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Abstract

We investigate the joint effects of feedback delay and channel estimation errors (CEE) over Nakagami-m fading channels for two-hop amplify-and-forward (AF) relaying systems with transmit beamforming (TB) and relay selection (RS). We derive new closed-form expressions for the system outage performance including the exact analysis and informative high SNR asymptotic approximations, which indicate that TB feedback delay reduces the achievable diversity order to one, regardless of Nakagami-m fading parameters. Whereas RS delay reduces the diversity order to a case without relay selection and CEE merely result in coding gain loss, numerical results verify the theoretical analysis and illustrate that the outage performance is more sensitive to the TB feedback delay.

1. Introduction

Transmit beamforming (TB) and relay selection (RS) are regarded as two promising techniques for multiple-antenna multiple-relay assisted networks to achieve full spatial diversity and have been investigated intensively [1]. Generally, to achieve coherent beamforming for a maximal ratio transmission or select one or a set of relays for signal forwarding, perfect knowledge of channel state information (CSI) is required at the transmitter, which is probably unavailable because of practical limitations such as feedback delay and channel estimation errors (CEE).

The performance of transmit beamforming systems under imperfect CSI has been widely investigated in the literature. The effects of feedback delay on the performance of a two-hop TB and amplify-and-forward (AF) relay network with and without interference over Rayleigh-fading channels are investigated in [2, 3], respectively. The undesirable effect of CEE with beamforming is quantified in [4]. When relay selection is considered, feedback delays or estimation errors can create critical challenges. The impact of outdated CSI on the performance of AF relays with the kth worst partial relay selection scheme is presented in [5]. Then, the analysis of the effect of delay on relay selection has been extended to different relay selection schemes or channel environments (see [6, 7] and the references therein). Apart from feedback delay, the effects of CEE on the outage probability of selection cooperation are studied in [8].

However, most existing related works limited their analysis to either single-relay or single-antenna systems and do not yet consider TB and RS simultaneously or investigate the joint effect of outdated and imprecise CSI. Furthermore, to the best of our knowledge, prior works have been confined to Rayleigh-fading channels, and analytical results for TB-RS systems with delay and CEE over Nakagami-m fading channels have not yet been reported. In this paper, we consider a two-hop TB-RS system and a general imperfect CSI model taking feedback delay and CEE together into account is assumed over Nakagami-m fading channels. New closed-form expressions for the system outage performance including the exact analysis and informative high SNR asymptotic approximations are derived. Numerical results verify the theoretical analysis and illustrate the decremental effects of feedback delay and CEE on the system performance.

2. System and Signal Model

2.1. System Description

Consider a cooperative wireless communications system where a source, S, equipped with $N_{t}$ antennas communicates with a destination, D, assisted by a set of $N_{r}$ relays $R_{i}$ , $i = 1, \dots, N_{r}$ . The direct communication link is assumed to be unavailable due to heavy shadowing between the source and the destination. We assume that each of the $S \to R_{i}$ and $R_{i} \to D$ channels experience independent and identically distributed Nakagami-m fading with parameters $(m_{1}, σ_{1}^{2})$ and $(m_{2}, σ_{2}^{2})$ , denoted by $h_{S R_{i}} = {[h_{S R_{i}, 1}, \dots, h_{S R_{i}, N_{t}}]}^{T}$ and $h_{R_{i} D}$ , respectively. The noises at all receivers are mutually independent zero-mean Gaussian random variables with equal variance $N_{0}$ . In addition, we consider a total data transmit power constraint in which $P = P_{s} + P_{r}$ , where $P_{s}$ and $P_{r}$ denote the transmit power at the source and relay, respectively.

