Conditional error probability
A closed-form expressions are derived in what follows for the case of N
=2 transmit antennas. A generalization to any number of transmit antennas can be obtained by using the union bounding technique as in (, Section III-B). Let us assume that at a particular time instant the active antenna index is ν. Then, the decision metrics can be rewritten as
where , , , and .
The instantaneous probability of error, P
) conditioned upon the channel impulse responses (f1,f2,hm,1,hm,2, and g
), can explicitly be written as follows
After a few algebraic manipulations, the instantaneous probability of error, given that transmit antenna one was active, is reduced to
where , which when conditioned upon the fading channels is a random variable with zero-mean and a variance of .
can readily be computed in closed form as follows [23, 28, 29]
Using similar analytical steps, can be obtained and is equivalent to (11). Substituting |P
|ℓ=1 and |P
|ℓ=2 in (9), the conditional error probability can be written as
where , , and with being the m th relay output energy.
Average error probability using moment generation function-based approach
In what follows, the average error probability will be computed by exploiting the moment-generation function (MGF)-based approach for performance analysis of digital communication systems over fading channels.
Let us define and , γ
|hm,2−hm,1|2/2 with , and . Note that γ
are random variables following exponential distribution given by and , respectively. The MGF of γs−dis 
The cumulative distribution function of is computed as follows [30, 31]
The integration in (14) can be evaluated to yield
(·) is the ν th-order modified Bessel function of the second kind. The probability density function (PDF) of can be computed from (15) and is given by,
The MGF of can be computed from the PDF in (16) and is given by [20, 30, 31]
where E1(·) is the exponential integral function.
Using the MGF, an exact closed form expression for the average error probability in a finite single integral can be computed as follows ,
To avoid numerical integration, this integral can be approximated as
Asymptotic analysis at high SNR analysis
A simpler form for the expression in (18), which offer insight into the effect of the system parameters, is derived in what follows. According to [28, 29], the asymptotic error and outage probabilities can be derived based on the behavior of γ
around the origin. By using Taylor’s series, can be written as
where H.O.T stands for higher-order terms. Therefore, the average error probability can be simplified to
A diversity gain of M is clearly seen in the above equation.
Arbitrary number of transmit antennas
So far, exact closed-form expressions for the average error probability when the source is equipped with two transmit antennas are provided. The framework is generalized in what follows to account for an arbitrary number of transmit antennas. The error performance is derived using the well-known union bounding technique. The average error probability for the system with N
transmit antennas is union bounded as (, pp. 261–262)
where is the number of error bits when choosing instead of ℓ as the transmitting antenna index and is the pairwise error probability (PEP) of deciding on given that x
was transmitted. The PEP for two transmit antennas can be computed as in (18) and substituted in (22) to obtain the error probability for an arbitrary number of transmit antennas.