent. This last statement is reinforced by the results reported by Kasami et al. [ 1 I]. To decrease the number of nearest neighbors they encode componentwise with block codes but the codes are related and hence encoding of components are dependent.

APPENDIX A

Proof of Lenimu 3: The butterfly structure will require that for = 0 . 1 ant1 A, = 1. 2. . . . . r / - 2

and

and

!I, ( k ) + q,+,(k) = fD (at,') + fD ( i M 2 + 2 t i 1 ) . (14)

By iterating (13), we find, for k = 1. 2.. . . , 71 - 1,

q , ( k ) + Q l + 2 ( X . ) = + Q 1 + 2 ( 1 )

= f D ( B 7 t o ' ) + fD(Pz+2ti1). (15)

since by definition qo(0) = 0. Combining (14) and (15) will give, for k = 0. 1 . . . . . r ) - 1.

fr, [(d, + P + * ) q ' ] = fD [(a, + P 2 + 2 ) t i 1 ] . (16)

Now, recall that t k l is the kth row of T-' and hence t i 1 cyclically shifted k steps. Similarly fp (pi!,') is f~ (Ot , ' ) cyclically shifted k times. There are only two n-tuples that will satisfy (16) for all k = 0, 1, . . . . 71 - 1, namely, [O 0 . . . 0] and [1 1 . . . 11. The first one must be excluded since from (15) we get q ! ( 1 ) = q l+*( l ) that implies parallel transitions for SI U SI+?. 0

Proof of Lemma 4: For the upper trellis we have, for k = 1. 2:... I ) - 1.

q0(I ; ) = [OO... 01 and ~ ~ ( k ) = [ l l . . . 11. (17)

which follows from (5), (6) , (1 I ) , and (15). For the lower subtrellis,

q 1 ( k ) + q , ( k ) = [11. . .1] . (18)

which is deduced from (5) and (15).

subtrellis such that, for I; = 1. 2 . . . . . n - 1. From Fig. 6, it follows that there is a path through the lower

A-1

,/ = 0

With (8) satisfied, q l ( k : ) # q o ( k ) ant1 q 2 ( k ) , and the same holds for q 3 ( k ) due to (18). In view of (18), q l ( k ) and q 3 ( k ) are different and all four states are unequal at all depths k = 1. 2 . . ' ' . / I - 1. 0

Proof ofLemmu 5: From (4), we know that q1 (n ) = q3(n) = 0. For the lower subtrellis by definition and Fig. 6,

q1 ( ' / I , ) = q1 (7, - 1) + f73 (ljltit1)

ql('/?,) = Q1( ' / " - 1 ) + fD(Lj&').

(20)

(21)

or

First, consider the case defined by (20) then a particular path through the subtrellis will give

I 1 - 1

Since, t;' is a cyclic shift of to ' , we have ,,-1

k=O

where < is the sum of the components of t i ' .

346 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993

Equation (22) can now be expressed as

q,(n,) = [fD(P11), fD(PlE1,. ’ . 1 fD(P101. (24)

Hence ql(n) = 0 will require f ~ ( P 1 t ) = 0 with

through the subtrellis will give

E 5’1.

Next, consider the case defined by (21). A particular path

n-2

cll(n) = fD(Pl t2) + f D ( P 3 t 2 1 ) k=O n-1

= fD(Plt,l) + [ 1 1 . . . 11, (25) k=O

where in the last equality relation (5) and (16) has been used. Using (22) and (23),

Ql(n,) = 11 + fD(PlE),l + fD(PlE), ‘ . . > 1 + fD(P11)I. (26)

P1 E s1. 0 Hence, ql(n) = 0 in this case will require f ~ ( P 1 t ) = 1 with

ACKNOWLEDGMENT

The authors are most grateful to Prof. T. Ericsson for comments he made as faculty opponent of the dissertation by M. Isaksson.

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