IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.11 NOVEMBER 2010

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INVITED PAPER

Special Section on Signal Design and its Application in Communications

Recent Advances in Single-Carrier Frequency-Domain
Equalization and Distributed Antenna Network
Fumiyuki ADACHI†a) , Fellow, Kazuki TAKEDA† , Tatsunori OBARA† , Tetsuya YAMAMOTO† ,and Hiroki MATSUDA† , Student Members

SUMMARY
Broadband wireless technology that enables a variety ofgigabit-per-second class data services is a requirement in future wirelesscommunication systems. Broadband wireless channels become extremely
frequency-selective and cause severe inter-symbol interference (ISI). Furthermore, the average received signal power changes in a random manner
because of the shadowing and distance-dependant path losses resulted fromthe movement of a mobile terminal (MT). Accordingly, the transmission
performance severely degrades. To overcome the performance degradation, two most promising approaches are the frequency-domain equalization (FDE) and distributed antenna network (DAN). The former takes advantage of channel frequency-selectivity to obtain the frequency-diversity
gain. In DAN, a group of distributed antennas serve each user to mitigate
the negative impact of shadowing and path losses. This article will introduce the recent advances in FDE and DAN for the broadband single-carrier(SC) transmissions.
key words: single -carrier, frequency-domain equalization, distributed antenna network, MIMO, signal detection

1.

Introduction

The future wireless communications systems are expected
to oﬀer a variety of broadband data services of close to oreven above 1 Gbps [1]. However, for such broadband dataservices, channels become extremely frequency-selective
and cause severe inter-symbol interference (ISI), thereby
severely degrading the bit error rate (BER) performance [2].
The use of an advanced equalization technique is indispensable. In general, there are two types of signal transmission:
multi-carrier (MC) and single-carrier (SC). An advantage
of SC signal transmissions over MC is its lower peak-toaverage power ratio (PAPR). This lower PAPR property of
SC is quite beneficial for the uplink (terminal-to-base) transmissions.
As for SC equalization, simple one-tap frequencydomain equalization (FDE) has been extensively studied[3]–[7]. Although the one-tap FDE can exploit the channel
frequency-selectivity to improve the BER performance, its
performance improvement is limited due to the residual ISI
after FDE. Recently, we proposed a new frequency-domain
block signal detection that incorporates the idea of multiinput/multi-output (MIMO) signal detection into FDE to effectively suppress the ISI [8].
Manuscript received February 12, 2010.
Manuscript revised June 6, 2010.
†
The authors are with the Dept. of Electrical and Communication Engineering, Graduate School of Engineering, Tohoku University, Sendai-shi, 980-8579 Japan.
a) E-mail: adachi@ecei.tohoku.ac.jp
DOI: 10.1587/transfun.E93.A.2201

In addition to the ISI problem of broadband SC signaltransmissions, the shadowing and distance-dependent pathlosses cause a severe power loss problem since the transmit
power is limited. The power loss problem is a crucial issue
as well as the ISI problem in order to make the frequency
and power eﬃcient broadband wireless communication systems a reality. Distributed antenna system or network (DASor DAN) [9]–[14] can solve the power loss problem. In
DAN, many antennas are spatially distributed over a coverage area of a DAN signal processing center (SPC), whichtakes the role of conventional base station (BS). There arealways some distributed antennas in the vicinity of a mobile terminal (MT) and they can cooperate to perform spatial diversity, beam forming, or spatial multiplexing. Since
diﬀerent antennas are located at diﬀerent positions, DAN
can exploit the random variations in the shadowing and pathlosses. We have been investigating the potential of DAN
downlink spatial diversity using maximal-ratio transmission(MRT) [14].
This article will introduce the recent advances in FDE
and DAN for the broadband SC signal transmissions.
2.

Broadband Wireless Channel

The broadband fading channel model is illustrated in Fig. 1.
There are several large obstacles between a BS and an MT
and also many local scatterers (such as neighboring buildings) in the vicinity of the MT. The reflection of the signal
by large obstacles creates the propagation paths with diﬀerent time delays; each path is a cluster of irresolvable multipaths created by reflection or diﬀraction, by local scatterers,of the transmitted signal reaching the surroundings of theFig. 1

Fading channel model.

c 2010 The Institute of Electronics, Information and Communication Engineers
Copyright IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.11 NOVEMBER 2010

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Fig. 3Fig. 2Frequency-selective channel.

MT.
The SC signal transmission with FDE is a block transmission. The channel transfer function severely changes
both in the frequency- and time-domains according to the
movement of an MT. However, the channel can be assumed to stay constant during the transmission of one block,so that, the time-selectivity can be neglected. Assuming
a symbol-spaced L-path channel, the channel impulse response h(τ) can be expressed as [15]h(τ) =L−1

hl δ(τ − τl ),

(1)

l=0

where
 τl are the complex-valued path gain with hl andL−1|hl |2 = 1 (E[.] denotes the expectation operation)
E l=0
and time delay of the lth path, respectively.
The channel transfer function (or the channel gain at
frequency f ) is the Fourier transform of h(τ) and is given by
H( f ) =L−1
hl exp(− j2π f τl ).

