In traditional relay communications, each source node takes advantage of a different time slot to transmit information, and each relay node also successively uses a different time slot to deliver information, which will result in poor real-time performance for information transmission [
30]. Network coding can greatly reduce time slots, and the excellent characteristics of this suggest network coding has a very promising future in wireless multicast networks [
31,
32]. The classification of network coding, different network coding performance evaluations, and the encoding and decoding derivation of CFNC in the two modes are provided by
Section 3.
3.1. The Classification of Network Coding
Based on the arithmetic mode, network coding can be divided into several categories, such as the binary field, the Galois field, complex field, and so on. The application of network coding in UAV cluster must consider the characteristics of UAV communication. With the application of new mission payloads, such as large-area and high-resolution digital aerial cameras, synthetic aperture radars, infrared imagers, etc., the information quantity detected by drones is growing exponentially. Saving on the return time of reconnaissance information implies a decrease in discovery probability. Next, we investigate which network coding scheme has the best real-time performance.
In general, network coding designs are based on the Galois field, which implements bit level operations. This coding scheme can improve throughput to some extent, but the advantage is diminished with an increasing number of source and relay nodes. A
Ns-source
Nr-relay single-destination structure with traditional network coding is depicted in
Figure 6. Assuming that each node is equipped with an antenna,
Ns sources (
) transmit information to the destination (
) directly and via the relays (
). To avoid interference, sources
, in the traditional relay format, transmit over orthogonal channels, e.g., via time division multiple access (TDMA) [
27]. To start with, source
transmits information symbols
to
and
simultaneously during channel use (CU) 1. Then, the relay
forwards
to
in CU 2, and
is the decoding output of
according to
. From CU 3 to CU (
Nr+1), the
relays send
to
successively. The information symbol
takes (
Nr+1) CU from source
to the destination
through relays
. For the information symbol sequence
, a total of
Ns(
Nr+1) channel uses are needed to deliver
Ns symbols with
Ns sources, and the throughput of this scheme is 1/(
Ns(
Nr+1)) symbol per source per channel use (sym/S/CU).
The relay scheme based on Galois field network coding (GFNC) is depicted in
Figure 7. In CU 1, source
transmits information symbol
to both
and
, the same as in a traditional relay. From CU 2 to CU
Ns, information symbols
are sent to
and
successively.
forwards the Galois field coded symbol
to
in CU (
Ns+1), where
denotes a bitwise exclusive XOR operation. Likewise,
forwards the Galois field coded symbol
to
in CU (
Ns+
Nr). From the above analysis, we can deduce that (
Ns+
Nr) channel uses are needed for information symbol sequence
transmission from
Ns sources to
. Thus, the throughput of a GFNC-based relay is 1/(
Ns+
Nr) sym/S/CU.
For improving the real-time performance, a CFNC is introduced in this paper. As illustrated in
Figure 8, before transmission in time slot 1, the source information
from
is multiplied by
, which is the
ith element of
. We assume that
is available at every node in the network. The choice for a diversity maximizing
value is not unique but is available for any
Ns. Among the different (parametric/non-parametric) choices for
, [
28] takes it to be any row of the Vandermonde matrix, i.e.:
where the so-called generators,
, have a unit modulus in complex field
. Relays
simultaneously receive information symbols
, transmitted by
in CU 1, and the agreed coefficients
drawn from
will be specified later. After detecting
as
,
forwards
to
in CU 2. Therefore, the throughput of CFNC is 1/2 sym/S/CU. The throughput comparison of the above three schemes is listed in
Table 1.
As can be seen from
Table 1, GFNC is superior to traditional coding in terms of throughput, and the advantage gradually decreases with the increasing number of source and relay nodes, but CFNC can naturally avoid such a problem. The unique coding method employed by CFNC makes the throughput increase to 1/2 sym/S/CU, which is beneficial to improving the real-time performance. Moreover, the XOR operation is usually adopted by the GFNC, which will cause one-to-one mapping to be impossible between the source information and the received information. By contrast, the received information
(
) and information symbol sequence
easily satisfy one-to-one mapping, unless
. Meanwhile, the mapping offers a method to detect
through the received information
.
3.2. Information Transmission Based on Complex Field Network Coding (CFNC) in Mixed Mode
Based on the theoretical analysis in the previous section, we have deduced that the CFNC obtains overwhelming superiority over other network coding schemes in terms of throughput when the source and relay nodes are of large quantities. Next, the information transmissions based on CFNC applied to the proposed topology structures is derived for the mixed and relay modes, respectively. According to the irregular topology structure
Ns-
Nr-1 for the mixed mode, as shown in
Figure 4, the information symbol transmission based on CFNC merely involves two channel uses. The received symbols at
and
after CU 1 are given as follows (see
Figure 9):
where for each subscript duplet,
denotes the channel coefficient and
denotes the AWGN term. The instantaneous and average signal-to-noise ratios (SNRs) are given respectively by
and
, where
and
denote the average transmission power of source symbol
, which is assumed to be drawn from a finite alphabet
with cardinality
[
27]. Here,
,
, and information symbol vector
, where
and
.
The design of
in Equations (4) and (5) is critical to CFNC. The design relates the linear complex field (LCF) encoder given in [
33] for multiple input multiple output (MIMO) systems. Based on the concept of Euler numbers and their properties, two systematic designs of these generators are provided in [
34]:
if
and
if
, where
indicates the
nth row of Vandermonde matrix. In other words,
if
and
if
, where
. However, the similarities with MIMO-LCF designs stop here. Notice that the coded symbol
in CFNC is transmitted through different nodes (sources) in the network simultaneously, instead of through multiple co-located antennas on one terminal [
33]. Therefore, a normalizing factor, as in ([
34], Eq. (3.68)), to meet the power constraint on one node is not necessary here [
28].
After
relay channels, the maximum likelihood (ML) of detection at relay
is given as follows:
The relaying node
re-encodes the demodulation results then sends it to the target node. The input/output (I/O) relationship in CU 2 is expressed as follows:
where
,
represents a link-adaptive scalar which controls the transmission power at
,
is an
vector designed as the above, i.e.,
. For
, the entries of
are given by
and
, and for
,
for any
.
The symbol rate is 1/2 sym/S/CU, because
Ns sources transmit
Ns signals over 2 channels. After passing through 2 channels, the ML detection result at
is given as follows:
where
.