Further Results on Space-Time Coding for Rayleigh Fading - CiteSeerX

Further Results on Space-Time Coding for Rayleigh Fading J. Grimm and M. P. Fitz School of Electrical Engineering, Purdue University, W. Lafayette, IN and School of Electrical Engineering, Ohio State University, Columbus, OH Abstract– transmitter diversity I just need to put these keywords into a coherent statement space-time codes geometric uniformity uniform distance spectrum code design design metrics effective product distance effective Hamming distance rank and determinant criteria delay diversity I. INTRODUCTION Trellis coded modulation (TCM) was introduced by Ungerboeck [1] as a form of low complexity, bandwidth efficient coding for AWGN channels. The performance of TCM on fading channels was first analyzed by Divsalar and Simon [2], and using the Chernoff bound they proved that for fading channels it is more important to optimize the Hamming and product distances than Euclidean distance. Numerous researchers have designed many trellis codes for independently fading channels [3]. Traditional code design for fading channels assumes interleaving can make the fading independent. However, if the mobile unit is not moving, infinite interleaving depth would be required. Transmitter diversity [4], [7], [5] can be used to achieve any desired diversity level without interleaving, even if the mobile is not moving. A powerful TCM scheme for transmitter diversity called space-time codes was proposed in [5] for two levels of diversity. Guey et al. [6], [7] derived the exact pairwise error performance for transmitter diversity, and were the first to prove that the diversity level is equal to the rank of the codeword difference matrices. They also pointed out that the squares of the singular values of the codeword difference matrices should be maximized. The authors in [5] derived a determinant criterion. In [8] it is shown how the rank and determinant criteria are related to the Hamming and product distances from [2]. To reduce encoding and decoding complexity, [9] and [10] use IQ TCM [11], where the inphase and quadrature components of the signal are encoded and decoded separately. The same Hamming distance as traditional TCM is achieved using the square root of the number of states and the square root of the number of branches from each state. The results in this work apply both to traditional and IQ TCM. The main thrust of this work is to extend space-time codes [5] to any level of diversity. It is also shown that under certain restrictions, geometrically uniform space-time codes can only achieve one level of diversity, so designing space-time codes with a uniform distance spectrum is difficult. Delay diversity [18], [5] is generalized to any throughput and diversity level, and it 1

is proven that delay diversity uses the minimal number of trellis states. Code search is used to find higher complexity codes with better distance properties. II. PROBLEM FORMULATION A. Signal Model For simplicity, the code design will be confined to non-time varying channels, since this is the worst case scenario. This means that the fading is 100% correlated in time, so if the mobile is stopped in a deep fade interleaving won't help. If the mobile unit is moving and time diversity is available, the codes designed here will have at least as good performance as in the non-time varying case, and most likely better performance on average. The design considers antenna (spatial) diversity and time diversity, hence the name space-time code. The system has Lt transmitter antennas and Lr receiver antennas, and it is assumed that the fading across the antennas is uncorrelated. Since the receiver antennas are uncorrelated, the total diversity level achieved will be Lr times the diversity of the rest of the system. This allows the code design to focus only on the transmitter antenna and time diversity channels, thus the received signal x(n) may be modeled as Lt

x(n) EsM d(n, l)c(l)  n(n), n 1:Nc l 1

(1)

where Es is the total received signal energy, and the transmitted symbol d(n,l) is normalized to unit average energy per information bit. The fading channel is c(l) and the AWGN is n(n). Equation (1) can be written more compactly in matrix notation: x3 Es Dc3 n3 (2) where x3 is an Nc×1 vector with elements x(n) , D is an Nc×L t matrix with elements d(n,l) , c3 is Lt×1 and n3 is Nc×1 . To summarize from [5], [8] the two most important design criteria for transmitter diversity systems are 1) effective Hamming distance (eH) 2) effective product distance (eP) In this work the fading is assumed to be mutually uncorrelated across the antennas (Cc=I), so these are defined as eP(D) ¹ N n(C1) (3) n