To exploit the multiantenna and multirelay diversity, transmit beamforming at the source-relay link and AF relay selection at the relay-destination link are employed. It should be noted that both the TB and RS processes are based on outdated and imperfect estimated CSI. Let $\hat{h} (t)$ and $\hat{h} (t - T_{d})$ represent the actual (used for data transmission) and the outdated (used for relay selection and beamforming vector calculation) channel estimates with a time delay $T_{d}$ . According to the minimum mean-squared error (MMSE) channel estimation model [9, 10] and the time-correlated outdated CSI model [11, 12], we may have a modified joint delay and error model as follows (it should be noted that although the traditional bivariate distributed expression is more popular (see [11] and the references therein), this error model is in line with [9] and can be regarded as a good approximation):

\begin{matrix} \hat{h} (t) = ρ \hat{h} (t - T_{d}) + v (t), \end{matrix}

(1)

where

v (t)

are approximated as Gaussian-distributed error with variance of

(1 - ρ^{2}) σ_{\hat{h}}^{2}

, and

\begin{matrix} ρ = \{\begin{cases} ρ_{e} ρ_{d}, & if ρ_{d} < 1 \\ 1, & if ρ_{d} = 1, \end{cases} \end{matrix}

(2)

where

ρ_{e} = δ η σ_{h}^{2} / (1 + δ η σ_{h}^{2})

and

ρ_{d} = J_{0} (2 π f_{d} T_{d})

denote the estimation and delay correlation coefficients, respectively. Besides,

σ_{h}^{2}

denotes the variance of the actual channel

h (t)

σ_{\hat{h}}^{2} = ρ_{e} σ_{h}^{2}

is the variance of the channel estimates,

η = P / N_{0}

is the data transmission SNR,

J_{0} (\cdot)

is the zero-order Bessel function of the first kind,

f_{d}

is the Doppler frequency, and

δ = P_{p i l o t} / P

is determined by the cost of obtaining CSI in terms of the training pilots’ power consumption and reflects the quality of channel estimation [10]. It should be noted that

ρ_{d} = 1

corresponds to the case in which the estimated CSI is not outdated.

2.2. TB and RS with Delay and CEE

The data transmission process is divided into two phases. During the first phase, S beamforms its signal $s (t)$ to a certain selected relay $R_{k}$ which achieves the largest receive SNR of the relay-destination link, and the received signal at $R_{k}$ can be written as

\begin{array}{l} y_{R} (t) = \sqrt{P_{s}} w (t | T_{d_{1}}) {\hat{h}}_{S R_{k}} (t) s (t) \\ + \sqrt{P_{s}} w (t | T_{d_{1}}) e_{S R_{k}} (t) s (t) + n_{S R_{k}} (t), \end{array}

(3)

where

{\hat{h}}_{S R_{k}} (t) = {[{\hat{h}}_{S R_{k}, 1} (t), \dots, {\hat{h}}_{S R_{k}, N_{t}} (t)]}^{T}

and

e_{S R_{k}} (t)

are the estimated channel and errors from S to

R_{k}

, and

n_{S R_{k}} (t)

is the AWGN at the relay. The beamforming vector

w (t | T_{d_{1}}) = {\hat{h}}_{S R_{k}}^{H} (t - T_{d_{1}}) / ‖{\hat{h}}_{S R_{k}} (t - T_{d_{1}})‖

is calculated from the outdated channel [9], where

T_{d_{1}}

is the time delay between the beamforming vector feedback and data transmission. Then we define the received SNR of the first hop as

\begin{array}{l} γ_{S R} = \frac{P_{s} {|w (t | T_{d_{1}}) {\hat{h}}_{S R_{k}} (t)|}^{2}}{(P_{s} σ_{e_{1}}^{2} + N_{0})} \\ = \frac{{|ρ_{1} w (t | T_{d_{1}}) {\hat{h}}_{S R_{k}} (t - T_{d_{1}}) + w (t | T_{d_{1}}) v_{S R_{k}} (t)|}^{2} {\bar{γ}}_{1}}{σ_{1}^{2}}, \end{array}

(4)

where

{\bar{γ}}_{1} = P_{s} σ_{1}^{2} / (P_{s} σ_{e_{1}}^{2} + N_{0})

ρ_{1} = ρ_{e_{1}} J_{0} (2 π f_{d} T_{d_{1}})

, and

v_{S R_{k}} (t)

is the joint delay and estimation error from (1).

