Impro ed Space-Time Con v olutional Co des for Quasistatic ...pdfs.semanticscholar.org/3f6d/1230a6d756a314ab5d13d62e13c541b59660.pdfed Space-Time Con olutional Co des for Quasistatic

Improved Space-Time Convolutional Codes forQuasistatic Slow Fading ChannelsQing Yan and Rick S. Blum �yAbstractSpace-time convolutional codes, that provide maximum diversity and coding gain,are produced for cases with PSK modulation and various numbers of states and anten-nas. The codes are found using a new approach introduced recently in a companionpaper. The new approach provides an e�cient method that allows a search for op-timum codes for many practical problems. The new approach also provides a simplemethod for augmenting the criteria of maximum diversity and coding gain with a newmeasure which is shown to be extremely useful for evaluating code performance with-out extensive simulations. To validate the approach, an extensive set of simulationresults are presented comparing the codes designed here to many other recently pro-posed space-time convolutional codes. The comparisons, given in terms of frame errorrate, indicate that our new method provides codes which yield excellent performance.The approach is especially useful for �nding a handful of good codes. Selection amongthese codes can be made with a limited number of simulations for frame error rate.Index Terms:convolutional coding, space-time coding, space-time modulation, transmit diversity.1 IntroductionSpace-time codes (STC) [1]-[19] have attracted considerable attention recently mainly dueto the signi�cant performance gains they can provide. A number of investigations [1]-[9]�This material is based on research supported by the Air Force Research Laboratory under agreementNo. F49620-01-1-0372 and by the National Science Foundation under Grant No. CCR-0112501.yR. S. Blum is with the Electrical Engineering and Computer Science Department, Lehigh University,Bethlehem, PA 18015 1

have shown the great promise of space-time convolutional codes in particular. In this paperwe present some new space-time convolutional codes designed to optimize the diversity gainand coding gain criteria proposed in [1, 10] augmented with a new performance measurewhich is easy to compute but extremely useful. We emphasize that optimizing based ondiversity gain and coding gain, even with the addition of our new performance measure, isnot equivalent to optimizing in terms of frame error rate, but optimizing based on diversitygain and coding gain is a fairly standard practice and has been found to be very useful.In the �rst investigation of space-time convolutional codes a few particular \handdesigned" codes were given in [1, 2, 3]. More recently a number of authors began studyingmore systematic design procedures for space-time convolutional codes [4, 5, 6, 8, 9] assumingcoherent detection. These studies produced a number of important results, but only [4, 6, 9]have attempted to �nd optimum codes, based on the criteria proposed in [1, 10]. The mostsigni�cant previous work is summerized in [6], with a more complete treatment in [7].In this paper we use the techniques in [9] to produce optimum codes for many casesof interest. Further, a new useful criterion is proposed in [9] to augment the criteria in[1, 10]. An e�cient method for designing codes using the augmented criteria is describedin [9] and the utility of this approach is demonstrated here. In comparison, the approachtaken in [4] is so complicated that it can �nd an optimum code only in one case, the simplestcase considered in [1]. The procedure in [9] is much more e�cient. We note that the focusof [9] was on the design procedure. Length constraints did not allow us to present theextensive code design examples we provide here. Further, we present extensive simulationresults comparing the codes we have designed to many other recently designed codes. Thecomparisons are in terms of frame error rate, which is not equivalent to the criteria proposedin [1, 10] so that these comparisons are very useful. These simulation results indicate thatthe method presented [9] provides an excellent approach for �nding a handful of good codeswhich all provide excellent performance. To �nd the best code among this handful (in termsof frame error rate) a few carefully chosen simulations can be very useful.2

In Section 2 the system model is described and the existing design criteria is brie yreviewed. In Section 3 the class of convolutional codes considered and the new design criteriaare outlined. Optimum space-time codes are produced in Section 4. Simulation resultscomparing their performance to other popular space-time codes are provided in Section 5.Section 6 gives conclusions.2 System Model and CriteriaHere we study communication systems employing n transmit antennas and m receive anten-nas. In particular a space-time coder is used to produce n streams of modulated constellationsymbols which will be transmitted using the n antennas. The baseband constellation symboltransmitted by antenna i during time slot k is denoted by pEsci(k), where ci(k) is normal-ized in magnitude so that Es is the average energy in each transmitted constellation symbol.During time slot k, the symbols pEsc1(k); : : : ;pEscn(k) are transmitted simultaneously.The observed signal at each receive antenna is a noisy superposition of the n transmittedsignals corrupted by quasistatic, at Rayleigh fading. Let rj(k) denote the received signalat antenna j and time slot k. Thus, at antenna j, the sampled version of one frame (` timeslots) of the received signal after matched �ltering isrj(k) = nXi=1 �ijqEsci(k) + nj(k) j = 1; : : : ; m k = 1; : : : ; ` (1)where nj(k); j = 1; : : : ; m; k = 1; : : : ` is a complex white Gaussian random sequence withV arfRefnj(k)gg = V arfImfnj(k)gg = N0=2 (Re denotes taking the real part and Imdenotes taking the imaginary part) and �ij; i = 1; : : : ; n; j = 1; : : : ; m is a white Gaussianrandom sequence with zero-mean and unit variance. Each �ij is constant over the frameduration `.Then [1, 10], the probability of transmitting the codewordc = c1(1); : : : ; cn(1); : : : ; c1(`); : : : ; cn(`)3

