Chapter 4

Propagation Model

“Definition of free space propagation: In a homogeneous space comparable to the vacuum, the radiated energy of an isotropic source propagates at light speed and uniformly spread over the surface of a sphere whose radius increases with time.”
Fundamentals of Radio Propagation,
Jacques Deygout, Eyrolles, Paris, 1994

4.1. Introduction

The aim of this chapter is to give a description of a baseband wireless optical link. First, we compare the modeling of the system to a radio system. Then, we describe the disruptive noises and define the electrical signal-to-noise ratio (SNR). Finally, an analysis of the diffuse channel is performed and different calculation models are presented.

4.2. Baseband equivalent model

In this section, the radio propagation principles are first remembered and compared to those of free-space optical ones. Finally, a baseband free-space optical link model is presented.

4.2.1. Radio propagation model

We can define a real signal f (t), located in a narrow frequency band around a carrier frequency f₀. When the frequency band occupied by f(t) is small compared to f₀, the signal can be called narrowband signal or passband signal. It can be written in terms of its complex envelope in the following form:

where f₀ is the carrier frequency, a_f(t) is the complex envelope of the signal, and p(t) and q(t) are quantities, respectively, called phase and quadrature components off (t). They are real values and low-pass-type signals, generated in order to carry digital information by choosing a given type of modulation (OOK, PSK, QAM, etc.).

The development of equation [4.1] can write the narrowband signal f(t) in the following two equivalent forms:

This approach is interesting because it can represent any narrowband or passband signal from its phase and quadrature components [PRO 00].

The cos(2f₀t) and sin(2f₀t) signals are produced on the emission side by a local oscillator at the frequency f₀ (Figure 4.1). At the reception, a similar local oscillator allows us to find the phase and quadrature components from the signal f (t) . It must be in that case phase and frequency synchronized with the received signal carrier.

The transmission channel is defined as the physical medium of propagation. For wireless links, it is, for example, the room itself. By neglecting the channel time variations and ignoring the noise, the received signal is then expressed as:

In this equation (equation [4.3]), c_k, t_k, and _k are, respectively, the amplitude, time delay, and phase change for the order k component of the path (kth path), and N is the total number of received paths.

The phase _k depends on the length of the covered path. For each distance equal to the signal wavelength, the phase change modulo is 2 so that the phase of the received signal can vary significantly and the resulting sum may cause signal loss. This is important in radio propagation.

For example, at the radio frequency of 1 GHz, signal fluctuations are separated in distance by 30 cm. The received signal on a small antenna spatially varies due to multipath and that antenna can be located in an area that causes a fading of the received signal. Thus, for each position of the equipment in a room, it is possible to model the channel as a linear filter.

4.2.2. Model of free-space optical propagation

In the field of free-space optic, it is expensive and difficult to control the phase of an optical carrier; moreover, in case of reflection, there is a partial or a total change of this phase. Furthermore, in reception, it is also difficult to estimate the phase in free-space optic. This is why most current equipments use techniques of light intensity modulation (IM) such as the On/Off Keying (OOK) and the pulse position modulation (PPM). The binary data are transmitted by the instantaneous optical power of the emission device.

This optical emission device is generally a light-emitting diode (LED) or a laser that converts an electrical current proportionally to an optical power.

Similarly, the reception is based on direct detection (DD) in which the photodetector produces a current proportional to the instantaneous received optical power. This type of transmission is commonly known as IM/DD.

Multiple reflections can lead to spatial variations in the amplitude and phase of the received optical signal. In the field of radio, multipath fading is known because the size of the detector is smaller than the wavelength. Although, in the field of free-space optic, multipath fading is present but the size of the detector being very large compared to the wavelength, we get signal integration.

Figure 4.2 [KAH 97] describes an infrared channel using IM/DD. It consists of an infrared transmitter and a significant photoreceptor area. The dimensions of a typical photodetector with an area of 1 mm² are several orders of magnitude compared to the optical wavelength of the signal, which is of the order of 1 ,m. The output signal of the photocell (Y (t)), which acts as an integrator, is an instantaneous spatial average of the received optical signal, including all impairments values.

Nevertheless, in time domain, since the emitted optical power is spread over several optical paths of different lengths, wireless optical communication can suffer from multipath distortion or intersymbol interference (ISI). From a certain data rate, from 20 Mbps to over 1 Gbps, function of the volume of the room, the location and orientation of the transmitter and receiver, this characteristic may be detrimental in the case of a diffuse communication.