During the second phase, the received signal $y_{R} (t)$ is multiplied by a variable-gain and retransmitted to the destination. After some mathematical manipulations, the end-to-end SNR at D can be written as [13]

\begin{matrix} γ_{eq} = \frac{γ_{S R} γ_{R D}}{γ_{S R} + γ_{R D} + 1}, \end{matrix}

(5)

where

γ_{R D} = P_{r} {|{\hat{h}}_{R_{k} D} (t)|}^{2} / (P_{r} σ_{e_{R D}}^{2} + N_{0})

denotes the received SNR of the second hop with the average value

{\bar{γ}}_{2} = P_{r} σ_{h_{2}}^{2} / (P_{r} σ_{e_{2}}^{2} + N_{0})

. As mentioned above, before data transmission, a partial relay selection process is performed based on the highest instantaneous SNR of the second hop in which

k = a r g \max_{i} \{{\hat{γ}}_{R_{i} D} = γ_{R_{i} D} (t - T_{d_{2}})\}

and

T_{d_{2}}

is the delay between the relay selection and data transmission.

3. Outage Analysis

3.1. Exact Analysis

The outage probability is defined as the probability in which $γ_{e q}$ drops below an acceptable SNR threshold $γ_{t h}$ . We have

\begin{matrix} P_{out} = \Pr (γ_{eq} < γ_{th}) = F_{γ_{eq}} (γ_{th}) . \end{matrix}

(6)

The CDF of $γ_{e q}$ can be written as a single-integral expression as [4]

\begin{array}{l} F_{γ_{eq}} (x) \\ = 1 - \int_{0}^{\infty} {\bar{F}}_{γ_{S R}} (\frac{x z + x (x + 1)}{z}) f_{γ_{R D}} (z + x) d z, \end{array}

(7)

where

\bar{F} (x)

denote the complementary CDF.

Lemma 1.

The exact PDF and complementary CDF of $γ_{S R}$ are given in the following:

\begin{matrix} f_{γ_{S R}} (x) = \frac{m_{1}^{m_{1} N_{t}}}{{\bar{γ}}_{1}^{m_{1} N_{t}}} \sum_{k = 0}^{m_{1} N_{t} - 1} (\begin{pmatrix} m_{1} N_{t} - 1 \\ k \end{pmatrix}) \frac{ρ_{1}^{2 (m_{1} N_{t} - 1 - k)} {({\bar{γ}}_{1} (1 - ρ_{1}^{2}))}^{k} x^{m_{1} N_{t} - 1 - k}}{(m_{1} N_{t} - 1 - k)! {[m_{1} - (m_{1} - 1) ρ_{1}^{2}]}^{2 m_{1} N_{t} - 1 - k}} e^{- m_{1} x / {\bar{γ}}_{1} [m_{1} - (m_{1} - 1) ρ_{1}^{2}]}, \end{matrix}

(8)

\begin{matrix} {\bar{F}}_{γ_{S R}} (x) = \sum_{k = 0}^{m_{1} N_{t} - 1} (\begin{pmatrix} m_{1} N_{t} - 1 \\ k \end{pmatrix}) \sum_{v = 0}^{m_{1} N_{t} - 1 - k} \frac{m_{1}^{k + v}}{{\bar{γ}}_{1}^{k + v}} \frac{ρ_{1}^{2 (m_{1} N_{t} - 1 - k)} {({\bar{γ}}_{1} (1 - ρ_{1}^{2}))}^{k} x^{v}}{v! {[m_{1} - (m_{1} - 1) ρ_{1}^{2}]}^{m_{1} N_{t} - 1 + v}} e^{- m_{1} x / {\bar{γ}}_{1} [m_{1} - (m_{1} - 1) ρ_{1}^{2}]} . \end{matrix}

(9)

Proof.

