Impact of LED transmitters&#x02019; radiation pattern on received power distribution in a generalized indoor VLC system

Yinlu Wang; Ming Chen; Jin-Yuan Wang; Jianfeng Shi; Zhaohui Yang; Yijin Pan; Rui Guan

doi:10.1364/OE.25.022805

1. Introduction

Visible light communication (VLC) has been widely investigated as a promising communication method due to its favorable features such as simultaneous illumination and communication [1], license-free spectrum [2], security [3], low implementation cost and low complexity [4]. It has been expected to play an important role in indoor positioning, machine-to-machine communication, and extreme scenarios(hospital, aircraft, gas station, etc.) where electromagnetic interference is strictly banned [5,6].

Received power distribution analysis is the foundation of design and performance estimation of VLC systems. There has been extensive work on received power analysis for indoor and outdoor VLC systems under various assumptions [1,7–9]. In [1], received power and channel direct current gain is investigated, and the influence of interference and field of view is discussed. In [7], received power distribution of the optical multi-input multi-output system with imaging and non-imaging receivers are analyzed respectively. In [8], an optimal lights layout scheme is provided for an indoor VLC system. In [9], a car-to-car VLC system is studied and an analysis on the received power and bit-error-rate performance is provided.

While both indoor and outdoor VLC enjoy considerable popularity, we restrict our attention to an indoor VLC scenario in this paper, due to its widespread applications in illumination-based communication, indoor positioning, internet of things and so on [10,11]. For indoor VLC systems, previous received power distribution analysis were generally based on three assumptions: Firstly, all the LEDs are placed on the ceiling [1]; Secondly, the LED transmitters are of Lambertian luminous intensity pattern [12]; Thirdly, the receiver has a small light receiving area (LRA) such that luminance on it can be considered uniformly distributed [13]. However, new applications may pose challenges to those traditional models. For example, [14] developed a VLC earphone system with transmitters on a vertical plane, which can correspond to a museum environment where LEDs are placed on the show window instead of the ceiling. Under this kind of scenarios (including but not limited to the museum case), simultaneous illumination and communication is achieved with LEDs mounted on the vertical wall. Also, radiation patterns of commercially available LEDs are not necessarily Lambertian. As pointed out by [15,16], the Lambertian model cannot accurately reproduce the luminous intensity pattern of the phosphor-coated multi-chip LEDs, the batwing LEDs, or the side-emitting LEDs. Moreover, some VLC systems use the solar panel(with an area of several hundred cm²) [17–19] instead of the photodiode(less than 1cm² sized) as the receiver, where the uniform incident light assumption no longer holds.

A generalized indoor VLC model is thus proposed in this paper to deal with these limitations. The generalization is threefold: Firstly, both ceiling-mounted and wall-mounted LED transmitters are considered; Secondly, a widely applicable sum-of-sine luminous intensity pattern (SSLIP) is adopted to model LEDs’ radiation curves instead of the traditional Lambertian one; Finally, receivers with large LRAs and non-uniform illuminance distribution are considered. Based on the proposed model, the closed form expression of received power at any position of the room is derived. The impact of LEDs’ radiation patterns on received power is further studied, to help decide what types of LEDs to choose when designing an indoor VLC system.

2. System model

Consider an indoor VLC scenario, where the room height is H and the square ceiling is of length L, as shown in Fig. 1. There is an (2N + 1) × (2N + 1) (assume N to be a nonnegative integer) LED transmitting array with an interval l between adjacent LEDs placed at the center of the ceiling, or along the central line of the wall, shown by Fig. 1(a) and 1(b) respectively. Define H₀ as the height of the central LED in the wall-mounted case. The optical receiver is carried by a user, so it can move around a plane with a general height of 1 ∼ 1.4m. It is assumed that H₀ > 1.4m so that illumination is also possible in the wall-mounted case.

