Performance of Cross-Layer Design with Antenna Selection and Imperfect Feedback Information in MIMO Systems

Abstract

By combining adaptive modulation and automatic repeat request, a cross-layer design (CLD) scheme for MIMO system with antenna selection (AS) and imperfect feedback is presented, and the corresponding performance is studied. Subject to a target packet loss rate and fixed power constraint, the variable switching thresholds of fading gain are derived. According to these results, and using mathematical manipulation, the average spectrum efficiency (SE) and packet error rate (PER) of the system are further derived. As a result, closed-form expressions of the average SE and PER are obtained, respectively. These expressions include the expressions under perfect channel state information as special cases and provide good performance evaluation for the system. Numerical results show that the proposed CLD scheme with antenna selection has higher SE than the existing CLD scheme with space-time block coding, and the CLD scheme with variable switching thresholds outperforms that with conventional-fixed switching thresholds.

1. Introduction

With the fast development of modern communication techniques, the demand for high data rate service is growing increasingly in the limited radio spectrum. For this reason, the future wireless communication systems will require spectrally efficient techniques to increase the system capacity. Cross-layer design (CLD), as a good work to improve the spectral efficiency (SE) and system throughput of future wireless systems while meeting the prescribed quality of service (QoS) requirements, has received much attention recently [1–3]. Especially, CLD combining adaptive modulation (AM) and automatic repeat request (ARQ) which is widely accepted as an efficient means to improve the overall performance of transmission in fading channels has been introduced in the improved 3G scheme, such as high-speed downlink packet access (HSDPA) [4–8]. Antenna selection (AS), which is a one promising approach to achieve the goal of providing performance benefits while significantly decreasing the hardware complexity and cost, has received considerable studies [9–11]. It can provide a good tradeoff between the performance, cost, and complexity and can be realized at both ends. Therefore, effective combination of cross-layer design and antenna selection techniques will receive much attention for practical purpose.

The transmit antenna selection scheme with maximal ratio combining (MRC) is an effective diversity scheme [9, 10]. Based on this antenna selection scheme, the performance analysis of discrete-rate adaptive M-ray quadrature amplitude modulation (MQAM) with AS is presented in [12], where the average frame error rate of the system is considered. The performance of adaptive MQAM with antenna selection is studied in [11, 13] for perfect and imperfect estimation, respectively, where the average BER and spectrum efficiency are analyzed. By combining adaptive modulation and coding (AMC) and selection of transmit diversity in VBLAST multiplexing multiple-input and multiple-output (MIMO) system, [14] investigates the average throughput of the combined systems through simulations. But the above systems do not consider the superiority of cross-layer design. For this reason, a CLD-combined AMC at the physical layer (PHY) and ARQ protocol at the data link layer (DLL) over single-input and single-output fading channel is developed in [4]. Based on the basic idea of CLD in [4], a cross-layer design scheme with space-time block coding (STBC) is presented in Rayleigh fading channel [5]. In terms of different space-time coding schemes, the performance of MIMO system with CLD is studied in [6]. The performance of CLD with STBC is analyzed in optical MIMO system in [7]. Based on perfect channel state information (CSI), a cross-layer approach to transmit antenna selection on the basis of a throughput-based criterion is presented in [3]. Under the feedback constraint, a CLD scheme for multiuser MIMO systems is analyzed in [8].

In all these studies, the system performances of AM with antenna selection or CLD with space-time coding are well analyzed. However, the CLD scheme with antenna selection and the corresponding performance study are not provided in the existing literature. Moreover, the above CLD schemes are basically based on perfect CSI, whereas in practice, the perfect CSI is difficult to obtain due to channel estimation error or feedback delay. Especially, the CLD scheme with variable switching thresholds is not studied, and thus the improvement of system performance is limited. Motivated by the reason above, we will develop a cross-layer design scheme for MIMO system with antenna selection and imperfect feedback information by combining the AM at the PHY and ARQ at the DLL. With ARQ correcting occasional packet errors at the DLL, the stringent error control requirement is alleviated for adaptive modulation at the PHY. Thus, the performance is obviously improved when compared to the usage of AM only at the PHY or incorporation of ARQ only with a fixed modulation. Based on the performance analysis under imperfect CSI, the probability density function (PDF) of effective signal-to-noise ratio (SNR) is presented. Subject to target packet loss rate (PLR) constraint, the variable switching thresholds are derived by using the maximum a posteriori (MAP-) optimal method. According to these results, the closed-form expressions of average packet error rate (PER) and SE of the system are obtained by means of the theoretical analysis and mathematical derivation. With these expressions, the system performance for imperfect CSI can be effectively assessed, and the theoretical expressions for perfect CSI are included. Numerical results show that the proposed CLD with AS has higher SE than the conventional CLD with STBC. Moreover, the derived variable thresholds can reduce the effect of imperfect CSI on the system performance. Namely, the system performance with variable thresholds is superior to that with conventional-fixed thresholds.