Conditioned on ${\hat{h}}_{S R_{k}} (t - T_{d_{1}})$ , $γ_{S R} (t | T_{d_{1}})$ can be approximated as complex Gaussian quadratic form $C N (ρ_{1} \sqrt{{\bar{γ}}_{1} / σ_{1}^{2}} w (t | T_{d_{1}}) {\hat{h}}_{S R_{k}} (t - T_{d_{1}}), (1 - ρ_{1}^{2}) γ_{1})$ , whose MGF can be derived as [14]

\begin{array}{l} Φ_{γ_{S R} (t | T_{d_{1}})} (s | {\hat{h}}_{S R_{k}} (t - T_{d_{1}})) \\ = \frac{e^{s {\bar{γ}}_{1} ρ_{1}^{2} {‖{\hat{h}}_{S R_{k}} (t - T_{d_{1}})‖}^{2} / [1 - s {\bar{γ}}_{1} (1 - ρ_{1}^{2})] σ_{1}^{2}}}{1 - s {\bar{γ}}_{1} (1 - ρ_{1}^{2})} . \end{array}

(10)

Reminding the assumption of Nakagami-m fading with identical parameter, the MGF of

{\hat{h}}_{S R_{k}} (t - T_{d_{1}})

Φ_{{‖{\hat{h}}_{S R_{k}} (t - T_{d_{1}})‖}^{2}} (s) = {(1 - s σ_{1}^{2} / m_{1})}^{- m_{1} N_{t}}

, using which we can obtain the MGF of

γ_{S R}

\begin{matrix} Φ_{γ_{S R}} (s) = \frac{{[1 - s {\bar{γ}}_{1} (1 - ρ_{1}^{2})]}^{m_{1} N_{t} - 1}}{{[1 - s {\bar{γ}}_{1} (1 - ((m_{1} - 1) / m_{1}) ρ_{1}^{2})]}^{m_{1} N_{t}}} . \end{matrix}

(11)

Obviously,

Φ_{γ_{S R}} (s)

has

m_{1} N_{t} - 1

zeros at

s_{1} = 1 / {\bar{γ}}_{1} (1 - ρ_{1}^{2})

and

m_{1} N_{t}

poles at

s_{2} = 1 / {\bar{γ}}_{1} [1 - (1 - 1 / m_{1}) ρ_{1}^{2}]

. Using (11) and the residue theorem, we have a closed-form PDF and the corresponding CDF results in (8) and (9).

The PDF and complementary CDF of the $γ_{R D}$ with outdated relay selection are given in (12) and (14) as [3]

\begin{array}{l} f_{γ_{R D}} (x) = N_{r} \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \frac{{(- 1)}^{n} Ψ}{Γ (m_{2})} \sum_{p = 0}^{Δ} (\begin{pmatrix} Δ \\ p \end{pmatrix}) \\ \cdot \frac{Γ (m_{2} + Δ) ρ_{2}^{p} {(1 - ρ_{2})}^{Δ - p} x^{m_{2} + p - 1}}{Γ (m_{2} + p) {({\bar{γ}}_{2} ς)}^{m_{2} + p} ς^{Δ}} e^{- ((1 + n) / ς {\bar{γ}}_{2}) x}, \end{array}

(12)

\begin{array}{l} {\bar{F}}_{γ_{R D}} (y) = \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \frac{{(- 1)}^{n} N_{r} Ψ}{Γ (m_{2})!} \sum_{p = 0}^{Δ} (\begin{pmatrix} Δ \\ p \end{pmatrix}) \\ \cdot \frac{ρ_{2}^{p} {(1 - ρ_{2})}^{Δ - p}}{{(1 + n)}^{m_{2} + p} ς^{Δ}} \sum_{i = 0}^{m_{2} + p - 1} \frac{Γ (m_{2} + Δ)}{Γ (i + 1)} {(\frac{1 + n}{ς {\bar{γ}}_{2} / m_{2}} y)}^{i} \\ \cdot e^{- ((1 + n) / (ς {\bar{γ}}_{2} / m_{2})) y}, \end{array}

(13)

where

ς = 1 + n (1 - ρ_{2})

Δ = \sum_{q = 1}^{m_{2} - 1} n_{q}

, and

\begin{matrix} Ψ = \sum_{n_{1} = 0}^{n} \sum_{n_{2} = 0}^{n_{1}} \dots \sum_{n_{m_{2} - 1} = 0}^{n_{m_{2} - 2}} \frac{n!}{n_{m_{2} - 1}!} \prod_{l = 1}^{m_{2} - 1} \frac{{(l!)}^{n_{l + 1} - n_{l}}}{(n_{l - 1} - n_{l})!} \end{matrix}

