[[abstract]]This paper presents a semiblind maximum-likelihood (ML) detector for the orthogonal space-time block coded orthogonal frequency division multiplexing (OSTBC-OFDM) system. Many existing blind/ semiblind OSTBC-OFDM receivers typically require that the channel is static over a multitude of OSTBC-OFDM blocks. The proposed method is specifically for detection over one OSTBC-OFDM block only, and hence is well suited to block fading channels. The presented identifiability analysis shows that the data can be uniquely identified in a probability one sense by using one pilot code only, in contrast to the pilot-based least-squares channel estimator which requires at least L pilot codes where L is the channel length. Simulation examples are then presented to show the efficacy of the proposed detector.[[fileno]]2030157030006[[department]]電機工程學

Chang,Tsung-Hui

Chi,Chong-Yung

Ma,Wing-Kin

National Tsing Hua University Institutional Repository

SEMIBLIND ML OSTBC-OFDM DETECTION IN BLOCK FADING CHANNELS
Tsung-Hui Chang, Wing-Kin Ma, and Chong-Yung Chi
Institute of Commun. Eng. & Dept. of Elect. Eng.
National Tsing Hua University,
Hsinchu, Taiwan 30013, R.O.C.
E-mail: {wkma, cychi}@ee.nthu.edu.tw
ABSTRACT
This paper presents a semiblind maximum-likelihood (ML) detector
for the orthogonal space-time block coded orthogonal frequency di-
vision multiplexing (OSTBC-OFDM) system. Many existing blind/
semiblind OSTBC-OFDM receivers typically require that the chan-
nel is static over a multitude of OSTBC-OFDM blocks. The pro-
posed method is speciﬁcally for detection over one OSTBC-OFDM
block only, and hence is well suited to block fading channels. The
presented identiﬁability analysis shows that the data can be uniquely
identiﬁed in a probability one sense by using one pilot code only,
in contrast to the pilot-based least-squares channel estimator which
requires at leastL pilot codes whereL is the channel length. Simula-
tion examples are then presented to show the efﬁcacy of the proposed
detector.
Index Terms— MIMO systems, Maximum likelihood detec-
tion, Identiﬁcation, Multipath channels.
1. INTRODUCTION
In recent years, the space-time block coded orthogonal frequency di-
vision multiplexing (STBC-OFDM) system has drawn a lot of atten-
tions because it provides a straightforward way of extending STBC
techniques to frequency selective fading channels [1–7]. In par-
ticular, the STBC-OFDM system based on the orthogonal STBCs
(OSTBCs) [9] maximizes the transmit diversity and has a simple
Maximum-likelihood (ML) decoder given channel state information
(CSI) at the receiver. To estimate CSI with high spectral efﬁciency, it
has been a great interest to develop blind/semiblind channel estima-
tion methods for OSTBC-OFDM systems [1–4]. However, most of
the existing methods are based on the second-order statistics (SOSs)
