This section provides the sum-rate analysis assuming that the average interference power is identically distributed, i.e. δ
i
= δ
i
= δ. Essentially, we consider the scenario that mobile users are distributed along the cell boundary. Meanwhile, the transmitted signal energies for both cells have been defined as Es 1= P1T, Es 2= P2T, in which T denotes the transmit time duration. Notice that in our later work, without loss of generality, we all assume P1 = P2 = 1. That is why all the power term has been dismissed in the later derivations.
4.1. Common analysis
We first present some statistical results on the ordered projection norm squares, which will be broadly applied in the later analysis. Noting that each component of vectors discussed in this section is modeled as i.i.d. complex Gaussian random variable with zero mean and unit variance.
Let us firstly consider the projection norm squares of K independent vectors h
i
, i = 1,..., K to a normalized vector w, i.e. , i = 1, 2,..., K. Since h
i
are independent, and a
i
are i.i.d. chi-square random variable with two degrees of freedom [22]. It follows that the probability density function (PDF) of the l th largest among totally K projection norm square al:K= rank
l
{a
i
}, i = 1, 2,..., K is given, after applying the basic ordered statistic result, by:
(6)
We now consider the project norm square of vector h onto B normalized vectors, w
j
, j = 1,..., B, i.e. b
j
= |hTw
j
|2, j = 1, 2,..., B, and focus again on the l th largest one among totally B projection norm square, i.e. bl:B. Since w
j
are not necessarily orthogonal with one another, the projection norm squares no longer constitute a set of independent random variables. To overcome such difficulty, we rewrite bl:Bas
(7)
It can be shown that follows i.i.d. beta distribution with parameters 1 and N - 1 [23], with PDF given by:
(8)
Now u becomes l th largest one of B i.i.d. beta random variables, whose PDF can be obtained as
(9)
where A = (N - 1)(B - 1 - i) + N - 2.
Noting that v = ||h||2 follows a modified distribution, with PDF given by:
(10)
the PDF of bl:Bcould be obtained as the product of two random variables [24]. After several steps of computation, we have
(11)
where
(12)
Note that this result can be broadly applied in other related analysis. In Figure 2, we plot the PDF of b1:B, and find that it matches perfectly with the simulation results.
4.2. Sum-rate analysis
In this part, we analyze the ergodic sum-rate performance of the beamforming transmission schemes under consideration. The sum-rate of the proposed dual-cell random beamforming system can be calculated as
(13)
where and are the PDF of received SINR of the selected users in cell 1 and 2, respectively. We now derive the exact statistics of the selected users' SINRs.
1) SRB: Due to the symmetry, let us consider the received SINR of selected user by BS1, as given in (2), which can be rewritten as
(14)
where ρ is the normalized noise power, equal to N0/E
s
. n
i
follows the chi-square distribution with 2 degrees of freedom, whose common PDF is represented as
(15)
And the PDF of was given in (6), with K changed to K1. Noting the independence of n
i
and , the PDF of γ1 can be calculated using the PDFs of n
i
and , as [22],
(16)
After carrying out the integration with proper substitutions, we have
(17)
2) IA-RB: Again due to symmetry, we consider PDF of the received SNR at the selected user by BS2, which was given in (4) as the maximum of K2 independent random variables, defined as
(18)
Note that p term follows i.i.d. distribution over , with PDF
and q
j
term are i.i.d. with distribution over , but with variance δ2, whose PDF is the same as (15).
Following the similar steps as for SRB, we can obtain the PDF of , as
(20)
It follows the CDF of , denoted by is given by
(21)
Finally, the PDF of γ2 is obtained as,
(22)
3) LC-RB: Based on the notation introduced in previous subsection, the first user's SINR, can be written as,
(23)
The PDF of both and bB:Bcan be obtained as the special case of the general result in (6) and (11), as
(24)
and
(25)
respectively. Note that the element of vector has variance δ here.
Consequently, the PDF of γ1 can be calculated in terms of PDFs of and bB:B, as
(26)
The statistics of the received SINR at the selected user by BS2 is exactly the same as that of IA-RB scheme presented previously, with PDF given in (22).
4.3. Numerical examples
In this section, we present and discuss selected numerical examples to illustrate the mathematical formalism on the sum-rate analysis of the proposed coordinated beamforming schemes. Noting that all the analytical results in this paper have been verified through Monte-Carlo simulation.
For comparison purpose, we also provide the simulation results of one of the popular conventional coordinated beamforming techniques with user selection, with CSI exchange between cells, which is called coordinated zero-forcing beamforming (CZF). Specifically, the CZF option relies on a simple multiuser scheduling method, i.e. to select the user with the largest channel vector norm square. After the full CSI sharing between two cells, the new 'super-BS' uses zero-forcing method to transform the interference channel into a MIMO broadcast channel [3–5]. Suppose that h1i*and h2j*are the two selected user's channel vectors respectively in cell 1 and 2. Then, the beamforming vector w1 needs to satisfy the orthogonality condition to cancel its interference for cell 2.
In Figure 3, we compare the single-cell achieved rates for three schemes under consideration as functions of the common number of users K = K1 = K2. The radius of each cell R is 1 km, the path loss exponent is 3.7, and both BSs are equipped with N = 4 antennas. Both analytical-and simulation-based curves have been provided. It will be firstly observed that at high SNR regime (20 dB), CZF provides the best rate performance, due to its interference cancelation. And the cell 1 for LC-RB outperforms all the others, and can nearly approach CZF, especially when the volume of users is large enough. More specifically, under LC-RB, rate for cell 1 performs better than that for cell 2, owing to the effective interference control from BS2 to cell 1. When the channel SNR is low (5 dB) and the system is noise limited, LC-RB is still the best, while CZF performs the worst, since interference effect is trivial now.