Improved Space-Time Convolutional Codes for Quasistatic Slow ...

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3] V. Tarokh, A. F. Naguib, N. Seshadri, A. R. Calderbank, "Space-time codes for high data rate wireless communication: performance criteria in the presence of ...
Improved Space-Time Convolutional Codes for Quasistatic Slow Fading Channels Qing Yan and Rick S. Blum y Abstract Space-time convolutional codes, that provide maximum diversity and coding gain, are produced for cases with PSK modulation and various numbers of states and antennas. The codes are found using a new approach introduced recently in a companion paper. The new approach provides an ecient method that allows a search for optimum codes for many practical problems. The new approach also provides a simple method for augmenting the criteria of maximum diversity and coding gain with a new measure which is shown to be extremely useful for evaluating code performance without extensive simulations. To validate the approach, an extensive set of simulation results are presented comparing the codes designed here to many other recently proposed space-time convolutional codes. The comparisons, given in terms of frame error rate, indicate that our new method provides codes which yield excellent performance. The approach is especially useful for nding a handful of good codes. Selection among these 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 Introduction Space-time codes (STC) [1]-[19] have attracted considerable attention recently mainly due to 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 agreement No. F49620-01-1-0372 and by the National Science Foundation under Grant No. CCR-0112501. y R. 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 paper we present some new space-time convolutional codes designed to optimize the diversity gain and coding gain criteria proposed in [1, 10] augmented with a new performance measure which is easy to compute but extremely useful. We emphasize that optimizing based on diversity gain and coding gain, even with the addition of our new performance measure, is not equivalent to optimizing in terms of frame error rate, but optimizing based on diversity gain 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 \hand designed" codes were given in [1, 2, 3]. More recently a number of authors began studying more systematic design procedures for space-time convolutional codes [4, 5, 6, 8, 9] assuming coherent 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 most signi 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 cases of interest. Further, a new useful criterion is proposed in [9] to augment the criteria in [1, 10]. An ecient method for designing codes using the augmented criteria is described in [9] and the utility of this approach is demonstrated here. In comparison, the approach taken in [4] is so complicated that it can nd an optimum code only in one case, the simplest case considered in [1]. The procedure in [9] is much more ecient. We note that the focus of [9] was on the design procedure. Length constraints did not allow us to present the extensive code design examples we provide here. Further, we present extensive simulation results comparing the codes we have designed to many other recently designed codes. The comparisons are in terms of frame error rate, which is not equivalent to the criteria proposed in [1, 10] so that these comparisons are very useful. These simulation results indicate that the method presented [9] provides an excellent approach for nding a handful of good codes which all provide excellent performance. To nd the best code among this handful (in terms of 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 y reviewed. In Section 3 the class of convolutional codes considered and the new design criteria are outlined. Optimum space-time codes are produced in Section 4. Simulation results comparing their performance to other popular space-time codes are provided in Section 5. Section 6 gives conclusions.

2 System Model and Criteria Here we study communication systems employing n transmit antennas and m receive antennas. In particular a space-time coder is used to produce n streams of modulated constellation symbols which will be transmitted using the n antennas. The baseband constellation symbol p transmitted by antenna i during time slot k is denoted by Esci(k), where ci(k) is normalized in magnitude so that Es is the average energy in each transmitted constellation symbol. p p During time slot k, the symbols Esc (k); : : : ; Escn(k) are transmitted simultaneously. The observed signal at each receive antenna is a noisy superposition of the n transmitted signals corrupted by quasistatic, at Rayleigh fading. Let rj (k) denote the received signal at antenna j and time slot k. Thus, at antenna j , the sampled version of one frame (` time slots) of the received signal after matched ltering is 1

n q X rj (k) = ij Esci(k) + nj (k) j = 1; : : : ; m k = 1; : : : ; ` i=1

(1)