of the received signal, which usually require the channel to remain
static over many OSTBC-OFDM blocks. When the channel is static
only for two OSTBC-OFDM blocks, a particularly convenient blind
scheme is the differentially encoded OSTBC-OFDM system [6]. It,
however, incurs a 3 dB performance loss in signal-to-noise ratio
(SNR).
In the paper, the channel is assumed to be block fading, i.e., the
channel coefﬁcients do not vary for one OSTBC-OFDM block in-
terval but can change from block to block. By using a time-domain
channel parameterization, we apply the deterministic ML criterion
[10, 11] to jointly detect the data and estimate the channel with one
OSTBC-OFDM signal block only. An important result shown here
is that the data can be uniquely identiﬁed with probability one when
there is only one pilot code transmitted. This results in a semiblind
ML OSTBC-OFDM detector that uses fewer pilot codes than the
This work is supported by National Science Council, R.O.C., under
Grant NSC 95-2221-E-007-036 and Grant NSC 95-2219-E-007-004.
conventional pilot-aided channel estimator [8]. Regarding the real-
ization problem, it is further shown that the proposed detector can
be recast as a Boolean quadratic program (BQP), which can be han-
dled effectively by various means. Simulation results in Sec. 4 fur-
ther demonstrate that the proposed detector can yield very promising
performance.
2. OSTBC-OFDM SIGNAL MODEL
Consider an OSTBC-OFDM system [5, 7] equipped with Nt trans-
mit antennas and Nr receive antennas. Let Nc denote the discrete
Fourier transform (DFT) size and T be the code length. Assuming
that the channel coefﬁcients are static for T OFDM blocks, the re-
ceived code matrix in subchannel n is given by [5]
Yn = C(sn)Hn + Wn, (1)
where n = 1, ..., Nc, and
Yn ∈ CT×Nr received code matrix for subchannel n;
sn ∈ {±1}K transmitted data vector for subchannel n where K
is the number of bits per code;
C(·) ∈ CT×Nt transmitted OSTBC (T ≥ Nt);
Hn ∈ CNt×Nr multiple-input-multiple-output (MIMO) channel
frequency response matrix for subchannel n;
Wn ∈ CT×Nr AWGN matrix for subchannel n where the aver-
age power per entry is σ2w.
In general, the OSTBCs can be represented by a linear disper-
sion form. Speciﬁcally, for BPSK/QPSK OSTBCs, they can be ex-
pressed as [9]
C(sn) =
K 
k=1
Xksn,k, (2)
where sn,k ∈ {±1} is the kth entry of sn, andXk ∈ CT×Nt are the
basis matrices. The basis matrices are specially designed such that
for any sn ∈ {±1}K ,
CH(sn)C(sn) = KINt , (3)
where INt is the Nt ×Nt identity matrix.
Because the time-domain channel coefﬁcients can usually be
modeled by a ﬁnite impulse response (FIR) whose length can be
much less than Nc, we can parameterize Hn using the FIR chan-
nel coefﬁcients. Let hm,j ∈ CL be a column vector containing the
time-domain channel coefﬁcients between the mth transmit antenna
and the jth receive antenna, where L is the channel length. Deﬁne
H =


