Water-filling for full-duplex multiple-hop MIMO relay system
- Pham Thanh Hiep^{1, 2}Email author and
- Ryuji Kohno^{1}
https://doi.org/10.1186/1687-1499-2014-174
© Thanh Hiep and Kohno; licensee Springer. 2014
Received: 19 December 2013
Accepted: 10 October 2014
Published: 24 October 2014
Abstract
A water-filling (WF) method is one of the techniques to improve a performance of multiple-input multiple-output (MIMO) systems. However, the application of the WF method to multiple-hop MIMO relay systems (MMRSs) has not been discussed, especially in amplify-and-forward (AF) schemes. In this paper, the WF method for MMRSs with AF scheme is proposed and evaluated in both perfect and imperfect channel state information (CSI). The effect of imperfect CSI on optimization of transmit power at all transmitters and on co-channel interference is taken into account. Compared to the average transmit power, the end-to-end channel capacity of the WF method is higher in the perfect CSI; however, it is subject to the effect of imperfect CSI. Additionally, half-duplex and full-duplex transmission methods are also discussed in this paper. The end-to-end channel capacity of full duplex is higher and more robust than that of the half duplex due to a double of allocation time and a half of delay time.
Keywords
1 Introduction
Multiple-input multiple-output (MIMO) relay systems have been discussed in several literatures [1–4]. According to transmission of MIMO schemes, a channel capacity of system was improved. Moreover, the allocation of transmit power for each antenna based on channel model, meaning water-filling (WF), also has been applied to improve the channel capacity of MIMO systems [5–8].
Additionally, in order to reduce the transmit power and/or improve the performance of MIMO system, a multiple-hop MIMO relay system (MMRS) has been analyzed [9–11]. In MMRSs, when the distance between the base station (Tx) and the final receiver (Rx) is fixed, the distance between the Tx to a relay (RS), a RS to a RS, a RS to the Rx called the distance between each transceiver, is shorten. Consequently, the signal-to-noise ratio (SNR) and the channel capacity are increased. However, according to the number of relays, the location, and the transmit power of each relay, the channel capacity of each hop is changed. In addition, the end-to-end channel capacity is restricted by the bottleneck node. Therefore, to obtain the upper bound of end-to-end channel capacity, the location of each relay node, meaning, the distance between each transceiver and the transmit power of each relay, should be optimized. We have analyzed the performance of MMRS with amplify-and-forward (AF) scheme [12, 13] and decode-and-forward (DF) scheme [14]. The distance between each transceiver as well as the transmit power for each relay are optimized to obtain the upper bound of end-to-end channel capacity. However, the transmit power of each relay was divided equally to all antennas in the relay.
Since the DF relay decodes the received data before forwarding, it is known as a regenerative strategy. Furthermore, since each hop is independent of the other hops in the DF scheme MIMO relay system, the original WF method of a MIMO system can be directly applied. On the other hand, the AF relay amplifies the received data only; therefore it is known as a non-regenerative strategy. The computational simplicity in AF relay makes it a highly attractive and a strong candidate for the real-time application. Therefore, in this paper, we focus on the AF scheme MIMO relay system.
There are many literatures that focus on the AF scheme MIMO relay network [9, 10]. In addition, the mutual information and the total mean square error criteria were selected as objective functions [15, 16]. The diversity multiplexing tradeoff for MIMO relays also was investigated [17]. There are some works on the beamforming design for the special type of half-duplex AF scheme MIMO relay system [18]. However, in these literatures, the WF method was not applied.
In short, the WF method was proposed for the original MIMO system and the relay system which has only one relay. The WF method can be directly applied to DF scheme MMRSs. Moreover, the perfect channel state information (CSI) and the half duplex were assumed. In this paper, the WF method for AF scheme MMRSs is proposed and compared to the original WF of MIMO system. The half-duplex as well as the full-duplex transmission schemes are analyzed based on the end-to-end channel capacity. Moreover, the perfect and the imperfect CSI are taken into account. Three schemes of channel model are proposed, and the end-to-end channel capacity is analyzed based on the proposed scheme by the theoretical calculation method.
The rest of the paper is organized as follows. We introduce the concept of MMRSs in Section 2. The WF method for AF scheme MMRSs is proposed in Section 3, and the numerical evaluation is described in Section 4. Finally, Section 5 concludes the paper.
2 AF scheme MMRSs
2.1 Description of MMRSs
2.2 Channel model
In order to easily describe, the Tx and Rx are also be denoted as the R S_{0} and R S_{m+1}, respectively. Since the path loss is taken into consideration, the channel matrix is a composite matrix, and we model as $\sqrt{{l}_{\mathit{\text{ii}}+1}}{\mathbf{\text{H}}}_{\mathit{\text{ii}}+1},i=0,\cdots \phantom{\rule{0.3em}{0ex}},m$, of which l_{i i + 1} and H_{i i + 1} represent the path loss and the K_{i + 1} × K_{ i } channel matrix between the R S_{ i } and the R S_{i + 1}, respectively. H_{i i + 1} is a matrix with independent and identical distribution (i.i.d.), zero mean, unit variance, circularly symmetric complex Gaussian entries.
