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Distributed transmit diversity (DTD) technique that combines cooperative communications and diversity techniques is a suitable solution in 5th-generation (5G) systems. In this paper, we investigate the effect of receiver phase compensation (RPC) on the performance of DTD. We introduce new expressions for the average error rate of DTD in the presence of RPC. The derived expressions are useful for a large number of modulation schemes. We obtain further insights by comparing the RPC effect on DTD with different spatially correlated collocated transmit diversity (CTD). The new observations include the following:

In the literature, transmit diversity (TD) systems are classified into distributed and collocated systems, which are referred to as distributed TD (DTD) and collocated TD (CTD), respectively. In CTD, multiple transmit (TX) antennas are located on a single transmitter. On the other hand, in DTD, these antennas are far apart and have different slow fading factors. Hence, in DTD, there is another diversity called macrodiversity, which is not available in CTD [

In cellular systems with DTD, user equipment (UE) is connected to individual antennas at various base stations (BSs). On the other hand, in cellular systems with CTD, the BS that has the highest mean channel gain (MCG) is connected to the considered UE. In the relevant literature, MCG is referred to as a large-scale fading factor, channel gain variance, or long-term fading factor. Unlike the DTD, where the MCGs are different among the connected TX antennas, all the TX antennas in CTD have the highest MCGs as the TX antennas are collocated, but there is a spatial correlation among the collocated TX antennas [

Because of the realization of signal-to-noise ratio (SNR) maximization and coherent CTD and DTD, receiver-side channel state information (CSI) is needed in the transmitter. This is prepared by the CSI feedback from the receiver side to the transmitter side [

As mentioned above, the feedback delay in TX diversity systems incurs undesirable self-noise term in the received signal. This term disturbs the magnitude and the phase of the combined signal component in front of the RX antenna and thus degrades the performance of CTD and DTD systems. Therefore, to improve the system’s performance, the compensation (derotation) for the disturbed phase is needed on the receiver side before detecting the signal. This compensation is to referred as RPC (receiver phase compensation) in this paper. Owing to price or space limitations, the RPC is sometimes excluded, for example, a very low complexity receiver such as machine-type communication (MTC) terminals, sensors, or Internet of Things (IoT) devices, which enables the receiver side to focus only on demodulation, while the transmitter side consists of all the preprocessing, such as precompensation of the phase rotation by the channel, by using CSI feedback from the receiver. At present, the joint effect of limited-rates and outdated feedback has been investigated in CTD systems without RPC [

In this paper, we investigate the effect of RPC on the DTD for the case when the transmit diversity weights are outdated due to feedback delay. By exploiting an accurate method for modeling the feedback delay [

The paper is organized as follows. In Section

We consider a MISO system with

System model.

Model of DTD and CTD with feedback delay

Model of DTD and CTD in a cell

In this model, the TD antennas are shared between the connected multiple UEs. The signals to the UEs are multiplexed in the code/time or frequency domains, as done in orthogonal frequency-division multiple access (OFDMA) or code-division multiple access (CDMA) and not in the spatial domain, as in multiuser (MU) multiple-input multiple-output (MIMO). In this study, the use of TD is to improve the link quality and not for multiplexing in the spatial domain. The same usages of TD in OFDMA and CDMA have been studied in [

Common to collocated and distributed TDs, the received signal in each UE is calculated as [

In the case involving delay, the term

In this section, we derive the average error rate of DTD systems with and without RPC, respectively, using the statistical characterization of the SNR.

With a feedback delay

By substituting (

In general, the delayed CSI feedback causes performance degradation. In this subsection, we present a new closed-form expression for the moment generating function (MGF) of SNR in DTD with RPC systems. In order to derive the MGF of the SNR, the conditional MGF,

The received SNR conditioned on the feedback loop delay can be calculated in a complex Gaussian quadratic form as follows:

We can treat

Note that although the overall channel fading factors (

According to the similar procedure of deriving a closed-form PDF expression in the received SNR as ([

By substituting (

The second term in (

Using the SNR statistical characterization, the average error rate of DTD without RPC is calculated in our previous work [

Let us now consider the case of DTD in an ideal environment without delay. Because we omit the error term of the delay in ideal TD systems, RPC does not affect their performance. The average error rate formula for the no-delay DTD system is calculated in our previous work as [

In this section, we perform Monte Carlo simulations to validate the analytical results derived in previous subsections, and we investigate the effect of RPC and the feedback loop delay on the performance of DTD and CTD systems. As previously mentioned, the channel model is considered as a composite Rayleigh fading that consists of small-scale fading and MCG (shadowing or path loss). Unless otherwise specified, for simplicity, we assume that

Figure

Simulated and analytical average error rate of DTD in the presence of RPC with various delay values and parameters:

Figure

Simulated and analytical average error rate of DTD in the absence of RPC [

Figure

Average error rate comparison of DTD and correlated CTD versus MCG (

Finally, to obtain further insights, we analyze in detail the effect of RPC on DTD compared to the correlated CTD. Figure

Average error rate comparison of DTD and correlated CTD in detail, according to RPC with

The various Doppler frequency and delay product with

The various SCC

In this paper, we investigated the impact of the feedback loop delay and RPC on the average error rate of DTD over Rayleigh fading channels. We presented the analytical error rate expressions and simulation results to show the effectiveness of the study based on DTD. The results show that not only the transmitter side PC (weighting factors) but also the receiver side PC is needed in the case of existing feedback delay. In the case where the minimized receiver side is preferred to the small improvements in the system performance, RPC is omitted in the small-size feedback loop delay. The results also present that as the difference between the MCGs increases (e.g., UE gets close to one of the BSs), the performance of DTD with RPC improves. However, this is opposite to the case without any RPC. We performed the comparison for the DTD performance with conventional correlated CTD, and we conclude that, in the case of existing feedback loop delay and excluding RPC, DTD is preferable over conventional correlated CTD, except for the cases where there are large differences between the MCGs. Finally, we conclude that, in the no-delay cases, RPC does not affect the performance of CTD and DTD systems.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This research was supported by Basic Science Research Program through the National Research Foundation (NRF) (2015R1D1A3A01015970) funded by the Ministry of Education, Republic of Korea.