A two-hop relay system is considered where there are
antennas at the source,
antennas at the destination, and a single antenna at the relay, as shown in Figure 1. To make the relay as simple as possible, an AF relaying protocol is employed. It is also assumed that a direct communication link between source and destination is not available, as is reasonable in the case where the communication link between source and destination is in a deep fading state and/or the separation distance between them is large. In addition, half-duplex transmission is assumed; that is, the relay cannot transmit and receive simultaneously in the same time slot or frequency band.
OSTBC transmission containing
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
where
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
where
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 [5] and is amenable to mathematical manipulation. As in most of the literature, the ideal relay gain is used to compute exact average SERs of the proposed system in this paper. Later, it will be observed from simulations that the ideal relay gain provides a tight lower bound on outage probability and average SER in the case of medium-to-high SNR. Substituting
into (2) leads to the received signal at the destination given by
Using maximum likelihood (ML) detection of OSTBCs for the case of spatially colored noise given in [11], the received SNR is obtained as follows.
Theorem 1.
Using the exact relay gain, the received SNR of two-hop AF OSTBC transmission is given by
When the ideal relay gain is utilized, the received SNR is given by
where
,
,
,
, and
denotes the OSTBC code rate.
Proof.
It can be observed from (3) that the noise at the destination is temporally white but spatially colored; that is, the columns of noise matrix
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 [11, 12] to decode each symbol in the OSTBC. The noise-whitening process is given by
where
,
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:
where
and
denotes the Frobenius norm of
. Following similar arguments to [11], the received SNR for each symbol can be written as
where
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
Using the Matrix Inversion Lemma
[13], (9) can be written as
Using
, (10) can be simplified to
Using the fact that
and
in OSTBCs, substituting the exact relay gain into (11) yields (4), and substituting the ideal relay gain into (11) yields (5).
Remark 2.
Using maximal ratio combining (MRC) at the destination, Theorem 1 can be generalized to the case of relay systems with a direct link by including a term corresponding to the SNR of the direct link that increases the SNR by
, 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.