H

where D D D, D and D are codewords, C1 DD and  is the set of all nonzero eigenvalues. The effective Hamming distance eH is the rank of C1 (which is equivalent to the rank of D ). The terms eH and rank are used interchangeably. Note that when a codeword is H full rank, eP(D) det(C1) . Since the nonzero eigenvalues of C1 DD are the same as the H nonzero eigenvalues of C2 DD , we can also use eP(D) N n(C2) . n

The metrics eH and eP operate on the eigenvalues of C1 in the same way the Hamming and product distances [2] operate on codeword differences directly. I need a really convincing sentence here to explain the word "effective". In [5], eH is called the rank criterion and eP is called the determinant criterion. However, to emphasize the relation to the original metrics in [2], we use eH and eP. The term "product distance" is somewhat of a misnomer that Divsalar and Simon [2] actually avoided using but has been in common use ever since. Neither the product distance nor eP satisfy 2

the triangle inequality, which is one of the three axioms of a norm. Consider A=B=1; then eP(A)=eP(B)=1, but eP(A+B)=4, so eP(A+B)> eP(A)+ eP(B), and the triangle inequality is violated! Therefore great caution should be exercised when using eP as a distance, because the desired properties may not hold. However, eP is intimately related to the BEP (as Euclidean distance is for AWGN channels), and it is convenient to use it as a "distance" when possible. This work assumes Nc  L t , so the rank is bounded by eH  L t . A code is said to achieve full diversity if all codeword differences are full rank, with eH=Lt. Then the diversity level achieved is LD=LtLr, the maximum diversity that can be achieved if the mobile unit is not in motion. (If time diversity were available and Nc > L t , LD=NcLr diversity levels could be achieved.) The worst case eP out of all codeword differences of a code is denoted ePmin . Likewise, the minimum eH among all codeword pairs is denoted eHmin . The goal of code design in this paper is to find codes with the best ePmin out of all codes with eHmin Lt . B. Terminology A TCM encoder for space-time coding is shown in Fig. 1. At each symbol time n, k information bits are fed into the convolutional encoder. The encoder has v memory elements, hence it has V=2v states. Every symbol time the encoder outputs m coded bits which are mapped to a signal constellation (see Fig. 4) with M=2m possible symbols. When IQ TCM is used, there are m bits in the I channel and m bits in Q, so the constellation has M2 symbols. In each symbol time, the output symbol vector d3(n, :) [d(n,1) d(n,2) à d(n,Lt)] has Lt symbols for the Lt transmitter antennas. Notice that all input bits are coded, because no diversity can be achieved with parallel transitions unless multiple TCM (MTCM) [12] is employed, and MTCM codes are not considered here. When MTCM is not used, the maximum attainable diversity level LD is bounded by LD  )v/k/ 1 [13], so full diversity codes require )v/k/  Lt 1 .

Fig. 1. TCM encoder block diagram.

Fig. 2. Signal constellations used in this work. The PAM constellations can be used either for PAM or IQ-QAM. A typical trellis is shown in Fig. 2. This is the Lt=2, k=2, v=3, m=2 TCM from [5]. States of the trellis are a function of the v previous input bits. The trellis is a graph that shows which symbols the convolutional encoder is allowed to transmit given the current state, and also which future state the encoder will enter when a particular symbol is transmitted. There are V 2v states and K=2k branches from each state. Each branch has Lt symbols, one for each transmitter 3

antenna. From each state, the uppermost branches correspond to input bits equal to zero, and the lower branches correspond to input bits equal to one.