where nj (k); j = 1; : : : ; m; k = 1; : : : ` is a complex white Gaussian random sequence with V arfRefnj (k)gg = V arfImfnj (k)gg = N =2 (Re denotes taking the real part and Im denotes taking the imaginary part) and ij ; i = 1; : : : ; n; j = 1; : : : ; m is a white Gaussian random sequence with zero-mean and unit variance. Each ij is constant over the frame duration `. Then [1, 10], the probability of transmitting the codeword 0

c = c (1); : : : ; cn(1); : : : ; c (`); : : : ; cn(`) 1

1

3

and deciding erroneously in favor of a di erent codeword

e = e (1); : : : ; en(1); : : : ; e (`); : : : ; en(`) 1

is bounded by

1

Yr !?m  Es ?rm P (c ! e)  i 4N i=1

(2)

0

where r is the rank of the matrix (a denotes the conjugate of a complex number a)

0P BB `k (c (k) ? e (k))(c (k) ? e (k)) BB P` (c (k) ? e (k))(c (k) ? e (k)) BB k A(c; e) = B ... BB @ P`  k (cn (k) ? en (k))(c (k) ? e (k)) =1

1

1

1

1

=1

2

2

1

1

1

1

=1

   P`k (c (k) ? e (k))(cn(k) ? en(k))    P`k (c (k) ? e (k))(cn(k) ? en(k)) ...

=1

1

1

=1

2

2

...    P`k (cn(k) ? en(k))(cn(k) ? en(k)) =1

and  ; : : : ; r are the nonzero eigenvalues of A(c; e). Further, A(c; e) = B (c; e)B (c; e)H where (AH denotes the Hermitian transpose of a matrix A) 1

1    c (`) ? e (`) CC    c (`) ? e (`) CCC CC : (4) ... CC A    cn(`) ? en(`) From (2), each pairwise error probability is determined by rm and (Qr  ) =r . For asymp0 BB c (1) ? e (1) c (2) ? e (2) BB c (1) ? e (1) c (2) ? e (2) BB B (c; e) = B ... ... BB @ cn(1) ? en(1) cn(2) ? en(2) 1

1

1

1

1

1

2

2

2

2

2

2

i=1 i

1

totically large signal-to-noise ratios, performance is determined by the largest pairwise error probabilities so we de ne  as the minimum value of (Qri i) =r over all codeword pairs. As per [1] the minimum value of rm over all codeword pairs is called the diversity gain (the slope of the pairwise error probability on a log-log plot) and  is called the coding gain (?rm is an o set on a log-log plot). Since the diversity gain comes into (2) as an exponent, it is clear that achieving maximum diversity gain is more important than achieving high coding gain at all but extremely low signal-to-noise ratios. Further, recall that performance may not be completely =1

1

1

1

We shall refer to (

Qr

i=1 i

)1=r for general c; e as the pairwise coding gain.

4

1 CC CC CC (3) CC CA

determined by  for smaller signal-to-noise ratios, other pairwise coding gains may need to be considered. Thus our approach will be to nd schemes that achieve maximum diversity gain rst and of these we prefer those that maximize coding gain . Such codes will be called optimum in the sequel. 2

3 A Class of Space-time Convolutional Codes First consider a set of convolutional codes which can be represented by (x (k); x (k) : : : ; xn(k)) = aG 1

where

0 BB BB BB BB G=B BB BB BB B@

(5)

2

g g g g g g g g ::: ::: gQR; gQR; 11

12

21

22

31

32

41

42

1

::: ::: ::: ::: ::: :::

2

gn gn gn gn ::: gQR;n 1

2

3

4

1 CC CC CC CC CC CC CC CC A

(6)

and

a = (ak; ; ak; ; : : : ; ak;R; ak? ; : : : ; ak? ;R; : : : ; ak?Q 1

2

11

1

;

+1 1

: : : ; ak?Q

;R):

(7)