For systems using light IM at the emission with DD at the reception (IM/DD), the propagation medium can be replaced by an equivalent baseband model [HAS 94, BAR 94, KAH 97]. The source is assumed to be linear and we can write:

where

– y(t) is the instantaneous current delivered by the photodetector (A).

– R is the photodetector responsivity or the spectral sensitivity (A/W).

– x(t) is the instantaneous optical power of the emitter (W). This optical power is determined with respect to the used modulation.

– h(t) is the channel impulse response invoking the N paths.

– ⊗ is the convolution operator.

– n(t) is the additive white noise. It can be regarded as a Gaussian noise independent of x(t).

From the model described above (equation [4.4]), all the channel coefficients represent the attenuations and their values are real and positive. It is shown that although the information is carried by an optical carrier, the whole system can be expressed as an equivalent baseband model (Figure 4.3).

In a confined space configuration, the channel is assumed static. This approach is generally appropriate to model this channel, since we have a change only when the transmitter or the receiver is moving, even when objects or people in the room move or are moved [KAH 97].

So the input–output relation of the channel is a convolution, with the particularity that the function x(t) represents an instantaneous power, so it is not negative (x(t) ≥ 0).

From this property, the temporal average of the signal is a positive quantity; it takes the form of the transmitted average optical power:

so that the received average optical power is proportional to the temporal integral of x(t) and is expressed as:

where H(0) represents the direct current (DC) channel gain (frequency equal to zero) or the useful average optical power attenuation. Its expression is:

It should be noted that from equation [4.7], the received average optical power is proportional to the average value of z(t), unlike the electricity domain where the average power of the useful signal to the output of a photodiode is [WOL 03 and LAL 07]:

The average of the signal x(t) is the average of the transmitted optical power Pt and the value is its variance.

Unlike most radio systems, the average of the useful signal is not equal to zero and it adds to the total electric transmitted power.

The last parameter is the noise n(t) that can be divided into three main sources:

– The thermal noise due to thermal effects in the receiver.

– The periodic noise, or optical origin electrical disturbance, is mainly the result of the variation of light due to the design method of the lamp using an electronic switching power supply. This produces a specific periodic signal with, most of the time, a fundamental frequency of 44 kHz and significant harmonics up to several hundred kilohertz. All ambient artificial light sources are modulated at low frequency (mains frequency: 50 or 60 Hz) or high frequency (in the case of fluorescent lamps).

– The shot noise or optical origin disturbances that are the result of photon arrivals from the useful signal and ambient light sources such as the lamp lights and the Sun.

In the receiving wireless optical device, the shot noise usually dominates when the device is well made [KAH 97, TAN 02, ALQ 03a]. It is defined by:

The B term is the modulation bandwidth; it can be defined from a percentage of power of the first zero frequency or from the Nyquist criterion. In the case of a binary modulation such as OOK modulation, the bandwidth B is equal to the bit rate.

The term is the single-sided power spectral density. Theoretically, the shot noise comes only from the useful incident beam and from ambient light entering the detector active area (photodiode). However, this noise from ambient light is often very high compared to the useful signal, despite the use of a suitable optical filter in the receiver. Therefore, it is possible to neglect the shot noise generated by a useful signal.

In addition, since most disturbing ambient light has a mainly continuous component, the integration of a dynamic shift or a suitable filter in the receiving device offers a first simple and inexpensive solution.

Finally, the induced shot noise from the ambient light can be modeled as a Gaussian noise [KAH 97, ALQ 03a]:

where

– q = 1.6 × 10⁻¹⁹ is the electron charge.

– I_inc is the current produced by the incident optical power.

– I_d is the dark current of the photoreceptor.

– I_u is the useful signal current.

– I_bg is the background light current.

– P_bg is the optical power of the ambient light.

Four main types of ambient light can interfere with wireless optical communication in domestic and professional environments:

– incandescent lamps;

– fluorescent lamps;

– devices with LED; and

– the Sun.

Several experiments were carried out. A synthetic example is presented (Figure 4.4), a representation of the disruptive effect in function of the wavelength [BOU 07], at 1 m distance from the source (except the Sun).

It appears that the Sun in line of sight is the most important disruptor, but its influence may be significant even for non-line of sight. Then, in decreasing order are the incandescent lamp (which are set to disappear in the near future), fluorescent lamps, and LEDs.