(2)

l=0

When H( f ) changes within the signal bandwidth, the channel is called the frequency-selective channel. A one shotobservation of H( f ) is illustrated in Fig. 2. A frequencyselective channel having an L = 16-path uniform power delay profile with E[|hl |2 ] = 1/L and τl = l×100 ns (i.e., the l-th
path length is l × 30 m) is assumed. The frequency-selective
channel severely distorts the received SC signal spectrum
and thus, the use of advanced equalization techniques is indispensable [2].
3.

Channel Capacity

The channel capacity analysis for broadband SC transmissions is an interesting study topic. In [16], the achievablechannel capacity of broadband SC signal transmissions with
FDE in a frequency-selective channel is theoretically examined and compared to that of orthogonal frequency divisionmultiplexing (OFDM). It is shown [16] that the OFDM withadaptive modulation provides always higher channel capacity than SC using one-tap FDE. However, below we will

SC signal transmission system model.

show that when block signal detection is utilized, the broadband SC transmission has a potential to achieve the samechannel capacity as OFDM.
The SC signal transmission system model is illustrated
in Fig. 3 (in this paper, for the simplicity purpose, we omit
the insertion and removal of cyclic prefix (CP)). An Nc symbol block d = [d(0), · · · , d(t), · · · , d(Nc − 1)]T is transmitted over a frequency-selective channel. The received
signal block r = [r(0), · · · , r(t), · · · , r(Nc − 1)]T can be expressed in the matrix form as

2E s
r=
hd + n,(3)Tswhere E s and T s are respectively the data-modulated symbolenergy and length, h is the channel impulse response matrix
given as⎡hL−1⎢⎢⎢ h0⎢⎢⎢⎢..
⎢⎢⎢ h1
.
⎢⎢⎢ .
..
⎢⎢⎢ .
. h0
0
⎢⎢⎢ .
⎢⎢⎢
.
..
⎢⎢⎢h
h1
⎢ L−1
h = ⎢⎢⎢⎢⎢
..
..
.. ..
⎢⎢⎢
.
.
.
.
⎢⎢⎢
..
⎢⎢⎢
. h0
hL−1
⎢⎢⎢
⎢⎢⎢
..
⎢⎢⎢
.
h1
⎢⎢⎢
⎢⎣
..
..
.
0
.

⎤
h1 ⎥⎥
.. ⎥⎥⎥⎥⎥
. ⎥⎥⎥
⎥⎥⎥
⎥
hL−1 ⎥⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥ , (4)
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
..
⎥⎥⎥
.
⎥⎥⎥
⎥⎥⎦
..
. h0

···
..
.

and n = [n(0), · · · , n(t), · · · , n(Nc − 1)]T is the noise
vector.
The frequency-domain representation R =[R(0), · · · , R(k), · · · , R(Nc −1)]T of the received signal blockis given as2E s
R = Fr =HFd + N,
(5)
Ts
where H = FhFH and N = Fn with [.]H denoting the Hermitian transpose operation and F being an Nc × Nc fast
Fourier transform (FFT) matrix given by
⎡
⎤
1
···
1
⎢⎢⎢ 1
⎥⎥⎥
− j2π 1×(NNcc −1) ⎥⎢⎢⎢⎢ 1 e− j2π 1×1
⎥⎥⎥
Nc
·
·
·
e
1 ⎢⎢⎢
⎥⎥⎥
F = √ ⎢⎢⎢ .
⎥⎥⎥ .
..
..
..
.
⎥⎥⎥
.
Nc ⎢⎢⎢ .
.
.
⎢⎣⎥c −1) ⎦
− j2π (NcN−1)×1− j2π (Nc −1)×(N
N
c
c
··· e
1e

(6)

H is the channel matrix, which is diagonal because of thecirculant property of h, and is given by

ADACHI et al.: RECENT ADVANCES IN SINGLE-CARRIER FREQUENCY-DOMAIN EQUALIZATION AND DISTRIBUTED ANTENNA NETWORK

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⎢⎢⎢
..
⎢⎢⎢
.
0
⎢
⎢
H(k)
H = FhFH = ⎢⎢⎢⎢⎢
⎢⎢⎢
..
⎢⎢⎢
.
0
⎢⎣

H(Nc − 1)

⎤
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥ .
⎥⎥⎥
⎥⎥⎥
⎥⎥⎦

4.1 Iterative FD-SIC&MMSED(7)

Equation (5) implies that the SC frequency-domain received signal has the same signal representation as an Nc ×Nc
MIMO multiplexing with channel matrix HF but with asymbol rate per stream reduced by a factor of Nc . Following [17], the channel capacity C of SC signal transmission
is given asEs1(HF) (HF)H
log2 det I +
C =
Nc
N0

N
c −1

1
Es
2
=
log2 1 +(8)|H(k)| ,Nc k=0N0which is identical to the channel capacity of OFDM with Nc
subcarriers [18].
4.

The frequency-domain block signal detection using iterative
FD-SIC&MMSED can achieve better BER performance
than the conventional one-tap FDE based on the MMSE criterion (MMSE-FDE).
The FD-SIC&MMSED is composed of i) ordering, ii)
ISI cancellation, and iii) signal detection. The transmitted symbol which has the highest signal-to-interference plus
noise power ratio (SINR) among undetected symbols is detected by performing MMSED [21]. However, in the case of
SC transmission, since the SINR is the same for all symbols
in a block, no ordering is necessary; the detection can be carried out simply from the first symbol (i.e., d(0)) in a block.
The replicas of the symbols except for the one to be detected
are generated and subtracted from the received signal (i.e.,
ISI cancellation). The MMSE weight matrix is updated forcarrying out MMSED. Until all of the transmitted symbols
are detected, the above interference cancellation and detection is repeated.