+1

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; ; ak; ; : : : ; ak;R. Thus R bits are input during each time slot. The state is given by ak? ; : : : ; ak? ;R; : : : ; ak?Q ; : : : ; ak?Q ;R. At each time slot, each component of the output vector from (5) is mapped into a constellation symbol. These symbols are transmitted simultaneously from n antennas. In this case the gij ; i = 1; : : : ; QR; j = 1; : : : ; n must be in an alphabet whose size is equal to the constellation 1

2

11

2

1

+1 1

+1

Other 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 and end of each frame. As discussed in [9], each of the codes which can be described by (6) will have a shortest length error event whose length we denote by L. If none of the rst or last R rows of G in (6) are zero then in non-degenerate cases L = Q. In this case the number of states is 2 Q? R . Thus, if R = 2 and Q = 2; 3; 4, the number of states is 4; 16; 64. By setting one of the rst or last R rows of G in (6) to zero, it is possible to obtain 8 and 32 state codes with 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)H considering just the rst dL=2e and the last bL=2c nonzero columns in (4). The subscript of AP (L) is explained by noting that all paths (AP) of length L or longer are considered in the bound. Besides producing the just mentioned lower bound, if this calculation yields a nonzero value, this is a sucient 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 in obtaining the exact value of . In computing our lower bound for  in [9], the procedure involves trying all possible starting states and all possible ending states prior to and following the dL=2e and bL=2c time slots where errors occur (nonzero columns in (4)), besides trying all possible inputs during these dL=2e and bL=2c time slots. Then the minimum result obtained over all these cases is used as the lower bound. A better measure of performance can be obtained by looking at the results from each of the cases considered (starting states, ending states, inputs) instead of just selecting the minimum value. In fact each of the results is a lower bound for some sets of codeword pairs. We summarize this information by giving the average of the respective (

1)

6

bounds, averaged over all the di erent cases, which we call AP (L) . 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 which yield L sequential nonzero columns in (4). The subscript of CP (L) is explained by noting that continuous paths (CP) of length L are considered in the calculation. There is a physical interpretation to these cases, which is discussed in [9], which makes this upper bound very useful. Likewise CP (L) gives the same quantity for L replaced by L. Thus CP (L) provides an 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 smallest upper 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 3

 = min fCP (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 to consider only L  L + 2.

4 Optimum Space-Time Codes Our 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 only 3

This was called LB for the case of L = Q in [9]

7

those codes satisfying the sucient conditions for maximum diversity gain. For these codes we computed the exact coding gain and noted those producing the largest coding gain. Then we compared these largest coding gains with the largest coding gain achieved by any code satisfying the necessary conditions for maximum diversity gain. In many cases this enabled us to nd a code satisfying the sucient conditions for maximum diversity gain which also produced maximum coding gain of all codes achieving maximum diversity gain. Typically we 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]. Next we 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 previously discussed q = 2 Q? R with Q = 2 or 3 if the number of states is 4 or 16, while [9] q = 2 Q? R with Q = 2 or 3 if the number of states is 8 or 32. Our results for this case are summarized in Table 1. Besides providing ; AP (L) and CP (L), for convenience we provide the \e ective product distance", ePmin, from [6] in Table 1. To help the reader understand the process taken to arrive at Table 1, we describe some of the details. In particular, we found that there are 1840 di erent 4-state codes (not counting permutations of the columns of G) which satis ed the sucient conditions for maximum diversity gain while simultaneously providing the highest coding gain,  = 2, of any codes satisfying the sucient conditions p for maximum diversity gain. We found 288 codes providing the largest coding gain of 8 of all codes satisfying the necessary conditions for maximum diversity gain. The 4 state case was 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 sucient conditions for 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 performance (

1)

(

2) +1

4

In the count of 1840 given, several of the codes counted could be considered to be equivalent. The count is just meant to give a rough idea of the number of codes found. 4

