Basestation–based cooperative communication is an asynchronous cooperative communication system. The challenge therein is to estimate
the relative delay between the transmitters in the system. A channel and delay estimation algorithm based on the distributed Alamouti scheme
has been previously discussed for the asynchronous cooperative system. The algorithm only makes accommodation for positive delay, that is,
when data from Transmitter one always arrives at the receiver before data from Transmitter two. In reality, the data from Transmitter one
does not always arrive at the receiver before data from Transmitter two in the asynchronous cooperative system, because of the constantly
changing mobile environment in the system. This paper extends the algorithm to accommodate both positive and negative delays, which is when
the data from Transmitter one arrives at the receiver before or after data from Transmitter two, in a Rayleigh block flatfading
channel. Simulation results show that the Cramér–Rao lower bound for channel and delay estimation is achieved for different delay
values. The symbol error rate performance is also achievable compared to the channel and the delay is known at the receiver.
Cooperative diversity is a relatively new transmission scheme that combines the ideas of relays and multiple antennas, while avoiding the
size limitations of a single mobile node. It provides spatial diversity by allowing multiple nodes, each with a single antenna (it is
possible to increase the number of antennas), to work together to form a virtual multipleinput–multipleoutput (MIMO) system.
Sendonaris et al.^{1,2} have shown that even with a noisy interuser channel, cooperative communication still leads to improved
performance, and is generally a more robust system. One real MIMO system is the spacetime block code (STBC) system, which achieves both
space and time diversity. If both STBC and cooperative diversity are incorporated together, a distributed STBC system is formed.
^{3,4} In this paper, we focus on cooperative communication using the Alamouti scheme,^{5} which is a specialcase, fullrate orthogonal
STBC with two transmit antennas and one receive antenna. A typical example of the system is a basestation–based cooperative
communication in macrocell downlink networks, which is proposed by Skjevling et al.^{6} In the basestation–based
cooperation system, all base stations are connected via a very fastwired local area network and are setup to transmit data to one
mobile user. Skjevling et al.^{6} discussed a precoded distributed STBC synchronous cooperative diversity system. But in
reality this scenario is unlikely. The challenge therein is to estimate the relative delay between the transmitters in the system.
This problem has been addressed, which resulted in a channel and delay estimation algorithm that achieved the lower Cramér–
Rao bound (CRB).^{7} Tourki and Deneire provided a simple maximum likelihood channel and delay estimation algorithm, but this
system was only derived for positive delays, that is, when Transmitter one’s data always arrives at the receiver before
Transmitter two’s data.^{7} The main motivation behind this paper is to extend the scheme in Tourki and Deneire^{7} to accommodate negative delays (i.e.
when Transmitter two’s data arrives at the receiver before Transmitter one’s data). This extension is significant,
for example, in the case of a macrocell, as considered in this paper. Specifically, the macrocell considered here consists of two base
stations transmitting the same data to one node. Since the node is mobile, it could be closer to either base station at any given time,
and it cannot be arbitrarily assumed that the first signal it receives is from base station one, as the Alamouti’s scheme^{5}
strictly dictates the role of each of the two transmitters in the transmission scheme. By extending the delay estimation search to negative
delays, the receiver can determine which transmitter’s data arrived first, and hence properly decode the received signals. This paper is structured as follows: in Section II, the system model of Tourki and Deneire^{7 }is extended to accommodate negative
delays, in Section III the channel and delay estimation algorithm for both positive and negative delays is derived, in Section IV the detection
scheme for a single receiver is shown, in Section V the symbol error probability is discussed, in Section VI the simulation results are
presented and, finally, in Section VII the conclusions are drawn.
The system model consists of two base stations (BS1 and BS2) transmitting data to one mobile node, each with a single transceiver.^{7}
The channel gains from BS1 and BS2 to the mobile node are h_{1} and h_{2} , respectively, where h_{1}
and h_{2} are assumed to be the complex scalar channel parameters. It is assumed that both base stations have
knowledge of both pilot sequences d_{}1 and d_{}2. In reality, this can be achieved via either a
wired, highspeed connection such as Ethernet, or a wireless transmission of the data between two base stations preceding the
transmission scheme presented here. The data sequences to be transmitted are replicated in space and time according to the timereversed block form of the
Alamouti’s scheme,^{5} allowing the mobile node to combine and decode the two received signals, using a simple
linear technique while enjoying the benefits of spatial diversity.