(14)

with

n_{0} = n

and

n_{m_{2}} = 0

To this end, substituting (9) and (12) into (7) and using [15, Eq. (3.471.9)], the integral in (7) can be solved to yield a closed-form expression for $P_{o u t}$ as in

\begin{array}{l} F_{γ_{eq}} (x) = 1 - 2 \sum_{k = 0}^{m_{1} N_{t} - 1} (\begin{pmatrix} m_{1} N_{t} - 1 \\ k \end{pmatrix}) \\ \cdot \sum_{v = 0}^{m_{1} N_{t} - 1 - k} \frac{m_{1}^{k + v}}{{\bar{γ}}_{1}^{k + v}} \frac{ρ_{1}^{2 (m_{1} N_{t} - 1 - k)} {({\bar{γ}}_{1} (1 - ρ_{1}^{2}))}^{k}}{v! {[m_{1} - (m_{1} - 1) ρ_{1}^{2}]}^{m_{1} N_{t} - 1 + v}} \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \\ \cdot \frac{{(- 1)}^{n} N_{r} Ψ}{Γ (m_{2})} \sum_{p = 0}^{Δ} (\begin{pmatrix} Δ \\ p \end{pmatrix}) \frac{Γ (m_{2} + Δ) ρ_{2}^{2 p} {(1 - ρ_{2}^{2})}^{Δ - p}}{Γ (m_{2} + p) {(ς {\bar{γ}}_{2} / m_{2})}^{m_{2} + p} ς^{Δ}} \sum_{u = 0}^{v} (\begin{pmatrix} v \\ u \end{pmatrix}) \\ \cdot \sum_{j = 0}^{Λ} (\begin{pmatrix} Λ \\ j \end{pmatrix}) x^{Λ - j} e^{- α γ_{th}} {(\frac{m_{1} ς {\bar{γ}}_{2} γ_{th} (γ_{th} + 1)}{m_{2} {\bar{γ}}_{1} [m_{1} - (m_{1} - 1) ρ_{1}^{2}] (1 + n)})}^{(j - v + 1) / 2} \\ \cdot K_{j - v + 1} (2 \sqrt{\frac{m_{1} m_{2} (1 + n) γ_{th} (γ_{th} + 1)}{ς {\bar{γ}}_{1} {\bar{γ}}_{2} [m_{1} - (m_{1} - 1) ρ_{1}^{2}]}}) \end{array}

(15)

with

Λ = m_{2} + p - 1 + u

and

α = m_{1} / {\bar{γ}}_{1} [m_{1} - (m_{1} - 1) ρ_{1}^{2}] + m_{2} (1 + n) / ς {\bar{γ}}_{2}

It should be noted that the outage result in (15) can be regarded as a generalized form where lots of previous works could be included, such as the TB without RS over Rayleigh-fading case in [2] by setting $N_{r} = 1$ and $m_{1} = m_{2} = 1$ and RS without TB case in [5, 6] by setting $N_{t} = 1$ and $m_{1} = m_{2} = 1$ .

3.2. Asymptotic Analysis

To gain further insights, we now look into the high SNR regime and derive the asymptotic expression for the outage probability, which enables the characterization of the joint effect of TB feedback delay, RS delay, channel estimation quality order, Nakagami-m fading parameters, the number of antennas and relays on the achievable diversity order, and array gain of the system. Assuming that $P_{s} = λ P$ , $μ = {\bar{γ}}_{2} / {\bar{γ}}_{1}$ , and when $η \to \infty$ , by substituting the related parameters we have

\begin{matrix} {\bar{γ}}_{1} = \frac{λ P (σ_{1}^{2} - σ_{1}^{2} / (1 + δ η σ_{1}^{2}))}{λ P σ_{1}^{2} / (1 + δ η σ_{1}^{2}) + N_{0}} ⟶ \frac{δ λ σ_{1}^{2} η}{1 + δ} . \end{matrix}

(16)

Obviously, according to the definition of δ that the training symbol power is scale to the data transmit power, we have $ρ_{e_{1}}, ρ_{e_{2}} \to 1$ in high SNR region. Thus, CEE have no effect on the system diversity gain and only lead to coding gain loss.