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 sucient conditions for maximum diversity gain while simultaneously providing the largest value of  = 4 of all codes satisfying the sucient conditions for maximum diversity gain. No codes were found which achieved 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 necessary conditions for maximum diversity and not the sucient conditions. Of the codes achieving  = 4 and satisfying the sucient conditions for maximum diversity, we found a group of codes which provided the largest AP (2) = 3:42 and the largest CP (2) = 4:00 and one such code is listed in Table 1. For the 16-state case, we found 96 codes which satis ed the sucient conditions for p maximum diversity gain while simultaneously providing the largest value of  = 32. All of 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 were also able to show that any codes satisfying the necessary conditions for maximum diversity p could not provide  = 32 without yielding AP (3) < 5:76. For the 32-state case we found that the maximum coding gain is  = 6 and one such code with best CP (3) = 6:33 and best CP (4) = 8:72 is put in Table 1. Further this code satis es the sucient conditions for maximum diversity gain. We note that a few of our calculations concerning the cases in Table 1 di er from p those in [1] and [4]. The coding gain for the 8-state case from [1] is  = 12 based on our p calculations, as opposed to  = 20 as stated in [1]. This correction for this code was also p made in [4]. However, the new code in [4] gives a coding gain of  = 12 also, instead of p  = 4 as stated in [4]. Based on our calculations we nd  = 12 for the 32-state code from p [1], instead of  = 28 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? with Q = 2; 3; 4; 5 cases in the terminology of (5). For these 1

9

q  of the code from [1] our  ePmin AP (L) CP (L) 4 8 16 32

p

8

3:54

3:80

16

3:42

4:00

32

32

5:76

6:24

6

36

5:63

6:33

8

2

p

12

4

p

p

12

p

12

our GT 1 0 2 0 1 2 CA B@ 2 2 2 1 1 0 B@ 0 2 1 0 2 CA 2 1 0 2 2 1 0 0 2 1 1 2 0 CA B@ 2 2 1 2 0 2 1 0 2 0 1 2 1 2 2 CA B@ 2 2 0 1 2 0 2

Table 1: Optimum q-state 2 b/s/Hz 4-PSK space-time codes (ePmin from [6]).

q our  ePmin AP (L) CP (L) 2 4 4 8 16

4

4

4.00

12

7.33

8

7.33

80

20

10.06

112

28

11.38

p

48

p

32

p p

our GT 1 0 0 1 CA B@ 4.00 1 1 1y 0 0 1 1 CA B@ 6.93 1 0 1 1 0 B@ 0 1 1 CA 7.73 1 1 1 1 0 B@ 1 0 1 1 CA 10.47 1 1 0 1 1 0 1 1 0 1 1 CA 14.27 B @ 0 1 1 1 1

Table 2: Optimum q-state 1 b/s/Hz BPSK space-time codes. A "y" denotes that the code is catastrophic. 10

cases, Table 2 presents one representative scheme we have found using an approach similar to that taken for the QPSK space-time code cases just discussed. In each case, the selected code satis ed the sucient conditions for maximum diversity gain while simultaneously providing maximum values of , AP (L) and CP (L). These values are listed in Table 2. In cases where the codes which provide best ; AP (L) and CP (L) are catastrophic, the best non-catastrophic codes are also presented.

n > 2 Cases Some optimum (maximum diversity and coding gain) space-time codes using 3 or 4 transmit antennas and BPSK modulations are provided in Table 3 and Table 4. The searching procedure for these cases is similar to that for the n = 2 cases. In each case, the selected code satis ed the sucient conditions for maximum diversity gain while simultaneously providing maximum values of , AP (L) and CP (L). These values are listed in Table 3 and Table 4. A 16-state QPSK space-time code using 3 transmit antennas is given in Table 5. The code listed in Table 5 satis ed the sucient conditions for maximum diversity gain. We conjecture it achieves near optimum (if not optimum) coding gain.