As part of the disruptor management, an indication of the levels can capture the preferred spectral ranges for wireless optical communication device.

4.2.3. The signal-to-noise ratio

To study the system performances and compare them to other systems based on different technologies of digital communications, it is possible to define a parameter that is the SNR, ratio of the useful electric power to the noise power:

Since the received useful signal current is directly proportional to the received optical power, the electric SNR commonly used in many studies in the wireless optical field [KAH 97, TAN 02, CAR 02a] only takes into account the square of the average received signal as an input signal and not the average of the square of the signal.

This expression considers only the variable part of the modulated signal and removes the DC component related to the fact that the received signal does not have a null average.

In the case of line of sight propagation, theoretically, the only noise taken into account is the shot noise from ambient light. It is equivalent to an additive white Gaussian noise.

For example, a typical value of the spectral radiance of ambient light for wavelengths approximately 850 nm, during a sunny day and inside a house, is 0.04 μW/(mm².sr.nm) [WOL 09]. The noise value will therefore depend strongly on the effective area of the optical receiver, the optical filtering, and the field of view (FOV) or solid angle.

In fact, it should also take into account the unwanted effects of electronic circuits, especially in the choice and implementation of amplifiers, preamplifiers, and (or) photodiode(s).

Nevertheless, in wireless optical transmission scheme, there is an important difference with radio systems. The electrical SNR depends on the square of the average transmitted power Pt and not only on Pt because it is evaluated after the quadratic detection (optical/electrical conversion). So when it is attenuated by 3 dB, the SNR is reduced by 6 dB, and conversely, when the optical power is increased by 3 dB, the SNR reaches 6 dB gain. This parameter is important in finding solution to increase the gain in a link budget.

4.3. Diffuse propagation link budget in a confined environment

In this section, we present ISI and optical reflection models on materials before discussing the various models proposed in diffuse propagation.

4.3.1. Intersymbol interference

In diffuse propagation, the receiver receives not only the optical beam of the direct path, but also the reflected path beams on different surfaces of the room. For example, in OOK modulation case, if the frequency of the symbols becomes very important, we will have an overlap of values that will produce an interference phenomenon; this phenomenon is called ISI.

For reliable communication (i.e. with an ISI close to zero), it is necessary that the entire system takes into account the impulse response requirements.

An example is given in Figure 4.5 with three different paths:

– the direct path (factor h = 0), “direct”;

– the one reflection path (factor h = 1), “1 reflection”;

– the two reflections paths (factor h = 2), “2 reflections”.

In this example and when the symbols’ frequency becomes very important, we get an overlap of values as shown in Figure 4.6. The impulse response of the first symbol (solid line) includes the direct path (H0), the path with one reflection (H1), and a third path with two reflections (H2).

The impulse responses of higher order (n>3) are generally considered as negligible. The impulse response of the first symbol can disrupt the second (dotted line) and the third (line and two points) symbols because despite the fact that it is a lower amplitude signal, its value may be sufficient to degrade the quality of transmission.

Figure 4.7 gives us an example of the impulse response of the incident path and the different reflections for two different rooms configurations (2 m × 2 m — line and 4 m × 4 m — dotted line); the respective distances between transmitter and receiver are the same.

One of the first impulse response models was proposed by Gfeller [GFE 79]:

Figure 4.8 shows the shape of the impulse response h(t) for two values of the FOV angle (1 = /6 or 30° (triangle) and 2 = /4 or 45° (square), where t₀ = 2.10⁻⁹ s.

The temporal width of the impulse response is proportional to FOV angle. This behavior is logical because when this angle (which characterizes the receiver) is decreasing, there are fewer reflected rays on the walls of the room or on the furniture that can be received (which arrive at the receiver under an angle with its axis below the receiving system angle (FOV)). When FOV increases, wide line of sight and diffuse propagation case, the reflected paths number and the reflected paths power arriving at the receiver increase proportionally. The response delay is then greater and it is possible to obtain the symbol T+1 in correspondence with reflections of the symbol T. In this case, then there is occurrence of interference between symbols; the result is a performance degradation.

This degradation appears to occur at a data rate between 50 Mbps [OBR 04] and 260 Mbps [GFE 79] depending on the room configuration and the respective positions of the transmitter and the receiver. However, solutions, such as dynamic equalization [KAH 95] and OFDM [ELT 09], from the radio channel or the optical fiber could help overcome this problem.