and deciding erroneously in favor of a di�erent codeworde = e1(1); : : : ; en(1); : : : ; e1(`); : : : ; en(`)is bounded by P (c! e) � rYi=1�i!�m � Es4N0��rm (2)where r is the rank of the matrix (a� denotes the conjugate of a complex number a)A(c; e) = 0BBBBBBBBB@

Pk̀=1(c1(k)� e1(k))(c1(k)� e1(k))� � � � Pk̀=1(c1(k)� e1(k))(cn(k)� en(k))�Pk̀=1(c2(k)� e2(k))(c1(k)� e1(k))� � � � Pk̀=1(c2(k)� e2(k))(cn(k)� en(k))�... ... ...Pk̀=1(cn(k)� en(k))(c1(k)� e1(k))� � � � Pk̀=1(cn(k)� en(k))(cn(k)� en(k))�1CCCCCCCCCA (3)

and �1; : : : ; �r are the nonzero eigenvalues of A(c; e). Further, A(c; e) = B(c; e)B(c; e)Hwhere (AH denotes the Hermitian transpose of a matrix A)B(c; e) = 0BBBBBBBBB@ c1(1)� e1(1) c1(2)� e1(2) � � � c1(`)� e1(`)c2(1)� e2(1) c2(2)� e2(2) � � � c2(`)� e2(`)... ... ...cn(1)� en(1) cn(2)� en(2) � � � cn(`)� en(`)

1CCCCCCCCCA : (4)From (2), each pairwise error probability is determined by rm and (Qri=1 �i)1=r. For asymp-totically large signal-to-noise ratios, performance is determined by the largest pairwise errorprobabilities so we de�ne � as the minimum value of (Qri=1 �i)1=r over all codeword pairs.As per [1] the minimum value of rm over all codeword pairs is called the diversity gain (theslope of the pairwise error probability on a log-log plot) and � is called the coding gain1(��rm is an o�set on a log-log plot).Since the diversity gain comes into (2) as an exponent, it is clear that achievingmaximum diversity gain is more important than achieving high coding gain at all but ex-tremely low signal-to-noise ratios. Further, recall that performance may not be completely1We shall refer to (Qri=1 �i)1=r for general c; e as the pairwise coding gain.4

determined by � for smaller signal-to-noise ratios, other pairwise coding gains may need tobe considered. Thus our approach will be to �nd schemes that achieve maximum diversitygain �rst and of these we prefer those that maximize coding gain2. Such codes will be calledoptimum in the sequel.3 A Class of Space-time Convolutional CodesFirst consider a set of convolutional codes which can be represented by(x1(k); x2(k) : : : ; xn(k)) = �aG (5)whereG =

0BBBBBBBBBBBBBBBB@g11 g12 : : : g1ng21 g22 : : : g2ng31 g32 : : : g3ng41 g42 : : : g4n: : : : : : : : : : : :gQR;1 gQR;2 : : : gQR;n

1CCCCCCCCCCCCCCCCA (6)and �a = (ak;1; ak;2; : : : ; ak;R; ak�1;1 : : : ; ak�1;R; : : : ; ak�Q+1;1 : : : ; ak�Q+1;R): (7)We partition �a into the input and the state where each ai;j; j = 1; : : : ; R; i = 1; 2; : : : ; `is binary. At time slot k, the input is ak;1; ak;2; : : : ; ak;R. Thus R bits are input duringeach time slot. The state is given by ak�1;1 : : : ; ak�1;R; : : : ; ak�Q+1;1 : : : ; ak�Q+1;R. At eachtime slot, each component of the output vector from (5) is mapped into a constellationsymbol. These symbols are transmitted simultaneously from n antennas. In this case thegij; i = 1; : : : ; QR; j = 1; : : : ; nmust be in an alphabet whose size is equal to the constellation2Other pairwise coding gains can also be considered as we will illustrate in the next Section.5

size. As per [1] the convolutional encoder starts and ends in state zero at the beginning andend of each frame.As discussed in [9], each of the codes which can be described by (6) will have ashortest length error event whose length we denote by L. If none of the �rst or last R rowsof G in (6) are zero then in non-degenerate cases L = Q. In this case the number of statesis 2(Q�1)R. Thus, if R = 2 and Q = 2; 3; 4, the number of states is 4; 16; 64. By setting oneof the �rst or last R rows of G in (6) to zero, it is possible to obtain 8 and 32 state codeswith R = 2 for example as discussed in [9]. In these cases L = Q� 1.Consider a code from (5) with a given Q;R; n; s. In [9] we provide a lower bound on �for such a code, given it provides maximum diversity gain. This lower bound, which we call�AP (L), computes the smallest possible value of the determinate of A(c; e) = B(c; e)B(c; e)Hconsidering just the �rst dL=2e and the last bL=2c nonzero columns in (4). The subscriptof �AP (L) is explained by noting that all paths (AP) of length L or longer are consideredin the bound. Besides producing the just mentioned lower bound, if this calculation yieldsa nonzero value, this is a su�cient condition for a code to provide maximum diversity gain[9]. Further, considering just the �rst dL=2e and the last bL=2c nonzero columns in (4) gives�AP (L), a lower bound on the pairwise coding gain for all c; e corresponding to length L(or longer) error events. We shall illustrate that this quantity is quite useful, for example inobtaining the exact value of �.In computing our lower bound for � in [9], the procedure involves trying all possiblestarting states and all possible ending states prior to and following the dL=2e and bL=2c timeslots where errors occur (nonzero columns in (4)), besides trying all possible inputs duringthese dL=2e and bL=2c time slots. Then the minimum result obtained over all these casesis used as the lower bound. A better measure of performance can be obtained by looking atthe results from each of the cases considered (starting states, ending states, inputs) insteadof just selecting the minimum value. In fact each of the results is a lower bound for some setsof codeword pairs. We summarize this information by giving the average of the respective6

bounds, averaged over all the di�erent cases, which we call ��AP (L)3.In [9] we also compute an upper bound on �. This upper bound, which we call �CP (L),computes the smallest value of the determinate of A(c; e) = B(c; e)B(c; e)H for all c; e whichyield L sequential nonzero columns in (4). The subscript of �CP (L) is explained by notingthat continuous paths (CP) of length L are considered in the calculation. There is a physicalinterpretation to these cases, which is discussed in [9], which makes this upper bound veryuseful. Likewise �CP (L) gives the same quantity for L replaced by L. Thus �CP (L) providesan upper bound on � by considering only c; e that di�er for exactly L consecutive columns in(4). Considering �CP (L) for L 6= L can be useful. For example, by considering the smallestupper bound for each of a set of error events of di�erent lengths, for example �CP (L); �CP (L+1); : : : we can provide a tighter bound on � since � < min[�CP (L); �CP (L+ 1); : : :]. Further,the condition that �CP (L) > 0 becomes a necessary condition for maximum diversity gain.From this [9] we see that we must have L � n in order to achieve maximum diversity gain.Similar to ��AP (L) (average over starting state, ending state, and input cases), we de�ne��CP (L) which is a better measure of performance.Finally, our upper and lower bounds can be used to obtain the exact value of �. Let�UB be some upper bound of �. For example, one choice is �UB = �CP (L). If L > L and�AP (L) � �UB then � = min f�CP (L); �CP (L+ 1); : : : ; �CP (L� 1)g (8)while, if �AP (L) = �CP (L) then � = �CP (L). In the cases we studied we typically needed toconsider only L � L+ 2.4 Optimum Space-Time CodesOur general approach towards �nding optimum space-time codes using the theory from [9]is to �rst formulate the case of interest using the terminology of (5). Next we consider only3This was called ��LB for the case of L = Q in [9] 7