The notations throughout this paper are as follows. Bold upper case and lower case letters denote matrices and column vectors, respectively. Some parameters are described as Table 1 for a better understanding.

Table 1 

Parameter description.

2. System Model

We consider a wireless multiantenna communication system with 𝑁_𝑡 transmit antennas and 𝑁_𝑟 receive antennas operating over a flat and quasistatic Rayleigh fading channel represented by a 𝑁_𝑟×𝑁_𝑡 fading channel matrix 𝐇={ℎ_𝑖𝑛}. The complex element ℎ_𝑖𝑛 denotes the channel gain from the 𝑛th transmit antenna to the 𝑖th receive antenna, which is assumed to be constant over a symbol and varied from one symbol to another. The channel gains are modeled as independent complex Gaussian random variables with zero mean and variance 0.5 per real dimension. Let 𝛼_𝑛=∑^𝑁𝑟_𝑖=1|ℎ_𝑖𝑛|² for 𝑛=1,…,𝑁_𝑡 denote the instantaneous channel power gain between the 𝑛th transmit antenna and all the receive antennas. For the analysis convenience, the power gains {𝛼_𝑛} are sorted in ascending order and denoted by {𝛽_𝑛} such that  𝛽₁≤𝛽₂≤⋯≤𝛽_𝑁𝑡. Thus,  𝛽_𝑁𝑡=max{𝛼_𝑛}. The basic operation of transmit AS with MRC scheme is to select the antenna corresponding to the maximum 𝛽_𝑁𝑡 for transmitting data and use MRC at the receiver for combining the received signals. Thus, the average received SNR after MRC can be expressed as 𝜌=𝑆𝜎²_𝑛𝛽_𝑁𝑡=𝜌𝛽_𝑁𝑡, (1) where 𝑆 is the average transmitted power;  𝜎²_𝑛  is noise power. 𝜌=𝑆/𝜎²_𝑛 is the average SNR per receive antenna.

For Rayleigh fading channel, 𝛼_𝑛 is a central 𝜒²-distributed random variable with 2𝑁_𝑟 degrees of freedom. From equation (2-1-110) in [15], the PDF of 𝛼_𝑛 is given by 𝑓(𝛼)=𝛼^𝑁𝑟−1()exp−𝛼Γ𝑁_𝑟 (2) and the corresponding cumulative distribution function (CDF) is expressed as 𝐹(𝛼)=1−exp(−𝛼)^𝑁𝑟−1_𝑖=0𝛼^𝑖𝑁_𝑟!. (3)

Using the order statistic [16] together with (1) and transformation of random variable, the pdf of 𝜌 can be obtained as follows: 𝑓𝜌=𝑁_𝑡𝜌Γ𝑁_𝑟𝜌𝜌^𝑁𝑟−1−𝜌×exp𝜌−𝜌1−exp𝜌^𝑁𝑟−1_𝑖=0𝜌/𝜌^𝑖𝑖!^𝑁𝑡−1=𝑁_𝑡𝜌Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛×−1_{𝑛(𝑁𝑟−1)𝑙=0}𝑤^𝑛_𝑙𝜌𝜌^{𝑁𝑟−1+𝑙}−()𝜌exp𝑛+1𝜌, (4)(5) where 𝑤^𝑛_𝑙 is the coefficient of 𝑥^𝑙,𝑙=0,1,…,𝑛(𝑁_𝑟−1) in the expansion of (∑^𝑁𝑟−1_𝑡=0(𝑥^𝑡/𝑡!))^𝑛.

3. Cross-Layer Design for MIMO Systems with Antenna Selection

In this section, we will give a cross-layer design scheme for MIMO system with antenna selection, and square MQAM is considered for modulation in the system due to its inherent SE and ease of implementation. For discrete-rate MQAM, the constellation size 𝑀_𝑘 is defined as {𝑀₀=0,  𝑀₁=2, and 𝑀_𝑘=2^2𝑘−2,  𝑘=2,…,𝐾}, where 𝑀₀ means no data transmission. The instantaneous SNR range is divided into 𝐾 fading regions with switching thresholds {𝜌₀,𝜌₁,…,𝜌_𝐾,𝜌_𝐾+1;  𝜌₀=0,𝜌_𝐾+1=+∞}. The MQAM of constellation size 𝑀_𝑘 is used for modulation when 𝜌 falls in the 𝑘th region [𝜌_𝑘,𝜌_𝑘+1). Hence, the data rate is 𝑑_𝑘=log₂𝑀_𝑘  bits/symbol with  𝑑₀=0.

