Using the notations
for all
, and assuming that the leakage power (The derivation of outage probability expression and its minimization become too involved if the noise power is not negligible. However, noting that the cellular systems such as UMTS with beamforming techniques can support a significant number of cochannel users per cell [21] (this number can be further increased if more scrambling codes can be allocated for each cell [22]), the assumption that the multiuser leakage power dominates the thermal noise power at each user is not a stringent one.) is large compared to the noise power, we get the SLR from (3) as
We first note that the rows of
are statistically independent, and each row has an
-variate complex Gaussian distribution with the mean vector
and the covariance matrix
. According to [23], in this case,
are complex Wishart distributed with the scaling matrix
and the degrees of freedom parameter
. For conciseness and simplified mathematical presentation, in the rest of this paper, we assume that
. Here, we also stress that our results can be easily extended to the general case where
are different. Mathematically, we can thus write
, where
represents the complex Wishart matrix of size
. Let us use the notations
and
. According to the results of [14] and since
, we get
. We note that for any
,
, because
is a positive semidefinite matrix. Since
is a Chi-square distribution, the random variable
has the following PDF:
where
,
and
is the Gamma function. Comparing the PDF of (7) to the standard form of Chi-square PDF [23],
can be alternatively expressed as
where
is the Chi-square distribution with degrees of freedom
. Using (8),
can be written as
It can be observed from (9) that
is a weighted sum of statistically independent Chi-square random variables, where the weights
since
are positive semidefinite. The exact and closed-form solution for the PDF of
is not known. However, according to [24] and the references therein, the PDF of
can be found by approximating
as a random variable with the Chi-square distribution having degrees of freedom
and the scaling factor
as
where
and
can be determined by equating the first- and second-order moments of the left-and right-hand sides of relation (10). (This approximation is very accurate and widely adopted in statistics and engineering. The accuracy of the approximation will be confirmed later through numerical simulation results.) Evaluation of the first-order moment (mean) of the both sides of (10) gives
Similarly by equating the second-order moment (variance) of the both sides of (10), we get
Solving (11) and (12),
and
can be expressed as
Like the PDF of
given in (7), the PDF of
is well known to be [23]
where again
. For the sake of better exposition, let
, where
is the ratio of two statistically independent random variables. The PDF of
can be thus written as
Applying (7) and (14) into (15) and after some steps, we get
With the help of [25, equation 3.38.4], (16) can be written in the closed-form as
The average of the SLR is thus given by
After substituting
from (17), applying [25, equation 3.194.3], and after some steps of straightforward derivations, we get
where
is the Beta function. Noting that
and
, (19) can be further simplified as
The outage probability of SLR is a parameter that shows how often the transmit beamformer is not capable of maintaining the ratio of the signal power to the leakage power above a certain threshold value. The outage probability for the
th user is defined as
where
is the system specific threshold value. Note that (21) represents the probability of the transmit beamformer failing to perform its beamforming task properly. Hence, the concept of the SLR outage is analogous to the probability of receiver failing to work properly but is only applicable from a transmitter's point of view. Since the PDF of SLR is already known, the outage probability of (21) can be expressed as
Using (17) and applying [25, equation 3.194.1], it can be shown that the outage probability (22) can be expressed as
where
and
is the Gauss hypergeometric function (see [25, equation 9.100]). Noting the transformation rule
(see [25, equation 9.131.1]) and the fact that
=
, and after some simple manipulations, (23) can also be expressed in the following alternative form:
Here, it is worthwhile to mention that for
,
(7) becomes exponentially distributed whereas
(9) becomes a weighted sum of independent exponentially distributed random variables. In this case, the outage probability expression of [15] can be easily derived. However, it cannot be analytically obtained by substituting
in (23) due to the approximation (10). Also, note that the proposed outage probability analysis can be applied to frequency-selective fading channels where we can consider that the orthogonal frequency division multiplexing (OFDM) is used as a modulation technique. In this context, the MIMO channel for each subcarrier can be considered to be a flat-fading channel. Considering that all users can access a given subcarrier and that the lengths of channel impulse responses for all receive-transmit antenna combinations of all users are shorter than the cyclic prefix [26], the SLR for the
th user and
th subcarrier can be expressed as
where
is the MIMO channel in frequency domain for the
th user and
th subcarrier, and
is the corresponding gain. Let
be the
th row and
th column entry of
, and be given by
where
is the total number of subcarriers,
is the number of independently fading channel-taps, and
is the impulse response for
th tap of the channel between
th receive and
th transmit antenna. If
are ZMCSCG, it is very easy to note that
is a ZMCSCG. Furthermore, if the average sum of the tap-powers for the channel between the
th receive and
th transmit antennas is same, that is, if
for all
, after some straightforward steps, we can easily verify that the distribution of
remains complex Wishart with the same scaling matrix
and the degrees of freedom parameter
. This shows that the statistics of the signal and leakage powers for a given subcarrier and user remain unchanged.