Base-station–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 flat-fading 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 multiple-input–multiple-output (MIMO) system. Sendonaris et al.^1,2 have shown that even with a noisy inter-user channel, cooperative communication still leads to improved performance, and is generally a more robust system. One real MIMO system is the space-time 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 special-case, full-rate orthogonal STBC with two transmit antennas and one receive antenna. A typical example of the system is a base-station–based cooperative communication in macrocell downlink networks, which is proposed by Skjevling et al.⁶ In the base-station–based cooperation system, all base stations are connected via a very fast-wired 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 Deneire⁷ 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 scheme⁵ 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 Deneire⁷ 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 h₁ and h₂ , respectively, where h₁ and h₂ are assumed to be the complex scalar channel parameters. It is assumed that both base stations have knowledge of both pilot sequences d1 and d2. In reality, this can be achieved via either a wired, high-speed 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 time-reversed 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.

Figure 1 Data transmission model

Figure 2 Block transmission

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, dn and dn₊₁, where n is the block number. Training symbols d1 and d2, each of length L, are then added at the end of dn and dn₊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 pre-multiplied by a precoding matrix F_p. Multiplying the vectors by F_p results in two (N + 2L) × 1 blocks, sn and sn₊₁.

The blocks are then transmitted according to the time-reversed 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 time-reversal matrix T ⁷ are given by

By defining τ = τ₂ - τ₁ as the difference between the arrival time of the two signals, where τ₁ and τ₂ are the arrival times of the first (BS1) and second (BS2) signal respectively, the received signal r can then be defined as⁷

and

For positive delay A(τ) is given by

It is worthwhile to note that for the synchronous case, Γ =O2_{N+4 L} and Ψ = I2_N+4L.

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 dashed-dotted 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 s1[n], but from BS2, the delayed τ symbols from the previous frame [s2 (n-1)] are received first and the last τ symbols of s2[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 dashed-dotted lines represent ones, and blank spaces represent zeros in the corresponding matrices.

Figure 3 & 4

A maximum likelihood (ML) estimator used by Tourki and Deneire⁷ and Sirbu⁸ is also used for the channel and delay estimation in this section. Firstly we summarise the estimation algorithm of Tourki and Deneire⁷ for τ ≥ 0, and then use the same approach to derive the estimation algorithm for τ < 0.

Let ts₁ and ts₂ be defined as

where T_s is a time-reversal 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) = [ss1(τ) ss2(τ)]. For convenience, ‘L₁:L₂’ is defined as a sequence from index L₁ to L₂.

Figure 5 shows S(τ) for τ ≥ 0, where ss₁ and ss₂ 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 ss₁ and ss₂ 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 that⁷

Figure: 5 & 6

where [.]^# represents the pseudo-inverse 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 Deneire⁷ and Sirbu⁸ 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.

While ignoring the noise terms, define ra and rb as⁷

where TN is a time-reversal 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 Deneire⁷. 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

Figure 7 & 8

Channel estimation performance

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

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.

Figure 9 & 10

SER performance

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.

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