According to [4], the PER of MQAM with two-dimensional Gray code over additive white Gaussian noise (AWGN) channel for the received SNR 𝜌 and constellation size 𝑀_𝑘 is approximately given by PER_𝑘𝜌≅1,𝜌<𝜌_𝑝𝑘,𝑎_𝑘exp−𝑔_𝑘𝜌,𝜌≥𝜌_𝑝𝑘, (6) where {𝑎_𝑘,𝑔_𝑘,𝜌_𝑝𝑘} are constellation and packet-size-dependent constants, and they can be obtained by fitting (6) to the exact PER. Specifically, their values can be found in Table 1 in [4].

In our CLD scheme, adaptive modulation at the physical layer and truncated ARQ protocol at the data link layer are employed for cross-layer design. We first define the target packet loss rate for the data link layer as 𝑃_loss. Since truncated ARQ is used at the data link layer, the packets in error may be retransmitted up to 𝑁_𝑟^max (maximum number of retransmissions). Hence, the target PER is 𝑃_𝑜=𝑃^{1/(𝑁𝑟max+1)}_loss at the physical layer, which is generally limited as 𝑃_𝑜<1. The switching thresholds {𝜌_𝑘} can be set to be the required SNR to achieve the target PER, 𝑃_𝑜, over an AWGN channel. By inverting the 𝑃_𝑜 in (6), we can obtain the switching threshold values as follows: 𝜌_𝑘=𝐵0𝑘=0,_𝑘𝑔_𝑘𝑘=1,…,𝐾,+∞𝑘=𝐾+1, (7) where 𝐵_𝑘=−ln(𝑃_𝑜/𝑎_𝑘) is a factor dependent on the choosing of the modulation mode 𝑘. With the above switching thresholds, the PER and SE performance of MIMO system with cross-layer design and antenna selection may be assessed effectively. Moreover, the system will operate with a PER below target PER 𝑃_𝑜 for perfect CSI when the switching thresholds are chosen according to (7).

4. Performance Analysis of MIMO System with CLD and AS

In this section, we will give the performance analysis of CLD with antenna selection (referred to as CLD-AS) under perfect feedback. Closed-form expressions of the average PER and spectrum efficiency are derived, which will be used to assess the performance of CLD-AS for perfect CSI.

Based on the switching thresholds described in (7), and using (5), we can calculate the probability that the SNR, 𝜌, falls in the 𝑘th region [𝜌_𝑘,𝜌_𝑘+1), denoted by 𝑃_𝑘, as 𝑃_𝑘=^𝜌𝑘+1_𝜌𝑘𝑓𝜌=𝑁𝑑𝜌_𝑡Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙(𝑛+1)^{𝑁𝑟+𝑙}×Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘𝜌−Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘+1𝜌. (8)

For discrete-rate adaptive scheme, the average SE at the physical layer is defined as the ensemble average of effective transmission rate. So using (5) and the incomplete gamma function Γ(⋅,⋅) [17], the average SE of the system at the PHY with adaptive MQAM and AS can be obtained as 𝜂_phy=^𝐾_𝑘=1𝑑_𝑘^𝜌𝑘+1_𝜌𝑘𝑓𝜌=𝑑𝜌_𝐾𝑘=1𝑑_𝑘𝑃_𝑘=𝑁_𝑡Γ𝑁_𝑟^𝐾_𝑘=1𝑑_𝑘×^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙(𝑛+1)^{𝑁𝑟+𝑙}×Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘𝜌−Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘+1𝜌. (9) This equation is a closed-form expression of the average SE of the system at the physical layer.

