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IEEE COMMUNICATIONS LETTERS, VOL. XX, NO. Y, MONTH 2000 1

Space-Time Block Codes versus Space-Time

Trellis Codes

Sumeet Sandhu, Robert W. Heath Jr., Arogyaswami Paulraj

The authors are with Information Systems Laboratory, Stanford University, Stanford, CA 94305. R. Heath was

supported by Ericsson Inc. through the Networking Research Center at Stanford University.

November 9, 2000 DRAFT

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Abstract

Transmit diversity schemes for the coherent multiple-antenna flat-fading channel range from space-

time block codes (STBC) to space-time trellis codes (STTC). In this letter we compare the performance

of STBC and STTC by means of frame error rate. We discover that a simple concatenation of STBCwith traditional AWGN (additive white Gaussian noise) trellis codes outperforms some of the best known

STTCs at SNRs (signal to noise ratios) of interest. Our result holds for small numbers of receive antennas

and trellis states, and may extend to greater numbers of antennas and states with improved block codes.

Keywords

Block codes, Diversity methods, MIMO systems, Modulation coding, Fading channels.

I. Introduction

The challenge of transmit diversity design for the multiple-antenna fading channel has been

met with several novel modulation and error correction techniques in the recent past. Prominent

among these space-time block codes ([1], [2]) and space-time trellis codes ([3], [4], [5], [6]).

Space-time block codes operate on a block of input symbols producing a matrix output over

antennas and time. Unlike traditional single-antenna AWGN block codes, full rate space-time

block codes do not provide coding gain. Their key feature is the provision of full diversity with

extremely low encoder/decoder complexity. The best known orthogonal block codes are provided

in [2].

Space-time trellis codes operate on one input symbol at a time producing a sequence of spatial

vector outputs. Like traditional TCM (trellis coded modulation) for the single-antenna channel,

space-time trellis codes provide coding gain. Since they also provide full diversity gain, their

key advantage over space-time block codes is the provision of coding gain. However, they are

extremely difficult to design and require an expensive encoder and decoder.

This paper proposes a compromise between STBC (known designs, simple ML (maximum

likelihood) decoding, no coding gain) and STTC (difficult to design, expensive ML decoding,

coding gain). We concatenate AWGN trellis codes with STBC in order to obtain coding gain.

Then we compare the performance of concatenated STBC with STTC by keeping transmit power

and spectral efficiency fixed. The results are somewhat surprising - with the same number of

trellis states (4 and 8), concatenated STBC outperforms STTC for 1 and 2 receive antennas.

Concatenation of TCM with STBC was also proposed in [7] but its performance was not com-

pared with STTC. A performance comparison of STTC and concatenated STBC was provided


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in [8], but it was not fair in terms of date rate and number of trellis states. A fair comparison

of STTC and concatenated STBC was provided in [9] for the fast fading channel but not for the

quasi-static channel.

II. Data and Channel Model

Consider a system with Mr receive antennas and Mt > 1 transmit antennas. The channel is

flat-fading and quasi-static. It is unknown at the transmitter but is known at the receiver. At

time nT, the channel output corresponding to the nth input block spanningTsymbol times is

YnT = HXnT+ VnT (1)

where the received signal YnT is MrT, the fading channel H is MrMt, the encoded codewordXnT isMtT, and receiver noise VnT isMrT. The entries ofH are i.i.d.(independent, identicallydistributed) circular complex Gaussian random variables with variance 0.5 in each dimension,

i.e. hi,j c(0, 1). The entries ofVnT are i.i.d., vnTi,j c(0,N0), and independent over n.The average power transmitted on Mt antennas is Es. The codeword XnT is encoded using

concatenated STBC or STTC, both of which are described in detail in the following sections.

III. Space-Time Block Codes

The input to the encoder is a stream of real or complex modulated symbols. The encoder

operates on a block ofKsymbols producing an MtT codeword XnTwhose rows correspondto transmit antennas and columns correspond to symbol times. At the receiver, ML decoding is

simplified by the orthogonal structure of the codewords.

The effective channel induced by space-time block coding of input symbols (before ML de-

tection) is an AWGN channel with receive SNR equal to ||H||2FEs

MtN0[10], [11]. This motivates the

concatenation of traditional single-antenna TCM with STBC. The system used here consists

of an outer TCM encoder/decoder concatenated with the STBC encoder/decoder. The overall

coding gain is only due to the outer TCM encoder since we consider full rate STBCs.Recently, new block codes were designed in [12] by maximizing average capacity, some of which

outperform the orthogonal codes in [2]. Here we will only consider the codes in [2], which are the

best orthogonal linear codes with respect to the union bound on symbol error probability [13].