System parameters of each transceiver
Tx | RS _{1} | ⋯ | RS _{ m } | Rx | |
---|---|---|---|---|---|
Transmit signal | S _{0} | S _{1} | ⋯ | S _{ m } | |
Received signal | R _{1} | ⋯ | R _{ m } | R _{m+1} | |
Noise vector | n _{1} | ⋯ | n _{ m } | n _{m+1} | |
Amplification factor | Γ _{0} | Γ _{1} | ⋯ | Γ _{ m } | |
Total transmit power | E _{0} | E _{1} | ⋯ | E _{ m } |
According to using the system channel matrix H_{m + 1}, the MMRS can be analyzed as same as a conventional MIMO system.
2.3 End-to-end channel capacity
λ_{ i }(i=1,2,⋯,M) is the i th eigenvalue of the correlation matrix ${\mathbf{H}}_{01}{\mathbf{H}}_{01}^{\mathrm{H}}$ (or ${\mathbf{H}}_{01}^{\mathrm{H}}{\mathbf{H}}_{01}$), ${\mathbf{e}}_{0{1}_{r,i}}$ is the eigenvector belonging to the eigenvalue λ_{ i } of correlation matrix ${\mathbf{H}}_{01}{\mathbf{H}}_{01}^{\mathrm{H}}$, and ${\mathbf{e}}_{0{1}_{t,i}}$ is the eigenvector belonging to the eigenvalue λ_{ i } of correlation matrix ${\mathbf{H}}_{01}^{\mathrm{H}}{\mathbf{H}}_{01}$. The SVD is applied similarly to the other channel matrices.
here, I_{ M } denotes the M×M unit matrix.
here, ${\mathbf{SNR}}_{i-1i}=\frac{{l}_{i-1i}{\mathbf{P}}_{i-1}}{{\sigma}^{2}}=\text{diag}\left({\gamma}_{i-1{i}_{1}},{\gamma}_{i-1{i}_{2}},\cdots \phantom{\rule{0.3em}{0ex}},{\gamma}_{i-1{i}_{M}}\right)$ represents the SNR of all paths of hop between the R S_{i-1} and the R S_{ i } regardless the other relays. Since the transmit power and the distance are fixed after optimization as optimization method in [12, 13] and notice that Tr(P_{ i })=W_{ i }, trace of S N R_{i-1i} is fixed as $\frac{{l}_{i-1i}{W}_{i-1}}{{\sigma}^{2}}={\text{snr}}_{i}$. The snr_{ i } is the ratio of received signal power to noise power where the signal is transmitted and received by one antenna.
3 WF method for AF scheme MMRS
3.1 WF method for AF scheme MMRS with perfect CSI
The WF method can be applied to the AF scheme MMRS by three schemes. Scheme 1: the WF method is applied to each hop in MMRS. Scheme 2: the WF method is applied to each process. Scheme 3: the WF method is applied to all channel models of MMRS. The detail of each scheme is explained as follows.
The channel capacity maximization problem is now changed to the choice of the maximal number of real, non-negative values ${\gamma}_{i-1{i}_{k}}$ subject to the power constraint ${\sum}_{k=1}^{M}{\gamma}_{i-1{i}_{k}}={\text{snr}}_{i}$.
A computational procedure for determining the optimal value of q is to compute for q=M,M-1,⋯ the value of ${\gamma}_{i-1{i}_{k}}$ until this quantity is greater than zero for all k from 1 to q. By substituting the optimal ${\lambda}_{i-1{i}_{k}}$ to (13) and (11), the end-to-end channel capacity can be obtained.
and the S N R_{i-1i} is optimized based on D_{ i } as mentioned above.
The WF method of all hops is dependent on D_{m+1}; it means that the WF method of all hops is the same.
Up to now, the perfect CSI at both the transmitter and the receiver has been assumed, and the WF method for MMRS has been analyzed based on the proposed scheme. However, in actuality, the perfect CSI assumption is not always practical due to channel estimation errors, feedback delay, and noise. Compared to channel estimation errors, the CSI imperfection introduced by feedback channel delay is sometimes more significant and inevitable.
3.2 WF method for AF scheme MMRS with imperfect CSI
where ${\widehat{\mathbf{H}}}_{i-1i}$ denotes the outdated channel matrix while ${\stackrel{\u0304}{\mathbf{H}}}_{i-1i}$ is the true one. Notice that there are two ways the outdated CSI affects the system. One is the effect on the separation of subchannel between each transceiver; the other one is the effect on optimization of transmit power of all transmitters. The latter one depends on the scheme of the WF method. The former one depends on the CSI between each transceiver regardless of other CSIs, and it is the same as the CSI of scheme 1 of the latter one. The CSI of the latter one is explained as follows. Notice that the full duplex is assumed, all terminals transmit and receive in the same time. The delay of half duplex is a double of full duplex.