h1,1 · · · h1,Nr
...
. . .
...
hNt,1 · · · hNt,Nr

 ∈ CLNt×Nr . (4)
III ­ 3091­4244­0728­1/07/$20.00 ©2007 IEEE ICASSP 2007
Let F ∈ CNc×L with the nth row given by
fTn =
1√
Nc
[1, e
−j 2π
Nc
(n−1)
, ..., e
−j 2π
Nc
(n−1)(L−1)
]. (5)
Then one can easily show that
Hn =
 
INt ⊗ fTn

H, (6)
where⊗ denotes the Kronecker product. By substituting (6) into (1),
the received signal can be rewritten as
Yn = C(sn)
 
INt ⊗ fTn

H + Wn. (7)
Let s = [sT1 , ..., sTNc ]
T and Y = [YT1 , ..., YTNc ]T . Then (7) can
be further re-expressed in a compact form as
Y = C(s)H + W , (8)
where W = [WT1 , ..., WTNc ]T and
C(s) =


C(s1)

INt ⊗ fT1

...
C(sNc)

INt ⊗ fTNc


	
 ∈ CNcT×LNt . (9)
In the paper, our interest lies in jointly estimating H and detecting
s from Y . As mentioned and clariﬁed in Sec. 1, the signiﬁcance of
this investigation is in the block fading scenario where H may not
be the same for each OSTBC-OFDM block.
3. SEMIBLIND ML DETECTION
We consider the deterministic semiblind ML criterion [10,11] where
part of s is known at the receiver. For simplicity, assume that the ﬁrst
M subchannels contain pilots (generalization to other pilot place-
ments is straightforward). Let
s = [sTp , s
T
d ]
T ∈ {±1}NcK , (10)
where sp = [sT1 , ..., sTM ]
T stands for the pilot data and sd =
[sTM+1, ..., s
T
Nc ]
T denotes the unknown data. Then the semiblind
ML detector is given by
{sˆd, Hˆ} = arg min
sd ∈ {±1}(Nc−M)K
H ∈ CLNt×Nr
||Y − C(s)H||2F , (11)
where || · ||F denotes the Frobenius norm. We present a data iden-
tiﬁability analysis for (11), followed by the implementation method
of the semiblind ML detector. It will be shown in Sec. 3.1 that the
data identiﬁability can be guaranteed in a probability one sense when
there is one pilot code transmitted, i.e., M = 1. Regarding the re-
alization issue, in the next subsection, it is shown that (11) can be
reformulated as a Boolean quadratic program (BQP) where efﬁcient
implementation is possible [12].
3.1. Data Identiﬁability
To investigate the data identiﬁability, the signal model in (8) in the
absence of noise is considered
Y = C(s)H. (12)
It is clear that, in the absence of noise, {sd,H} is a solution pair
of (11). Suppose that there exist s′d ∈ {±1}(Nc−M)K and H′ ∈
C
LNt×Nr so that {s′d,H′} is also a solution pair of (11). Then
C(s′)H′ = C(s)H, (13)
where
s′ = [sTp , s
′T
d ]
T ∈ {±1}NcK . (14)
To ensure the data identiﬁability, we analyze (13) and look for con-
ditions under which (13) holds only if {s′d,H′} = {sd,H}.
We start our analysis by considering that there are at leastL pilot
codes (i.e., M ≥ L). Let
Cp(sp) =


C(s1)

INt ⊗ fT1

...
C(sM )

INt ⊗ fTM


	
 ∈ CMT×LNt . (15)
From (13) and (15), one can obtain
Cp(sp)H′ = Cp(sp)H. (16)
It is not hard to show that Cp(sp) is of full column rank for M ≥ L.
Hence, (16) results in H′ = H, i.e., the channel can be uniquely
identiﬁed. However, channel identiﬁcation does not imply data iden-
tiﬁcation because, if Hn = 0 for some n, then the data sn can never
be recovered even when full CSI is available. However, the probabil-
ity that this channel nullity occurs is quite low for an MIMO fading
channel. Let us consider the following channel assumption:
A1) The channel H is Gaussian distributed and at least one column
of H has a positive deﬁnite covariance matrix.
One can see that independent identically distributed (i.i.d.) Rayleigh
fading channels satisfy A1). Under A1), one can show that the
probability of the event {Hn = 0} is of measure zero. Thus we
conclude that the data can be identiﬁed with probability one if there
are at least L pilot codes. In practice, given M ≥ L, the channel can
be estimated by a least-squares (LS) channel estimator [10, 15]
Hˆ =
 