those codes satisfying the su�cient conditions for maximum diversity gain. For these codeswe computed the exact coding gain and noted those producing the largest coding gain. Thenwe compared these largest coding gains with the largest coding gain achieved by any codesatisfying the necessary conditions for maximum diversity gain. In many cases this enabledus to �nd a code satisfying the su�cient conditions for maximum diversity gain which alsoproduced maximum coding gain of all codes achieving maximum diversity gain. Typicallywe found many such codes. In such cases we present those giving the largest ��AP (L) and��CP (L) values for L = L or for L just larger than L. This criterion is justi�ed in [9]. Nextwe describe some speci�c cases.First consider the q-state 2 b/s/Hz 4-PSK space-time code cases considered in [1].These are n = 2, R = 2 and s = 4 (QPSK) cases in the terminology of (5). As previouslydiscussed q = 2(Q�1)R withQ = 2 or 3 if the number of states is 4 or 16, while [9] q = 2(Q�2)R+1with Q = 2 or 3 if the number of states is 8 or 32. Our results for this case are summarized inTable 1. Besides providing �; ��AP (L) and ��CP (L), for convenience we provide the \e�ectiveproduct distance", ePmin, from [6] in Table 1. To help the reader understand the processtaken to arrive at Table 1, we describe some of the details. In particular, we found thatthere are 1840 di�erent 4-state codes (not counting4 permutations of the columns of G)which satis�ed the su�cient conditions for maximum diversity gain while simultaneouslyproviding the highest coding gain, � = 2, of any codes satisfying the su�cient conditionsfor maximum diversity gain. We found 288 codes providing the largest coding gain of p8 ofall codes satisfying the necessary conditions for maximum diversity gain. The 4 state casewas atypical in this respect, it being the only case where we could not �nd optimum codes,with the largest possible diversity and coding gain, which satis�ed the su�cient conditionsfor maximum diversity. From these 288 codes, we selected one of the 24 codes with largest��AP (2), ��CP (2) and ��AP (3) to put in Table 1. By examining the slope of the FER performance4In the count of 1840 given, several of the codes counted could be considered to be equivalent. The countis just meant to give a rough idea of the number of codes found.8

curve, it was seen that this code achieved maximum diversity gain.For the 8-state case, we found 84 codes which satis�ed the su�cient conditions formaximum diversity gain while simultaneously providing the largest value of � = 4 of all codessatisfying the su�cient conditions for maximum diversity gain. No codes were found whichachieved a larger � of those satisfying the necessary conditions for maximum diversity gain.An additional 188 codes were found that achieved � = 4 which satis�ed only the necessaryconditions for maximum diversity and not the su�cient conditions. Of the codes achieving� = 4 and satisfying the su�cient conditions for maximum diversity, we found a group ofcodes which provided the largest ��AP (2) = 3:42 and the largest ��CP (2) = 4:00 and one suchcode is listed in Table 1.For the 16-state case, we found 96 codes which satis�ed the su�cient conditions formaximum diversity gain while simultaneously providing the largest value of � = p32. Allof the 96 codes provided the same ��AP (3) of 5:76, and 24 of them gave best ��CP (3) of 6:24.Thus we selected the �rst one we encountered with ��CP (3) = 6:24 to put in Table 1. We werealso able to show that any codes satisfying the necessary conditions for maximum diversitycould not provide � = p32 without yielding ��AP (3) < 5:76. For the 32-state case we foundthat the maximum coding gain is � = 6 and one such code with best ��CP (3) = 6:33 andbest ��CP (4) = 8:72 is put in Table 1. Further this code satis�es the su�cient conditions formaximum diversity gain.We note that a few of our calculations concerning the cases in Table 1 di�er fromthose in [1] and [4]. The coding gain for the 8-state case from [1] is � = p12 based on ourcalculations, as opposed to � = p20 as stated in [1]. This correction for this code was alsomade in [4]. However, the new code in [4] gives a coding gain of � = p12 also, instead of� = 4 as stated in [4]. Based on our calculations we �nd � = p12 for the 32-state code from[1], instead of � = p28 as reported in [1].Next consider q-state 1 b/s/Hz BPSK space-time codes. These are n = 2, R = 1,s = 2 (BPSK) and q = 2Q�1 with Q = 2; 3; 4; 5 cases in the terminology of (5). For these9

q � of the code from [1] our � ePmin ��AP (L) ��CP (L) our GT4 2 p8 8 3:54 3:80 0B@ 2 0 1 22 2 2 1 1CA8 p12 4 16 3:42 4:00 0B@ 0 2 1 0 22 1 0 2 2 1CA16 p12 p32 32 5:76 6:24 0B@ 0 2 1 1 2 02 2 1 2 0 2 1CA32 p12 6 36 5:63 6:33 0B@ 2 0 1 2 1 2 22 2 0 1 2 0 2 1CATable 1: Optimum q-state 2 b/s/Hz 4-PSK space-time codes (ePmin from [6]).q our � ePmin ��AP (L) ��CP (L) our GT2 4 4 4.00 4.00 0B@ 0 11 1 1CA4 p48 12 7.33 6.93 0B@ 0 1 11 0 1 1CAy4 p32 8 7.33 7.73 0B@ 0 1 11 1 1 1CA8 p80 20 10.06 10.47 0B@ 1 0 1 11 1 0 1 1CA16 p112 28 11.38 14.27 0B@ 1 1 0 1 10 1 1 1 1 1CATable 2: Optimum q-state 1 b/s/Hz BPSK space-time codes. A "y" denotes that the codeis catastrophic. 10