Fig. 3. AT&T's 8 state, 4-PSK, Lt=2 TCM. The all-zeros path is the path taken when the transmitted information bits are all zero. For most useful trellises, when the encoder is in state zero and the information bits are zero, the encoder remains in state zero. A TCM codeword is the set of symbols along a given path through the trellis. The all-zeros codeword is the set of transmitted symbols along the all-zeros path. An error event path (EEP) is a possible received path through the trellis that begins in the same state as the transmitted path and ends in the same state as the transmitted path, but diverges in between. Since the optimal decoder must only choose paths that follow the trellis, the only way it can make an error is by following an EEP. A codeword difference is the transmitted codeword minus the codeword along an EEP. In a typical communications network data is transmitted in frames with a fixed number of symbols. The codeword length Nc could be defined to be this frame length. Alternatively, the codeword length could be defined as Nc=Le, the length of an error event path. The analysis in this work applies to either definition. The distance spectrum (DS) is an enumeration of the distance of all EEPs from a transmitted codeword. The DS is measured with respect to (wrt) a specified metric, such as Euclidean distance or the effective product distance. For the purpose of code analysis, it is desirable to use codes where the DS is the same regardless of the transmitted codeword. This will be called a uniform distance spectrum (UDS). A catastrophic code has EEPs where a finite number of symbol decoding errors result in an infinite number of bit errors. Necessary and sufficient conditions for obtaining noncatastrophic feedforward convolutional (FFC) codes with k inputs and Lt outputs are given in [14]. These same conditions hold for the codes in this work. C. Space-Time TCM The structure of FFC TCM codes may be generalized to support space-time codes as shown in Fig. 3. Such an encoder is required for each of the binary input bits Ii(n), i 1:k. The encoder memory for each input may be written I3i(n) [Ii(n) Ii(n 1) à Ii(n vi)]t . The tap weights gi(l, -) and the output symbol labels si (n, l) are elements of GF(2m) . The encoder for each input bit k may have different memory length vi , and the total memory of the encoder is v M i 1 vi . To keep the coding gain as uniform as possible for each bit, we only consider )v/k/  vi  5v/k; .

4

Fig. 4. Encoder for ith input bit.

The FFC encoder may be represented by k generator matrices gi(1, 0) g i(1, 1) à g i(1, vi) G i

gi(2, 0)

g i(2, 1) à g i(2, vi)







, i 1:k

(4)



gi(L t, 0) gi(L t, 1) à gi(L t, vi) The output symbols are the sum over all encoders: t

k

s3(n, :) mod

3 M Gi I i(n) , M i 1

(5)

d3(n, :) map s3(n, :) where d3(n, :) [d (n, 1) d (n, 2) à d (n, Lt)] . An example is the Lt 2 , 4-PSK, 8 state, two input bit code from [5]. The trellis is shown in Fig. 2 and the generator matrices are 0 1 2 0 2 (6) G1

, G2

1 0 2 2 0 FFC TCM are completely described by the k impulse responses which correspond to the k generator matrices, one for each input bit. Alternatively, some FFC TCM may be represented more compactly using a single encoder like Fig. 3 where the single input is I(n)  GF(2k) , rather than k inputs Ii(n)  GF(2) . In this case there is only a single generator matrix G, so v/k must be an integer. The symbols are given by 3 :) map mod GI(n) 3 t, M (7) d(n, where k

i 1 3 3 I(n)

M 2 I i(n) i 1

The symbol labels can be rewritten 5

(8)

k

s3(n, :) mod

M i 1

t

G2i 1I3i(n)