Frequency-Domain Block Signal Detection

The frequency-domain received signal representation is
given by Eq. (5). The task of the receiver is to estimate
the transmit signal block d for the given received signal
R. Recently, we proposed a new SC signal detection,
called frequency-domain block signal detection, based on
the frequency-domain received signal representation given
by Eq. (5) [8]. The conceptual structure of SC transmission
using frequency-domain block signal detection is illustrated
in Fig. 4.
We incorporate an iterative frequency-domain successive interference cancellation & minimum mean square error detection (iterative FD-SIC&MMSED) and frequencydomain maximum likelihood detection (MLD) employing
QR decomposition/M-algorithm (FD-QRM-MLD), which
are modified versions of the Vertical-Bell Laboratories layered space-time architecture (V-BLAST) detection [19] andthe QRM-MLD [20], respectively, developed for the signal detection for MIMO multiplexing. The QRM-MLD is
known to achieve the BER performance fairly close to the
MLD but with quite reduced complexity. The algorithms of
iterative FD-SIC&MMSED and FD-QRM-MLD are illustrated in Fig. 5.

Fig. 4 Conceptual structure of SC transmission using frequency-domain
block signal detection.

Fig. 5Iterative FD-SIC&MMSED and FD-QRM-MLD.

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The single use of FD-SIC&MMSED cannot suppressthe ISI suﬃciently, in particular for those symbols whichhave been detected earlier. To further improve the BER performance, iterative processing can be used [22]. After allof transmitted symbols are detected, the FD-SIC&MMSED
is carried out again. This is repeated a suﬃcient number oftimes. Below, the detection of the nth symbol (n = 0∼Nc −1)in the ith iteration stage is presented.
(1) ISI cancellation
(i,n)
for
The frequency-domain received signal vector R̃
the detection of the nth symbol in the ith iteration stage (notethat the symbols with n = 0∼n − 1 have been detected) is
given by

2E s
(i,n)
(i,n)
R̃
= R−
(9)
H̄ d̂ ,
Ts(i,n)=
where H̄ = HF = [ H̄0 , · · · , H̄n , · · · , H̄ Nc −1 ] and d̂[d̂(i) (0), · · · , d̂(i) (n − 1), 0, d̂(i−1) (n + 1), · · · , d̂(i−1) (Nc − 1)]T
is the soft decision replica vector. The interference replicas{ H̄n d̂(i) (n ); n = 0∼n − 1} are generated using a sequenceof the log likelihood ratios (LLRs) of the present iteration
stage associated with {d(n ); n = 0∼n − 1}. The symbols
with indices n = n∼(Nc − 1) are undetected and hence, theirreplicas are generated using LLRs of the previous iteration
stage.
(2) Signal detection
The ISI still remains after ISI cancellation and therefore, the MMSE weight matrix needs to be updated taking
into account the residual ISI. The MMSE weight matrix updating proposed in [23] can be applied.
The MMSE weight matrix taking into account theresidual ISI for the detection of the nth symbol in the ith
iteration stage is given by⎡−1 ⎤−1⎥⎥Es⎢⎢⎢ (i,n) H
H⎢(i,n)= H̄n ⎢⎣ H̄ρ H̄ +I⎥⎥⎥⎦ ,
(10)
W
N0
(i)
where ρ(i,n) = diag[ρ0(i) , · · · , ρn(i) , · · · , ρ(i)Nc −1 ] with ρn =E[|d(n ) − d̂(i) (n )|2 ] representing the extent to which the ISI
remains on the n’th symbol. In the case of quadrature phaseshift keying (QPSK) data modulation, ρn(i) is given as [24]
⎧
(i)
(i)
⎪
2eλn (1)2eλn (0)
⎪
⎪

⎪
+
⎪
2  (i)2 , if n < n
 (i)
⎪
⎪
⎪
λ
(0)
λ
(1)
⎪
⎪
e n + 1
e n + 1
⎪
⎪
⎪
⎪
⎨
(i)
1,
if n = n ,(11)ρn = ⎪
⎪
⎪
(i−1)
(i−1)
⎪
λn (0)λn (1)
⎪
⎪
2e
2e
⎪

⎪
⎪
⎪
2 +  (i−1)
2 , if n > n (i−1)
⎪
⎪
⎪
λ
(0)
λ
(1)
⎪


⎩ en
+1
+1
en

where λn(i) (x) is the log-likelihood ratio (LLR) of the xth bit
in the n’th symbol (x = 0∼1 and n = 0∼Nc − 1) obtained in
the ith iteration stage.
When i = 0, the symbols with n > n are undetected andhence, ρn(i) is replaced by 1. After ISI cancellation using soft

replicas as shown by Eq. (9), MMSE detection on the nth
symbol is performed by multiplying the frequency-domain(i,n)
received signal vector R̃ by 1 × Nc MMSE weight matrix
(i,n)
as
W
d̃(i) (n) = W (i,n) R̃

(i,n)

.