We define ensemble average PER for MIMO system with CLD and antenna selection as ∑PER=^𝐾𝑘=1𝑑_𝑘PER_𝑘∑^𝐾_𝑘=1𝑑_𝑘∫^𝜌𝑘+1_𝜌𝑘𝑓𝜌=∑𝑑𝜌_𝐾𝑘=1𝑑_𝑘PER_𝑘𝜂_phy, (10) where PER_𝑘 denote the average PER for constellation size 𝑀_𝑘. Using (5) and (6), PER_𝑘 can be obtained as PER_𝑘≅^𝜌𝑘+1_𝜌𝑘PER_𝑘𝜌𝑓𝜌=𝑁𝑑𝜌_𝑡𝑎_𝑘Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙𝜌𝑏_𝑛,𝑘^{𝑁𝑟+𝑙}×Γ𝑁_𝑟+𝑙,𝑏_𝑛,𝑘𝜌_𝑘−Γ𝑁_𝑟+𝑙,𝑏_𝑛,𝑘𝜌_𝑘+1, (11) where 𝑏_𝑛,𝑘=𝑔_𝑘+(𝑛+1)/𝜌. Substituting (11) and (9) into (10), the average PER can be evaluated as follows: 𝑁PER≅^𝑡Γ𝑁_𝑟𝜂_phy^𝐾_𝑘=1𝑑_𝑘𝑎_𝑘^𝑁𝑡−1_𝑛=0(−1)^𝑛×𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙𝜌𝑏_𝑛,𝑘^{𝑁𝑟+𝑙}×Γ𝑁_𝑟+𝑙,𝑏_𝑛,𝑘𝜌_𝑘−Γ𝑁_𝑟+𝑙,𝑏_𝑛,𝑘𝜌_𝑘+1, (12) where 𝜂_phy is defined as (9). Equation (12) is a tightly closed-form expression of the average PER for MIMO system with CLD and AS. With (12), the average PLR at the data link layer with the maximum number of retransmissions 𝑁_𝑟^max is PLR=PER^{𝑁𝑟max+1}=𝑝^𝑁𝑟max_𝑟⁺¹, (13) where 𝑝_𝑟=PER. Thus, the average number of transmissions per packet can be calculated as 𝑁=1+𝑝_𝑟+𝑝²_𝑟+⋯+𝑝^𝑁𝑟max_𝑟=1−𝑝^𝑁𝑟max_𝑟⁺¹1−𝑝_𝑟. (14) Using (9) and (14), the overall average SE of the MIMO system with CLD and AS can be attained as 𝜂𝜂=^phy𝑁=𝑁_𝑡𝑁Γ𝑁_𝑟^𝐾_𝑘=1𝑑_𝑘^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛×−1_{𝑛(𝑁𝑟−1)𝑙=0}𝑤^𝑛_𝑙(𝑛+1)^{𝑁𝑟+𝑙}×Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘𝜌−Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘+1𝜌. (15) When 𝑁_𝑟^max is set to be zero, then 𝑁=1, and (15) is reduced to 𝜂_phy, which corresponds to the average SE for AM at the physical layer only.

5. Performance of CLD with Antenna Selection under Imperfect Feedback

5.1. Variable Switching Thresholds

In the above section, we analyze the performance of CLD-AS scheme assuming the CSI is perfectly available at the receiver and the transmitter. In practice, the channel state information is imperfect due to imperfect feedback. So in this section, we investigate the effect of imperfect feedback (delayed feedback) on the performance of CLD-AS system. The channel is assumed to be perfectly known at the receiver and is fed back to the transmitter with time delay 𝜏.  𝐇 is the 𝜏 time-delayed version of  𝐇, and it is drawn from the same Gaussian process as  𝐇. The entries of 𝐇, {ℎ_𝑖𝑛} are correlated with {ℎ_𝑖𝑛} with correlation coefficient 𝑐=𝐽₀(2𝜋𝑓_𝑑𝜏) [18], where 𝐽₀(⋅)  is the zero-order Bessel function of the first kind [17], and 𝑓_𝑑 is the maximum Doppler frequency [15]. At the transmitter, the transmitter will use the delayed CSI, 𝜌=𝜌𝛽_𝑁𝑡, to adjust the modulation mode. According to the above analysis, 𝜌 will have the same PDF as the perfect CSI 𝜌. Thus, the PDF of 𝜌 can be obtained as 𝑓𝜌=𝑁_𝑡𝜌Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛×−1_{𝑛(𝑁𝑟−1)𝑙=0}𝑤^𝑛_𝑙𝜌𝜌^{𝑁𝑟−1+𝑙}−()𝜌exp𝑛+1𝜌. (16)