t

k

mod

, M

i 1

3 M mod G2 , M I i(n) , M i 1

From this it may be seen that any I(n)  GF(2k) code can be written as an Ii(n)  GF(2) using Gi mod (2i 1G, M), i 1:k (10) However, the converse is not true: not every Ii(n)  GF(2) can be converted to an equivalent I(n)  GF(2k) code. This is not the most general configuration for FFC TCM, and in fact, many of the optimal codes for k >1 are not in this sub-class. III. SPACE-TIME TCM WITH UNIFORM DISTANCE SPECTRUM Performance analysis of TCM is greatly simplified when the DS is uniform, because then it is sufficient to consider the DS only from the all-zeros path, rather than from all paths. Linear convolutional codes have a UDS, but unfortunately, bandwidth efficient TCM are not linear [15]. However, numerous researchers have shown that codes with weaker conditions than linearity still have a UDS, as summarized in [16]. To date, no one has shown a method for finding codes where the DS wrt eP is uniform. This work has found several codes with a UDS wrt eP, but no general method has been developed. Several of the challenges encountered in this endeavor are noted here. One of the most restrictive types of symmetry that provides a UDS is geometric uniformity (GU) [17]. GU codes were originally defined wrt Euclidean distance, but the concept can be extended to GU wrt eP or eH. An isometry is defined as Definition 1 [17] An isometry u of Euclidean N-space ÜN is a transformation u: ÜNÜN that preserves Euclidean distances, ||u(x) u(y) ||E ||x y ||E, ~ x, y  ÜN (11) where u(x) and u(y) are the images of x and y under the transformation u. This will be called an isometry wrt Euclidean distance. Similarly, an isometry wrt eP or eH L ×L L ×L is a transformation of a complex space u: Ù c tÙ c t that preserves eP or eH, respectively. For Euclidean distance all isometries are composed of translations and unitary transformations of the form [17] uR, -(x) Rx  (12) N

L

The Euclidean distance for space-time codes is defined as ||D||E r M n c 1 M l t 1 |d(n,l)|2 , hence L ×L L ×L for space-time codes Euclidean isometries are a transformation of u: Ù c tÙ c t . Translations and unitary transformations are also isometries wrt both eP and eH [8]. It may be shown that R must be unitary if (12) is an isometry wrt eP. Hence uR, - is an isometry wrt both L ×L L ×L L ×L eP and eH when x  Ù c t , R  Ù c c is unitary, R HR I and - is an arbitrary element of Ù c t . Forney defines GU as follows: Definition 2 [17] 6

A signal set S is geometrically uniform if, given any two points s and s in S, there exists an isometry us, s that transforms s to s while leaving S invariant: us, s (s) s , (13) us, s (S) S





The signal set is GU wrt the metric the isometry preserves; i.e. if the isometry is wrt eP, the signal set is GU wrt eP. The signal set could be a constellation or an entire code. Theorem 1 in [17] proves that GU codes have a UDS. It is stated in [17] that for GU codes with PSK constellations, the - in (12) must equal zero. This statement combined with the following theorem severely limits the usefulness of GU wrt eP codes. Theorem 1 Geometrically uniform wrt eP FFC TCM codes described by isometries of the form uR, -(x) Rx achieve at most one level of diversity. Proof: Let D be the all-zeros codeword, which has rank one because FFC codes have an all-zeros path with all rows identical. By hypothesis and Definition 2, for every codeword D there exists a unitary matrix R such that D RD . Thus D D D (I R)D . From linear algebra, the rank of a product of two matrices is less than or equal to the minimum of their ranks. Hence rank(D) eH(D)  1 , and at most one level of diversity can be achieved.  The possible ways to overcome this problem are not appealing. There are trellis codes that don't have a transition from state zero back to state zero, and these can have an all-zeros codeword with rank greater than one. However, this type of code is not widely used and is more difficult to work with. Another potential solution is to use different isometries than (12), if they exist for eP (which is not likely). Alternatively, a weaker condition than GU must be used to obtain codes with a UDS. Even this is difficult. Exhaustive computer search has shown that for Lt=3, k=1 and BPSK constellations, no four or eight state FFC TCM codes with a UDS wrt eP achieve more than one level of diversity. The most likely path of future research is to develop codes that have a DS that is "close enough" to uniform. The following theorem justifies using the UDS analysis of the TCM in some cases where the DS is not uniform. It also suggests that codes may exist that don't have a UDS, which are better than the best codes with a UDS. Theorem 2 If an Lt=2, BPSK full rank TCM is GU wrt Euclidean distance, the DS wrt eP from any codeword at least as good as DS wrt eP from the all-zeros codeword. Proof: For Lt=2 the codeword difference matrix of an EEP may be written D [d3(:,1) d3(:,2)] where d3(:, l), l 1:Lt is an Nc× 1 vector with elements d(n, l), n 1:Nc . Since the constellation is BPSK, the effective product distance depends on c11 c12 t C2 DD