Fig. 1 Indoor VLC system scenario with (a) ceiling-mounted LED transmitters and (b) wall-mounted LED transmitters.

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Assume the LRA to be a circle with radius R that R > πd/360° (d being the receiving distance), which suggests a field angle larger than 1° so that illuminance incident on it cannot be considered uniformly distributed. Take the central LED transmitter as the origin and build a Cartesian coordinate. Suppose the LRA is vertical. Define the coordinate of the center of the LRA as (x, y, z). Define the angle between the receiving plane and the X − Z plane as α (0 ≤ α < π/2). That is, the LRA deviates from the neutral position (parallel to the X − Z plane) by a rotational angle α. The wall-mounted and ceiling-mounted settings are illustrated in Fig. 1(a) and 1(b) respectively.

To characterize the illumination properties of the transmitting LEDs, the first thing to be determined is the luminous intensity pattern, which describes the luminous flux radiated by the LED in a given direction [20]. In most literatures, the radiated luminous flux per solid angle, I_v[lm/sr], at a given emergent angle θ is given by the Lambertian model as [12]

I_{v} (θ) = [(m + 1) / 2 π] {cos}^{m} θ,

where m = − ln 2/ ln(cos ϕ_1/2) with ϕ_1/2 being the half-power angle. However, as pointed out in [16], some types of LEDs cannot be accurately modeled by the traditional Lambertian distribution. In light of this, a cosine-power function is proposed in [16] that is widely applicable for most LEDs currently available on the market, given by:

I_{v} (θ) = \sum_{i = 1}^{P} c_{1 i} {cos}^{c_{3 i}} (| θ | - c_{2 i}),

where P is the order of the sum to be determined by required precision, and c_1i, c_2i, c_3i are parameters to be determined by the luminous intensity curve. Note that powers of cosine functions can be expanded into a series of cosine functions [21]

\begin{array}{l} {cos}^{2 n} x = \frac{1}{2^{2 n}} [\sum_{k = 0}^{n - 1} 2 (\begin{matrix} 2 n \\ k \end{matrix}) cos 2 (n - k) x + (\begin{matrix} 2 n \\ x \end{matrix})], \\ {cos}^{2 n - 1} x = \frac{1}{2^{2 n - 2}} \sum_{k = 0}^{n - 1} (\begin{matrix} 2 n - 1 \\ k \end{matrix}) cos (2 n - 2 k - 1) x, \end{array}

where

(\begin{matrix} n \\ k \end{matrix}) = \frac{n!}{k! (n - k)!},

and that cosine functions and sine functions only differ by a constant π/2, we can turn Eq. (1) into the following equation by changing the powers of cosine functions into a series of sine functions

I_{v} (θ) = \sum_{k = 1}^{Q} a_{k} sin (b_{k} | θ | + c_{k}) 0 \leq θ \leq π / 2,

where Q is a positive integer, and a_k, b_k, c_k are parameters determined by the actual luminous intensity curve. In this paper, Q is set to be 3 since the expression has been accurate enough in most cases. Equation (2) is defined as sum-of-sine luminous intensity pattern (SSLIP) in this paper.

Figure 2 shows the curve fitting results of two types of LEDs. SSLIP and Lambertian fitted curves are compared with the original luminous intensity curve provided in [22, 23]. The parameters determined by curve-fitting are provided in Table 1.

Fig. 2 Curve fitting results of (a) OSRAM’s LUWCN5M type LED and (b) OSRAM’s LEUWS2LN type LED, by SSLIP and Lambertian model.