5 Probability of Frame Error Performance Figure 1 shows the frame error rate of the space-time codes listed in Table 1 for cases with 2 transmit and 2 receive antennas. Figure 1 illustrates the gain achieved by increasing the constraint length of the codes. Figures 2 through 4 compare the performance of our codes and 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 130 transmissions 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. The 11

q our  ePmin AP (L) CP (L) 4

64 27

5.08

8 256 31

256 27

7.65

8 192 31

64 9

7.65

512 27

10.00

16

4

8

0 our G 1 BB 0 1 1 CC BB 1 0 1 CC 5.08 CA B@ 1 1 1 1y 0 BB 1 0 0 1 CC BB 1 0 1 0 CC 7.05 CA B@ 01 1 1 1 1 BB 1 1 1 0 CC BB 1 1 0 1 CC 8.32 CA B@ 0 1 0 1 1 1 BB 1 0 0 1 1 CC 10.18 B BB 1 1 0 1 0 CCC A @ T

1 1 1 0 1

Table 3: Optimum q-state 1 b/s/Hz BPSK 3-space-time codes. A "y" denotes that the code is catastrophic.

12

q our  ePmin AP (L) CP (L) 8

4

1

5.97

16 1280 41

5

8.11

16 1024 41

4

7.99

32 4352 41

17

9.80

0 our G 1 BB 0 1 0 1 CC BB 0 1 1 1 CC C BB 5.97 BB 1 0 1 0 CCC CA B@ 1 1 1 0 1y 0 BB 1 0 0 0 1 CC BB 1 0 1 1 1 CC C BB 8.37 BB 1 1 0 1 1 CCC CA B@ 01 1 1 1 0 1 BB 0 1 1 0 1 CC BB 1 1 0 0 1 CC C BB 9.32 BB 1 1 1 1 0 CCC CA B@ 0 1 1 1 1 1 1 BB 1 0 0 0 0 1 CC BB 1 0 1 1 1 1 CC CC BB 10.38 B BB 1 1 1 0 1 0 CCC A @ T

1 1 1 1 0 0

Table 4: Optimum q-state 1 b/s/Hz BPSK 4-space-time codes. A "y" denotes that the code is catastrophic.

q our  ePmin AP (L) CP (L) 0 our GT 1 0 2 1 2 2 0 CC BB 1 C B 16 32 3 3.90 4.72 B B@ 1 2 2 0 0 2 CCA 2 2 0 2 1 2 256 27

Table 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 codeword pair that makes (2) the largest, is very useful, but it does not completely determine the FER of 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 in Figure 2 as an example. Consider the 4 state code

1 0 2 0 1 2 CA GT = B @ 2 2 2 1

(9)

p

given in Table 1. Recall this code provides a coding gain of 8. Contrast this with the code given in [4]

1 0 2 0 1 3 CA GT = B @

p

2 2 0 1

(10)

which also provides a coding gain of 8. 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 2 verify that (9) outperforms (10) . The results in Figure 2 also verify that the code in (9) outperforms the code

1 0 0 2 2 3 CA GT = B @ 2 3 0 2

discussed 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] which provides  = 2 and AP (2) = 2:59 and CP (2) = 2:67. From the above example, one might expect that the coding gain is more important than the average values AP (L) or CP (L) for the code to perform well, but this is not always true. Figure 5 compares the simulation results of our codes with optimum coding gain (from Table 2) with the codes in [5] which are constructed from the known binary convolutional codes that achieve optimal values of free distance dfree. Only in one case 14

do our codes with optimum coding gain outperform the codes in [5], although generally the di erences appear to be small in every case we tested. Again, this may be explained by using the average values CP (L) and CP (L + 1). In particular, the 4-state code in [5] provides better AP (2) = CP (2) = 7:73 and outperforms the code in Table 2 (with AP (2) = 7:33 and 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 performs better. In all the other cases we compared, we found that the codes in [5] always provide larger CP (L) with smaller coding gain . It is observed that as we increased the number of states the FER performance comparison between our new codes and those provided in [5] changes with a clear trend. For cases 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, the number 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 cases with a larger number of states, the pool of candidate codes that maximize the rst two performance metrics, diversity gain and coding gain, is larger. Therefore, we have a good chance of being able to choose codes that also provide large AP (L) and CP (L). Although in this paper 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 by optimum we mean optimum diversity gain and optimum coding gain), this will not always be the best ordering. To avoid the possible pitfall that can occur for cases with a small number of states, one may put more emphasis on maximizing AP (L) and CP (L), instead of emphasising maximizing the worst case coding gain, and the general method provided in this paper is still valid. 5

p

p

p

Figure 5 considers BPSK cases from [5]. For q = 4; 8; 16 cases with 2 antennas,  = 32; 48; 96, respectively. For q = 8; 16 cases with 3 antennas,  = 128 31 and 256 31 , respectively. 5

