Space-Time Transmit Diversity Combined with Frequency-domain Equalizationfor Single-carrier Transmission
Kazuaki TAKEDA+, Takeshi ITAGAKI+, and Fumiyuki ADACHI++
Dept. of Electrical and Communication Engineering, Graduate School of Engineering, Tohoku University
05 Aza-Aoba, Aramaki, Aoba-ku, Sendai, 980-8579 Japan
E-mail: +{takeda, itagaki}@mobile.ecei.tohoku.ac.jp, ++adachi@ecei.tohoku.ac.jp
Abstract
In a frequency-selective fading channel, the transmission
performance of single-carrier (SC) transmission significantly
degrades due to severe inter-symbol-interference (ISI). In thispaper, the joint use of space-time transmit diversity (STTD),
receive antenna diversity and frequency-domain minimum
mean square error (MMSE) equalization in SC transmission isstudied. Space-time encoding and decoding for
frequency-domain MMSE equalization is proposed. The bit
error rate (BER) performance of SC transmission is evaluatedby computer simulation. It is found that joint use of STTD,
receive antenna diversity and frequency-domain MMSE
equalization can significantly improve the BER performancein a severe frequency-selective fading channel.
Keywords
Single carrier transmission, space-time transmit diversity,
frequency-domain equalization, frequency-selective fading
1.

Introduction

Wireless channel is composed of many propagation paths
with different time delays, producing frequency-selective
multipath fading [1]. In a frequency-selective fading channel,
the bit error rate (BER) performance of single-carrier (SC)transmission significantly degrades due to severeinter-symbol-inference (ISI). Direct-sequence code divisionmultiple access (DS-CDMA) can exploit the channel
frequency-selectivity by the use of rake receiver that resolves
the propagation paths having different time delays andcoherently combines them to achieve the path diversity effect[2]. Wideband DS-CDMA has been adopted as a wirelessaccess technique in the 3rd generation mobile communications
systems, known as IMT-2000 systems, for data transmissionsof up to a few Mbps [3]. However, in the case of broadband
wireless data transmissions more than a few Mbps using
DS-CDMA, the transmission performance may significantly
degrade due to large inter-path interference (IPI) even if
coherent rake combining is used.
Recently, multi-carrier code division multiple access(MC-CDMA) has been attracting much attention forbroadband wireless data transmissions in a severe
frequency-selective channel [4~6]. In MC-CDMA, thefrequency diversity effect is attained by one-tap
frequency-domain equalization, resulting in a better BER
performance than DS-CDMA using coherent rake combining[7]. However, the MC-CDMA has a problem of large
peak-to-average power ratio (PAPR) and thus, a linear
transmit power amplifier with a large peak power is required.
Quite
recently,SCtransmission
using
one-tap
frequency-domain equalization has been gaining an increasing
popularity [8]. SC transmission has an advantage that theproblem of high PAPR can be avoided and the computational

complexity of frequency-domain equalization does notdepend on the degree of channel frequency-selectivity.
Transmit/receive antenna diversity is a well-known
effective technique to improve the transmission performance[9]. In this paper, we apply two-antenna space-time transmit
diversity (STTD) [10] to further improve the BER
performance of SC transmission with frequency-domain
equalization. Space-time encoding and decoding combinedwith frequency-domain equalization is proposed. Remainderof this paper is organized as follows. Section 2 presents
space-time encoding and decoding combined withfrequency-domain minimum mean square error (MMSE)
equalization. In Section 3, the achievable BER performance ina frequency-selective Rayleigh fading channel is evaluated bycomputer simulation. The BER performance achievable with
STTD combined with frequency-domain MMSE equalizationis compared with that of simple delay transmit diversity(DTD) [11]. Section 4 offers some conclusions and futurework.
2.