(14) c21 c22 t

where cl l d3(:, l1) d3(:, l2) is the correlation of the column vectors. It is clear that the 12

7

autocorrelations c11 and c22 do not depend on the signs of the elements of D and that c12=c21. 2 Since by hypothesis the TCM is full rank, eP(D) det(C2) c11c22 c12 , and the sign of c12 is not important. Without loss of generality assume that the symbols of the all-zeros codeword are positive. Then when the all-zeros codeword is transmitted the signs of the elements of D are all the same, N so d(n, lt)  0, d , where d is a positive number. Hence c 2 abs(c12) abs(M n 1 d(n,1)d(n,2)) rd , where r is the number of rows of D with d in both columns. Since the TCM is GU wrt Euclidean distance, when any other codeword is transmitted there is a corresponding EEP with codeword difference matrix D˜  . The elements of D˜  are the same as the elements of D except the phases may differ. Then c˜11 c11, c˜22 c22 , and Nc 2 abs(c˜12) abs(M n 1 (±d(n,1))(±d(n,2)))  abs(c12) . Since eP(D˜) det(C˜2) c˜11c˜22 c˜12 , the DS from the other codeword is at least as good as the DS from the all-zeros codeword.  This theorem does not always hold for larger constellations or higher levels of diversity. An example non-GU code with a better DS than the best possible GU code is given in [8]. In summary, it is not as easy to design codes with a UDS wrt eP as it is for Euclidean distance. It is doubtful if full diversity GU wrt eP codes exist. Also, Theorem 2 shows that codes with a UDS may not be optimal. Without a UDS, performance analysis is extremely computationally intensive, so finding some notion of linearity that applies to TCM for transmitter diversity is an important topic of future research. The most hopeful fact is that Theorem 2 implies the existence of codes where the DS from any codeword is at least as good as the DS from the all-zeros codeword. For these codes, an upper bound of the BEP can still be made using only the DS from the all-zeros codeword. There are also many codes where ePmin is the same for every transmitted path. IV. TRELLIS CODE DESIGN FOR TRANSMITTER DIVERSITY This section begins with a discussion and generalization of design rules proposed in [5]. Also a repetition coded delay diversity scheme is proposed which uses the fewest states possible to achieve full diversity. These results serve as a lower bound on ePmin for a given throughput and diversity level. Finally, a search procedure is used to find codes with better ePmin . A. Design Rules for Achieving Full Diversity I.AT&T Rules (Lt=2) In [5] the researchers at AT&T proposed the following two design rules to guarantee trellis codes with two levels of diversity. Design Rule 1: Transitions departing from the same state differ in the first1 symbol. Design Rule 2: Transitions arriving at the same state differ in the second symbol. As stated, these rules are not sufficient to guarantee full diversity. The code in Fig. 5 meets these design rules but the highlighted path is not full rank.

1

In the AT&T paper, the words "first" and "second" in rules 1 & 2 are reversed, but both ways are equivalent. Here the rules are stated so the delay diversity codes are on the forward diagonal of the codeword matrix instead of the backward diagonal. 8

Dα Dβ

Fig. 5. A two state BPSK TCM which meets AT&T's design rules but does not achieve full diversity. The codeword difference matrix of the highlighted error event path is clearly not full rank: 2 2 D D D

1 2

0 0

(15)

2 2

However, this is a minor technicality. It is implicit in these rules that transitions departing from the same state are the same in the second symbol, and transitions arriving at the same state are the same in the first symbol. When this is followed, the codeword difference matrices always have the form y(1) 0 D D D  0