cases, Table 2 presents one representative scheme we have found using an approach similarto that taken for the QPSK space-time code cases just discussed. In each case, the selectedcode satis�ed the su�cient conditions for maximum diversity gain while simultaneouslyproviding maximum values of �, ��AP (L) and ��CP (L). These values are listed in Table 2. Incases where the codes which provide best �; ��AP (L) and ��CP (L) are catastrophic, the bestnon-catastrophic codes are also presented.n > 2 CasesSome optimum (maximum diversity and coding gain) space-time codes using 3 or 4transmit antennas and BPSK modulations are provided in Table 3 and Table 4. The search-ing procedure for these cases is similar to that for the n = 2 cases. In each case, the selectedcode satis�ed the su�cient conditions for maximum diversity gain while simultaneously pro-viding maximum values of �, ��AP (L) and ��CP (L). These values are listed in Table 3 andTable 4.A 16-state QPSK space-time code using 3 transmit antennas is given in Table 5. Thecode listed in Table 5 satis�ed the su�cient conditions for maximum diversity gain. Weconjecture it achieves near optimum (if not optimum) coding gain.5 Probability of Frame Error PerformanceFigure 1 shows the frame error rate of the space-time codes listed in Table 1 for cases with2 transmit and 2 receive antennas. Figure 1 illustrates the gain achieved by increasing theconstraint length of the codes. Figures 2 through 4 compare the performance of our codesand those from [1, 4, 6]. Clearly, the codes in Table 1 are better than the codes from [1, 4, 6]when judged in terms of frame error rate. In all our simulations, each frame consists of 130transmissions from of each transmit antenna (` = 130).The improvements demonstrated in Figures 2 through 4 are partially attributed to our�nding codes with larger � and partially to our use of an improved performance estimate. The11

q our � ePmin ��AP (L) ��CP (L) our GT4 4 6427 5.08 5.08 0BBBBB@ 0 1 11 0 11 1 1 1CCCCCA8 256 13 25627 7.65 7.05 0BBBBB@ 1 0 0 11 0 1 01 1 1 1 1CCCCCAy8 192 13 649 7.65 8.32 0BBBBB@ 1 1 1 01 1 0 11 0 1 1 1CCCCCA16 8 51227 10.00 10.18 0BBBBB@ 1 0 0 1 11 1 0 1 01 1 1 0 1 1CCCCCATable 3: Optimum q-state 1 b/s/Hz BPSK 3-space-time codes. A "y" denotes that the codeis catastrophic.

12

q our � ePmin ��AP (L) ��CP (L) our GT8 4 1 5.97 5.97 0BBBBBBBBB@ 0 1 0 10 1 1 11 0 1 01 1 1 01CCCCCCCCCA

16 1280 14 5 8.11 8.37 0BBBBBBBBB@ 1 0 0 0 11 0 1 1 11 1 0 1 11 1 1 1 01CCCCCCCCCAy

16 1024 14 4 7.99 9.32 0BBBBBBBBB@ 0 1 1 0 11 1 0 0 11 1 1 1 01 1 1 1 11CCCCCCCCCA

32 4352 14 17 9.80 10.38 0BBBBBBBBB@ 1 0 0 0 0 11 0 1 1 1 11 1 1 0 1 01 1 1 1 0 01CCCCCCCCCATable 4: Optimum q-state 1 b/s/Hz BPSK 4-space-time codes. A "y" denotes that the codeis catastrophic. q our � ePmin ��AP (L) ��CP (L) our GT16 32 13 25627 3.90 4.72 0BBBBB@ 0 2 1 2 2 01 2 2 0 0 22 2 0 2 1 2 1CCCCCATable 5: A q-state 2 b/s/Hz 4-PSK space-time code using 3 transmit antennas.13

