Abstract
A two-hop amplify and forward (AF) relay system is considered where source and destination are each equipped with multiple antennas while the relay has a single antenna. Orthogonal space-time block coding (OSTBC) is employed at the source. New exact expressions for outage probability in Rayleigh fading as well as symbol error rate (SER) expressions for a variety of modulation schemes are derived. The diversity order of the system is evaluated. Monte Carlo simulations demonstrate the accuracy of the analyses presented. Results that can be extended to relay systems with a direct source-destination link are also highlighted. To put the results in context, the two-hop system performance is then compared to that of a MIMO point-to-point system. Finally, the new analysis is applied to evaluate two-hop system performance as a function of relay location.
antennas at the source, 
antennas at the destination, and a single antenna at the relay, as shown in Figure 
symbols 
and block length of 
is utilized at the source to achieve space diversity. During the first time slot, the 
received vector signal at the relay can be written as
denotes a 
OSTBC transmission matrix, 
denotes the 
Rayleigh fading complex channel gain vector from the source to the relay, and 
is the 
independent and identically distributed (i.i.d) complex Gaussian noise vector at the relay. During the second time slot, the 
received signal at the destination can be written as
denotes the 
Rayleigh fading channel gain vector from the relay to the destination, 
is the 
i.i.d noise matrix at the destination where 
denotes the noise at the 
th receive antenna during the 
th symbol period, and 
denotes the 
signal vector sent by the relay where 
is a relay gain. As categorized in the literature, relay gains may be fixed or variable. In this paper, variable relay gain is considered. Relay gains can be further classified as exact or ideal. We denote 
as the exact relay gain, where 
and 
are average power constraints at the source and relay, respectively. If we ignore the noise at the relay, 
which is denoted as the ideal relay gain [
into (2) leads to the received signal at the destination given by
, 
, 
, 
, and 
denotes the OSTBC code rate.
are independent and Gaussian with covariance matrix 
. Also, note that system (3) under consideration is equivalent to a conventional MIMO system with an effective channel gain matrix 
. We first whiten the colored noise and then employ the ML method given in [
, 
and 
. After ML detection, the MIMO system in (6) can be transformed into the following 
parallel and independent single-input and single-output (SISO) systems:
and 
denotes the Frobenius norm of 
. Following similar arguments to [
denotes trace of a matrix, 
is inverse of matrix 
, and 
denotes average power of a symbol where 
is assumed for OSTBCs. By substituting for 
and 
defined above, (8) can be written as
[
, (10) can be simplified to
and 
in OSTBCs, substituting the exact relay gain into (11) yields (4), and substituting the ideal relay gain into (11) yields (5).
, where 
is the 
Rayleigh fading complex channel gain matrix from source to destination and 
denotes the received noise variance during the source-to-destination time slot. It is assumed that relay-to-destination communication occurs in a separate second time slot, that is, half-duplex communication.
in which 
denotes outage capacity and 
denotes the modified Bessel function of second kind and order 
. When the ideal relay gain is utilized, the outage probability is given by
and 
are complex Gaussian distributed, where 
and 
are elements of 
and 
, respectively, it is readily found that 
and 
are exponentially distributed. Setting rate parameters of 
and 
to be equal to 
and 
, respectively, the cumulative distribution function (CDF) and PDF of 
and 
are, respectively, [
and 
, in terms of the constant parameter 
, is given by
denotes the CDF of 
with the exact relay gain and 
denotes the CDF of 
with the ideal relay gain. Setting 
and substituting (14) and (17) into (18) yield
into (20), and through straightforward mathematical manipulations (12) with 
and (13) with 
are yielded.
[
[
denotes the derivative of 
and 
denotes the CDF of 
for the ideal relay gain. Substituting (13) with 
into (22) yields
is the Gauss hypergeometric function, 
for binary phase-shift keying (BPSK), 
for binary frequency-shift keying (BFSK), and 
for M-ary pulse amplitude modulation (MPAM).
, and
is given by the following theorem.
is given by
is first presented in the following lemma.
is given by
with respect to 
and using the expression for the modified Bessel function derivative in [
, where 
is the rate parameter of the exponentially distributed source-to-destination gain 
.
when an OSTBC strategy is employed at the source.
, then when 
, 
, it can be claimed that 
by ([
). Substituting (15) and (17) into the previous expression yields
:
.
and outage capacity 
. In Figures 
from the source antennas to the relay and the variance of Rayleigh fading channels 
from the relay to the destination antennas are set to 0.8 and 0.9, respectively. Also, OSTBCs with the highest code rate and minimum decoding delay are employed as given in [
.
including BPSK, BFSK, and 4 PAM as well as with QPSK and 16 QAM. Clearly, the simulations very closely match the analyses. Again, the ideal relay gain provides a tight lower bound on average SER even in the case of medium SNR.
. The path loss exponent is set to 4, and it is assumed that shadowing effects are the same for the source-relay and relay-destination links, with a standard deviation 
, a value typically assumed in urban cellular environments. For a fair comparison, the transmit SNR as well as the OSTBC transmit strategy is assumed to be identical for both systems.
, is larger than that of the relaying system for the same antenna configuration at both source and destination. In other words, as transmit SNR increases, the MIMO point-to-point system has an advantage over the relaying system. However, it can also be observed that the relaying system outperforms the MIMO point-to-point system in lower SNR regimes: for the case of 
, the relay system outperforms point-to-point MIMO systems at transmit SNRs below 6 dB. Adding a transmit antenna to the MIMO system lowers this threshold to -1 dB, while at the same time, the code rate for OSTBCs becomes 3/4. On the other hand, adding a receive antenna to the MIMO system lowers this threshold to -4 dB. Of course, the MIMO point-to-point system would be able to achieve similar system performance gain by adding transmit or receive antennas, but this would increase complexity.