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Table 1. LED Parameters Determined by Curve Fitting

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Now we know the characteristics of the transmitters and the receiver, we can model the transmission relationships between them. Suppose that all the LED transmitters are identical with the same transmitting power P_t. They transmit the same intensity modulated signal s simultaneously as introduced in [12]. All the signals’ time delay can be neglected because of the short transmission distance. Similar to [7], reflected light is neglected and the line-of-sight links are considered exclusively. Model channel noise by Additive White Gaussian Noise (AWGN) [8], the received signal of the receiver centered at (x, y, z) with a rotational angle α can be written as [12]

Y (x, y, z, α) = γ P_{t} s \sum_{i, j = - N}^{N} h_{i j} (x, y, z, α) + Z,

where Z is the AWGN, γ is the optoelectronic conversion factor, h_ij (x, y, z, α) is the gain of the link between the receiver and the LED in the i^th row and the j^th column.

3. Received power distribution analysis

In this section, the expression of h₀₀(x, y, z, α) is derived under both ceiling and wall-mounted scenarios. Then it is generalized to h_ij (x, y, z, α) and the expression of the received power is derived.

3.1. Calculation of h₀₀(x, y, z, α)

3.1.1. Ceiling-mounted scenario

If the radiant flux (W) falling upon the receiving plane is Φ, then the channel gain of the link between the central LED and the receiver, h₀₀(x, y, z, α), is [12]

h_{00} (x, y, z, α) = \frac{n^{2} T_{s} Φ}{P_{t}},

where n is the refractive index of the concentrator and T_s is the filter gain.

The received radiant flux Φ can be calculated by integrating the radiant intensity (W/sr) over the solid angle (sr) subtended by the transmitter to the receiving plane [24, Chapter 25], [25, Chapter 1]

Φ = \int_{Ω} I (ρ) d ω = \iint I (ρ) sin φ d φ d ψ,

where Ω is the solid angle subtended by the transmitter to the receiving plane, I(ρ) is the radiant intensity in the direction of view angle ρ, dω is the differential solid angle, φ is the meridian angle and ψ the sagittal angle of the integral elemental area on LRA. I(ρ) is given by [12]

I (ρ) = \frac{P_{t} I_{v} (ρ)}{2 π \int_{0}^{π / 2} I_{v} (ρ) sin ρ d ρ},

where I_v (ρ) is the luminous intensity pattern of the LED transmitters as defined in Section 2, and the denominator is a normalization factor. From Eq. (5), we see that the keys to the expression of h₀₀(x, y, z, α) are: the view angle ρ (denoted by ρ_c in the ceiling-mounted case and ρ_w in the wall-mounted case) and the range of integration.

In terms of the view angle, we need to calculate ρ_c of every integral elemental area on the LRA given its φ and ψ. This requires a deep and detailed investigation of the spatial relationship. Thus we extract the central LED and an integral elemental area on the LRA from Fig. 1 (a), and draw Fig. 3. Note that Fig. 3 is in fact an upside-down version of Fig. 1 for a clearer view. Take the central LED as the origin O, and center of the LRA as point A. Draw a plane ϒ orthogonal to the X − Y plane through A. Build the spherical coordinate taking the ray $\vec{OA}$ as the zenith direction Z′. In plane ϒ, construct a line perpendicular to Z′ through A as the direction X′. Draw a plane Π orthogonal to Z′ through X′. For a given integral elemental area on the LRA, its solid angle equals that of its projection on Π. So we consider an integral elemental area whose projection on Π is centered at point B. Its meridian angle φ and sagittal angle ψ are illustrated in Fig. 3.

Fig. 3 Using the spherical coordinate to calculate received power when LED transmitters are placed on the ceiling.

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Construct BC perpendicular to X′ on plane Π. Denote the length of the line segment $\bar{OA}$ as l₁ and that of $\bar{OC}$ as l₂. On plane ϒ, construct AD perpendicular to the X − Y plane with foot D. Now we define ∠BOC ≜ ξ, ∠DOC ≜ μ, ∠DOA ≜ β. In the ceiling-mounted scenario, ρ_c is the view angle between the transmitting light (OB) and LED’s principal direction (Z axis). According to the minimal angle theorem, we have