criterion proposed in [1], to minimize the pairwise error probability in (2) for that codewordpair that makes (2) the largest, is very useful, but it does not completely determine the FERof space-time codes. To compensate, the quantities ��AP (L) and ��CP (L) were introduced in[9]. We discuss the signi�cance of augmenting � with ��AP (L) and ��CP (L), using the cases inFigure 2 as an example. Consider the 4 state codeGT = 0B@ 2 0 1 22 2 2 1 1CA (9)given in Table 1. Recall this code provides a coding gain of p8. Contrast this with the codegiven in [4] GT = 0B@ 2 0 1 32 2 0 1 1CA (10)which also provides a coding gain of p8. The code in (9) provides ��AP (2) = 3:54 and��CP (2) = 3:80 while (10) provides ��AP (2) = 2:26 and ��CP (2) = 2:83. From this we expect(9) may perform better when compared in terms of FER performance. The results in Figure 2verify that (9) outperforms (10) . The results in Figure 2 also verify that the code in (9)outperforms the code GT = 0B@ 0 2 2 32 3 0 2 1CAdiscussed in [9], which has a coding gain of 2 and provides ��AP (2) = 3:58 and ��CP (2) = 4:16.Further Figure 2 also shows that the code in (9) outperforms the code proposed in [1] whichprovides � = 2 and ��AP (2) = 2:59 and ��CP (2) = 2:67.From the above example, one might expect that the coding gain is more importantthan the average values ��AP (L) or ��CP (L) for the code to perform well, but this is notalways true. Figure 5 compares the simulation results of our codes with optimum codinggain (from Table 2) with the codes in [5] which are constructed from the known binaryconvolutional codes that achieve optimal values of free distance dfree. Only in one case14

do our codes with optimum coding gain outperform the codes in [5], although generally thedi�erences appear to be small in every case we tested. Again, this may be explained by usingthe average values ��CP (L) and ��CP (L + 1). In particular, the 4-state code in [5] providesbetter ��AP (2) = ��CP (2) = 7:73 and outperforms the code in Table 2 (with ��AP (2) = 7:33and ��CP (2) = 6:93). In the 8-state case, the code in [5] provides ��CP (3) = 11:68 and��CP (4) = 9:66, while our 8-state code gives ��CP (3) = 10:47 and ��CP (4) = 14:26 and performsbetter. In all the other cases we compared, we found that the codes in [5] always providelarger ��CP (L) with smaller coding gain5.It is observed that as we increased the number of states the FER performance com-parison between our new codes and those provided in [5] changes with a clear trend. Forcases with a small number of states, the codes in [5] are somewhat better for the BPSK cases.For cases with a large enough number of states, our new codes provide better (or very close)FER performance. This is understandable since for cases with a small number of states, thenumber of candidate codes that provide maximum diversity gain and coding gain is small,which gives us limited choices for further selection based on ��AP (L) or ��CP (L). For caseswith a larger number of states, the pool of candidate codes that maximize the �rst two per-formance metrics, diversity gain and coding gain, is larger. Therefore, we have a good chanceof being able to choose codes that also provide large ��AP (L) and ��CP (L). Although in thispaper we followed the rule of maximizing diversity gain, coding gain and (��AP (L); ��CP (L))in a decreasing order of preference (recall we restrict attention to optimum codes where byoptimum we mean optimum diversity gain and optimum coding gain), this will not alwaysbe the best ordering. To avoid the possible pitfall that can occur for cases with a smallnumber of states, one may put more emphasis on maximizing ��AP (L) and ��CP (L), insteadof emphasising maximizing the worst case coding gain, and the general method provided inthis paper is still valid.5Figure 5 considers BPSK cases from [5]. For q = 4; 8; 16 cases with 2 antennas, � = p32;p48;p96,respectively. For q = 8; 16 cases with 3 antennas, � = 128 13 and 256 13 , respectively.15

6 ConclusionsSpace-time convolutional codes which provide maximum diversity and coding gain werefound for some practical cases with 2, 3 and 4 antennas, various number of states and variousnumbers of input bits per time slot. The optimum codes produced here are found using a newprocedure given in [9], which is much more e�cient than previous approaches. Further, anew performance measure is suggested to augment coding gain and diversity gain. Extensivesimulation results showing frame error rate, justify the utility of the approach. The resultsshow the performance of each code we produce is the best or very close to the best we foundby trying many codes. The overall approach is very useful for selecting a handful of goodcodes. Limited simulation results can then be used to choose among the handful of codes.We emphasize that even the augmented performance measure suggested here has limitationsand that simulations to verify the selection of a space-time code are always prudent. Theinvestigations here have been limited to cases with relatively small array sizes. For largerarrays, the considerations in [21, 22] should be considered.References[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data ratewireless communication: Performance criteria and code construction," IEEE Trans.Inform. Theory, Mar. 1998, pp. 744-764.[2] A. F. Naguib, V. Tarokh, N. Seshadri, A. R. Calderbank, "A space-time coding modemfor high-data-rate wireless communications," IEEE J. Select. Commun., vol. 16, No. 8,Oct. 1998, pp. 1459-1478.[3] V. Tarokh, A. F. Naguib, N. Seshadri, A. R. Calderbank, "Space-time codes for highdata rate wireless communication: performance criteria in the presence of channel es-16