(12)

4.2 FD-QRM-MLD
In the case of SC transmissions, all symbols have the same
SINR and hence, no ordering is necessary; signal detection
can be carried out simply from the last symbol d(Nc − 1). H̄
is decomposed by using the QR decomposition asH̄ = QU,(13)

where Q is an Nc × Nc unitary matrix (i.e., QH Q = I with I
representing the identity matrix) and U is an Nc × Nc upper
triangular matrix given by⎡⎤⎢⎢⎢ U0,0 U0,1 · · · U0,Nc −1 ⎥⎥⎥⎢⎢⎢⎢U1,1 · · · U1,Nc −1 ⎥⎥⎥⎥⎥⎥⎥ .
U = QH H̄ = ⎢⎢⎢⎢⎢
(14)
..
..
⎥⎥⎥
⎢⎢⎢
.
.
⎥
⎥
⎣
⎦
0
U Nc −1,Nc −1
The transformed frequency-domain received signal R̂ is obtained as
2E s
H
Ud + QH N.
(15)
R̂ = Q R =TsThe M-algorithm [25] is composed of Nc stages with
the nth stage corresponding to the (Nc − 1 − n)th symbol ina block (n = 0∼Nc − 1). In each stage, the branch metricdefined as the squared Euclidian distance from R̂ is computed for all candidate symbols (4 symbols for QPSK and16 symbols for 16QAM) connected to each of M surviving symbols in the previous stage and then, only a total of
M symbols having the smallest accumulated branch metrics
are selected as the surviving symbols in the present stage
and others are discarded. This process is repeated until thelast stage (i.e., the first symbol d(0) in a block). Finally, thesymbol which has the smallest accumulated branch metricis chosen and a sequence of Nc detected symbols is obtainedby tracing back the surviving symbols until the first stage.
4.3 BER Performance
We assume a block transmission of Nc = 64 symbols, where
two data modulation schemes, QPSK and 16QAM, are considered. The channel is assumed to be a frequency-selective
block Rayleigh fading channel having symbol-spaced L =16-path uniform power delay profile. Ideal channel estimation is assumed. The average BER performances ofSC block signal detection using iterative FD-SIC&MMSEDand FD-QRM-MLD are compared in Fig. 6 as a function of
the average received signal energy per bit-to-noise power
spectrum density ratio Eb /N0 . For FD-QRM-MLD, three

ADACHI et al.: RECENT ADVANCES IN SINGLE-CARRIER FREQUENCY-DOMAIN EQUALIZATION AND DISTRIBUTED ANTENNA NETWORK

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Fig. 6

BER performance of SC block signal detection.

cases of the number M of surviving symbols in each stage
are plotted, i.e., M = 16, 64, and 256. Also plotted are the
BER performances achievable by the one-tap MMSE-FDE
and the matched filter (MF) bound. For comparison, theBER performance of MMSE-FDE when L = 1 is also plotted.
MMSE-FDE provides better BER performance when
L = 16 than when L = 1 because of the frequency diversity
eﬀect. However, due to the residual ISI, a big performance
gap from the MF bound still exists. On the other hand, iterative FD-SIC&MMSED and QRM-MLD can achieve muchbetter BER performance than MMSE-FDE. It can be seenfrom Fig. 6 that as the number i of iterations increases, theBER performance of iterative FD-SIC&MMSED improves.
This is because the residual ISI is eﬀectively reduced byusing the MMSE weight taking into account the residual
ISI and because the error propagation can be prevented bysoft ISI cancellation. It can be seen that the use of 2 iterations (i = 2) provides suﬃcient performance improvement. For QPSK (16QAM), the Eb /N0 gap from the theoretical lower bound for the average BER = 10−4 is reduced
by 1.5 (3.2) dB when i = 2. When QPSK is used, both iterative FD-SIC&MMSED and FD-QRM-MLD provide almost identical BER performance. However, when 16QAM
is used, FD-QRM-MLD with either M = 64 or 256 provides
better BER performance than iterative FD-SIC&MMSED.
Since this paper is to discuss the fundamental performances achievable by the frequency-domain block signal
detection, we assume no channel coding. OFDM without
channel coding provides BER performance similar to the
SC case of L = 1 irrespective of L. As seen from Fig. 6,
the SC frequency-domain block signal detection can takeadvantage of channel frequency-selectivity to obtain the frequency diversity gain and can achieve much better BER performance than OFDM. However, the introduction of channel coding to OFDM can significantly improve the BER performance since the channel coding across the subcarrier dimension can also take the advantage of channel frequencyselectivity. The coded OFDM can achieve slightly better
BER performance than the SC transmission using MMSEFDE as indicated in [26] (note that coded multicode DS- and
MC-CDMA are considered in [26] and those with spreading factor (SF) = multiplexing order (U) = 1 correspond tocoded SC and OFDM, respectively).
It should be noted that iterative FD-SIC&MMSEDand FD-QRM-MLD can improve the BER performance at
the cost of increased complexity compared to the MMSEFDE and OFDM. For example, in the case of 16QAM
and Nc = 64, the computational complexity (defined as thenumber of complex multiply operations) of iterative FDSIC&MMSED (i = 2) and FD-QRM-MLD (M = 256) isabout 4.0 × 104 times and 1.0 × 104 times higher than thatof the MMSE-FDE. This is because FD-SIC&MMSED requires (i+1)×Nc times matrix inversion of an Nc ×Nc matrix
and QRM-MLD requires QR decomposition of an Nc × Nc
matrix and a number of squared Euclidean distance calculations. Various complexity reduction algorithms for the matrix inversion have been studied [27]–[29]. We have alsostudied some complexity reduction schemes for FD-QRMMLD [30], [31]. Further reducing the complexity of the proposed frequency-domain block signal detection is left as animportant future study.
5.