cos ρ_{c} = cos ξ cos (π / 2 - μ) = cos ξ sin μ,

where cos ξ and sin μ can be calculated as

cos ξ = \frac{l_{2}}{\sqrt{l_{1}^{2} + {(l_{1} tan φ)}^{2}}},

\begin{array}{l} sin μ & = sin (β - ∠ COA) \\ = sin (β - arctan \frac{\bar{CA}}{\bar{OA}}) \\ = sin (β - arctan \frac{l_{1} tan φ cos ψ}{l_{1}}) \\ = sin [β - arctan (tan φ cos ψ)], \end{array}

with β, l₁ and l₂ having the following expressions

\begin{array}{l} β = arccos \frac{\sqrt{x^{2} + y^{2}}}{\sqrt{x^{2} + y^{2} + z^{2}}}, \\ l_{1} = \sqrt{x^{2} + y^{2} + z^{2}}, \\ l_{2} = \sqrt{l_{1}^{2} + {(l_{1} tan φ cos ψ)}^{2}} . \end{array}

So the expression of ρ_c is derived by substituting Eqs. (8) and (9) into Eq. (7) as

ρ_{c} = arccos | sin β cos φ - cos β sin φ cos ψ | .

By now, we have completed the calculation of view angle ρ_c.

Next, consider the integral area in Eq. (5). As mentioned before, solid angle of every point on the LRA can be converted to that of its projection on Π. So we should integrate the incident radiant flux over LRA’s projection on Π. Figure 4(a) shows that the circular receiving area on plane Λ is first rotated on the horizontal plane to plane Γ and then in the axial direction to plane Π. Figure 4(b) illustrates the first step: rotation of a circular area on plane Λ to an ellipse on plane Γ by an angle ||δ| − α|. Here δ = arctan(x/y) is the deviation angle of LRA’s center from the Y axis. Figure 4(c) illustrates the second step: rotation of the ellipse on plane Λ in Fig. 4(b) to that on plane Π by an angle β. It can be seen that the projected area on Π is an ellipse with major semi-axes R cos(|δ| − α) and minor semi-axes R cos β. Since integral on the ellipse is too complex under this situation, for simplicity and without loss of accuracy, we can integrate I(ρ) over the circular area of radius R cos β and then multiply the result by cos(|δ| − α)/ cos β. By now, we have determined the range of integration.

Fig. 4 Projection of the receiving area on plane Π with (a) illustrating the two-step projection, (b) illustrating the projection from plane Λ to plane Γ, and (c) illustrating the projection from plane Γ to plane Π.

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The explicit expression of Φ is now derived as

Φ = \frac{cos (| δ | - α) P_{t} \int_{0}^{2 π} \int_{0}^{U} I (ρ_{c} (φ, ψ)) sin φ d φ d ψ}{2 π cos β \int_{0}^{\frac{π}{2}} I (θ) sin θ d θ},

where

U = arctan \frac{R \sqrt{x^{2} + y^{2}}}{x^{2} + y^{2} + z^{2}} .

h₀₀(x, y, z, α), channel gain of the central LED to the receiver, is then given by Eq. (4) as h₀₀(x, y, z, α) = n²T_sΦ/P_t, where Φ has the representation of Eq. (11).

3.1.2. Wall-mounted scenario

Build the same coordinate as in Fig. 3, the wall-mounted scenario is shown in Fig. 5. This scenario differs from the ceiling-mounted one in that the view angle ρ_w becomes the angle between the transmitting light ray and the Y axis instead of the Z axis. Note that Fig. 5 is an extended version of Fig. 3, thus all the points and lines that have been defined in 3.1.1 have the same meanings here. To calculate ρ_w, we use some auxiliary points and lines: Through point C, draw CE perpendicular to line OD with foot E. Through point B, construct BF perpendicular to the X − Y plane with foot F. The plane determined by points O, B and F is defined as plane Θ. Obviously plane Θ is perpendicular to the X − Y plane, so again we use the minimal angle theorem to get

cos ρ_{w} = cos η cos δ_{1},

where ∠BOF ≜ η and δ₁ is the angle between

\bar{OF}

and the Y axis. To calculate cos η in Eq. (13), note that

cos η = \frac{\bar{OF}}{\bar{OB}},

and that

\bar{OF} = \frac{\bar{EF}}{sin δ_{2}} = \frac{\bar{OB} sin φ sin ψ}{sin δ_{2}},

we have

cos η = \frac{sin φ sin ψ}{sin δ_{2}} .