The data transmission model is shown in Figure 1. To begin with, the data set to be transmitted is parsed into two blocks of
N symbols each, d_{}n and d_{}n_{+1}, where n is the block number.
Training symbols d_{}1 and d_{}2, each of length L, are then added at the end of
d_{}n and d_{}n_{+1}, respectively, to form two (N + L) ×
1 vectors. To insert a cyclic prefix of training symbols between any two successive blocks, these vectors are premultiplied by a
precoding matrix F_{p}. Multiplying the vectors by F_{p} results in two (N + 2L) ×
1 blocks, s_{}n and s_{}n_{+1}. The blocks are then transmitted according to the timereversed block form of the Alamouti’s scheme shown in Figure 2.
In Figure 2, (.)^{*} indicates the complex conjugate operator. The precoding matrix F_{p}_{ }and timereversal matrix T ^{7} are given by
where O_{}L_{×}_{N} is a (L × N) matrix of zeros,
I_{}L and I_{}N+L are identity matrices of dimensions (L × L) and
(N + L) × (N + L), respectively.By defining τ = τ_{2}  τ_{1} as the difference between the arrival time of the two signals,
where τ_{1} and τ_{2} are the arrival times of the first (BS1) and second (BS2) signal respectively,
the received signal r can then be defined as^{7}
and
In [Eqn 3] to [Eqn 5], r is a symbol vector of length (2N + 4L); τ є [(L1), L1]
is the relative delay between the two received signals; h_{i}, i = 1, 2, are the zero mean complex
scalar channel parameters. They are also assumed to be Rayleigh block flatfading, that is, the amplitude of the fading envelope
follows a Rayleigh distribution which stays constant for the duration of each frame, but varies independently from frame to frame.
In [Eqn 3], b is assumed to be zero mean complex additive white Gaussian noise (AWGN), with each entry having a variance of
N_{0}/2 per dimension, while X is the data matrix where x denotes symbols from the previous frame
which are treated as ‘don’tcares’. The matrix A(τ), denotes the delay matrix.For positive delay A(τ) is given by
where · is the absolute value operator.It is worthwhile to note that for the synchronous case, Γ =O_{}2_{N}_{+4}_{
L} and Ψ = I_{}2_{N}_{+4}_{L}. To extend the received signal model to accommodate negative delays, one needs only to modify [Eqn 6]. To understand this extension,
one first needs to understand how [Eqn 6] accounts for the asynchronicity of the system. Figure 3 shows the illustration of A(τ)
for τ ≥ 0. In Figure 3, the dasheddotted lines represent ones and blank spaces represent zeros in the corresponding matrices.
When A(τ) is multiplied by the data matrix X, for τ ≥ 0 it can be seen that the received signal from BS1
will just be s_{}1[n], but from BS2, the delayed τ symbols from the previous frame [s_{}2
(n1)] are received first and the last τ symbols of s_{}2[n] will be received in the next frame. For negative delays, only the symbols from BS1 will be delayed, and hence the resulting equation for A(τ) is
Figure 4 shows the illustration of A(τ) for τ < 0. Once again, the dasheddotted lines represent ones, and blank
spaces represent zeros in the corresponding matrices.
III. Channel and Delay Estimation Algorithm


A maximum likelihood (ML) estimator used by Tourki and Deneire^{7 }and Sirbu^{8} is also used for the channel and delay
estimation in this section. Firstly we summarise the estimation algorithm of Tourki and Deneire^{7} for τ ≥ 0, and then
use the same approach to derive the estimation algorithm for τ < 0. Let ts_{1} and ts_{2} be defined as
where T_{s} is a timereversal matrix of size L (i.e. T_{s} is basically a small version of
the matrix T). The section of the received signal where the pilot symbols from each transmitter overlap are defined as S(τ) =
[ss_{}1(τ) ss_{}2(τ)]. For convenience, ‘L_{1}:L_{2}’ is
defined as a sequence from index L_{1 }to L_{2}. Figure 5 shows S(τ) for τ ≥ 0, where ss_{1} and ss_{2} are defined as
Figure 6 shows S(τ) for V<0. From Figure 6, it is observed that the symbols from BS1 are shifted.