Oversampling MMSE-FDE

In most of spectrum-eﬃcient wireless communication systems, a square-root Nyquist filter is used at the transmitterand receiver to limit the signal bandwidth while maximizing the received signal-to-noise power ratio (SNR). How-

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Fig. 7

Oversampling MMSE-FDE receiver.

ever, when timing oﬀset exists between the transmitter andreceiver, the spectrum of the received signal sequence sampled at the symbol rate is distorted and hence, the BER performance degrades as the filter roll-oﬀ factor α increases.
This is because adjacent frequency-shifted spectra (which
are the copies of the original spectrum and are given diﬀerent phase rotations due to the timing oﬀset) overlap whenα  0.
To solve the timing oﬀset problem, recently, we proposed an oversampling MMSE-FDE [32]. Its receiver structure is illustrated in Fig. 7 (for the simplicity purpose, weomit the insertion and removal of CP). First, the received
signal is oversampled at 2 times higher rate than the symbol
rate to prevent the spectrum overlapping. Then, the received
signal sample sequence is transformed by a 2Nc -sample FFT
into the frequency domain signal {R(k); k = −Nc ∼Nc − 1}.
R(k) can be expressed as
2E sR(k) =
H̃(k, Δ)S (k) + N(k) + Π(k),(16)
Tswhere H̃(k, Δ) is the overall transfer function of the transmit plus propagation channel including the phase rotation
due to the timing oﬀset Δ, and S (k), N(k), and Π(k) are thesignal, the inter-block interference (IBI), and the noise, respectively.
One-tap MMSE-FDE is applied to simultaneouslycompensate both the phase rotation due to the timing oﬀset
and the spectrum distortion due to the channel frequencyselectivity asR̂(k) = R(k)W(k),(17)

where W(k) is the MMSE-FDE weight given as∗H̃ (k, Δ)
W(k) = H̃(k, Δ)2 + Λ−1 (k, Δ)(18)

with Λ(k, Δ) being the signal-to-inter block interference plus
noise power ratio (SINR), given asΛ(k, Δ) =2E s /T s
E[|N(k)|2 + |Π(k)|2 ]
E 
s

=

N

0
⎧
⎫.
2
−1   
⎪
⎪

⎪
⎪
2
l

⎪
⎪


⎪
⎪
+
Δ
h
|l
|

 2
⎪
E ⎪
Nc 
⎪
⎪
⎨
⎬
l
=−∞


1 + N0s ⎪⎪⎪∞   
2


⎪
⎪
⎪
2
l



⎪
⎪
⎪
⎪
⎪
⎩ + Nc l =2Ng h 2 + Δ  (l − 2Ng )⎪⎭(19)

Since H̃(k, Δ) includes the transfer function of the transmit

Fig. 8Spectrum combining after one-tap MMSE-FDE.

Fig. 9BER performance in the presence of timing oﬀset.

filter, the MMSE-FDE weight also takes the role of a receive
filter matched to the transmit filter (this is the reason why noreceive filter is necessary in the receiver structure of Fig. 7).
After MMSE-FDE, the spectrum combing (or thefrequency-domain down-sampling) is performed (see Fig. 8)
to restore the ISI-free condition (note that the perfectrestoration of the ISI-free condition cannot be achieved in
the frequency-selective channel case) asR̃(k) = R̂(k − Nc ) + R̂(k) + R̂(k + Nc )(20)

for k = −Nc /2∼Nc /2 − 1. After spectrum combining, an Nc point inverse FFT (IFFT) is applied to the frequency-domain
signal {R̃(k); k = −Nc /2∼Nc /2 − 1} for succeeding data demodulation.
The BER performances of the proposed oversampling
MMSE-FDE and of the conventional symbol rate sampling MMSE-FDE are compared for the frequency-selective
Rayleigh fading channel having L = 16-path uniform power
delay profile and the frequency-nonselective Rayleigh fading channel (L = 1), respectively. It is seen from Fig. 9 thatthe proposed oversampling MMSE-FDE provides a good
BER performance which is insensitive to α in the presenceof the timing oﬀset regardless of the degree of the channel
frequency-selectivity.

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6.