We now mainly focus on the feedback delay on the asymptotic outage analysis and present a separate treatment for the following three cases.

Lemma 2.

Case 1. It represents TB and RS feedback delay; that is, $ρ_{1} < 1$ and $ρ_{2} < 1$ ; the asymptotical approximation of the outage probability is given by

\begin{matrix} P_{o u t}^{\infty} \approx \{\begin{cases} \frac{m_{1}^{m_{1} N_{t}} {(1 - ρ_{1}^{2})}^{m_{1} N_{t} - 1}}{{[m_{1} - (m_{1} - 1) ρ_{1}^{2}]}^{m_{1} N_{t}}} (\frac{1 + δ}{δ λ σ_{1}^{2}} \frac{γ_{t h}}{η}), & m_{2} > 1 \\ [\frac{m_{1}^{m_{1} N_{t}} {(1 - ρ_{1}^{2})}^{m_{1} N_{t} - 1}}{{[m_{1} - (m_{1} - 1) ρ_{1}^{2}]}^{m_{1} N_{t}}} + \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \frac{{(- 1)}^{n} N_{r} (m + 1)}{[m (1 - ρ_{2}^{2}) + 1] μ}] (\frac{1 + δ}{δ λ σ_{1}^{2}} \frac{γ_{t h}}{η}), & m_{2} = 1 . \end{cases} \end{matrix}

(17)

Case 2. It represents RS feedback delay only; that is, $ρ_{1} = 1$ and $ρ_{2} < 1$ ; we have

\begin{matrix} P_{o u t}^{\infty} \approx \{\begin{cases} \frac{{((m_{1} (1 + δ) / δ λ σ_{1}^{2}) (γ_{t h} / η))}^{m_{1} N_{t}}}{(m_{1} N_{t})!}, & m_{1} N_{t} < m_{2} \\ [\frac{m_{1}^{m_{2}}}{m_{2}!} + \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \frac{{(- 1)}^{n} N_{r} Ψ}{Γ (m_{2})} \frac{Γ (m_{2} + Δ) {(1 - ρ_{2})}^{Δ} m_{2}^{m_{2}}}{Γ (m_{2} + 1) ς^{Δ + m_{2}} μ^{m_{2}}}] {(\frac{1 + δ}{δ λ σ_{1}^{2}} \frac{γ_{t h}}{η})}^{m_{2}}, & m_{1} N_{t} = m_{2} \\ \sum_{n = 0}^{N_{r} - 1} (\begin{pmatrix} N_{r} - 1 \\ n \end{pmatrix}) \frac{{(- 1)}^{n} N_{r} Ψ}{Γ (m_{2})} \frac{Γ (m_{2} + Δ) {(1 - ρ_{2})}^{Δ}}{Γ (m_{2} + 1) ς^{Δ + m_{2}}} {(\frac{m_{2} (1 + δ)}{μ δ λ σ_{1}^{2}} \frac{γ_{t h}}{η})}^{m_{2}}, & m_{1} N_{t} > m_{2} . \end{cases} \end{matrix}

(18)

Case 3. It represents no feedback delay; that is, $ρ_{1} = 1$ and $ρ_{2} = 1$ ; we have

\begin{matrix} P_{o u t}^{\infty} \approx \{\begin{cases} \frac{{((m_{1} (1 + δ) / δ λ σ_{1}^{2}) (γ_{t h} / η))}^{m_{1} N_{t}}}{(m_{1} N_{t})!}, & m_{1} N_{t} < m_{2} N_{r} \\ [\frac{m_{1}^{m_{1} N_{t}}}{(m_{1} N_{t})!} + \frac{m_{2}^{m_{1} N_{t}}}{{(m_{2}!)}^{N_{r}} μ^{m_{1} N_{t}}}] {(\frac{1 + δ}{δ λ σ_{1}^{2}} \frac{γ_{t h}}{η})}^{m_{1} N_{t}}, & m_{1} N_{t} = m_{2} N_{r} \\ \frac{{((m_{2} (1 + δ) / μ δ λ σ_{1}^{2}) (γ_{t h} / η))}^{m_{2} N_{r}}}{{(m_{2}!)}^{N_{r}}}, & m_{1} N_{t} > m_{2} N_{r} . \end{cases} \end{matrix}

(19)

Proof.