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IV. Space-Time Trellis Codes

Space-time trellis codes encode the input scalar symbol stream into an output vector symbol

stream. Unlike space-time block codes, space-time trellis codes map one input symbol at a time

to an Mt1 vector output. Decoding is performed via ML sequence estimation.Code performance is quantified by the diversity advantageand the coding advantage [4]. For

a given number of transmit antennas, the design objective is to construct the largest possible

codebook with full diversity advantage (= Mt) and the maximum possible coding advantage.

A number of hand-crafted codes with full diversity advantage were provided in [4]. Codes with

greater coding advantage than those in [4] were reported in [5] and [14] after exhaustive computer

searches over a feedforward convolutional coding (FFC) generator. A structured method of code

construction that ensures full diversity was provided in [15] along with new designs. Recently

new codes with better distance spectrum properties and performance were reported in [6].

V. Performance Analysis

Note that any space-time code can be analyzed in terms of the criteria presented for space-

time trellis codes, namely diversity advantage and coding advantage. These criteria affect the

performance curve in different ways. Diversity advantage affects the asymptotic slope of the

FER (frame error rate) versus SNR graph - greater the diversity, the more negative the slope.

Coding advantage affects the horizontal shift of the graph - greater the coding advantage, thegreater the shift to the left.

Since all codes considered here are full diversity, the asymptotic slopes of their FER graphs

will be the same. The difference in coding advantage can be quantified as follows. At high SNRs

the logarithm of the minimum PEP for the kth code isPk MrMtskMrMtck whereMrMtisthe diversity advantage, sk =log(

Es4N0

) is the SNR term, and ck is the coding advantage. Defining

P =PkPl, c = ckcl, and s = sksl for codes k and l we have

P MrMts MrMtc (2)

At a given SNR, s = 0, and if c > 0 then the PEP for k is lower than that for l by

P MrMtc. Therefore the effect of improved coding advantage increases linearly with thenumber of receive antennas, as also observed in [14].


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A. Coding Advantage

We will only consider full diversity codes for 2 transmit antennas at a bit rate of 2 bits per

symbol time. We will focus on a small number of trellis states, equal to 4 and 8, both for STTCs

and concatenated STBCs. In order to predict the performance of different codes, we computed

their coding advantages by means of a computer search and listed them in Table I. These values

agree with those found in published literature [16], [5], [6]. Note that the values in Table I are

normalized by Mt = 2.

The block code labeled STBC stands for the complex Mt= 2 Alamouti code [1] and has a

4-PSK constellation. The codes labeled STBC-TCM represent the Alamouti code concatenated

with outer AWGN trellis codes and have 8-PSK constellations. The outer codes are both rate 2/3

and are the best 4-state (parallel transition) and 8-state trellis codes in [16]. The codes labeled

STTC denote space-time trellis codes, of which the 4 and 8 state codes in [4] (STTC-Tarokh),

[5] (STTC-Grimm), and [6] (STTC-Yan) are shown, all of which have 4-PSK constellations. The

4-state Grimm and Yan codes have the best possible coding advantage in the class of FFC codes

[5].

As demonstrated in [6], although the 4 and 8 state Yan codes have the same coding advantage

as the Grimm codes, they outperform the Grimm codes because of better distance spectrum

properties. Since the Grimm codes were shown to outperform the Tarokh codes [14], [6], we will

only consider the Grimm and Yan codes in the sequel.

VI. Simulations

In this section Monte Carlo simulations of the codes in Table I are presented. Performance

is measured in terms of FER for a burst of 130 symbols. The Rayleigh channel described

in (1) is used with two transmit antennas. Channel realizations are i.i.d.from burst to burst.

Performance is plotted for one, two and three receive antennas in Figures 1, 2 and 3 respectively.

The performance of STBC is plotted in all the figures for reference.

In Figure 1 (a) note that STBC by itself performs as well or better than all the STTCs, even

though it provides no coding gain. This is explained by the multidimensional structure of STBC

that spans two time symbols.

With one receive antenna, concatenated STBC performs dramatically better than all the 4-

state STTCs. The performance gap reduces with increasing receive antennas in Figures 2 and 3.


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The performance gap is also less noticeable with 8-state codes for all receive antennas.

With three receive antennas and 8 trellis states, in fact, STTC-Yan outperforms STBC-TCM.

The loss in performance of STBC-TCM with increasing receive antennas is explained by the

capacity loss incurred by STBC [11]. Considering the capacity-optimized block codes in [12], we

expect that those block codes will outperform STTC-Yan for even greater numbers of receive

antennas and trellis states.