Joint Transmit/Receive MMSE-FDE

If the channel state information (CSI) is available at thetransmitter, transmit FDE (or called pre-FDE) can be jointly
used with the receive FDE to compensate the signal spectrum distortion due to the channel frequency-selectivity. The
frequency-domain representation of the transmission system using joint transmit/receive MMSE-FDE is illustrated
in Fig. 10 (for the simplicity purpose, we omit the insertionand removal of CP).
Before transmitting the signal, an Nc -symbol block d
is transformed by an Nc -point FFT into a frequency-domain
signal block D = Fd. One-tap transmit FDE weight ismultiplied to each element (frequency component) of D.
Denoting an Nc × Nc transmit FDE weight matrix by W t= diag{Wt (0), . . . , Wt (k), . . . , Wt (Nc − 1)}, the time-domain
signal block s after carrying out Nc -point IFFT can be expressed as
s = C · FH W t Fd,(21)
where C = Nc /tr[W t W tH ] is the transmit power normalization coeﬃcient. After transforming the frequency-domain
signal block by an Nc -point IFFT into the time-domaintransmit signal block, the signal block is transmitted.
The received signal block is transformed intofrequency-domain signal block R = [R(0), . . . , R(k), . . . ,R(Nc − 1)]T by Nc -point FFT as
⎞ 
⎛

⎟⎟
⎜⎜ 2E s2E s 
hs + n⎟⎟⎠ =R = F ⎜⎜⎝
FhFH (Fs) + Fn
Ts
Ts

2E s
=
C · HW t Fd + N.
(22)Ts

an equivalent channel W r HW t for both SC and MC (notethat W t and W r can be generalized to include spreading andde-spreading). Therefore, W t and W r can be optimized regardless of transmit signal waveforms (i.e., SC and MC).
For example, the choice of waveforms can be made basedon the peak-to-average power ratio (PAPR).
If W t and W r are chosen so as to satisfy W r HW t =I, the channel frequency-selectivity problem can be completely removed. However, since most of the transmit poweris allocated to the frequencies with lower channel gains, lesspower is allocated to other frequencies in order to keep thetransmit power the same. Therefore, the received signal
power is significantly reduced, thereby significantly degrading the BER performance. In [33], we proposed the joint
transmit/receive MMSE-FDE for SC transmissions. Since
it is quite diﬃcult to find the exact MMSE solution of (W t ,W r ), we took a sub-optimal approach: W r is derived firstfor the given W t and then, W t is derived assuming that thederived W r is used. By treating the concatenation HW t of
the transmit FDE and channel as an equivalent channel, W r
is given as
W r = W tH H H [HW t W tH H H + C −1 γ−1 · I]−1 ,(25)

where γ = (E s /N0 ). Then, W t is derived as [27]W t = diag{|Wt (0)|, . . . , |Wt (k)|, . . . , |Wt (Nc − 1)|} (26)with"|Wt (k)|2 = max

#
1
1
1
−
,
0
,
√
μ γ|H(k)| γ|H(k)|2(27)

where μ is the constant which satisfies tr[W t W tH ] = Nc .

At the receiver, one-tap receive FDE is carried out and then,Nc -point IFFT is performed to transform the frequencydomain signal block back into the decision variable blockd̂. Denoting an Nc × Nc receive FDE weight matrix by W r =diag{Wr (0), . . . , Wr (k), . . . , Wr (Nc − 1)}, d̂ is expressed as2E sd̂ = FH W r R =C · FH (W r HW t ) Fd+ FH W r N. (23)Ts
In the case of MC transmission (e.g., OFDM), the decisionvariable block d̂ MC is expressed as
2E sd̂ MC =C · (W r HW t ) d + W r N.
(24)
Ts
It can be understood from Eqs. (23) and (24) that the useof transmit and receive FDE transforms the channel H into

Fig. 10 Frequency-domain representation of transmission system using
joint transmit/receive MMSE-FDE.

Fig. 11 BER performance of SC using the joint transmit/receive
MMSE-FDE.

IEICE TRANS. FUNDAMENTALS, VOL.E93–A, NO.11 NOVEMBER 2010

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Figure 11 plots the achievable BER performance of SC
using the joint transmit/receive MMSE-FDE. For comparison, the BER performance of SC using the conventional
receive MMSE-FDE is also plotted. It can be seen from
Fig. 11 that the joint transmit/receive MMSE-FDE provides
much better BER performance than the conventional receive
MMSE-FDE.
One of remaining problems for implementing the joint
transmit/receive MMSE-FDE is to estimate the CSI at thetransmitter side. Many studies for estimating the CSI at thetransmitter can be found [34]–[37]. How the proposed joint
transmit/receive MMSE-FDE is sensitive to the CSI error isan important future study topic.
7.