To calculate cos δ₁ in Eq. (13), use the relationship

cos δ_{1} = cos (| δ | - δ_{2}) = cos δ cos δ_{2} + sin | δ | sin δ_{2},

where δ is the deviation angle of line OE from the Y axis and ∠FOE ≜ δ₂. Substituting Eqs. (14) and (15) into Eq. (13), we have

cos ρ_{w} = sin φ sin ψ sin | δ | + sin φ sin ψ cos δ cot δ_{2},

where

cot δ_{2} = \frac{\bar{OE}}{\bar{EF}} = \frac{\bar{OB} cos φ cos β + \bar{OB} sin φ cos ψ sin β}{\bar{OB} sin φ sin ψ} = \frac{cos φ cos β + sin φ cos ψ sin β}{sin φ sin ψ} .

Thus angle ρ_w has the expression of

ρ_{w} = arccos (sin β sin φ cos ψ cos δ + cos β cos φ cos δ + sin φ sin ψ sin | δ |) .

h₀₀(x, y, z, α) in this scenario has the same expression as given by Eq. (4): h₀₀(x, y, z, α) = n²T_sΦ/P_t, where Φ is given by Eq. (11) with ρ_c (φ, ψ) replaced by ρ_w (φ, ψ).

Fig. 5 Using the spherical coordinate to calculate received power when LED transmitters are placed on the wall.

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3.2. Deduction of received power expression

3.2.1. Ceiling-mounted scenario

The total received power of the LRA centered at (x, y, z) with a rotational angle α can be written as

P_{r} (x, y, z, α) = P_{t} \sum_{i, j = - N}^{N} h_{i j} (x, y, z, α),

Note that the calculation of h_ij (x, y, z, α) is almost the same as that of h₀₀(x, y, z, α) with only a change of the origin from LED(0, 0) to LED(i, j). By shifting the coordinates in Eq. (4), we have

h_{i j} (x, y, z, α) = h_{00} (x - j l, y - i l, z, α) .

So the total received power P_r (x, y, z, α)can be expressed as

P_{r} (x, y, z, α) = P_{t} \sum_{i, j = - N}^{N} h_{00} (x - j l, y - i l, z, α) .

3.2.2. Wall-mounted scenario

In the wall-mounted case, given h₀₀(x, y, z, α), calculation of h_ij (x, y, z, α) requires a shift in the x and z coordinates instead of the x and y coordinates as in the ceiling-mounted case. So h_ij (x, y, z, α) has the form of

h_{i j} (x, y, z, α) = h_{00} (x - j l, y, z - i l, α) .

And the total received power is

P_{r} (x, y, z, α) = P_{t} \sum_{i, j = - N}^{N} h_{00} (x - j l, y, z - i l, α) .

4. Impact of LEDs’ radiation pattern on received power performance

In this section, we use some approximation methods to derive the simplified expressions of received power in both ceiling-mounted and wall-mounted scenarios. Inference can then be made on how the transmitting LEDs’ radiation pattern affects received power performance. Here we consider the single-input single-output case, i.e. N = 0, to exclude the impact of the number of the transmitters.

