Interference-Free Transmission for X channels - Interference Cancellation Using Space-Time Processing and Precoding Design - page 84

5.5 Precoding Design for General N and M

In the last 3 sections, we have provided the precoding and decoding scheme for
2 transmitters each with 6 transmit antennas and 2 receivers each with 4 receive
antennas. In this section, we will extend our scheme to a general case with any N
and M .

5.5.1 M ≥ 4

When M 6, we have already provided a scheme in Sect. 5.2 .From

Eqs. ( 5.30 - 5.35 ), we know that in order to form the orthogonal structure as shown
in Fig. 5.3 , we need at least 6 transmit antennas. Because there are 6 equations
to be solved, 6 transmit antennas will lead to 6 unknown parameters which can be
solved. When N

= 4 and N =

6, we will have 6 equations and more than 6 unknown parameters.
We can always find the solution to form the orthogonal structure. Also with more
degree of freedoms, we can achieve better coding gain.
When M

>

4 instead
of 4. However, in order to achieve the orthogonal structure, we still have at most 6
equations as shown in Eqs. ( 5.30 - 5.35 ). So we need 6 transmit antennas for each user
because each transmit antenna will lead to one unknown parameter in the precoder
matrix. And the precoder design procedures are exactly the same as that of M

> 4, the dimension of each signal vector at the receiver is M >

= 4

in Sect. 5.2 .

5.5.2 M = 3

A special case is when receivers have M 3 antennas resulting in a 3-dimensional

signal vector space, the signal vector space at the receiver. In this case, we still have
4 signal vectors. Two of these 4 vectors are useful signals and the other two are
interference. So we cannot create the orthogonal structure shown in Fig. 5.3 . Instead,
we have to align the two interference vectors along the same direction. In this way,
we will have 3 different signal directions in this 3-dimensional space. What we need
to do is to make the 2 useful signal vectors orthogonal to each other and orthogonal
to the interference direction as shown in Fig. 5.5 . This is the main idea to achieve
interference-free transmission in this case. We call this method Scheme II.
Now we show that this idea is achievable and calculate the minimum number of
needed transmit antennas. Note that we design the precoders for C 1 , C 2 , S 1 , S 2 one
by one. Like before, designing precoders for the last codeword, S 2 , has to satisfy the
most number of constraints and results in determining the number of needed transmit
antennas. If we have enough transmit antennas to successfully design the precoder
for S 2 , we are guaranteed to be able to design precoders for C 1 , C 2 , S 1 as they need to

=