For delayed feedback, according to [18, 19], the conditional PDF of 𝜌 given 𝜌 is expressed as 𝑓𝜌=1𝜌∣𝜎²_𝑒𝜌𝜌𝑐²𝜌^{(𝑁𝑟−1)/2}𝐼_𝑁𝑟−12𝑐𝜌𝜌𝜌𝜎²_𝑒×exp−𝜌+𝑐²𝜌𝜌𝜎²_𝑒, (17) where 𝜎²_𝑒=1−𝑐², and 𝐼_𝑣(⋅)  is the  𝑣th order-modified Bessel function of the first kind [17]. With (16) and (17), the joint pdf of 𝜌 and 𝜌 can be obtained as 𝑓_𝜌,𝜌𝜌=𝑁𝜌,^𝑡𝜌Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛×𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙𝜌𝜌^{𝑁𝑟−1+𝑙}−()𝜌×exp𝑛+1𝜌1𝜎²_𝑒𝜌𝜌𝑐²𝜌^{(𝑁𝑟−1)/2}×𝐼_𝑁𝑟−12𝑐𝜌𝜌𝜌𝜎²_𝑒exp−𝜌+𝑐²𝜌𝜌𝜎²_𝑒. (18)

Considering that the switching thresholds in (7) are fixed, they can not adapt to the change of delayed feedback information. Thus, the performance loss will happen when the feedback has a delay. For this, we will present a variable threshold method for imperfect feedback by employing the Bayes’ theorem [20] and MAP criterion [21, 22]. With the variable thresholds, the system performance will be effectively improved and outperforms that with fixed thresholds (7).

In order to develop the threshold adaptation algorithm for switching thresholds, the maximum a posterior-based PER estimation will be adopted. Namely, the estimated instantaneous PER, PER^′_𝑘(𝜌), will be used to derive the variable thresholds. PER^′_𝑘(𝜌) can be obtained by maximizing the following conditional PDF: 𝑓_PER𝑘∣𝜌PER_𝑘∣𝜌=𝑓_PER𝑘,𝜌PER_𝑘,𝜌𝑓𝜌. (19) According to the functional relationship between PER_𝑘(𝜌) and 𝜌 in (6), using the analysis method in [21], the joint PDF of PER_𝑘(𝜌) and 𝜌, 𝑓_PER𝑘,𝜌(PER_𝑘,𝜌), is derived as 𝑓_PER𝑘,𝜌PER_𝑘,𝜌=𝜕𝜌𝜕PER_𝑘𝑓_𝜌,𝜌PER_𝑘⁻¹𝜌,𝜌=𝐴𝑔_𝑘𝑎_𝑘𝜉²𝑔_𝑘^{(𝑁𝑟−1)/2}𝐼_𝑁𝑟−1𝜉𝜇𝜌×exp𝜉²−𝜉²𝑔_𝑘𝜌𝜎²_𝑒, (20) where 𝜉=ln(𝑎_𝑘/PER_𝑘), PER_𝑘⁻¹(𝜌) is the inverse function of PER_𝑘(𝜌), and thus we have 𝜌=PER_𝑘⁻¹(𝜌)=𝜉²/𝑔_𝑘.  𝜇_𝑘=2𝑐√𝑔_𝑘/𝜈_𝑘, 𝜈_𝑘=𝑔_𝑘𝜌𝜎²_𝑒, and 𝐴 is written as 𝑁𝐴=^𝑡𝑐^1−𝑁𝑟𝜎²_𝑒Γ𝑁_𝑟^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛×−1_{𝑛(𝑁𝑟−1)𝑙=0}𝑤^𝑛_𝑙𝜌^{1.5(𝑁𝑟−1)+𝑙}𝜌^{𝑁𝑟+1+𝑙}exp−𝑛+1𝜌+𝑐²𝜎²_𝑒𝜌𝜌. (21) According to (20) and omitting the terms not related to 𝜉, we may define the following MAP function as 𝐹_MAP=𝜉^𝑁𝑟−1𝐼_𝑁𝑟−1𝜉𝜇exp𝜉²−𝜉²𝜈_𝑘. (22)

Using the approximate expression 𝐼_𝑛(𝑧)≈exp(𝑧)/√2𝜋𝑧 for large  𝑧  [23], (22) can be approximated as 𝐹_MAP=𝜉^{𝑁𝑟−1.5}2𝜋𝜇𝜌exp𝜉𝜇𝜌+𝜉²−𝜉²/𝜈_𝑘, (23) where 𝐹_MAP denotes the approximate form of the MAP function 𝐹_MAP for large SNR 𝜌. Based on (23), by solving 𝜕𝐹_MAP/𝜕PER_𝑘=(𝜕𝐹_MAP/𝜕𝜉)⋅(𝜕𝜉/𝜕PER_𝑘)=0, we can obtain the approximated expression of the estimated PER as PER^′_𝑘𝜌≅𝑎_𝑘exp−𝑐²𝑔_𝑘𝜌𝑔_𝑘𝜎²_𝑒−1². (24)