The modified equations for ss_{1} and ss_{2} for (τ)<0 are given by
From Figures 5 and 6, one can define z(τ) as z(τ) = S(τ)h = r
(N + L + τ + 1: N + 3L). It then follows that^{7}
where [.]^{#} represents the pseudoinverse operator and h is the linear least squares estimate of h
for a given value of τ. Since delay search space has been extended to accommodate negative delay values, the ML estimator used by Tourki and
Deneire^{7} and Sirbu^{8} has to be modified to
where ĥ is the final estimate of h based on the delay estimate τ^. The minimum square error (MSE) of the channel estimation is given by
where E[.] is the expectation operator and H represents the conjugate and transpose operation.
The data symbols are extracted and decoded as follows:While ignoring the noise terms, define r_{}a and r_{}b as^{7}
where T_{}N is a timereversal matrix of size N. For a clearer understanding, r_{a} and r_{b} are illustrated in Figures 7 and 8. A block form of the decoding scheme has been derived here, and uses the same orthogonal system properties as the scheme in Tourki and
Deneire^{7}. As will be shown, the following block form also allows for a simple modification to accommodate negative delays.
Define r_{w}, r_{x} and r_{n} as
where T_{}r is a square timereversal matrix of size N + τ, and r_{}a and r_{}b
are defined in [Eqn 20] and [Eqn 21], respectively.
The symbol error rate (SER) of orthogonal STBC over Rayleigh block flatfading channels can be evaluated using the wellknown approach
of averaging the conditional SE R, P(Eγ), over the probability density function, p_{}γ(γ),
of the instantaneous signaltonoise ratio (SNR) per symbol, γ, as shown below
The analysis done by Shin and Hong Lee^{9} yields the closed form SER of STBC for MPSK signals transmitted over a
Rayleigh block flatfading channels as
with E_{s} being the symbol energy and R being the transmission rate, the numbers n_{T}
and n_{R} are the numbers of transmit and receive antennas, respectively, Γ(.) represents the gamma
function, and represent the Gauss and Appell hypergeometric functions respectively and φ is the moment generating function
for the instantaneous SNR given by
In our simulations, each block contains 140 symbols of which 112 are used for data and L = 14 for the pilot sequence. When
choosing the pilot sequence length, there is a tradeoff between estimation performance and bandwidth efficiency. With this in mind,
a sequence length of 14 was chosen as a reasonable balance between the two. The 4PSK modulation was used, and the channel and noise
parameters used were as described from [Eqn 3]. The channels were assumed to be unknown and hence needed to be estimated. Unless
otherwise stated, the delay was assumed to remain constant over each frame, but was allowed to vary randomly from frame to frame.
The delays were uniformly distributed between (L1) and (L1). The SNR values that were used refer to the ratio between
symbol and noise energy. The singleinput–singleoutput (SISO) equaliser used was a minimum mean square error (MMSE) equaliser.
Channel estimation performance
Channel and delay estimation was normally implemented using a training sequence or pilot sequence. Tourki and Deneire^{7 }did
not discuss the method to design the pilot sequence. To estimate channel and delay, it is not merely enough to simply transmit a known
set of symbols as pilot sequences. These sequences have to be carefully designed in order to achieve the best channel estimation
performance possible. The design of the pilot sequence in frequency flatfading channels for asynchronous cooperative communication
systems has been discussed by Padayachee^{10,11}. To derive optimal pilot sequences, the base station has to have information
about the channel delay. In Padayachee^{10,11}, a packet transmit scheme was proposed to enable the base stations to use an
estimate of the delay to select the corresponding optimal pilot sequences for that specific delay. The scheme can be divided into three phases. In the first phase, the base stations transmit x pilot frames consisting of pilot
symbols only. The mobile receiver then uses these frames to obtain an average estimate of the channel delay, and then transmits the
estimated delay back to the base stations via the feedback channel in phase two. In phase three, the base stations use this delay
estimate to select the corresponding optimal pilot sequences for channel estimation, to be used in the next y normal frames
consisting of data and pilot symbols. It is assumed that the delay remains constant over each round of the above three phases, but
is allowed to vary from round to round. In our simulations, we also use the two transmit schemes. Figure 9 shows the MSE performance of the channel estimation. It has the
following plots: • MSE of the channel estimation when the channel and delay are unknown. • MSE of the channel estimation when the channel and delay are unknown and where the packet scheme is used. After every 200 data
frames, five pilot symbol frames are sent at an SNR of 10 dB. • CRB for channel estimation. The CRB serves as a fundamental lower bound on the performance of any unbiased estimator. The CRB for channel estimation as derived
by Berriche et al.^{12} is given by
where tr(.) represents the trace operator. It is important to note, however, that this bound was derived assuming that the
delay was a known parameter, therefore it is only used here in order to obtain a tractable comparison.It can be observed from Figure 9 that, for low SNRs, the channel estimation of the normal scheme significantly differs from the CRB,
due to the relatively high error rate of the delay estimation. However, using the packet scheme, which has a relatively low error rate,
the channel estimation performance practically achieves the CRB. This is because feedback is used in the packet transmit scheme to choose
an optimal pilot sequence.