Assume that α = 0, x = 0, i.e. we analyze the received power along the Y axis where the performance is representative. The conclusions can be easily generalized to the entire plane. The received power under this scenario, P_r (0, y, z, 0), is given by letting N = 0, α = 0, x = 0 in Eqs. (19) and (21) for ceiling and wall-mounted scenarios respectively

P_{r} (0, y, z, 0) = \frac{n^{2} T_{s} P_{t} \int_{0}^{2 π} \int_{0}^{U} I (ρ (φ, ψ)) sin φ d φ d ψ}{2 π cos β \int_{0}^{\frac{π}{2}} I (θ) sin θ d θ},

with ρ given by Eqs. (10) and (16) for ceiling-mounted and wall-mounted scenarios respectively. Note that when integrating Φ around a circle with meridian angle φ, the effect of ψ to the polar angle ρ can be neglected. So by mean value theorem, we have

\int_{0}^{2 π} \int_{0}^{U} I (ρ (φ, ψ)) sin φ d φ d ψ \approx 2 π \int_{0}^{U} I (ρ (φ, 0)) sin φ d φ .

Substituting Eq. (23) into Eq. (22), and applying the expression of I_v (θ) in Eq. (2), the approximated analytical expression of received power is

P_{r} (0, y, z, 0) = \frac{n^{2} T_{s} P_{t} \sum_{k = 1}^{3} [\frac{a_{k} (sin ϕ_{1} - sin ϕ_{2})}{2 (b_{k} - 1)} - \frac{a_{k} (sin ϕ_{3} - sin ϕ_{2})}{2 (b_{k} + 1)}]}{cos β \sum_{k = 1}^{3} \frac{a_{k} [sin c_{k} + b_{k} cos (c_{k} + π b_{k} / 2)]}{1 - b_{k}^{2}}},

where

\begin{array}{l} ϕ_{1} = (b_{k} - 1) U + b_{k} (π / 2 - β) + c_{k}, \\ ϕ_{2} = b_{k} (π / 2 - β) + c_{k}, \\ ϕ_{3} = (b_{k} + 1) U + (π / 2 - β) + c_{k}, \end{array}

for the ceiling-mounted scenario, and

\begin{array}{l} ϕ_{1} = - (b_{k} - 1) U + b_{k} β + c_{k}, \\ ϕ_{2} = b_{k} β + c_{k}, \\ ϕ_{3} = - (b_{k} + 1) U + b_{k} β + c_{k}, \end{array}

for the wall-mounted scenario.

Still, this expression is too complex to arrive at an explicit conclusion about the received power performance. Note that Eq. (24) can be rewritten as

P_{0} (0, y, z, 0) = W \sum_{k = 1}^{3} a_{k} \frac{b_{k} (sin ϕ_{1} - sin ϕ_{3}) + sin ϕ_{1} + sin ϕ_{3} - 2 sin ϕ_{2}}{2 (b_{k}^{2} - 1)},

where

W = \frac{n^{2} T_{s} P_{t}}{cos β \sum_{k = 1}^{3} \frac{a_{k} [sin c_{k} + b_{k} cos (c_{k} + π b_{k} / 2)]}{1 - b_{k}^{2}}},

and that U is generally a small angle less than 0.1rad, and b_k is no more than 5 in most application cases, Eq. (25) can be simplified applying the following approximations: cos(b_k + 1)U ≈ cos(b_k − 1)U ≈ cos b_kU ≈ 1, sin b_kU ≈ b_kU and sinU ≈ U. We finally get the simplified received power expression below which reveals the relationship between the LEDs’ SSLIP (determined by parameters a_k, b_k, and c_k) and the received power

\begin{array}{l} P_{r} (0, y, z, 0) & \approx W \sum_{k = 1}^{3} \frac{a_{k} b_{k}^{2} U^{2} sin (b_{k} β^{*} + c_{k})}{b_{k}^{2} - 1} \\ \approx W U^{2} \sum_{k = 1}^{3} a_{k} sin (b_{k} β^{*} + c_{k}), \end{array}

where β^* = π/2 − β for the ceiling-mounted scenario and β^* = β for the wall-mounted scenario.