15

6 Conclusions Space-time convolutional codes which provide maximum diversity and coding gain were found for some practical cases with 2, 3 and 4 antennas, various number of states and various numbers of input bits per time slot. The optimum codes produced here are found using a new procedure given in [9], which is much more ecient than previous approaches. Further, a new performance measure is suggested to augment coding gain and diversity gain. Extensive simulation results showing frame error rate, justify the utility of the approach. The results show the performance of each code we produce is the best or very close to the best we found by trying many codes. The overall approach is very useful for selecting a handful of good codes. 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 limitations and that simulations to verify the selection of a space-time code are always prudent. The investigations here have been limited to cases with relatively small array sizes. For larger arrays, the considerations in [21, 22] should be considered.

References [1] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate wireless 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 modem for 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 high data rate wireless communication: performance criteria in the presence of channel es-

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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 Coded Modulation", IEEE Communication Letters, Vol.4, No.1, 2000. [5] A. R. Hammons, H. E. Gammal, \On the Theory of Space-Time Codes for PSK Modulation", 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 for Rayleigh fading", in Proc. 1998 Allerton Conference. [7] J. Grimm,\Transmitter diversity code design for achieving full diversity on Rayleigh channels", 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," submitted to IEEE Trans. on Information Theory and similar work in the proceedings of Conference 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 Diversity 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 wireless communications: performance results," IEEE J. Select. Commun., vol. 17, No. 3, March 1999, pp. 451-460. [12] V. Tarokh, H. Jafarkhani, A. R. Calderbank, "Space-time block codes from orthogonal designs," IEEE Trans. on Information Theory, vol. 45, No. 5, July 1999, pp. 1456-1467. 17

[13] A. Wittneben, \A new bandwidth ecient transmit antenna modulation diversity scheme for linear digital modulation," IEEE ICC, pp. 1630-1634, May 1993. [14] V. Weerackody, \Diversity for direct-sequence spread spectrum system using multiple antennas." IEEE ICC, pp. 1775-1779, May 1993. [15] N. Seshadri and J. H. Winters, \Two signaling schemes for improving the error performance 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 fading environment 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," IEEE Transactions on Communications, COM-46, pp. 357-366, March 1998. [19] B. L. Hughes, \Di erential space-time modulation", in Proc. 1999 Wireless Communications 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 scheme on slow Rayleigh fading channels," in Proc. of International Conference on Communications, Helsinki, Finland, June 11-14, 2001. [22] E. Biglieri and A. M. Tulino, \Designing space-time codes for a large number of receiving antennas," In Proc. of the 35th Annual Conference on Information Sciences and Systems, Baltimore, MD, March 21-23, 2001. 18

Qing Yan received the B.S. and M.S. degrees in electrical engineering from University of Science and Technology of China (USTC), Hefei, in 1995 and 1998, respectively. Since 1998, he has been pursuing the Ph.D. degree at the Electrical Engineering and Computer Science Department, Lehigh University, Bethlehem, PA. He is currently a Research Assistant with the Signal Processing and Communications Research Laboratory, Lehigh University. His research interests include the application of signal processing theory to the design and analysis of communication systems. Rick S. Blum (M'91) received a B.S. in Electrical Engineering from the Pennsylvania State University in 1984 and his M.S. and Ph.D in Electrical Engineering from the University of Pennsylvania in 1987 and 1991. From 1984 to 1991 he was a member of technical sta at General Electric Aerospace in 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 Department at Lehigh University in Bethlehem, Pennsylvania where he is currently an Associate Professor and holds a Class of 1961 Professorship. His research interests include signal detection and estimation and related topics in the areas of signal processing and communications. He is currently an associate editor for the IEEE Transactions on Signal Processing and for IEEE Communications Letters. He is also a member of the Signal Processing for Communications Technical Committee of the IEEE Signal Processing Society. Dr. Blum is a member of Eta Kappa Nu and Sigma Xi, and holds a patent for a parallel signal and image processor architecture. He was awarded an ONR Young Investigator Award in 1997 and an NSF Research Initiation Award in 1992.