(16)

y(2)

where y(1), y(2)  Ù are nonzero values and the unspecified part of the matrix is arbitrary. These rules are not necessary to achieve full diversity, and in some cases they rule out optimal codes. Still, they are a good starting point because of their simplicity, and because they make it possible to hand design many codes which turn out to be optimal. II. Zeros Symmetry ( Lt  2 ) The AT&T rules can be extended to any level of diversity, Lt, by constraining the codeword difference matrices to be both upper- and lower-triangular, as illustrated in (17) for Lt=3. This will be referred to as zeros symmetry (ZS). y(1) 0 0 x(2) y(2)  D x(5)

x(3) x(1) 

x(8) x(6) 0 0

0 x(4)

(17)



y(3) x(7) 0

y(4)

The y(n)  Ù can be any nonzero value and the x(n)  Ù can be any value. This restriction is sufficient for full rank, but clearly not necessary. Some examples for ZS codewords are given in (18) for various levels of diversity and codeword length.

9

3 0 0 0 1 0 0 0 1 0 , 0 0 1

1 0 0

2 1 0 0

0 1 0

,

0 1 0 0 0 1

0 0 2 0

(18)

0 1 1 0 0 0 1 0 0 0 0 2

The generator matrices Gi of a FFC TCM are said to have ZS if the upper- and lowerdiagonal symbols are the all-zeros symbol and the yi(n)  1 : M 1 are not the all-zeros symbol. Note that Gi is transposed from D , as shown in (19) yi(1) xi(2) à xi(5) xi(8) 0 0 Gi

0 0

yi(2) xi(3) 0

à

xi(1) xi(4)

xi(6) yi(3) à

0

(19)

xi(7) yi(4)

It is a simple matter to ensure a FFC has ZS, as shown in the following theorem. Theorem 3 If k=1 and the generator matrix G1 has ZS, the entire code has ZS, hence is full rank. Proof: For any two codeword matrices D and D , denote the input bit vectors I3(n) and I3(n) , respectively. From (?) any codeword difference at time n along an EEP may be written k

d3(n, :) d3(n, :) map mod

3 3 M G I (n) I (n) i 1

t

, M

(20)

The vector I3(n) I3(n) I3(n) may be viewed as a sum of shifted impulses, thus all the codeword difference matrices are just a modulo sum of shifted versions of the generator matrix. Because of the shifts, the leading and trailing edges (positions of the y1(n) elements) are never canceled by the modulo sum. Hence all codeword difference matrices have ZS and the code is full rank.  For k>1, two extra restrictions on Gi are required to ensure the code has ZS. First, for every n each yi(n) must be different for all i 1:k . Otherwise if D was the impulse response of one bit and D was the impulse response of a different bit, the codeword difference matrix D D D would have one of the y(n)'s equal to zero. Secondly, for every n all possible k modulo sums mod (M i 1 yi(n), M) must not equal zero. Otherwise if D was the all-zeros codeword, the codeword difference matrix could have a y(n) equal to zero because both codewords could have symbol zero in that position. Unfortunately, since ZS is not a necessary condition for full rank it is overly restrictive, and full rank codes with better ePmin do exist. However, this is still a good starting point because it greatly reduces the number of codes to search, and in many cases it does result in the best code. In addition, since TCM with ZS are guaranteed to be full rank, it is not necessary to test each code in the search for full diversity. It is interesting to note that delay diversity codes [18] have ZS. B. Delay Diversity TCM 10