For scheme 1: we assume that transmitters add the CSI symbol in data packets and transmit to receivers; hence, the receiver knows both ${\stackrel{\u0304}{\mathbf{\text{H}}}}_{i}$ and ${\widehat{\mathbf{\text{H}}}}_{i}$. After receiving the CSI symbol, the receiver feedbacks the CSI to the transmitter immediately. However, the transmitter receives the CSI after a delay. Therefore, the CSI that is used at the transmitter is the outdated CSI. It means that the transmitter only knows the outdated channel model, ${\widehat{\mathbf{\text{H}}}}_{i}$, while the receiver knows both ${\stackrel{\u0304}{\mathbf{\text{H}}}}_{i}$ and ${\widehat{\mathbf{\text{H}}}}_{i}$. The delay at each hop is assumed to be the same as τ. Therefore, in scheme 1, the delay of CSI at each transmitter is the same as τ.
For scheme 2: the CSI symbol that was added at the Tx is assumed to be forwarded at all relays; therefore, all relays and the Rx can know the CSI from the Tx to itself. However, when a relay receives the CSI symbol, the CSI of the previous hop becomes the outdated. Notice that the delay of forward link (from the Tx to the Rx) and backward link (from the Rx to the Tx) was assumed to be the same as τ. The R S_{i-1} starts to optimize the transmit power after receiving the feedback CSI from the R S_{ i }; therefore, although the R S_{i-1} receives the CSI of hop between the Tx and the R S_{1} after the delay of (i-2)τ, however, it should wait to receive the feedback from the R S_{ i }; hence the R S_{i-1} starts to optimize the transmit power when the delay is i τ. Similarly, the delay of channel model between the R S_{j-1} and the R S_{ j } becomes (i-j+1)τ. As a result, the delay of CSI of H_{i-1i},⋯,H_{01} is 1τ,⋯,i τ, respectively.
For scheme 3: we assumed that a relay (R S_{ i }) not only feedbacks the CSI to the previous relay (R S_{i-1}) but also forwards the CSI received from the next relay (R S_{i+1}) to the previous relay (R S_{i-1}). The process is repeated until the Tx receives the CSI from the Rx. All transmitters optimize their transmit power every time the CSI is updated; however, as analyzed in the previous section, at all relays, the transmit power is optimized based on the same multiplication of channel models. Therefore, the delay of CSI is calculated based on the same channel models. When the Tx transmits a signal, the delay of CSI of H_{01},⋯,H_{m m+1} is τ,⋯,(m+1)τ, respectively. Compared to the transmission of the Tx, the R S_{ i } transmits a signal after the delay of i τ; therefore, the delay of CSI of H_{01},⋯,H_{m m+1} is (i+1)τ,⋯,(i+m+1)τ, respectively.
Compared to the system has the perfect CSI at both the transmitter and the receiver (13), in the function f(λ) of the system that has the outdated CSI at the transmitter (29), the co-channel interference by the outdated CSI is added.
4 Numerical evaluation
4.1 Effects of WF method
4.2 System with outdated CSI
5 Conclusions
In this paper, we have analyzed the end-to-end channel capacity of MMRSs according to the proposed WF method. The WF method is divided into three schemes. In scheme 1, the WF method is applied to each hop. In scheme 2, the transmit power is optimized based on the multiplication of channel models from the Tx to each relay, and in scheme 3, transmit power of all transmitters is the same and optimized based on multiplication of channel models from the Tx to the Rx. The end-to-end channel capacity of the proposed WF method is higher than that of the average transmit power, and the end-to-end channel capacity of scheme 3 is the highest in the perfect CSI. However, when the imperfect CSI is taken into account, the outdated CSI not only affects the gap between the optimized transmit power and channel models but also is effective in the co-channel interference. In the outdated CSI system, the end-to-end channel capacity decreases when the term f_{ D }τ increases, especially in the high-SNR region due to the high power of co-channel interference. Additionally, the end-to-end channel capacity of the WF methods decreases close to or drops below that of average transmit power. The average transmit power system is the most robust, and scheme 3 is the most sensitive. The half duplex and the full duplex also have been discussed in this paper. Since the full duplex has a double of allocation time and a half of delay time, the full-duplex system is more robust than the half-duplex system, and the end-to-end channel capacity of the full-duplex system is higher than that of the half-duplex system. In this paper, since the full duplex has been taken into consideration, at least four antennas are necessary. The full duplex has more advantages when the number of antennas is greater than or equal to four, whereas the half duplex has more advantages when the number of antennas is less than four. Additionally, the greater the number of antennas is, the higher the end-to-end channel capacity achieves. Moreover, the signal processing of full duplex is considered to be more complicated than that of the half duplex.
Since the distance and the transmit power have been optimized in other literatures, in this paper, the same SNR at all transmitters was assumed. However, the optimization of distance and transmit power based on each WF method should be considered to improve the performance of MMRS. Additionally, the transmission of all transmitters in both the full duplex and the half duplex was assumed to be controlled on MAC layer. However, the detail of control should be discussed. We leave them to the future works.
Declarations
Authors’ Affiliations
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