timation errors, mobility, and multiple paths," IEEE Trans. on Commun., vol. 47, No.2, Feb. 1999, pp. 199-207.[4] S. Baro, G. Bauch, and A. Hansmann, \Improved Codes for Space-Time Trellis CodedModulation", IEEE Communication Letters, Vol.4, No.1, 2000.[5] A. R. Hammons, H. E. Gammal, \On the Theory of Space-Time Codes for PSK Mod-ulation", IEEE Trans. on Information Theory, vol. 46, No. 2, March 2000, pp.524-542.[6] J. Grimm, M. P. Fitz, and J. V. Krogmeier,\Further results in space-time coding forRayleigh fading", in Proc. 1998 Allerton Conference.[7] J. Grimm,\Transmitter diversity code design for achieving full diversity on Rayleighchannels", Ph.D. dissertation, Purdue University, Dec. 1998.[8] Y. Liu, M. P. Fitz and O.Y. Takeshita, \A Rank Criterion for QAM Space-Time Codes",submitted to IEEE Trans. on Information Theory.[9] R. S. Blum, \New analytical tools for designing space-time convolutional codes," sub-mitted to IEEE Trans. on Information Theory and similar work in the proceedings ofConference on Information Sciences and Systems, Princeton University, Princeton, NJ,March 1999.[10] J-C Guey, M. P. Fitz, M. R. Bell, and W-Y Kuo, "Signal Design for Transmitter Diver-sity Wireless Communication Systems Over Rayleigh Fading Channels," IEEE Trans.on Communications, vol. 47, No. 4, April 1999, pp. 527-537.[11] V. Tarokh, H. Jafarkhani, A. R. Calderbank, "Space-time block coding for wirelesscommunications: performance results," IEEE J. Select. Commun., vol. 17, No. 3, March1999, pp. 451-460.[12] V. Tarokh, H. Jafarkhani, A. R. Calderbank, "Space-time block codes from orthogonaldesigns," IEEE Trans. on Information Theory, vol. 45, No. 5, July 1999, pp. 1456-1467.17

[13] A. Wittneben, \A new bandwidth e�cient transmit antenna modulation diversityscheme for linear digital modulation," IEEE ICC, pp. 1630-1634, May 1993.[14] V. Weerackody, \Diversity for direct-sequence spread spectrum system using multipleantennas." IEEE ICC, pp. 1775-1779, May 1993.[15] N. Seshadri and J. H. Winters, \Two signaling schemes for improving the error perfor-mance of frequency division duplex (FDD) transmission system using antenna diversity,"Int. J. Wireless Infom. Networks, Vol. 1, No. 1, 1994.[16] S. M. Alamouti, "A simple transmit diversity technique for wireless communications,"IEEE J. Select. Commun., vol. 16, No. 8, Oct. 1998, pp. 1451-1458.[17] G. J. Foschini, \Layered space-time architecture for wireless communication in a fadingenvironment when using multi-element antennas," Bell Labs Technical Journal. Vol. 1,No. 2, pp. 41-59, Autumn 1996.[18] G. Raleigh and J. M. Cio�, \Spatio-temporal coding for wireless communication," IEEETransactions on Communications, COM-46, pp. 357-366, March 1998.[19] B. L. Hughes, \Di�erential space-time modulation", in Proc. 1999 Wireless Communi-cations and Networking Conference, New Orleans, LA, Sept. 22-29, 1999.[20] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Univ. Press,1988.[21] Z. Chen, J. Yuan, and B. Vucetic, \Improved space-time trellis coded modulation schemeon slow Rayleigh fading channels," in Proc. of International Conference on Communi-cations, Helsinki, Finland, June 11-14, 2001.[22] E. Biglieri and A. M. Tulino, \Designing space-time codes for a large number of receiv-ing antennas," In Proc. of the 35th Annual Conference on Information Sciences andSystems, Baltimore, MD, March 21-23, 2001.18

Qing Yan received the B.S. and M.S. degrees in electrical engineering from Universityof Science and Technology of China (USTC), Hefei, in 1995 and 1998, respectively. Since1998, he has been pursuing the Ph.D. degree at the Electrical Engineering and ComputerScience Department, Lehigh University, Bethlehem, PA. He is currently a Research Assistantwith the Signal Processing and Communications Research Laboratory, Lehigh University.His research interests include the application of signal processing theory to the design andanalysis of communication systems.Rick S. Blum (M'91) received a B.S. in Electrical Engineering from the PennsylvaniaState University in 1984 and his M.S. and Ph.D in Electrical Engineering from the Universityof Pennsylvania in 1987 and 1991.From 1984 to 1991 he was a member of technical sta� at General Electric Aerospacein Valley Forge, Pennsylvania and he graduated from GE`s Advanced Course in Engineering.Since 1991, he has been with the Electrical Engineering and Computer Science Departmentat Lehigh University in Bethlehem, Pennsylvania where he is currently an Associate Professorand holds a Class of 1961 Professorship. His research interests include signal detection andestimation and related topics in the areas of signal processing and communications. He iscurrently an associate editor for the IEEE Transactions on Signal Processing and for IEEECommunications Letters. He is also a member of the Signal Processing for CommunicationsTechnical Committee of the IEEE Signal Processing Society.Dr. Blum is a member of Eta Kappa Nu and Sigma Xi, and holds a patent for aparallel signal and image processor architecture. He was awarded an ONR Young InvestigatorAward in 1997 and an NSF Research Initiation Award in 1992.