This estimated PER includes imperfect CSI, such as channel correlation coefficient related to the time delay, and thus it can be used to obtain the switching thresholds adaptation to feedback delay. Let PER^′_𝑘(𝜌) be target PER  𝑃₀, then with (24), we can obtain the variable thresholds dependent on the feedback delay as 𝜌_𝑘≅−ln𝑃₀/𝑎_𝑘𝑔_𝑘𝑔_𝑘𝜎²_𝑒−1²𝑐²=𝐵_𝑘/𝑔_𝑘𝑔_𝑘𝜎²_𝑒−1²𝑐². (25)

Unlike the fixed switching thresholds in (7) (which do not consider the impact of delayed feedback at the transmitter), the variable switching thresholds in (25) take the effect of delayed feedback into account when performing rate adaptation at the transmitter. Thus, with (25), the impact of feedback delay may be reduced, and reliable PER performance will be obtained. When the feedback is perfect (i.e., no delay), 𝑐²=1, (25) is reduced to (7), and so the variable switching thresholds include the fixed switching thresholds as special cases.

5.2. Performance Analysis

In this subsection, we will use the derived variable thresholds for performance analysis of the system. With (16) and (25), we can obtain the average SE of the MIMO system with adaptive MQAM and antenna selection at the PHY as 𝜂_phy=𝑁_𝑡Γ𝑁_𝑟^𝐾_𝑘=1𝑑_𝑘^𝑁𝑡−1_𝑛=0(−1)^𝑛×𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙(𝑛+1)^{𝑁𝑟+𝑙}×Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘𝜌−Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘+1𝜌. (26)

Equation (26) is a closed-form expression of average SE of the system with imperfect CSI at the PHY. Using this expression, we can evaluate the average SE at the PHY without ARQ protocol in the presence of imperfect CSI. With (16), (17), (25), and (26), the average PER of the CLD-AS with imperfect CSI can be expressed as 1PER=𝜂_phy^𝐾_𝑘=1𝑑_𝑘𝜌_𝑘+1𝜌_𝑘^∞₀PER_𝜌∣𝜌𝑓𝜌𝜌𝑑𝜌≅1𝜌∣𝑑𝜌𝑓𝜂_phy^𝐾_𝑘=1𝑑_𝑘𝜌_𝑘+1𝜌_𝑘^∞₀𝑎_𝑘×exp−𝑔_𝑘𝜌𝑓𝜌∣𝜌𝑑𝜌𝑓𝜌𝑑𝜌. (27)

Substituting (16) and (17) into (27), and solving the double integrals, we have 𝑁PER≅^𝑡Γ𝑁_𝑟𝜂_phy^𝐾_𝑘=1𝑑_𝑘𝑎_𝑘1+𝑔_𝑘1−𝑐²𝜌^{𝑀𝑁𝑡−1}_𝑛=0(−1)^𝑛×𝑁_𝑡𝑛−1^{𝑛(𝑁𝑟−1)}_𝑙=0𝑤^𝑛_𝑙𝜌𝑏̂_𝑛,𝑘^{𝑁𝑟+𝑙}×Γ𝑁_𝑟+𝑙,𝑏̂_𝑛,𝑘𝜌_𝑘−Γ𝑁_𝑟+𝑙,𝑏̂_𝑛,𝑘𝜌_𝑘+1, (28) where 𝑏̂_𝑛,𝑘=(𝑛+1)/𝜌+𝑐²𝑔_𝑘/[1+𝑔_𝑘𝜌(1−𝑐²)]. The above equation is a closed-form average PER expression of CLD-AS with imperfect feedback.