From the simplified expression Eq. (26), we find that the received power is mainly influenced by two entries: U² and $\sum_{k = 1}^{3} a_{k} sin (b_{k} β^{*} + c_{k})$ . At a fixed height z₀, U² is a function of y, which can be expressed as $U^{2} (y) = {arctan}^{2} [R y / (y^{2} + z_{0}^{2})]$ by substituting x = 0, z = z₀ into Eq. (12). The entry $\sum_{k = 1}^{3} a_{k} sin (b_{k} β^{*} + c_{k})$ can be derived by replacing θ with β^* in the expression of I_v (θ). We denote it as I_v (β^*). Figure 6 depicts the qualitative curves of U²(y), I_v (β^*) and P_r (received power) versus y under both scenarios. Note that the vertical axis is unlabeled because the values of the curves are scaled for better illustration, and that we are not making analytical analysis but just providing an intuitive perspective of the characteristics of the received power curve. Their effect is analyzed in detail as follows.

Fig. 6 Qualitative analysis of the effect of U²(y) and I_v (β^*) on received power distribution.

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Firstly, as can be seen from Fig. 6, the shape of the received power curve is mainly determined by U²(y). It can be easily verified that U²(y) achieves its optimum at the point $y^{*} = \sqrt{R z_{0} + z_{0}^{2}}$ by letting dU²(y)/dy = 0. This maximal power point is entirely determined by the z₀ coordinate, and the value of y^* increases with z₀. That is to say, if the LEDs’ radiation pattern is already determined, the receiver at a lower position (larger z₀) will achieve the maximal power at a farther distance. This conclusion is the same for both scenarios.

Secondly, the entry I_v (β^*) has different influence under two scenarios. Under the ceiling-mounted scenario, note that since I_v (θ) is generally a decreasing function of θ as shown in Fig. 2, I_v (β^*) is then a decreasing function of β^*. Since $β^{*} = π / 2 - β = π / 2 - arccos (y / \sqrt{y^{2} + z^{2}})$ increases with y, I_v (β^*) is thus a decreasing function of y. The curve of I_v (β^*) versus y under the ceiling-mounted scenario is depicted in Fig. 6. As can be seen from the figure, When U²(y) is multiplied by I_v (β^*), the maximal power point is moved from y^* on the U²(y) curve to $y_{c}^{*}$ on the received power curve to its left. In other words, the function I_v (β^*) narrows the received power curve and reduces the value of y^*. This effect is more obvious when the luminous intensity pattern is sharp, and less obvious when it is flat.

Under the wall-mounted scenario, however, the curve of I_v (β^*) versus y is close to a flipped version of I_v (θ) because $β^{*} = β = arccos (y / \sqrt{y^{2} + z^{2}})$ decreases with y. The curve of I_v (β^*) under the wall-mounted scenario is depicted in Fig. 6. As shown in the figure, multiplying U²(y) by I_v (β^*) in the wall-mounted scenario will move the maximal power point from y^* to $y_{w}^{*}$ on the received power curve to its right. That is, I_v (β^*) keeps U²(y) rising after y^*, flattening the curve and increasing the value of y^*. Again, it is more obvious when the luminous intensity pattern is sharp, and less obvious when it is flat.

It can be concluded that for both ceiling and wall-mounted scenarios, to achieve the biggest received power, the receiver at a lower position should go farther away from the central LED horizontally. When the receiver is at a fixed height under the ceiling-mounted scenario, a sharp radiation pattern leads to a closer optimal receiving position in the horizontal plane. Also, it will lead to a bigger power variation over the entire receiving plane, and vice versa. In contrast, for the wall-mounted scenario, a sharp luminous intensity pattern of the transmitter leads to a farther optimal receiving position in the horizontal plane and a smaller power variation.