A straightforward method to design full rank codes proposed in [18] is called delay diversity. Delayed copies of the transmitted symbol are sent consecutively across the Lt antennas, instead of all in the same symbol interval. Trellis coded delay diversity was demonstrated in [5] for Lt=2. The idea may be extended to any level of diversity, simply by using Gi 2i 1I, i 1:k , where I is an Lt× L t identity matrix. This means that the number of memory elements per information bit is vi L t 1 and the total number of memory elements is v k(Lt 1) . This fits the form of (10) so delay diversity can also be written in the compact notation where I(n)  GF(2k) and G=I. Theorem 4 This delay diversity scheme uses the fewest states possible to achieve full diversity. Proof: The standard Hamming distance of any trellis code is bounded by dH  )v/k/ 1 . To achieve full diversity, it is required that Lt  d H , thus Lt  )v/k/ 1 kLt  k)v/k/ k  k(v/k)  k

(21)

v  k(L t 1) Since delay diversity TCM has v k(Lt 1) , it uses the fewest states possible.



Delay diversity TCM has zeros symmetry because the generator matrix has zeros symmetry, thus it is always full rank. The worst case product distance for repetition coded delay diversity occurs on the shortest error event path from the all-zeros path, where the code symbols of the two paths are nearest neighbors in the constellation. Thus 2L ePmin t (22) where is the minimum Euclidean distance between constellation points. Hence IQ TCM has the same error performance of PAM, but the throughput is doubled. The worst case exact pairwise SEP is displayed in Figs. 6-8. By comparing the plots for 16-PSK and 4-PAM, it may be seen that 16-IQ TCM is about 4 dB better than 16-PSK for this repetition coded scheme. For some of these codes, the worst case eP could be improved by using a better code than a repetition code, as was done in [5] for Lt=2. This tends to decrease eP of the longer error event paths, so the average BEP is not improved by as much as might at first be expected.

11

Fig. 6. Worst case SEP for k=3 (8-PSK), and Lt=1:6.

Fig. 7. Worst case SEP for k=4 (16-PSK), and Lt=1:6.

Fig. 8. Worst case SEP for k=2 (4-PAM), and Lt=1:6.

12

C. Search for More Powerful Codes A code search based on the principles discussed so far was performed; the details of the search algorithm are in [8]. Generator matrices for some of the best known codes are presented in Tables I and II. More codes are presented in [8]. The worst case SEPs for the k=1 codes are shown in Figs. 9 and 10. Notice that for k=1 no minimal complexity codes were found with better performance than the delay diversity TCM. This is not true for k>1, as seen in Table II. Table I Best known k=1, m=1 codes Generator Lt v ePmin Matrix 2

1

G1 10 01

4.000 †*

2

3

G1 1110 0101

20.000 †*

2

6

G1 1011110 0111011

56.000 *^

3

2

100 G1 010 001

2.370 *

3

4

11100 G1 01010 00111

18.963 *

3

6

1111100 G1 0101110 0011011

56.889

3

1000 G1 0100 0010 0001

1.000 *

6

1011000 G1 0110100 0011110 0001101

32.000

4

10000 01000 G1 00100 00010 00001

0.328

6

1100000 0111000 G1 0010100 0001010 0000011

5.243

4

4

5

5

† best possible noncatastrophic TCM for the given Lt, k, v and m. * best possible noncatastrophic FFC TCM for the given Lt, k, v and m. 13

^ a catastrophic code with better ePmin exists.

L1 = 2

L1 = 5 L1 = 3

Fig. 9. Worst case SEP for the Lt=2, 3, and 5 codes from Table I.

Fig. 10. Worst case SEP for the Lt=4 codes from Table I. Table II Best known k=2, 4-PSK codes Generator Lt v ZS ePmin Matrices 2 2 yes G1 10 , G2 20 01 02