STTD Combined with Frequency-domain
Equalization

2.1 Space-time encoding suitable for frequency-domain
MMSE equalization
Figure 1 illustrates the transmitter and receiver structures
for SC transmission with STTD combined withfrequency-domain equalization. Throughout the paper, a
discrete time representation of the signal is used. The data
symbol sequence to be transmitted is transformed into asequence of data blocks, each with Nc symbols. Nc-data
symbol sequence of the q-th block is denoted by{d (t ); t = qN c ~ (q + 1) N c − 1} . The SC signal samplesequence {s (t ); t = qN c ~ (q + 1) N c − 1} of the q-th block isexpressed using the equivalent lowpass representation ass(t ) = 2 Es / T d (t ) , (1)where Es is the transmit signal energy per symbol and T is thesymbol length.
We extend Alamouti’s STTD encoding [10] to the SC
transmission using frequency-domain equalization. The signal
blocks {se(t)} and {so(t)} which represent the even (q=2u) andodd (q=2u+1) signal blocks, respectively, are transmitted
simultaneously after insertion of guard interval (GI) from two
transmit antennas during the time interval of two consecutive(even and odd) data blocks. Let the subcarrier components ofthe even (q=2u) and odd (q=2u+1) signal blocks be denotedby {Se(k)} and {So(k)}, respectively (note that although SC
transmission does not use subcarriers for modulation, theterminology “subcarrier” is used for explanation purposeonly). They are given byTwo data blocks­§t ·° Se ( k ) =¸se (t ) exp¨¨ − j 2πk°N c ¸¹©
°t =0®
N c −1°§t ·¸so (t ) exp¨¨ − j 2πk°S o ( k ) =¸N°¯c©
¹t =0
N c −1¦(2)

STTD encoded signal blocks

evenodd

se(t)

so(t)

t

Nc symbols

STTDencoding¦− so∗ ( N c − t )

even

odd

so(t)

se∗ ( N c − t )GIn=1se∗ ( N c − t )

so(t)

t

Nc +Ng symbols

Fig.2 STTD encoding process for transmission from two
transmit antennas.
Table 1Time (in block)

STTD encoding
Antenna index n01

Even (q=2u)

Se(k)

So(k)

¦Odd (q=2u+1)− So∗ (k )Se∗ (k )¦Table 2

N c −1­§k · *° 1¸ = se ( N c − t )Se* (k ) exp¨¨ j 2πt° Nc
N c ¸¹©
°k =0(3)®
N −1° 1 c *§k · *¸ = so ( N c − t )So (k ) exp¨¨ j 2πt°
N c ¸¹°¯ N c k = 0©

Equivalent time-domain STTD encoding
Antenna index nTime (in block)01

for t=0~Nc-1, where ( )* denotes the complex conjugate.
Hence, STTD encoding of Table 1 can be replaced bytime-domain STTD encoding shown in Table 2. STTD
encoding process is shown in Fig.2. The last Ng symbols ineach STTD encoded signal block are copied and inserted into
the beginning of each block as the GI. The GI-inserted signal
sample streams are simultaneously transmitted from twoantennas over a frequency-selective fading channel.

Datamodulation

Data

STTD
encoding

n=0
n=1

Insertionof GI

S/P conversion

IFFT

FFT

S/P conversion

. . .

. . .

η0(t)

. . .
. . .

…

RemovalRemovalof GIof GI
ηNr-1(t)STTD decoding withMMSE equalization(a) Transmitter
m=Nr-1Data
Data
de-modulation(b) ReceiverFig.1 Transmitter and receiver with joint STTD andfrequency-domain equalization.

tAntennas

Insertionof GI

presented. Applying Nc-point IFFT to {Se∗ ( k )} and {So∗ ( k )} ,we obtainInsertion
of GI

t

n=0− s o∗ ( N c − t ) tse(t)for k=0~Nc-1. The STTD encoding using {Se(k)} and {So(k)}is carried out subcarrier-by-subcarrier as shown in Table 1.
This subcarrier-by-subcarrier STTD encoding allows
decoding to be combined with frequency-domain equalization(this is described in Sect 2.2).
The above STTD encoding requires the Nc-point fast
Fourier transform (FFT) and inverse FFT (IFFT) operations toobtain {Se(k)} and {So(k)} and transform them back totime-domain signals. Below, an equivalent time-domain
STTD encoding that requires no FFT and IFFT operations is

m=0

se(t)

Even (q=2u)

se(t)

so(t)

Odd (q=2u+1)− so∗ ( N c − t )se∗ ( N c − t )2.2 Joint STTD decoding and frequency-domain MMSEequalization
The STTD encoded signal sample streams transmitted fromthe two antennas are assumed to be received by Nr antennas ata receiver. The propagation channel is assumed to be asymbol-spaced L-path frequency-selective fading channel,
each path being subjected to independent fading. Block fading,where the path gains stay constant over two (even andodd)-block interval, has been assumed. After removal of GI,
the even and odd block signal sequences received on the m-thantenna are decomposed by Nc-point FFT into Nc subcarriercomponents {Re,m(k); k=0~Nc-1} and {Ro,m(k); k=0~Nc-1},
respectively. Re,m(k) and Ro,m(k) can be written as­ R ( k ) = H ( k ) S ( k ) + H ( k ) S ( k ) + N (k )e1, moe, m0, m° e, m,(4)®
∗∗°̄Ro, m (k ) = − H 0, m (k ) So (k ) + H1, m (k ) Se (k ) + N o, m (k )where {H0,m(k)} and {H1,m(k)} represent respectively theNc-point Fourier transforms of the channel gains associatedwith the 0-th and the 1st transmit antennas, and {Ne,m(k)} and{No,m(k)} respectively represent the Fourier transforms ofnoise processes. One-tap frequency-domain equalization iscarried out jointly with STTD decoding and receive antenna~~
diversity combining to obtain Se (k ) and So (k ) :{}{}