CHp (sp)Cp(sp)
−1 CHp (sp)YM , (17)
where YM = [YT1 , ...,YTM ]T .
In the paper, instead of usingL pilot codes, we show that a single
pilot code is sufﬁcient to ensure probability one data identiﬁability.
Let us look at the following lemma:
Lemma 1: Assume that A1) holds and that there is no noise. Sup-
pose that C(s) satisﬁes the following condition:
C1) Let s′ = [sTp , s′Td ]T ∈ {±1}NcK . There does not exist a
matrix U ∈ CLNt×LNt such that
C(s′)U = C(s), (18)
for any s′d = sd.
Then the data sd can be uniquely identiﬁed with probability one.
The proof of Lemma 1 is given in the Appendix. One can see that
the aforementioned LS method (which uses at least L pilot codes) is
an approach to make C1) hold and thereby can have the probability
one data identiﬁability. However, by exploiting the special structure
III ­ 310
of F, we can show that a single pilot code is sufﬁcient to achieve
C1). Consider the following lemma.
Lemma 2: Let s, s′ ∈ {±1}NcK . There exists a matrix U ∈
C
LNt×LNt such that
C(s′)U = C(s) (19)
if and only if there exists a matrix Q ∈ CNt×Nt such that
C(s′n)Q = C(sn), (20)
for all n = 1, ..., Nc.
Due to space limit, the proof of Lemma 2 (which will be reported
in [13]) is omitted here. However, a special case of Lemma 2 where
Nt = 1 and T = 1 (uncoded SIMO OFDM system) has been im-
plicitly proved in [14]. Consider the case of M = 1, i.e., sp = s1 =
s′1. If (20) holds, we obtain Q = INt and subsequently, s
′
n = sn
for n = 2, ..., Nc, i.e., s′d = sd. Hence, by Lemma 2, (19) holds
only if s′d = sd, which means that C1) is achieved. Therefore, the
data identiﬁability can be guaranteed by merely transmitting one pi-
lot code. We can now formally summarize the above analysis as
follows:
Theorem 1: Assume that A1) holds and that there is no noise. If
any one of the subchannels carriers a pilot code, then the data sd
can be uniquely identiﬁed with probability one.
3.2. Realization of ML Detector
In this subsection, we show how the proposed detector can be real-
ized. For simplicity, we consider M = 1 which is the focus of this
paper (generalization to M > 1 is straightforward). Recall from
(11) that the semiblind ML detector for M = 1 is given by
{sˆd, Hˆ} = arg min
sd ∈ {±1}(Nc−1)K
H ∈ CLNt×Nr
||Y − C(s)H||2F , (21)
By using the same reformulation as in [12], one can turn (21) to the
following BQP
sˆd = arg max
sd∈{±1}(Nc−1)K
sTdGsd + 2s
T
1
 Gsd, (22)
where
G =