SER performance
The SER derivation [Eqn 33] was done assuming that the channel and delay were known parameters, as obtaining a closed form expression
of the SER would otherwise prove rather complex. This assumption was made in order to obtain a tractable comparison, and its result is
plotted in Figure 10 as the ‘ideal’. From Figure 10, it can be observed that at low SNRs, the normal scheme performance is between 1 dB and 2 dB worse than the ‘
ideal’, whereas the packet scheme is approximately 0.5 dB off the ‘ideal’. This performance demonstrates that if
the packet scheme is used, relaxing the known channel and delay assumption only results in a mere 0.5 dB difference. Also, if the
channel is assumed to be known, the delay estimation exhibits very low error rate and the SER performance achieves the ‘ideal’,
demonstrating that the channel estimation dominates the system performance.
In this paper, the scheme proposed by Tourki and Deneire^{7} was extended to accommodate negative delays in basestation–
based cooperative communication systems. The various concepts and modifications were explained in detail. A block form of the decoding
scheme was also presented, which allowed for a simple modification to accommodate negative delays. Both the normal scheme and the packet
scheme were simulated. It was shown via the simulations that the packet scheme outperformed the normal scheme by achieving the CRB for
channel estimation. As far as the SER is concerned, the packet scheme demonstrated that performance gains were achievable at minimal
expense in bandwidth efficiency. The only limitation for the packet scheme is that the delay remains constant over the three phases.
Possible future work would be to extend the scheme presented in this paper into frequencyselective fading channels.
1. Sendonaris A, Erkip E, Aazhang B. User cooperation diversity. Part I: System description. IEEE Trans Commun. 2003;51(11):1927–1938. 2. Sendonaris A, Erkip E, Aazhang B. User cooperation diversity. Part II: Implementation aspects and performance analysis. IEEE Trans Commun. 2003;51(11):1939–1948. 3. Laneman JN, Wornell GW. Distributed spacetime coded protocols for exploiting cooperative diversity in wireless networks. IEEE Trans Inf Theory. 2003;49(10):2415–2425. 4. Li X. Spacetime coded multitransmission among distributed transmitters without perfect synchronisation. IEEE Signal Process Lett. 2004;11(12):948–951. 5. Alamouti SM. A simple transmit diversity technique for wireless communications. IEEE J Sel Areas Commun. 1998;16(8):1451–1458. 6. Skjevling H, Gesbert D, Hjørungnes A. Precoded distributed spacetime block codes in cooperative diversitybased downlink. IEEE Trans Wireless Commun. 2007;6(12):4209–4214. 7. Tourki K, Deneire L. Channel and delay estimation algorithm for asynchronous cooperative diversity. Wireless Pers Commun. 2006;37:361–369. 8. Sirbu M. Channel and delay estimation algorithms for wireless communication systems. PhD thesis, Helsinki University of Technology, Helsinki; 2003. 9. Shin H, Hong Lee J. Exact symbol error probability of orthogonal spacetime block codes. Paper presented at: IEEE Globecom 2002. Proceedings of IEEE Globecom conference; 2002 Nov 17–21; Taipei, Taiwan. New York: IEEE Communications Society; 2002, p. 1197–1201. 10. Padayachee L. A channel and delay estimation algorithm for asynchronous cooperative diversity with pilot symbol design. Paper presented at: SATNAC 2008. Proceedings of Southern African Telecommunication Networks and Applications Conference; 2008 Sep 7–10; Wild Coast Sun, Eastern Cape, South Africa. Pretoria: Telkom; 2008. 11. Padayachee L. Asynchronous cooperative communication with channel and delay estimation. MSc thesis, University of KwaZuluNatal, Howard College campus, Durban; 2009. 12. Berriche L, AbedMeraim K, Belfiore JC. CramérRao bounds for MIMO Channel Estimation. Paper presented at: IEEE International Conference on Acoustics, Speech, and Signal Processing 2004. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing; 2004 May 17–21; Montreal, Canada. New York: IEEE Signal Processing Society; 2004, p. 397–400.