5. Numerical results

In this section, we calculate the received power along the Y axis using the analytical expression in Eq. (24) for both ceiling and wall-mounted systems. Assume that the receiver takes three different heights (z = 10cm, 20cm, 30cm), and that there are two types of transmitting LEDs. The radiation patterns of the chosen LED types: LUWCN5M and LEUWS2LN, are given in Fig. 2.

Figure 7 shows the received power in the ceiling-mounted scenario, while relative standard deviations (RSDs) of the received power and optimal receiving distances are listed in Table 2. Here the RSD is defined as the ratio of the standard deviation to the mean, which shows the variation of the received power. It can be seen that LUWCN5M, with a sharp SSLIP, has closer optimal receiving positions and larger RSDs than LEUWS2LN whose curve is relatively flat.

Fig. 7 Received power of (a) LUWCN5M and (b) LEUWS2LN at heights 10cm, 20cm and 30cm in ceiling-mounted scenario.

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Table 2. RSD of the Received Power/Optimal Receiving Distances of Two Types of LED at Three Different Heights under Ceiling-mounted Scenario

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Figure 8 shows the received power in the wall-mounted scenario, with the RSD of the received power and optimal receiving distance listed in Table 3. Results show that under the wall-mounted scenario, LUWCN5M (whose SSLIP is sharper) has farther optimal receiving positions and smaller RSDs than LEUWS2LN (whose SSLIP is flatter). These numerical results agree well with analysis in Section 4.

Fig. 8 Received power of (a) LUWCN5M and (b) LEUWS2LN at heights 10cm, 20cm and 30cm in wall-mounted scenario.

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Table 3. RSD of the Received Power/Optimal Receiving Distances of Two Types of LED at Three Different Heights under Wall-mounted Scenario

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6. Conclusion

In this paper, a generalized model for an indoor VLC system has been proposed. It is widely applicable for transmitters with any kind of radiation pattern, mounted on the ceiling or the wall; and for receivers with non-uniform incident light over its LRA. The expressions of received power from ceiling-mounted and wall-mounted transmitting LEDs have been derived respectively. The expressions have shown dependence on LEDs’ luminous intensity pattern. Further analysis reveals that sharp LED patterns lead to larger power variations and closer optimal receiving locations for the ceiling-mounted case. While in the wall-mounted case, sharp LED patterns lead to smaller power variations and farther optimal receiving locations. There is a good agreement between the numerical results and the analysis. Hence these conclusions would be helpful when choosing LEDs according to the desired power performance metrics.

Appendix

Some major notations used in this paper are listed in Table 4 below.

Table 4. Major notations

View Table | View all tables in this article

Funding

National Science and Technology Major Project (2016ZX03001016-003); National Natural Science Foundation of China (NSFC) (61372106, 61221002, 61701254);Natural Science Foundation of Jiangsu Province (BK20170901); the Funds for International Cooperation and Exchange of the National Natural Science Foundation of China (61720106003); the open research fund of National Mobile Communications Research Laboratory, Southeast University (2017D06); the open research fund of Key Lab of Broadband Wireless Communication and Sensor Network Technology (Nanjing University of Posts and Telecommunications), Ministry of Education (JZNY201706); NUPTSF (NY216009).

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	a ₁	b ₁	c ₁	a ₂	b ₂	c ₂	a ₃	b ₃	c ₃	m
LUWCN5M	1.93	0.36	2.68	0.20	4.67	0.75	0.06	8.33	−0.46	5.81
LEUWS2LN	2.12	1.12	0.71	1.41	1.32	3.42	0	0	0	4.69

	a ₁	b ₁	c ₁	a ₂	b ₂	c ₂	a ₃	b ₃	c ₃	m
LUWCN5M	1.93	0.36	2.68	0.20	4.67	0.75	0.06	8.33	−0.46	5.81
LEUWS2LN	2.12	1.12	0.71	1.41	1.32	3.42	0	0	0	4.69

Impact of LED transmitters’ radiation pattern on received power distribution in a generalized indoor VLC system

Abstract

1. Introduction

2. System model