The next step is the reflection modeling of an optical beam and the presentation of different models to characterize the impulse response including the most accurate and the most complete elements mentioned above. Several models exist and they are mentioned in the following sections.

4.3.2. Reflection models

The phenomenon of reflection occurs when a wave comes against a surface with large dimensions compared to the wavelength (floor, wall, ceiling, furniture, etc.).

The reflection characteristics of any surface depend on several factors:

– the material surface (smooth or rough);

– the wavelength of the incident radiation; and

– the angle of incidence.

4.3.2.1. Specular reflection

The roughness of the surface of a structure compared to the wavelength of the incident signal constitutes an important parameter of the shape of the reflection pattern.

A smooth surface reflects the incident radiation in one direction like a mirror and Descartes’ law is applied, the reflection is called specular reflection.

But, unlike a radio channel for which the reflections on the surfaces are mostly specular type, in optics field, the dominant reflections are diffuse type.

4.3.2.2. Diffuse reflection

In the case of a rough surface, the incident radiation will be reflected in all directions. A surface is considered as rough, according to the Rayleigh criterion, if the following relationship is satisfied:

where

- ς: maximum height of surface irregularities;

- : wavelength of incident radiation;

- _i: angle of incidence.

For optical radiation of 1,550, 850, and 550 nm wavelengths, assuming normal incidence, a surface is rough if the maximum height of the surface irregularities ç is, respectively, greater than 0.19, 0.11, and 0.07 µ.

These values mean that most of the surfaces found inside buildings may be considered as rough for optical radiation. In this case, the reflection pattern presents a high diffuse component, the reflected wave is scattered in multiple directions, and this reflection is called diffuse reflection.

To integrate this parameter, two models are commonly used to represent the reflection of optical radiation: Lambert’s and Phong’s models.

4.3.2.3. Lambert’s model

Most of the surfaces are very irregular and reflect the optical radiation in all directions, independently of the incident radiation. Such surfaces are known as diffuse and can be approximated using Lambert’s model. Figure 4.9 represents an experimental reflection pattern for an incident angle of 45° of a raw cement surface (before and after white painting) and their approximations according to Lambert’s model).

This model is very simple and easy to implement using computational software, and the relation is:

where

– : surface reflection coefficient;

– R_i: incident optical power;

– ₀: observation angle.

Table 4.1 provides example of reflection coefficient values of an infrared beam for various material surfaces [YAN 00].

Table 4.1. Reflection coefficient

Material	Reflection coefficient
Painted wall	0.184
Painted paper	0.184
Wooden floor	0.128
Brown wood shelf	0.0884
Clear glass	0.0625
White ceramic	0.0517
Plastic	0.1018

For example, if we consider a painted wall ( = 0.184), a receiver located 20° from the normal to the wall, and an incident power equal to 10 mW, the reflected power value (R ()) will be:

4.3.2.4. Phong’s model

The reflection pattern of several rough surfaces is well represented by Lambert’s model, but it is less valid under certain materials incidences on which specular reflection has a significant component. Phong’s model considers the reflection pattern as the sum of two components: the diffuse and specular component. The percentage of each component mainly depends on the surface characteristics and is a parameter of the model. The diffuse component is modeled by Lambert’s model and is incorporated into Phong’s model. The specular component is modeled by a function that now depends on the angle of incidence _i and the observation angle (or the reflection angle) _i. The Phong’s model is described by the relation:

where

– : surface reflection coefficient;

– R_i: incident optical power;

– r_d: percentage of incident signal that is diffusely reflected (it is a value ranging between 0 and 1);

– m: parameter that controls the directivity of the specular component of the reflection;

– _i: incidence angle;

– ₀: observation angle.

NOTE:— Lambert’s model appears as a special case of the Phong’s model taking r_d = 1.

Many reflection pattern measurements are found in the literature. In Figure 4.10, which represents the reflection patterns of a varnished wood surface for angles of incidence equal to 0° and 45° and their approximation by Lambert’s and Phong’s methods, authors [LOM 98] present reflection graphs of the different materials in polar diagram form.

Other reflection patterns exist in the literature and result from experimental measurements carried out by many authors [NIC 77, PHO 75, YAN 00]. The Phong’s model equations and reflection coefficients are given in Table 4.2 for different types of materials: painted paper, wood, etc.