Following the derivation approaches under perfect CSI, the overall average SE of CLD-AS can be obtained by using (26) and (28), that is, 𝜂𝜂=^phy𝑁=𝑁_𝑡𝑁Γ𝑁_𝑟^𝐾_𝑘=1𝑑_𝑘^𝑁𝑡−1_𝑛=0(−1)^𝑛𝑁_𝑡𝑛×−1_{𝑛(𝑁𝑟−1)𝑙=0}𝑤^𝑛_𝑙(𝑛+1)^{𝑁𝑟+𝑙}×Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘𝜌−Γ𝑁_𝑟()𝜌+𝑙,𝑛+1_𝑘+1𝜌, (29) where 𝑁=(1−PER^{𝑁𝑟max+1})/(1−PER), and PER is from (28). When the feedback has no delay, 𝑐²=1. Using 𝑐²=1 in (28) and (29), (28) and (29) will be reduced to (12) and (15) under perfect feedback, respectively. Hence, the derived SE and PER expressions under imperfect feedback include those under perfect feedback as special cases. With (28) and (29), we can assess the impact of time delay on the PER and SE performance of the CLD-AS. When the correlation coefficient 𝑐 decreases due to the impact of feedback delay, the variable switching thresholds will increase according to (25). Correspondingly, the PER performance is improved, which will bring about the decrease of the average number of transmission 𝑁. Thus, the overall average SE is effectively enhanced in terms of (29) when 𝜂_phy is fixed. Based on this, the CLD scheme with the variable thresholds will be superior to that with the fixed thresholds.

6. Numerical Results and Theoretical Evaluation

In this section, we will use the derived performance formulae to evaluate the average SE and PER performance of the MIMO system with CLD and AS over Rayleigh fading channel. The channel is assumed to be quasistatic flat fading and Gray code is employed to map the data bits to MQAM constellations. The set of MQAM constellations is {𝑀_𝑘}_{𝑘=0,1,…,5}={0,2,4,16,64,256}. The target packet loss rate in the data link layer is 𝑃_loss=10⁻³, and the maximum number of ARQ retransmissions is 𝑁_𝑟^max=2. Thus, the target PER, 𝑃_𝑜, is equal to 0.1. In the following figures, 𝑥T𝑦R denotes a multiple-antenna system with 𝑥 transmit antennas and 𝑦 receive antennas.

In Figure 1, we plot the average spectrum efficiency of the CLD-AS for different transmit antennas and different receive antennas under perfect CSI. The CLD with STBC (CLD-STBC) system in [5] is used for comparison, where G₂, H₃ codes are employed for STBC schemes. The average SE is calculated by using (15) with the switching thresholds defined in (7). As shown in Figure 1, CLD is able to increase SE with SNR. For the same number of receive antennas, the SE of CLD with three transmit antennas is higher than that with two transmit antennas because the former has greater diversity than the latter; that is, 3T1R and 3T2R systems outperform the corresponding 2T1R and 2T2R systems, respectively. Due to the same reason, the SE of CLD with two receive antennas is higher than that with single receive antenna for the same number of transmit antennas. Moreover, we can see that the SE of CLD-AS is much higher than CLD-STBC. It is because the proposed CLD-AS can make full use of the antenna with maximal channel power gain to transmit data so that the received signal power at the receiver is maximized. Especially when transmit antenna is larger than two, the SE superiority of the proposed CLD scheme becomes more significant because the rate of STBC is less than 1 in the case of more than two transmit antennas. Hence, the SE of CLD-AS with 3 transmit antennas is much higher than that of CLD-STBC with H₃ (3/4-rate). The above results show that the derived formulae are valid.

Figure 1 

Average SE of CLD-AS with different transmit antennas and one receive antenna.

In Figure 2, we plot the average PER of the CLD-AS system with two transmit antennas and two receive antennas, where perfect feedback (zero-delay feedback, i.e., 𝑓_𝑑𝜏=0) and imperfect feedback (i.e., 𝑓_𝑑𝜏=0.1) are considered. The fixed switching thresholds (FST) defined in (7) and variable switching thresholds (VST) defined in (25) are used for comparison. The average PER is calculated by (12) for perfect feedback and (28) for imperfect feedback. It is shown in Figure 2 that the average PER values are below the target PER (i.e., 𝑃_𝑜) for perfect feedback, which means that the CLD is successful in increasing SE while the target PER is maintained. When the feedback is imperfect, however, the PER performance degrades due to the impact of feedback delay, and the resultant PER starts to fail to meet the target PER, especially for FST method. Besides, the system PER performance with FST is worse than that with VST. This is because the latter can adapt to the delayed feedback. Thus, it can utilize the feedback information to reduce the effect of feedback delay on the performance. Whereas for the former, it is fixed for different feedback, and thus the impact of feedback delay is difficult to overcome. Based on this, the presented VST method is more effective than the FST method against the feedback delay, especially at high SNR. This is because the proposed VST method is obtained based on the approximation under high SNR. Namely, the approximate expression of the modified Bessel function of the first kind 𝐼_𝑛(𝑥)≈exp(𝑥)/√2𝜋𝑥 is used for deriving the VST method, and this approximation is much more effective for very large 𝑥. When 𝑥 is small, however, this approximation is not feasible; that is, it can not reflect the actual values of 𝐼_𝑛(𝑥). Thus, our VST scheme is suitable for high SNR, while for low SNR, it will no longer be valid for adaptation to the delayed feedback. When feedback is perfect, the VST is identical to the FST, and correspondingly, their PER curves are the same.