N r −1­~° Se ( k ) =w0∗, m (k ) Re, m (k ) + w1, m (k ) Ro∗, m (k )°°m =0®
N r −1°~w1∗, m (k ) Re, m (k ) − w0, m (k ) Ro∗, m (k )° So ( k ) =°¯m =0

¦{}

¦{}

Table 3
Simulation parameters
Modulation
BPSK(5)
Transmitterfor k=0~Nc-1, where w0,m(k) and w1,m(k) are the STTD
decoding weights. Direct application of decoding presented in[10] gives­w( MRC ) (k ) = H 0, m (k )° 0, m

H 0, m ( k )­ ( MMSE )(k ) =°w0, m−1
1 N r −1°2 §¨ 1 Es ·¸+|H(k)|nm,°¨2 N ¸0¹©
°
n = 0 m =0,(7)®
H1, m (k )°w( MMSE ) (k ) =−1° 1, m
1 N r −1
§ 1 Es ·°¸| H n, m (k ) |2 + ¨¨¸°
© 2 N0 ¹
n = 0 m =0¯¦¦¦¦

where Es/N0 is the average received signal energy per
symbol-to-AWGN power spectrum density ratio per receive
antenna. Removing the second term in the denominator of Eq.
(7) leads to zero-forcing (ZF) weights:
H 0, m ( k )­ ( ZF )°w0, m (k ) = 1 N r −1°| H n, m (k ) |2°°n =0 m =0(8)®
H1, m (k )°w( ZF ) (k ) =° 1, m
1 N r −1°
| H n, m (k ) |2°n =0 m =0¯¦¦¦¦~~
Nc-point IFFT is applied to {Se (k )} and {So (k )} toobtain the time-domain soft decision sample sequences used
for the data demodulation.

3.

Computer simulation

The average BER performance of SC transmission withjoint STTD and frequency-domain equalization is evaluatedby computer simulation. The simulation parameters are given
in Table 3. Binary phase shift keying (BPSK) data modulation,
Ng=32,andasymbol-spaced
L-pathNc=256,frequency-selective block Rayleigh fading having an
exponential power delay profile with decay factor α dB areassumed. Ideal sampling timing and ideal channel estimation
are also assumed at the receiver.

Nc=256GI

Ng=32(symbols)

Transmit diversity

STTD

FadingFrequency -selectiveblock Rayleigh fading

ChannelPower delay profile, (6)® ( MRC )°̄w1, m (k ) = H1, m ( k )which are the maximal-ratio combining (MRC) weights.
Another solution is that the weights are chosen so that the~mean square error (MSE) between Se (k ) and Se(k) and that~between So (k ) and So(k) are jointly minimized. Following [6],we can obtain

Number of FFT
points

Receiver

L=16-path exponential
power delay profile
Decay factor
α=0,2,8,∞(dB)Number of receiveantennas

