88 Journal of Communications and Information Networks
nal expressed as follows:
e
x
n,l
= s
n,l
e
jϕl
, (2)
where ϕ (0 6 ϕ < 2π) is the continuous azimuthal angle.
Here, the discrete azimuthal angle in terms of u turns to be
the continuous azimuthal angle in terms of ϕ along with the
OAM signal transmitting (This is a spatial digital to analog
(DA) process, which is defined as the process by which a dis-
crete OAM signal is converted to be a continuous OAM signal
through a UCA antenna).
We denote by h
v
m
,u
n
the channel gain from the uth array-
element on the nth transmit UCA to the vth array-element
on the mth receive UCA. Then, h
v
m
,u
n
can be written as
follows
[10]
:
h
v
m
,u
n
=
β λ e
−j
2π
λ
d
v
m
,u
n
4πd
v
m
,u
n
, (3)
where v
m
represents the vth array-element on the mth receive
UCA. The parameter β denotes all relevant constants such as
attenuation and phase rotation caused by antennas and their
patterns on both sides. We denote by s the distance between
the projection of the uth array-element on the nth transmit
UCA in the receive plane and the vth array-element on the
mth receive UCA. Then, we have
s =
q
r
2
n
+ R
2
m
−2r
n
R
m
cos(φ
u
+ nδ −ψ
v
−mα), (4)
where ψ
v
= 2π(v −1)/V is the basic angle for each receive
UCA, α = 2π/MV is the minimal interval for the angle of ro-
tation corresponding to the receive UCAs, and ψ
v
+ mα is the
azimuthal angle of the vth array-element on the mth receive
UCA. Then, we can write d
v
m
,u
n
as follows:
d
v
m
,u
n
=
p
d
2
+ s
2
=
q
d
2
+ r
2
n
+ R
2
m
−2r
n
R
m
cos(φ
u
+ nδ −ψ
v
−mα). (5)
Because of d
v
m
,u
n
r
n
and d
v
m
,u
n
R
m
, we can make ap-
proximation for d
v
m
,u
n
at the denominator and numerator of
Eq. (3). For the denominator, we use d
v
m
,u
n
≈ d. For the nu-
merator, d
v
m
,u
n
can be rewritten according to
√
1 −x ≈1−x/2
as follows:
d
v
m
,u
n
=
q
d
2
+ r
2
n
+ R
2
m
s
1 −
2r
n
R
m
cos(φ
u
−ψ
v
+ nδ −mα)
d
2
+ r
2
n
+ R
2
m
≈
q
d
2
+ r
2
n
+ R
2
m
−
r
n
R
m
cos(φ
u
−ψ
v
+ nδ −mα)
p
d
2
+ r
2
n
+ R
2
m
. (6)
Then, using the above-mentioned approximation, we can
rewrite h
v
m
,u
n
as follows:
h
v
m
,u
n
=
β λ e
−j
2π
λ
√
d
2
+r
2
n
+R
2
m
4πd
·
exp
"
j2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
cos(φ
u
−ψ
v
+ nδ −mα)
#
. (7)
Before spatially sampling at the receiver, the signal received
at the vth array-element of the mth receive UCA, denoted by
r
m,v
, can be derived as follows:
r
m,v
=
U/2
∑
l=(1−U)/2
N
∑
n=1
U
∑
u=1
h
v
m
,u
n
1
√
U
s
n,l
e
j
h
2π(u−1)
U
+nδ
i
l
=
U/2
∑
l=(1−U)/2
N
∑
n=1
h
v
m
,n,l
s
n,l
, (8)
where h
v
m
,n,l
is the channel gain for the lth OAM-mode corre-
sponding to the nth transmit UCA and the vth array-element
on the mth receive UCA, and can be given as follows:
h
v
m
,n,l
=
U
∑
u=1
β λ e
−j
2π
λ
√
d
2
+r
2
n
+R
2
m
4πd
√
U
e
j[
2π(u−1)
U
+nδ ]l
·
exp
"
j2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
cos(φ
u
−ψ
v
+ nδ −mα)
#
.
(9)
For convenience, we denote by ϑ = ψ
v
−nδ + mα. Then, we
have
j
l
U
U
∑
u=1
exp[j(φ
u
−ϑ)l]exp
"
j2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
cos(φ
u
−ϑ)
#
≈
j
l
2π
Z
2π
0
exp
h
j
e
φ
u
−ϑ
l
i
·
exp
"
j2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
cos
e
φ
u
−ϑ
#
d
e
φ
u
=
j
l
2π
Z
2π−ϑ
−ϑ
exp
h
j
e
φ
u
−ϑ
l
i
·
exp
"
j2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
cos
e
φ
u
−ϑ
#
d(
e
φ
u
−ϑ) =
J
l
2πr
n
R
m
λ
p
d
2
+ r
2
n
+ R
2
m
!
, (10)
where φ
u
= 2π(u −1)/U and
e
φ
u
is a continuous variable of φ
u
ranging from 0 to 2π, because when u = 1, φ
u
= 0 and when
u = U, φ
u
is 2π −2π/U and very close to 2π. The approxi-
mation error, denoted by Er, between the Bessel function and