Figure 2 

Average PER of CLD-AS with two transmit antennas and two receive antennas.

In Figure 3, we plot the average SE of the CLD-AS system with different transmit antennas and one receive antenna for perfect feedback (𝑓_𝑑𝜏=0) and imperfect feedback (𝑓_𝑑𝜏=0.1), where the FST and VST methods are used for performance evaluation. The average SE is calculated by (15) and (29) for perfect and imperfect cases, respectively. From Figure 3, we can see that the CLD scheme using 4T1R performs better than that using 2T1R because the former has greater diversity than the latter. Due to the effect of feedback delay, the system with imperfect CSI is worse than that with perfect CSI. Besides, the SE with VST is obviously higher than that with FST, especially at high SNR. This is because the application of VST can lower the system PER (as shown in Figure 3), which will bring about the decrease of the average number of transmissions 𝑁 according to (14). As a result, the increase of overall average SE is obtained. For perfect feedback (where VST is the same as FST), the average SE of the system using VST and FST will be identical. The above results further verify the effectiveness of the presented VST method.

Figure 3 

Average SE of CLD-AS with different transmit antennas and one receive antenna.

We plot the average PER versus normalized time delay (𝑓_𝑑𝜏) in Figure 4 for the CLD-AS with three transmit antennas and different receive antennas. The average SNR is set equal to 22 dB. The results show that the CLD-AS can tolerate the normalized time delay up to about 0.01 with a slight degradation in the average PER. But when 𝑓_𝑑𝜏 increases beyond 0.01, the PER performance will degrade increasingly. The 3T1R system fails to meet the target PER (𝑃₀=0.1) at 𝑓_𝑑𝜏=0.056 for FST and at 𝑓_𝑑𝜏=0.062 for VST, while the 3T2R system fails at 𝑓_𝑑𝜏=0.088 for FST and at 𝑓_𝑑𝜏=0.107  for VST. From these results, we can see that the system with the presented VST method can tolerate higher time delay than the conventional FST method. This is because the derived VST method is obtained based on the delayed feedback information, and correspondingly, the impact of delayed feedback on performance is relatively small, while the latter does not consider the delayed information. As a result, the former is superior to the latter. When the time delay is smaller (<0.01), however, their performance is almost the same. Besides, it is found that the PER performance of the 3T1R system is more sensitive to time delay than the 3T2R system. This is due to the fact that 3T2R system employs multiple receive antennas and has greater diversity than 3T1R system.

Figure 4 

Effect of normalized time delay on average PER of CLD-AS with three transmit antennas and different receive antennas.

7. Conclusions

We have studied the performance of cross-layer design with antenna selection for both perfect and imperfect CSIs over Rayleigh fading channel. Subject to the target PER constraint and delayed feedback, we have derived the variable switching thresholds by means of the maximum a posteriori method and approximate PER expression. As a result, closed-form variable thresholds are obtained, and they include the conventional-fixed thresholds as special cases. Based on the obtained switching thresholds, the closed-form expressions of the average PER and SE are derived for both perfect and imperfect CSIs. Using these expressions, the performance of CLD-AS system is effectively assessed. The impact of delayed CSI on the system performance is analyzed, and the analysis shows that CLD-AS systems are insensitive to the time delay <0.01. Moreover, the presented variable thresholds method can adapt to the delayed feedback information and reduce the effect of feedback delay. Thus, the system performance is superior to that with the conventional-fixed thresholds. Numerical results show the presented CLD with AS can obtain higher SE than the existing CLD with space-time coding the PER and SE performance of the system with variable thresholds both perform better than those of the system with fixed thresholds, especially for large time delay and/or large SNR.

Appendix

In this appendix, some parameters are provided in Table 1 for understanding the derived equations easily.

Acknowledgments

The authors would like to thank the anonymous reviewer and the editor for their valuable comments. This work was supported in part by National Natural Science Foundation of China (61172078, 61172077), Doctoral Fund of Ministry of Education of China (20093218120021), and the Fundamental Research Funds for the Central Universities (NS2012075).