Interference-Free Transmission for X channels - Interference Cancellation Using Space-Time Processing and Precoding Design - page 87

need at least as many transmit antennas as we needed in Sect. 5.2 . In what follows,
we will show this fact.
Let us assume we use Scheme II for the case of M

4. After designing the
precoders for C 1 , C 2 , S 1 , we consider designing precoder B 2

≥

for S 2 .Asshown

in Fig. 5.5 , we need D 14 || D 23 . So we will have exactly the

same equations as Eqs. ( 5.58 - 5.61 ). Note that ( 5.58 ) contains M equations since it
includes M

D 13 , D 24 ⊥ D 21 , D 24 ⊥

× 1 vector. So, we will have M + 3 equations. It is easy to show that

we must have N ≥ M + 2 transmit antennas which will leads to M + 3 unknown

parameters including ω . When M = 4, in order to align the interference along the

same direction, we need N 6 which is exactly the same with the number of

needed transmit antennas in Sect. 5.2 . However, when M

≥

> 4, in order to align the

interference along the same direction, we need N 6 while our scheme

proposed in Sect. 5.2 only needs 6 transmit antennas. Therefore, when M

≥ M + 2 >

≥ 4, we

prefer Scheme I over Scheme II.

5.5.3 M < 3

When M 3, the signal vector space at the receiver is 2-dimensional. But we have 4

signal vectors including 2 useful signal vectors and 2 interference signal vectors. Even
if we align the 2 interference vectors along the same direction, we still have 3 signal
vectors in this 2-dimensional space. Therefore, we cannot achieve interference-free
transmission in this case.
In summary, when there are 2 transmitters each with N transmit antennas and 2
receivers each with M receive antennas, we can achieve interference-free transmis-
sion and full diversity simultaneously for each user if N and M satisfy the following
conditions:

<

1. When M 5, we can achieve our goal using Scheme II, i.e.,

by putting all interference in the same direction and making all useful signal
vectors orthogonal to this interference direction.
2. When M

= 3, as long as N ≥

6, we can achieve our goal using Scheme I, i.e.,
by putting all interference in a subspace which is orthogonal to the useful signal
vectors as shown in Sect. 5.2 .

≥ 4, as long as N ≥

5.6 Extension to J t Transmitters Each with N Antennas and J r
Receivers Each with M Antennas

In this section, we will extend our previous results to a more general case, i.e., J t
transmitters each with N transmit antennas and J r receivers each with M receive
antennas. First, we provide our main result: