892 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 36, NO. 4, APRIL 2018
Fig. 1. Schematic diagram of RSCS-OFDM-DM communication system.
For the convenience of implementation of both transmitter and
receiver, all frequency shifts are designed to be orthogonal
with each other in this paper. Thus, the subcarriers in OFDM
systems are a natural choice as shown in Fig. 2. Suppose there
are N subcarriers in our OFDM systems, the set of subcarriers
is the following
S
sub
={f
m
|f
m
=f
c
+mΔf, (m =0, 1, ···,N − 1)}, (1)
where f
c
is the reference frequency, and Δf is the subchannel
bandwidth with Δf =1/T
s
. Here, T
s
denotes the period
of OFDM symbol. Fig. 2 sketches two different kinds of
RSCS patterns: block-level and symbol-level.. NΔf f
c
is assumed in this paper. In the system, the total bandwidth
is B = NΔf. The corresponding total subcarrier index set is
defined as follows
S
N
= {0, 1, 2, ···,N − 1}, (2)
then N
T
random subcarriers are chosen from S
N
, allocated to
N
T
transmit antennas individually, and represented as S
N
T
S
N
T
⊆ S
N
. (3)
The cardinality of the set S
N
T
is N
T
. Now, a chosen subcarrier
index function η(•) is defined as a mapping from the set
of transmit antenna indices {1, 2, ··· ,N
T
} to the chosen
subcarrier set
S
N
T
= {η(n)|n ∈{1, 2, ···,N
T
}}, (4)
where η(n) ∈ S
N
. The first pattern shown in Fig. 2 (a)
is called block-level pattern. Every block is made up of
several or even one thousand OFDM symbols. Within one
block, the same RSCS pattern is used but the RSCS pattern
will vary from one block to another. The second pattern shown
in Fig. 2 (b) is a symbol-level type. In this type, the RSCS
changes from one OFDM symbol to another. For desired
receivers, the latter provides a more secure protection but also
place a large computational complexity and uncertainty on the
receive active subcarrier test. The former can strike a good
balance among receiver complexity, security and performance
by choosing a proper block size. A large block size means
a high successful detection probability of active subcarriers
and less security because of less randomness. In other words,
a better performance can be achieved. Determining the block
size is a challenging problem beyond the scope of our paper
due to the length limit. Letting ΔT be the sampling interval,
we have the next definition
T
s
= NΔT = N/B, (5)
ΔT =1/B.
With the above definitions, the transmit RF signal vector from
N
T
transmit antennas is expressed in vector form as follows
s(t)=[s
1
(t),s
2
(t),...,s
n
(t), ··· ,s
N
T
(t)]
T
, (6)
with
s
n
(t)=x
k
e
j(2πf
n
t+φ
n
)
, (n =1, 2, ···,N
T
) (7)
where f
n
is randomly chosen from subcarrier set S
sub
, x
k
is the kth transmitted complex digital modulation symbol
with E{x
∗
k
x
k
} =1, φ
n
is the initial phase, and the time
variable t ∈ ((k − 1)T
s
,kT
s
). In far-field scenario, after the
transmit signal s
n
(t) from element n experiences the line-of-
propagation (LoP) channel, the corresponding receive signal
at an arbitrary position (θ, R),whereR is the distance with
respect to the first element of transmit antenna array, θ is the
spatial direction, and the first element is chosen as a reference
antenna, can be expressed as
s
n
(θ, R; t)=x
k
e
j
2πf
n
t−
R
(n)
c
+φ
n
= x
k
e
j
2π(f
c
+Δf
η(n)
)
t−
R
(n)
c
+φ
n
, (8)
with
R
(n)
= R − (n − 1)d cos θ, (9)
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