2.2.1. Universal Multicast Transmission Strategy Based on Ring Planning
The existing research results basically focus only on a specific model. However, in reality there are more hybrid transmission models. Due to the change of channel transmission model, the signal receiving model of the whole system changes. Optimization criteria are also different. In practical application scenarios, the channel quality is different due to the different geographical location and path loss of different nodes. In order to maximize the use of channel capacity, communication nodes adaptively choose different code rates and modulation methods according to the channel quality. Thus, an asymmetric transmission channel is formed. Therefore, aiming at the complex communication scenario of asymmetric/symmetric coexistence, a generalized method to systematically solve the heterogeneous information transmission under the condition of multicast is proposed.
Message transmission is expressed by adjacency of nodes in a weighted directed graph. Each directed edge represents a message flow. The degrees of freedom of the message are represented by the weights of the edges in the directed graph, and the directed edge is omitted when the weight is 0. Take a 4-node multicast channel as an example. Assuming that the number of relay antennas
N is 6, DoF is a tuple
. The information interaction among nodes is shown in
Figure 2. If the information exchange occurs at nodes 1 to 2, 2 to 3, and 3 to 1, it is said to constitute a closed loop
. For
K-node multicast relay channels, there may be a closed loop with length of 2 (
) to a closed loop with length of
K (
). As an example, it can be seen in
Figure 2a, there is a ring with a length of 2 (
) and (
). There is a ring of length 3 (
) and (
).
One-way transmission: When the one-way strategy is adopted, the relay uses decoding and forwarding to realize the one-way data communication of the point-to-point channel. The transmitted signal occupies a one-dimensional resource of the relay signal space. If this transmission method is used to realize the message transmission with tuple in DoF, must be satisfied. However, since , the condition cannot be obtained for one-way transmission at this time.
Two-way transmission: There is a problem with using a bidirectional strategy to solve 2-ring. The one-dimensional signals and interacted by nodes 1 and 2 are aligned to the same signal space of the relay with the signal space alignment (SSA) algorithm. Then, relay can decode to obtain a linear combination of the two signals . After the physical layer network coding, this information is sent to nodes 1 and 2 at the same time in the BC phase. Each node uses the message sent by itself as side information to decode the desired signal.
After and transmissions are implemented using the above method, the remainder of the DOF tuples become . If one-way transmission is used to complete the rest of the information interaction, dimensional space is required. Synthesize the above two steps to obtain DOF tuple , which needs to meet . However, due to , although the use of bidirectional transmission cuts down the two-dimensional signal space requirements, is still not available.
Multi-ring transmission: DOF tuples
cannot be obtained by using the combination of one-way transmission and two-way transmission. As shown in
Figure 2b, when the relay uses two-dimensional space to transmit the bidirectional signals of nodes 1 and 2, 2 and 3, the remaining DOF tuples become
. Align the signals
and
sent by node 1 and node 2 respectively to the one-dimensional signal space of the relay. Align the signals
and
sent by nodes 2 and 3 to another dimension space of the relay. The relay can obtain a linear combination of these signals
and
. The signal
is sent twice and aligned to different directions. Each node decodes the received signal using its own transmitted signal in the MAC phase. The relay realizes the transmission of 3-rings (
) in two-dimensional space. As shown in
Figure 2c.
Using the multi-ring strategy, the relay only needs a
dimensional relay space to transmit one 3-ring signal. It saves one-dimensional resource than using one-way mode to transmit signals. Through the combination of bidirectional transmission and multi-ring transmission, the minimum space resource required by the transmission system is: (10).
It can be seen from formula (10) that after the initial tuple
is processed by ring elimination, it is similar to a directed graph with no rings. It can be concluded that the key to studying the optimal communication scheme of a
K-node multidirectional relay channel is the ring elimination technique in directed graph. For a certain DoF tuple
, one-way transmission, two-way transmission and multi-ring transmission should be used in combination. By optimizing the ring elimination strategy, the DoF upper bound of message transmission can be realized. For a 4-node multicast channel the DoF domain is represented by the following inequality:
where
is the ordered permutation for
and
is
i-th element.
2.2.2. Resource Allocation Optimization of Space Compression
Each source node
i and the set
composed of other nodes exchange independent messages,
. The message sent by node
i to node
j is
, which contains
independent information flows. The sum of the number of independent information flows sent in the system is
. When
, the relay cannot decode all data streams independently. When the physical layer network coding is used, the relay simply decodes the network coding symbols
. The element of vector
is composed of
. When
N ≥ 2
M,
cannot be directly obtained by signal space alignment using precoding matrices
and
,
must be obtained by designing the relay compression matrix and aligning the desired interactive signal to the relay compression subspace. suppose that
-dimensional matrix
is a compressed matrix and full rank, the compressed relay received signal is:
Thus, the subspace alignment condition of a compressed signal is obtained as follows:
It is required that the compression matrix should not affect the expected decoded signal and that the designed precoding matrix has full rank. The equivalent condition is: .
Align each pair of signals to the same signal subspace. The relay will receive
alignment signals. In order to ensure the complete decoding of signal
by the relay,
must be guaranteed. The compressed signal processed by the compression matrix is:
Considering the space spanned by signal dimension,
can be obtained. If
, the intersection subspace is empty between
and
when the compression method is not exploited. After the compression matrix processing, SSA can be realized only when the intersection subspace exists. This is shown in
Figure 3b.
If the transmission channel between communicating nodes satisfies the independent Gaussian distribution, the basis vector of the intersection subspace of a pair of source node equivalent channel matrices and cannot be located in the intersection subspace of the other pair of source node equivalent channel matrices. Therefore, the matrix is a full rank matrix with full probability, which ensures the decidability of the signal by the relay.
The subspace alignment condition of the compressed signal (Equation (13)) holds if and only if there are at least basis vectors in the signal space formed by , which are located in the null space of the communication pair . Row of matrix is located in null space. Therefore, the null space dimension of is not less than . The precoding matrices and can always be found such that Equation (13) holds. This condition also makes true. That is, is full rank.