G2,2 · · · G2,Nc
...
. . .
...
GNc,2 · · · GNc,Nc

 , (23)
 G = [G1,2, G1,3, . . . , G1,Nc ], (24)
[Gm,n]k, = Re

tr

γm,nY
H
mXkX
H
 Yn
		
, (25)
for k,  = 1, ..., K, where tr(·) denotes the trace of a matrix and
γm,n = f
T
mf
∗
n . Therefore, the semiblind ML detector can be imple-
mented by solving this BQP. In recent years, there have been several
efﬁcient methods proposed for handling the BQP [12]; for example,
the sphere decoder, the norm relaxation method and the semideﬁnite
relaxation (SDR) algorithm [12]. To understand how these methods
can be applied, please refer to [12].
Although the aforementioned methods can be used to imple-
ment the ML detector, the complexity can still be an issue when the
DFT size Nc is very large. To tackle this problem, the subchannel
grouping method in [14] can be used. The application of subchan-
nel grouping to the problem here is straightforward, and is currently
being investigated.
4. SIMULATION RESULTS
This section presents two simulation examples to justify the efﬁcacy
of the proposed semiblind ML detector. The coefﬁcients of H are
zero-mean i.i.d. complex Gaussian distributed, and change from one
OSTBC-OFDM block to another. The SNR per subchannel is de-
ﬁned as
SNR = K
E{‖H‖2F }
TNcσ2w
.
The DFT size was 32 (Nc = 32) and the number of receive antennas
was two (Nr = 2). The signal constellation was QPSK. For the
proposed detector, the pilot code was placed at subchannel 1, and the
associated BQP in (22) was handled by the SDR algorithm [12]. The
detector performance was measured in terms of the average bit error
rate (BER). We compared the proposed detector to the coherent ML
detector (with perfect CSI) and the pilot-based LS channel estimator
mentioned in Sec. 3.1. For the LS channel estimator, L pilot codes
were used. Let S be the subset of subchannel indices where the pilot
codes for LS method are placed. In the simulation, it was set to be
S = 
1, 1 + Nc
L
, 1 + Nc
L
· 2, . . . , 1 + Nc
L
· (L− 1) . (26)
It can be shown [10,15] that, for ﬁxed L pilot codes, the pilot place-
ment in (26) leads to the minimummean-squared channel estimation
error. The differential OSTBC-OFDM scheme [6] was also consid-
ered for comparison, which was obtained by applying the differential
OSTBC scheme [16] to each subchannel. There were 15, 000 trials
performed in our simulation examples.
Figure 1 presents the results when the complex Alamouti code
(T = 2, Nt = 2) was used and L = 4. One can see, from this
ﬁgure, that the proposed detector outperforms both the pilot-based
LS method and the differential scheme. Moreover, the performance
difference between the proposed detector and the coherent ML de-
tector at BER= 10−4 is around 1 dB only. Similar results can also
be observed in Fig. 2 where the complex 4× 3 OSTBC (Eqn. (120)
of [17]) (T = 4, Nt = 3) was used and L = 8.
5. CONCLUSIONS
In the paper, we have proposed a semiblindML detector for OSTBC-
OFDM in block fading channels. The proposed detector, using the
FIR channel parameterization, can perform data detection in one
OSTBC-OFDM signal block. Our analysis shows that data iden-
tiﬁability with probability one is guaranteed as long as one pilot
code is used. It is also shown that the proposed detector can be efﬁ-
ciently realized by solving a BQP. The presented simulation results
further reveal that the proposed detector outperforms the pilot-based
LS method and the differential method.
6. APPENDIX
PROOF OF LEMMA 1
Let s′ = [sTp , s′Td ]
T ∈ {±1}NcK , s′d = sd and H′ ∈ CLNt×Nr ,
H′ = H. We show that if C1) holds, then the probability that
{s′d,H′} is a solution pair of (11) is of measure zero.
Assuming that {s′d,H′} is a solution pair of (11), one can have
(13). Let
U =

CH(s′)C(s′)
−1 CH(s′)C(s). (27)
III ­ 311
Premultiplying
 CH(s′)C(s′)−1 CH(s′) on both sides of (13) re-
sults in
H′ = UH. (28)
Substituting (28) into (13) yields C(s)− C(s′)UH = 0. (29)
Let Φ = C(s) − C(s′)U. Under C1), we have Φ = 0. The proba-
bility that (29) holds is given by
Pr {ΦH = 0} = Pr