19

List of Tables 1 2 3 4 5

Optimum q-state 2 b/s/Hz 4-PSK space-time codes (ePmin from [6]). . . . . Optimum q-state 1 b/s/Hz BPSK space-time codes. A "y" denotes that the code is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum q-state 1 b/s/Hz BPSK 3-space-time codes. A "y" denotes that the code is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimum q-state 1 b/s/Hz BPSK 4-space-time codes. A "y" denotes that the code is catastrophic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A q-state 2 b/s/Hz 4-PSK space-time code using 3 transmit antennas. . . . .

20

10 10 12 13 13

List of Figures 1

Performance comparison of some best 2 b/s/Hz, QPSK, q-state Space-time Codes with 2 transmit and 2 receive antennas (SNR per receive antenna = nEs=N ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance comparisons of some 2 b/s/Hz, QPSK, 4-state Space-time Codes with 2 transmit and 2 receive antennas. . . . . . . . . . . . . . . . . . . . . . Performance comparisons of some 2 b/s/Hz, QPSK, 8-state Space-time Codes with 2 transmit and 2 receive antennas. Zero symmetry (ZS) is de ned in [6]. Performance comparisons of some 2 b/s/Hz, QPSK, 16-and 32-state Spacetime Codes with 2 transmit and 2 receive antennas. . . . . . . . . . . . . . . Performance of some 1 b/s/Hz, BPSK, q-state Space-time Codes with 2 transmit and 2 receive or 3 transmit and 3 receive antennas (performance of the best non-catastrophic codes similar). . . . . . . . . . . . . . . . . . . . . . . 0

2 3 4 5

21

22 23 24 25

26

0

10

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

−1

Frame Error Probability

10

−2

10

−3

10

5

5.5

6

6.5

7 7.5 8 SNR per receive antenna (dB)

8.5

9

9.5

10

Figure 1: Performance comparison of some best 2 b/s/Hz, QPSK, q-state Space-time Codes with 2 transmit and 2 receive antennas (SNR per receive antenna = nEs=N ). 0

22

0

10

Frame Error Probability

q=4 q=4 q=4 q=4 q=4

[1] [4] [6] [8] optimum

from [4] from [6]

−1

10

−2

10

5

5.5

6

6.5

7 7.5 8 8.5 SNR per receive antenna (dB), q=4

9

9.5

10

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

23

0

10

Frame Error Probability

[6], ZS [1] [4] [6],non−ZS optimum

−1

10

from [4]

from [1]

−2

10

5

5.5

6

6.5

7 7.5 8 8.5 SNR per receive antenna (dB), q=8

9

9.5

10

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

24

0

10

q=16 q=16 q=16 q=32 q=32

in [1] in [4] optimum in [1] optimum

−1

Frame Error Probability

10

−2

10

−3

10

5

5.5

6

6.5

7 7.5 8 SNR per receive antenna (dB)

8.5

9

9.5

10

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

25

0

0

10

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

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

−1

−1

10

Frame Error Probability

Frame Error Probability

10

−2

10

q=16 in [6]

−3

−2

10

−3

10

10 q=16 opt η

−4

10

−4

10

5 6 7 8 9 10 SNR per receive antenna (dB), Tx=2, Rx=2

0 1 2 3 4 5 SNR per receive antenna (dB), Tx=3, Rx=3

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

26