For the three different cases, based on the PDF results in (8) and (12), we can expand the exponential function using Maclaurin series and choose the smallest order terms of x to obtain a high SNR approximation. Then using a lower bound in which $P_{o u t}^{l o w} = 1 - {\bar{F}}_{γ_{S R}} (γ_{t h}) {\bar{F}}_{γ_{R D}} (γ_{t h})$ we may obtain the asymptotic outage results. The details of the proof are omitted here due to the lack of space.

Lemma 2 shows that TB feedback delay plays the dominate role in reducing the achievable diversity order of the system to one (details in Case 1), regardless of Nakagami-m fading parameters and the number of antennas and relays. TB is so sensitive to CSI delay that we may obtain no diversity gain but only loss even compared with single-antenna communication link as $m_{1}$ . This is a new finding that differs from the intuitionistic result that RS delay will reduce the diversity order to a case without relay selection as $m_{2}$ (details in Case 2). When $ρ_{d_{1}}$ and $ρ_{d_{2}}$ approach one, the system achieves the full diversity order of the two-hop TB-RS system, as $G_{d} = m i n \{m_{1} N_{t}, m_{2} N_{r}\}$ .

4. Numerical Results

This section presents the numerical and Monte Carlo simulation results of the detrimental effect of delay and CEE on the system performance over Nakagami-m fading channels. Without loss of generality we set $P = 1$ , $σ_{1}^{2} = σ_{2}^{2} = 1$ , $N_{t} = 2$ , $N_{r} = 3$ , $m_{1} = m_{2} = 2$ , and $λ = 1 / 2$ .

Figure 1 illustrates the outage probability versus transmit SNR for various delays and channel estimation qualities. It can be clearly seen from the figure that analytical and simulated outage probability curves match excellently, which confirm the accuracy of our mathematical analysis and the tightness of the asymptotic analysis as well as the lower bound (in the proof of Lemma 2) in high SNR regime. As expected, the outage performance of the system is aggravated significantly, and we can draw same conclusions as described in Lemma 2.

Figure 1

Outage probability versus the transmit SNR.

Figure 2 plots the outage probability versus feedback delay (including TB feedback delay and RS delay) under perfect channel estimation. On the whole, we see that as $f_{d} T_{d}$ increases, the outage probability at first degrades significantly but then approaches a limit and will fluctuate as delays increase, where the nonmonotonic change fits the behavior of $ρ_{d}$ . Besides, it can be observed that, for the same assumptions on the two hops, the outage performance is more sensitive to the TB vector feedback delay compared with the RS delay.

Figure 2

Outage probability versus feedback delay with $δ = \infty$ .

5. Conclusions

We investigate the joint effect of outdated and imprecise CSI on the performance of the two-hop TB-RS system over Nakagami-m fading. Both analytical and simulated results indicate that TB feedback delay reduces the achievable diversity order to one, while RS delay reduces the diversity order to a case without relay selection. CEE will merely cause coding gain loss. These results will be helpful to predict practical relaying system performance with CEE and feedback delays.

Footnotes

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61371122 and 61301162), the China Postdoctoral Science Foundation under a Special Financial Grant no. 2013T60912, and the China Equipment Development Foundation Grant no. 9140A05020114JB91064.

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San Diego, Calif, USA

Academic Press

MR1773820

Outage Analysis of Transmit Beamforming and Relay Selection with Outdated Channel Estimates over Nakagami- m Fading Channels

Abstract

1. Introduction

2. System and Signal Model

2.1. System Description

2.2. TB and RS with Delay and CEE

3. Outage Analysis

3.1. Exact Analysis

Lemma 1.

Proof.

3.2. Asymptotic Analysis

Lemma 2.

Proof.

4. Numerical Results

5. Conclusions

Footnotes

Conflict of Interests

Acknowledgments

References