Nr
j=1
ΦHej = 0

≤ Pr {ΦHei = 0} , (30)
for some i, where ei ∈ RNr denotes a unit vector with the ith en-
try equal to unity. According to A1), let the ith column of H be
Gaussian distributed with a positive deﬁnite covariance matrix. Then
one can show that Pr {ΦHei = 0} is of measure zero. Hence, (13)
holds with probability zero. Therefore, the data sd can be uniquely
identiﬁed with probability one. 
7. REFERENCES
[1] Z. Liu, G. B. Giannakis, S. Barbarossa, and A. Scaglione, “Transmit-
antenna space-time block coding for generalized OFDM in the pres-
ence of unknown multipath,” IEEE J. Sel. Areas Commun., vol. 19, no.
7, pp. 1352-1364, July 2001.
[2] S. Zhou, B. Muquet, and G. B. Giannakis, “Subspace-based (semi-)
blind channel estimation for block precoded space-time OFDM,” IEEE
Trans. Signal Process., vol. 50, no. 5, pp. 1215-1228, May 2002.
[3] J. Guo, L. Jiang, Y. Gou, and X. Meng, “Blind channel estimator
for MIMO-OFDM systems based on linear non-redundant precoding,”
Proc. IEEE Wireless Commun. and Networking Conf., vol. 1, New Or-
leans, LA, Sept. 23-26, 2005, pp. 44-47.
[4] Y. Zeng, W. H. Lam, and T. S. Ng, “Semiblind channel estimation
and equalization for MIMO space-time coded OFDM,” IEEE Trans.
Circuits and Systems- I, vol. 53, no. 2, pp. 463-474, Feb. 2006.
[5] M. Uysal, N. Al-Dhahir, and C. N. Georghiades, “A space-time block-
coded OFDM scheme for unknown frequency-selective fading chan-
nels,” IEEE Commun. Lett., vol. 5, no. 10, pp. 393-395, Oct. 2001.
[6] S. N. Diggavi, N. Al-Dhahir, A. Stamoulis, and A. R. Calderbank,
“Differential space-time coding for frequency-selective channels,”
IEEE Commun. Lett., vol. 6, no. 6, pp. 253-255, June 2002.
[7] D. M. Teran, R. P. T. Jimenez, “Channel estimation for STBC-OFDM
systems,” Proc. IEEE Workshop Signal Process. Advances Wireless
Commun., Lisbon, Portugal, July 11-14, 2004, pp. 283-287.
[8] Z. Wu, J. He, and G. Gu, “Design for optimal pilot-tones for channel
estimation in MIMO-OFDM systems,” Proc. IEEE Wireless Commun.
and Networking Conf., vol. 1, New Orleans, LA, Sept. 23-26, 2005,
pp. 12-17.
[9] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block
codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45,
no. 5, pp. 1456-1467, July 1999.
[10] E. G. Larsson, P. Stoica, and J. Li, “Orthogonal space-time block
codes: Maximum likelihood detection for unknown channels and un-
structured interferences,” IEEE Trans. Signal Process., vol. 51, no. 2,
pp. 362-372, Feb. 2003.
[11] E. G. Larsson, P. Stoica, and J. Li, “On maximum-likelihood detec-
tion and decoding for space-time coding systems,” IEEE Trans. Signal
Process., vol. 50, no. 4, pp. 937-944, Apr. 2002.
Figure 1. Performance (BER vs. SNR) of the proposed detector for
complex Alamouti code and L = 4.
Figure 2. Performance (BER vs. SNR) of the proposed detector for
complex 4× 3 OSTBC and L = 8.
[12] W.-K. Ma, B.-N. Vo, T. N. Davidson, and P.-C. Ching, “Blind
ML detection of orthogonal space-time block codes: Efﬁcient high-
performance implementations,” IEEE Trans. Signal Process., vol. 54,
no. 2, pp. 738-751, Feb. 2006.
[13] T.-H. Chang, W.-K. Ma, and C.-Y. Chi, “Orthogonal space-time block
coded OFDM in unknown block fading channels: Blind/semiblind
ML, scheme designs, and identiﬁability analysis,” in preparation.
[14] T.-H. Chang, W.-K. Ma, and C.-Y. Chi, “Group-wise blind OFDM
ML detection for complexity reduction,” Proc. 14th European Signal
Process. Conf., Florence, Italy, Sept. 4-8, 2006.
[15] R. Negi and J. Ciofﬁ, “Pilot tone selection for channel estimation in a
mobile OFDM system,” IEEE Trans. Consumer Electronics, vol. 44,
no. 3, pp. 1122-1128, Aug. 1998.
[16] G. Ganesan and P. Stoica, “Differential detection based on space-
time block codes,”Wireless Perosnal Commun., Norwell, MA: Kluwer,
2002, pp. 163-180.
[17] X.-B. Liang, “Orthogonal designs with maximal rates,” IEEE Trans.
Inform. Theory, vol. 49, no. 10, pp. 2468-2503, Oct. 2003.
III ­ 312


Semiblind ML OSTBC-OFDM detection in block fading channels

http://nthur.lib.nthu.edu.tw/dspace/bitstream/987654321/13419/1/2030157030006.pdf

Semiblind ML OSTBC-OFDM detection in block fading channels

Abstract

Similar works

Full text

Available Versions

National Tsing Hua University Institutional Repository