4.000

02 G1 10 02 , G2 12

8.000 *

2 3 yes G1 130 , G2 20 011 02

12.000

2 3 no G1 102 , G2 12 012 20

16.000

2 2 no

V. CONCLUSION 14

The codes in Table II show that AT&T rules are not optimal for k>2, because better codes exist. REFERENCES [1]G. Ungerboeck, "Channel Coding with Multilevel/Phase Signals", IEEE Trans. on Info. Theory, Vol. IT-28, No. 1, pp. 55-67, Jan. 1982. [2]D. Divsalar and M. K. Simon, "The Design of Trellis Coded MPSK for Fading Channels: Performance Criteria," IEEE Trans. Com., Vol. 36, No. 9, pp. 1004-1012, Sep. 1988. [3]C.-E. W. Sundberg, N. Seshadri, "Coded Modulations for Fading Channels: An Overview," European Trans. Telecom., Vol. 4, No. 3, pp. 309-323, May-June 1993. [4]W. C. Jakes, "A Comparison of Specific Space Diversity Techniques for Reduction of Fast Fading in UHF Mobile Radio Systems," IEEE Trans. on Vehic. Tech., Vol. VT-20, No. 4, pp. 81-92, Nov. 1971. [5]V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-Time Codes for High Data Rate Wireless Communication: Performance Criterion and Code Construction," IEEE Trans. on Info. Theory, Vol. 44, No. 2, pp. 744-765, March 1998. [6]J.-C. Guey, M. P. Fitz, M. R. Bell and W.-Y. Kuo, "Signal Design for Transmitter Diversity Wireless Communication Systems over Rayleigh Fading Channels," Proceedings IEEE VTC '96, pp. 136-140. [7]J.-C. Guey, M. P. Fitz, M. R. Bell and W.-Y. Kuo, "Signal Design for Transmitter Diversity Wireless Communication Systems over Rayleigh Fading Channels," accepted by IEEE Trans. Com. [8]J. Grimm, "Transmitter Diversity Code Design for Achieving Full Diversity on Rayleigh Fading Channels," Ph.D. Dissertation, Purdue University, Dec. 1998. [9]P. K. Ho, J. K. Cavers and J. L. Varaldi, "The effects of Constellation Density on Trellis Coded Modulation in Fading Channels," IEEE Trans. on Vehic. Tech., Vol. 42, No. 3, pp. 318-325, Aug. 1993. [10] S. A. Al-Semari and T. E. Fuja, "I-Q TCM: Reliable Communication Over the Rayleigh Fading Channel Close to the Cutoff Rate," IEEE Trans. on Info. Theory, Vol. 43, No. 1, pp. 250-262, Jan. 1997. [11] A. J. Viterbi, J. K. Wolf, E. Zehavi, and R. Padovani, "A Pragmatic Approach to TrellisCoded Modulation," IEEE Communications Magazine, pp. 11-19, July 1989. [12] D. Divsalar and M. K. Simon, "Multiple Trellis Coded Modulation (MTCM)," IEEE Trans. on Comm., Vol. COM-36, pp. 410-419, Apr. 1988. [13] R. van Nobelen and D. P. Taylor, "Geometrically Uniform Trellis Codes for the Rayleigh Fading Channel," GLOBECOM, pp. 1820-1824, 1995.

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[14] J. L. Massey and M. K. Sain, "Inverses of Linear Sequential Circuits," IEEE Trans. on Computers, Vol. C-17, No. 4, pp. 330-337, Apr. 1968. [15] M. Rouanne and D. J. Costello, Jr., "An Algorithm for Computing the Distance Spectrum of Trellis Codes," IEEE JSAC, Vol. 7, No. 6, pp. 929-940, Aug. 1989. [16] S. Benedetto, M. Mondin, and G. Montorsi, "Performance Evaluation of Trellis-Coded Modulation Schemes," Proceedings of the IEEE, Vol. 82, No. 6, pp. 832-855, June 1994. [17] G. D. Forney, Jr., "Geometrically Uniform Codes," IEEE Trans. on Info. Theory, Vol. 37, No. 5, pp. 1241-1260, Sep. 1991. [18] N. Seshadri and J. H. Winters, "Two Signaling Schemes for Improving the Error Performance of Frequency Division Duplex (FDD) Transmission Systems Using Transmitter Antenna Diversity", International Journal of Wireless Information Networks, Vol. 1, No. 1, pp. 49-60, 1994.

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