Note that in all these comparisons, the encoding/decoding complexity of STTCs is much

higher than that of concatenated STBCs for large numbers of trellis states and multiple receive

antennas. A comparison that is fair with respect to complexity would entail concatenation of

more sophisticated codes with improved block codes, potentially extending their performance

advantage to multiple receive antennas.

VII. Conclusions

We have shown that for a small number of receive antennas and trellis states, a simple concate-

nation of space-time block codes with traditional AWGN trellis codes can significantly outperform

space-time trellis codes. The concatenated scheme separates spatial modulation from temporal

error correction and is therefore attractive from the aspect of computational complexity as well.

References

[1] S. M. Alamouti, A simple transmitter diversity scheme for wireless communications, IEEE J. Select. Areas

in Comm., vol. 16, pp. 14511458, October 1998.

[2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, Space-time block codes from orthogonal designs, IEEE

Trans. Information Theory, vol. 45, pp. 14561467, July 1999.

[3] J. C. Guey, M. P. Fitz, M. R. Bell, and W. Y. Kuo, Signal design for transmitter diversity wireless commu-

nication systems over Rayleigh fading channels, in Proc. VTC, vol. I, pp. 136140, 1996.

[4] V. Tarokh, N. Seshadri, and A. R. Calderbank, Space-time codes for high data rate wireless communication

I: performance criterion and code construction, IEEE Trans. Information Theory, vol. 44, pp. 74465, March

1998.

[5] J. Grimm, Transmitter diversity code design for achieving full diversity on Rayleigh fading channels. PhDthesis, Purdue University, 1998.

[6] Q. Yan and R. S. Blum, Optimum space-time convolutional codes, in Proc. WCNC, 2000.

[7] S. M. Alamouti, V. Tarokh, and P. Poon, Trellis-coded modulation and transmit diversity : design criteria

and performance evaluation, in Proc. ICUPC, 1998.

[8] J.-C. Guey, Concatenated coding for transmit diversity systems, inProc. VTC, vol. 5, pp. 25002504, 1999.


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[9] A. Yongacoglu and M. Siala, Space-time codes for fading channels, in Proc. VTC, vol. 5, pp. 24952499,

1999.

[10] G. Ganesan and P. Stoica, Space-time diversity using orthogonal and amicable orthogonal designs, in

Proc. ICASSP, 2000.

[11] S. Sandhu and A. Paulraj, Space-time block codes : a capacity perspective. Accepted IEEE Comm. Letters,

June 2000.

[12] B. Hassibi and B. M. Hochwald, High-rate codes that are linear in space and time, in Proc. 38th Annual

Allerton Conference on Communication, Control and Computing, 2000.

[13] S. Sandhu, R. Heath, and A. Paulraj, Union bound for linear space-time codes, in Proc. 38th Annual

Allerton Conference on Communication, Control and Computing, 2000.

[14] S. Baro, G. Bauch, and A. Hansmann, Improved codes for space-time trellis-coded modulation, IEEE

Comm. Lett., vol. 4, pp. 2022, January 2000.

[15] A. R. Hammons and H. E. Gamal, On the theory of space-time codes for PSK modulation,IEEE Trans.

Information Theory, vol. 46, pp. 524542, March 2000.

[16] E. Biglieri, D. Divsalar, P. J. McLane, and M. K. Simon, Introduction to Trellis-Coded Modulation with

Applications. Macmillan, 1991.


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Table I : Coding Advantage : 2 transmit antennas, 2 bits/symbol time

Figure 1 : One receive antenna

Figure 2 : Two receive antennas

Figure 3 : Three receive antennas


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TABLE I

Code States Coding Advantage in dB

STBC 1 1 0

STBC-TCM 4 2 3.01

STTC-Tarokh 4 1 0

STTC-Grimm 4

2 1.51

STTC-Yan 4

2 1.51

STBC-TCM 8

4.5858 3.31

STTC-Tarokh 8

3 2.39

STTC-Grimm 8 2 3.01

STTC-Yan 8 2 3.01

Fig. 1.

10 12 14 16 1810

3

102

101

100

(a) 4 state codes

SNR in dB

FER

STBCSTBC + 4state TCMGrimm 4stateYan 4state

10 12 14 16 1810

3

102

101

100

(b) 8 state codes

SNR in dB

FER



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Fig. 2. Two receive antennas

4 6 8 10 1210

3

102

101

100

(a) 4 state codes

SNR in dB

FER


4 6 8 10 1210

3

102

101

100

(b) 8 state codes

SNR in dB

FER



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Fig. 3. Three receive antennas

0 2 4 6 810

3

102

101

100

(a) 4 state codes

SNR in dB

FER


0 2 4 6 810

3

102

101

100

(b) 8 state codes

SNR in dB

FER



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