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List of Tables1 Optimum q-state 2 b/s/Hz 4-PSK space-time codes (ePmin from [6]). . . . . 102 Optimum q-state 1 b/s/Hz BPSK space-time codes. A "y" denotes that thecode is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Optimum q-state 1 b/s/Hz BPSK 3-space-time codes. A "y" denotes that thecode is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Optimum q-state 1 b/s/Hz BPSK 4-space-time codes. A "y" denotes that thecode is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A q-state 2 b/s/Hz 4-PSK space-time code using 3 transmit antennas. . . . . 13

20

List of Figures1 Performance comparison of some best 2 b/s/Hz, QPSK, q-state Space-timeCodes with 2 transmit and 2 receive antennas (SNR per receive antenna =nEs=N0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Performance comparisons of some 2 b/s/Hz, QPSK, 4-state Space-time Codeswith 2 transmit and 2 receive antennas. . . . . . . . . . . . . . . . . . . . . . 233 Performance comparisons of some 2 b/s/Hz, QPSK, 8-state Space-time Codeswith 2 transmit and 2 receive antennas. Zero symmetry (ZS) is de�ned in [6]. 244 Performance comparisons of some 2 b/s/Hz, QPSK, 16-and 32-state Space-time Codes with 2 transmit and 2 receive antennas. . . . . . . . . . . . . . . 255 Performance of some 1 b/s/Hz, BPSK, q-state Space-time Codes with 2 trans-mit and 2 receive or 3 transmit and 3 receive antennas (performance of thebest non-catastrophic codes similar). . . . . . . . . . . . . . . . . . . . . . . 26

21

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010

−3

10−2

10−1

100

SNR per receive antenna (dB)

Fram

e Er

ror P

roba

bilit

y

q=4, opt q=8, opt q=16, optq=32, opt

Figure 1: Performance comparison of some best 2 b/s/Hz, QPSK, q-state Space-time Codeswith 2 transmit and 2 receive antennas (SNR per receive antenna = nEs=N0).

22

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010

−2

10−1

100

SNR per receive antenna (dB), q=4

Fram

e Er

ror P

roba

bilit

y

from [4]

from [6]

q=4 [1] q=4 [4] q=4 [6] q=4 [8] q=4 optimum

Figure 2: Performance comparisons of some 2 b/s/Hz, QPSK, 4-state Space-time Codes with2 transmit and 2 receive antennas.

23

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010

−2

10−1

100

SNR per receive antenna (dB), q=8

Fram

e Er

ror P

roba

bilit

y

from [4]

from [1]

[6], ZS [1] [4] [6],non−ZS optimum

Figure 3: Performance comparisons of some 2 b/s/Hz, QPSK, 8-state Space-time Codes with2 transmit and 2 receive antennas. Zero symmetry (ZS) is de�ned in [6].

24

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 1010

−3

10−2

10−1

100

SNR per receive antenna (dB)

Fram

e Er

ror P

roba

bilit

y

q=16 in [1] q=16 in [4] q=16 optimum q=32 in [1] q=32 optimum

Figure 4: Performance comparisons of some 2 b/s/Hz, QPSK, 16-and 32-state Space-timeCodes with 2 transmit and 2 receive antennas.

25

5 6 7 8 9 1010

−4

10−3

10−2

10−1

100

SNR per receive antenna (dB), Tx=2, Rx=2

Fram

e E

rror

Pro

babi

lity

q=16 in [6]

q=16 opt η

q=4 opt η q=4 in [6] q=8 opt η q=8 in [6] q=16 opt ηq=16 in [6]

0 1 2 3 4 510

−4

10−3

10−2

10−1

100

SNR per receive antenna (dB), Tx=3, Rx=3

Fram

e E

rror

Pro

babi

lity

q=8 opt η q=8 in [6] q=16 opt ηq=16 in [6]

Figure 5: Performance of some 1 b/s/Hz, BPSK, q-state Space-time Codes with 2 transmitand 2 receive or 3 transmit and 3 receive antennas (performance of the best non-catastrophiccodes similar).

26