Nr=1,2,4

Frequency-domain
Equalization

MMSE, MRC, ZF

Channel estimation

Ideal

3.1 BER performance evaluation
How STTD combined with frequency-domain MRC,MMSE or ZF equalization improves the average BER
performance is discussed. Figure 4 plots the average BERperformance with the decay factor α dB as a parameter as afunction of the average signal energy per information
bit-to-AWGN power spectrum density ratio Eb/N0, which isgiven by Eb/N0=(1+Ng/Nc)(Es/N0). It is clearly seen that
MMSE equalization gives the better BER performance than
MRC and ZF equalization. Using MRC, BER floors are seen
when α=0, 2 and 8dB due to the large ISI produced by theenhanced frequency-selectivity. MMSE equalization cansuppress the noise enhancement while almost restoring thefrequency non-selective channel. Hence, irrespective of thedegree of channel frequency-selectivity, the BER performancewith MMSE equalization is always better than ZF
equalization. Hence, in the following, we use the MMSEequalization only.
The BER performances with STTD and without STTD
are compared in Fig. 5 with the decay factor α dB as aparameter. Since frequency-domain MMSE equalization canachieve frequency diversity effect by taking advantage offrequency-selective fading, α=0 dB gives better BER
performance than the case of α=8 dB. Joint STTD andfrequency-domain MMSE equalization gives further
improved BER performance. An STTD gain of 3.2dB atBER=10-4 is obtained when α=0 dB.
An additional use of receive antenna diversity can further
improve the BER performance. Figure 6 plots the average
BER performance as a function of the average Eb/N0 with αand the number Nr of receive antennas as parameters. The useof receive antenna diversity is always beneficial irrespectiveof the degree of channel frequency-selectivity. When α=0 dB
and Nr=2, a combined STTD/receive diversity gain of 4.4dB
is obtained at BER=10-4 compared with Nr=1. When Nr=4, acombined diversity gain of as much as 8 dB is obtained.

1.E-01

1.E-01

N r =1
1.E-02

1.E-03L =16

1.E-04

α=0dB
α=2dB
α=8dB
× α=∞dB

Average BER

Average BER

1.E-02

MMSE
MRC
ZF

1.E-05

α=0dB
α=2dB
α=8dB
× α=∞dB
L =162

1.E-034

1.E-04

1.E-05

1.E-06051015202530

Average received Eb/N0 (dB)
Fig.4

L =16N r =1
STTD
no STTD

Average BER

1.E-03

1.E-04

α=0dB
α=2dB
α=8dB× a=∞dB

1.E-06051015205101520Fig 6. Effect of joint STTD and receive antenna diversity.

1.E-01

1.E-050Average received Eb/N0 per receive antenna (dB)

Simulated BER performances with joint STTD andMMSE, MRC and ZF. No antenna diversity (Nr=1).

1.E-02

1.E-0625

Average received Eb /N0 (dB)
Fig.5 The BER performance with joint STTD and MMSE
equalization.

30

3.2 Comparison with delay transmit diversity (DTD)
Another simple transmit diversity is the so called delaytransmit diversity (DTD) [11]. We compare the BER
performances achievable by STTD and DTD. Figure 7
compares the BER performances achievable by STTD andDTD with the decay factor α dB as a parameter. For
comparison, the BER performance for no transmit diversity isalso plotted. As the channel frequency-selectivity increases (αdecreases), all the BER performances of STTD, DTD and notransmit diversity improve. When α=8 dB, the channel
frequency-selectivity is strong and hence, the BER
performance with DTD improves. An additional gain of2.5dB at BER=10-4, however, is attained by STTD compared
to DTD, since STTD doubles the equivalent number ofreceive antenna branches. When α=0 dB, on the other hand,
the DTD gain is only 0.4dB because the channel
frequency-selectivity is already strong enough. However, a
STTD gain of as much as 3.2dB is still obtained compared tono transmit diversity.

1.E-01

STTDDTDNo transmit
diversity

Average BER

1.E-02

1.E-03

Reference

α=0dB
α=8dB
α=∞dB

1.E-04

L =16N r =1

1.E-05

1.E-06051015202530

Average received Eb/N0 (dB)Fig.7therefore, DTD gain becomes smaller as the propagation
channel frequency-selectivity is already strong enough.
In this paper, ideal channel estimation and block fading,where the channel gains stay constant over two-signal blockinterval, were assumed. The BER performance using practicalchannel estimation in a fast fading environment is left for
future study.

BER performance comparison between STTD andDTD.
4.

Conclusion

In this paper, STTD combined with frequency-domain
equalization was presented for SC transmission in afrequency-selective fading channel and the BER performancewas evaluated by computer simulation. The STTD encoding
suitable for frequency-domain equalization was proposed, inwhich Alamouti’s STTD encoding [10] is applied to eachsubcarrier component (obtained by FFT) of the transmitting
signal sequence. The direct application of Alamouti’s STTD
encoding requires FFT and IFFT operations. In this paper, theSTTD encoding that does not require FFT and IFFT wasproposed.
Performance comparison of STTD and DTD has shownthat STTD combined with frequency-domain MMSEequalization always gives a better BER performance thanDTD irrespective of the degree of the channel
frequency-selectivity. This is because STTD withfrequency-domain MMSE equalization is equivalent toreceive antenna diversity combining with two-branch MRC
while DTD expects the increased frequency diversity effectby increasing the equivalent number of propagation paths and[1] W. C. Jakes, Jr., Ed., Microwave mobile communications,Wiley, New York, 1974.
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