Distributed Antenna Network

In DAN, many antennas are spatially distributed arounda BS so that with a high probability, there are some distributed antennas in the vicinity of an MT. Distributed antennas are connected to a SPC by means of optical fiberor wireless links, as illustrated in Fig. 12. They can cooperate to perform spatial diversity, beam forming, or spatial
multiplexing. The distributed antenna spatial diversity can
mitigate the received signal power loss problem caused bydistance-dependent path loss and shadowing loss as well as
the frequency-selective fading.
One of promising distributed antenna diversity for
DAN downlink is MRT [38] that exploits the CSI at the
network transmitter side. In a frequency-selective channel,
MRT is incorporated into the transmit FDE. This is called
MRT-FDE. As seen in Fig. 13, the signal block is simultaneously transmitted from Nt distributed antennas in the vicinity of an MT after multiplying MRT-FDE weight {Wm (k);k = 0∼Nc − 1}, m = 0∼Nt − 1, so that the same frequency
components transmitted from Nt distributed antennas are coherently combined at the MT, where Nt is the number of
distributed antennas involved in MRT-FDE.
The signal block s transmitted from Nt distributed antennas can be expressed using the matrix form as
s = [s0 , · · · , sm , · · · , sNt −1 ]T





 T
= FH W 0 F d, · · · , FH W m F d, · · · , FH W Nt −1 F d .
(28)

Fig. 12Distributed antenna network.

where W m = diag {Wm (0), · · · , Wm (k), · · · , Wm (Nc − 1)} is
the MRT-FDE weight matrix associated with the mth antenna, given by⎞− 1⎛ N −1 N −1tc⎜⎜⎜ 1 2 ⎟⎟⎟ 2 ∗⎜
Hm (k ) ⎟⎟⎟⎠ Hm ,W m = ⎜⎜⎝
Nc m =0 k =0(29)

where Hm = Fhm FH is given by
Hm = Fhm FH⎡⎢⎢⎢⎢ Hm (0)
⎢⎢⎢
..
⎢⎢⎢
.
0
⎢⎢⎢
Hm (k)
= ⎢⎢⎢
⎢⎢⎢
..
⎢⎢⎢
.
0
⎢⎣

Hm (Nc − 1)

⎤
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥
⎥⎥⎥ .
⎥⎥⎥
⎥⎥⎥
⎥⎥⎦

(30)

The received signal block r = [r(0), · · · , r(t), · · · , r(Nc −1)]T at an MT can be expressed as

2E s
r=
hs + n,(31)
Tswhere h = [h0 , · · · , hm , · · · , hNt −1 ], hm is the channel impulse response matrix of size Nc × Nc associated with themth antenna, given as⎤⎡
hm,L−1 · · · hm,1 ⎥⎥⎢⎢⎢ hm,0⎢⎢⎢.. ⎥⎥⎥⎥⎥..
⎢⎢⎢ hm,1 . . .
.
. ⎥⎥⎥
⎥⎥⎥
⎢⎢⎢⎢ .
..
⎥
⎢⎢⎢ .
.
hm,0
0
hm,L−1 ⎥⎥⎥⎥
⎢⎢⎢ .
⎥⎥⎥
⎢⎢⎢
..
⎥⎥⎥⎥
⎢⎢⎢h
.
h
m,1
⎥⎥⎥
⎢⎢⎢ m,L−1
hm = ⎢⎢⎢⎥⎥⎥ , (32)
..
..
.. ..
⎥⎥⎥
⎢⎢⎢
.
. .
.
⎥⎥⎥
⎢⎢⎢
..
⎥⎥⎥
⎢⎢⎢
. hm,0hm,L−1
⎥⎥⎥
⎢⎢⎢
⎥⎥⎥
⎢⎢⎢
.
.
⎥⎥⎥
⎢⎢⎢
..
..
h
m,1
⎥⎥⎥
⎢⎢⎢
⎥⎦
⎢⎣
..
..
..
.
.
0
.
hm,0
and n = [n(0), · · · , n(t), · · · , n(Nc − 1)]T is the noise vector.
The frequency-domain representation R of the received
signal block is given as

Fig. 13DAN downlink using SC MRT-FDE.

ADACHI et al.: RECENT ADVANCES IN SINGLE-CARRIER FREQUENCY-DOMAIN EQUALIZATION AND DISTRIBUTED ANTENNA NETWORK

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8.

Fig. 14 Channel capacity distribution of distributed antenna diversity
using SC MRT-FDE.


R = Fr =

⎛N −1
⎞
t
⎟⎟⎟
2E s ⎜⎜⎜⎜ ⎜⎝⎜
HmW m ⎟⎟⎠⎟ Fd + N,
T s m=0This article introduced the recent advances in FDE andDAN for the broadband SC signal transmissions. To realize the high quality broadband SC signal transmissions,
there are two crucial issues to overcome: the channel
frequency-selectivity and the power loss due to the shadowing and distance-dependant path losses. The frequencydomain block signal detection that incorporates the idea of
MIMO signal detection into FDE to eﬀectively suppress the
ISI was introduced. In addition, the oversampling one-tap
MMSE-FDE that can mitigate the problem arising from thetiming oﬀset between the transmitter and receiver was introduced. Finally, DAN was presented that can mitigate
the received signal power loss problem caused by distancedependent path loss and shadowing loss while exploiting thefrequency-selective fading.
References(33)

where N = Fn.
The MRT-FDE maximizes the received SNR at eachfrequency k, but enhances the ISI in the case of SC transmissions. Therefore, the ISI cancellation is necessary in thesignal detection.
Similar to Eq. (8), the channel capacity C of SC DAN
using MRT-FDE is given, from Eq. (33), as⎛⎛N −1⎞ ⎛N −1
⎞H ⎞
t
t
⎜⎜⎜
⎜⎜⎜ 
⎟⎟⎟ ⎜⎜⎜ ⎟⎟⎟ ⎟⎟⎟
1
E
s
⎜
⎜⎜⎝⎜
log2 det ⎜⎜⎜⎝ I +HmW m ⎟⎟⎠⎟ ⎜⎜⎝⎜
HmW m ⎟⎟⎠⎟ ⎟⎟⎟⎟⎠C =Nc
N0 m=0
m=0
⎛
2 ⎞
N
t −1
⎜⎜⎜
⎟⎟⎟
2
|Hm (k)|
N
⎜⎜⎜⎜
⎟⎟⎟⎟
c −1

