Page 152 - Proceedings of the 2017 ITU Kaleidoscope

2017 ITU Kaleidoscope Academic Conference

the same condition, the lens with a smaller σ value of the Where P(x j , y j ) = g jj · P j (t).
stray light PSF has a higher imaging performance because The additive noise in the MIMO-IS VLC system includes the
the imaging light is more concentrative and the imaging re- constant stray light noise, the thermal noise, the shot noise
sult suffers less from the stray light. and the readout noise. Since the noise of the latter three is far
less than the constant stray light noise, the additive noise in
2.3. Channel model system of the i channel is expressed as δ i = K. As a conse-
quence, the received current signal of ith channel is obtained
Assuming S(t) = (s 1 (t), ..., s i (t), ..., s n×n (t)), (1 ≤ i ≤ as
n × n) is a frame of the original serial digital data and ac-
n×n
cording to the equation (1), the output optical signal of the I i (t) = ξt e g ii · P i (t) + X s ij · P j (t) + K o (8)
n
T
LED array is P = (P 1 (t), ..., P i (t), ..., P n×n (t)) . There-
j=1,j6=i
fore, the received current signal of the image sensor I =
T
(I 1 (t), ..., I i (t), ..., I n×n (t)) can be written as
3. THE SIMULATION AND ANALYSIS
I = ξH × P + N (4)
3.1. The SNR and BER
where ξ is the photoelectric conversion coefficient of the im-
age sensor, H is the channel gain matrix of n × n order, Supposed that the optical active area of the LED is circular
T
N = (δ 1 , ..., δ n×n ) is the system noise matrix of n × n
and the radius is Ra, the projective radius of the LED opti-
order.
cal active area on the imaging plane can be given from the
Considering the channel as a line of sight (LOS) link, the 0 f · Ra
MIMO channel gain matrix is composed of the link gain G equation (6) as Ra = . Combined with the equation
u
and the SISI gain S as well as H = G + S. Both of them is (2)(7)(8), the received SISI noise of the system (0 ≤ ψ i ≤
n × n order. In addition, the link gain matrix G is a diagonal ψ c ) is obtained as
matrix and the SISI gain matrix is a matrix that the diagonal
elements is zero. n×nn×nZ 2π Z Ra 0 0 2 !
XX g jj · P j (t) r i
dθ √ exp − dr
Fig.3 shows a geometric graph of the MIMO-IS-VLC sys- I SISIN = ξt e 0 0 σ 2π 2σ 2
tem. The focal length of lens is f. u is the object distance and i=1 j=1
j6=i
v is the image distance. d is the spacing distance of two adja- (9)
cent LEDs in the array. On the basic of the imaging system, The SISI noise component is an additive noise to the system.
there is u > 2f and f < v < 2f so that a real and miniature The equation (9) indicates that SISI noise component mainly
image of the object is acquired on the imaging plane. depends on the spacing distance of two adjacent LEDs in the
0
The LED light radiation follows the Lambertian model. Thus imaging plane r ij . In other words, three major factors, the
the diagonal elements of G is written as [9] communication distance u, the lens’ focal length f and the
spacing distance of two adjacent LEDs r ij in the LED ar-

A ψ c ray influence the system BER performance jointly according

 2 R(φ) · cos(ψ i ), 0 ≤ ψ i ≤
g ii = u 2 (5) to the equation(6). As a result, with given communication
ψ c distance and lens, the probability of received SISI noise in-

0, ψ i > 0

2 tensity is the function of r ij .
0
m
where R(φ) = [(m + 1)/2π]cos φ, φ is the LED irra- The average expectation value of r ij can be calculated by
permutation and combination theory. Under the circum-
diance angle. ψ i is the incident angle of ith LED. m =
1
−ln2/ln(cosψ 1/2 ), m is the order of the Lambertian radi- stance that P[s(t) = 0] = P[s(t) = 1] = , the average
ation. ψ 1/2 is the emission angle at half power of the LED. 2
expectation intensity of SISI noise is able to be estimated.
ψ c is the angle of view of the lens. 0
For a n × n channels MIMO-IS-VLC system, r ij is inte-
Establishing the rectangular coordinate system based on one ger times of the LED spacing distance d or not the integer
of the LED in the array, the distance of ith and jth LED can times. Based on the permutation and combination theory, the
p
2
be expressed as r ij = d · |x i − x j | + |y i − y j | . Under average expectation value of r ij is calculated as
2
0
the condition that the object distance u is far more larger than
the image distance v, there is v ≈ f. Hence, the spacing  2 1 1 n n−q qdf
1
 C C C
0
distance of two adjacent LEDs in the imaging plane can be  2 · , r = qd


 C n×n u
expressed as 0 
r (n, q) =  C C C 1 ( t P C 1
1
1
0
r ij = f · r ij (6)  2 2 n−q p=1 n−p ) p q + p df
2
2
0
u  · , r 6= qd


C n×n u
 2
From equation (1), the off-diagonal elements of S is written
(10)
as
Where q, p ∈ N, and 0 < q ≤ n. Combined with
ZZ
0
s ij = P(x j , y j ) · s(r ij ) dx dy (7)
equation(9), the total expectation received SISI optical in-
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