1
Es
m=0
⎜⎜⎜
⎟⎟⎟
log2 ⎜⎜1 +
=
(34)
⎟⎟⎟ .
⎜⎜⎜
t −1 Nc −1
Nc k=0N0 1 N⎟⎠⎟ )|2 ⎟
(k
|H
⎝⎜
m
N
c

Conclusions

m=0 k =0

The channel capacity changes according to the movement of an MT because of the frequency-selective fading, shadowing loss, and distant-dependent path loss. The
channel capacity distribution of SC MRT-FDE is plotted inFig. 14. Due to higher frequency diversity gain, the 1% outage channel capacity when L = 16 is higher than that when
L = 1.
Another approach is the use of distributed antennas asrelay nodes. Multiple relay nodes in the vicinity of an MTcooperate to relay the signal transmitted from the MT to a
SPC. Multiple relay nodes can perform the antenna beam
forming or MRT-FDE to improve the SC signal transmission
performance. Note that in the case of OFDM, adaptive subcarrier allocation over multiple 2-hop routes constructed byrelay nodes can be introduced to make use of the frequency
and spatial diversity [39].

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Fumiyuki Adachi
received the B.S. and
Dr.Eng. degrees in electrical engineering fromTohoku University, Sendai, Japan, in 1973 and1984, respectively. In April 1973, he joined
the Electrical Communications Laboratories of
Nippon Telegraph & Telephone Corporation(now NTT) and conducted various types of research related to digital cellular mobile communications. From July 1992 to December 1999,
he was with NTT Mobile Communications Network, Inc. (now NTT DoCoMo, Inc.), where heled a research group on wideband/broadband CDMA wireless access for
IMT-2000 and beyond. Since January 2000, he has been with Tohoku University, Sendai, Japan, where he is a Professor of Electrical and Communication Engineering at the Graduate School of Engineering. His researchinterests are in CDMA wireless access techniques, equalization, transmit/receive antenna diversity, MIMO, adaptive transmission, and channel
coding, with particular application to broadband wireless communications
systems. He is a program leader of the 5-year Global COE Program “Centerof Education and Research for Information Electronics Systems” (2007–2011), awarded by the Ministry of Education, Culture, Sports, Science andTechnology of Japan. From October 1984 to September 1985, he was a
United Kingdom SERC Visiting Research Fellow in the Department ofElectrical Engineering and Electronics at Liverpool University. He is an
IEICE Fellow and was a co-recipient of the IEICE Transactions best paperof the year award 1996, 1998, and 2009 and also a recipient of Achievement award 2003. He is an IEEE Fellow and was a co-recipient of the IEEE
Vehicular Technology Transactions best paper of the year award 1980 andagain 1990 and also a recipient of Avant Garde award 2000. He was a recipient of Thomson Scientific Research Front Award 2004, Ericsson Telecommunications Award 2008, and Telecom System Technology Award 2010.

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Kazuki Takeda
received his B.S. and M.S.
degree in communications engineering from Tohoku University, Sendai, Japan, in 2006 and2008. Currently he is a Japan Society for the
Promotion of Science (JSPS) research fellow,
studying toward his Ph.D. degree at the Department of Electrical and Communications Engineering, Graduate School of Engineering, Tohoku University. His research interests include
precoding and channel equalization techniques
for mobile communication systems. He was arecipient of the 2010 IEICE RCS (Radio Communication Systems) Active
Research Award.

Tatsunori Obara
received his B.S. degree
in Electrical, Information and Physics Engineering in 2008 and M.S. degree in communications
engineering, in 2010, respectively, from Tohoku
University, Sendai Japan. Currently he is a graduate student, studying toward his Ph.D. degree
at the Department of Electrical and Communications Engineering, Graduate School of Engineering, Tohoku University. His research interests include channel equalization techniques formobile communication systems.

Tetsuya Yamamoto
received his B.S. degree in Electrical, Information and Physics Engineering in 2008 and M.S. degree in communications engineering, in 2010, respectively, fromTohoku University, Sendai Japan. Currently heis a Japan Society for the Promotion of Science (JSPS) research fellow, studying toward his
Ph.D. degree at the Department of Electrical andCommunications Engineering, Graduate Schoolof Engineering, Tohoku University. His researchinterests include frequency-domain equalizationand signal detection techniques for mobile communication systems. He
was a recipient of the 2009 IEICE RCS (Radio Communication Systems)
Active Research Award.

Hiroki Matsuda
received his B.S. degree
in Electrical, Information and Physics Engineering in 2008 and M.S. degree in communications engineering, in 2010, respectively, fromTohoku University, Sendai Japan. Since April
2010, he has been with Sony Corporation. His
research interests include antenna diversity techniques for distributed antenna network.