Table 4.2. Reflection coefficient of different materials

Materials	Phong’s equation
Painted wall	P(0) = cos(0)
Paint paper	P(0) = cos(0)
Glass	P(i, 0) = 0.001cos(0)+(1 − 0.001) cos13(0 − i)
White ceramic	P(i, 0) = 0.06 cos(0)+(1 − 0.06) cos1(0 − i)
Varnished wood	P(i, 0) = 0.3cos(0)+(1 − 0.3) cos97(0 − i)
Formica	P(i, 0) = 0.14 cos(0)+(1 − 0.14) cos112(0 − i)
Plastic	P(i, 0) = 0.55 cos(0)+(1 − 0.55) cos3(0 − i)

4.3.3. Modeling

As in all telecommunications systems, an analysis of the propagation channel is essential. As part of an optical communication in a confined space, the channel is characterized by the room and its constituent elements. Several methods are proposed to simulate and evaluate the behavior of the diffuse optical signal and, in particular, the impulse response.

To calculate the impulse response h(t) of the channel, taking into account the diffuse component, it is necessary to resort to numerical simulations to take account of the light beams that reach the receiving device from multiple reflections on the different surfaces of the room and different furniture.

In 1979, Gfeller and Bapst [GFE 79] proposed a first single reflection propagation model. This model estimates the impairment errors introduced by using the reflection Lambert’s model. The reflecting surface is divided into a large set of small elementary reflective areas called reflector elements. The spatial and temporal distributions of the transmitted signal are evaluated for each of these elements. Each of these elements is then regarded as a point source that emits the collected signal affected according to its reflection coefficient. The reflection pattern of each element is represented by Lambert’s model described above. The received signal is the sum of the signals that arrive at the receiver after reflection on the different elements. Owing to the difference in propagation length of the various optical paths, the received signal is dispersed over time. In this model, only the effects of the reflected model on the path loss without considering the signal propagation delays are evaluated. The accuracy of the model increases with the reduction in the size of the reflective area of each element. The principle remains the same for the following models but they introduce in the simulation several consecutive reflections.

In 1993, Barry [BAR 93] proposed a recursive simulation model for multiple reflections. This model is limited to an empty rectangular room and is based on the reflection of Lambert’s model. In 1997, Perez-Jimenez et al. presented a statistical model to estimate the impulse response [PER 97], while Lopez-Hernandez proposed DUSTIN algorithm [LOP 97]. In 1998, the Monte Carlo model, integrating the production of random radiation, was proposed for the simulation of a multipath wireless optical system in confined space [PER 98a, PER 98b]. Carruthers and Kannan have proposed a new iterative model based on an estimate of the impulse response [CAR 02b].

A comparison [SIV 03] of these techniques has shown that for a similar processing time, the Monte Carlo model seems more accurate than the DUSTIN algorithm. The Monte Carlo model is able to generate a simulation considering up to 40 reflections, which offers better accuracy. In these methods, the simulation is performed in three steps: (1) the ray generation, (2) the walls characterization, and (3) the photodiode response determination. The rays are randomly generated from the transmitting device and according to its transmission pattern. The impact points of each ray are then calculated. These impact points are either on the receiver effective area or on one wall of the room.

If the impact point is on a wall, then this point is considered as a new secondary light source and a new ray is generated based on the same method as before. If the point of impact reaches the receiver, then the incident power is saved as a vector with an appropriate position, which is calculated by estimating the total delay of the optical path between the transmitter and the receiver. In the modified Monte Carlo model [PER 98b], the contribution of the direct path between the source (wall or transmitter) and the receiver is calculated each time (Figure 4.11) and this greatly speeds up the processing time.

Nettle’s model is slightly different from previous models; this model [NET 99] has its origin in the process of iterative calculation of a diffuse surface [GOR 84]. Radiosity is a lighting technique used in some three-dimensional (3D) computer models. This is called global illumination because the illumination of each elementary surface can be calculated separately from the others. The system modeling all the lighting or energy transfers can be globally solved. Radiosity uses the light radiative transfer physical formulas between the different elementary diffuse surfaces that compose a 3D scene.

For example, a simple scene can be modeled using a polygonal mesh, e.g. a cubic empty room divided into elementary surface elements. These basic elements used by the radiosity are patches (plane surface elements). Each of the faces of the cube and the room is a patch (Figure 4.12).

Each of these patches receives energy from the other sides, absorbs a certain amount (depending on the material properties of the patch), and returns the other part to the other patches. The energy transmitted from a patch A to a patch B is a function of the following parameters:

– the respective value of the normal to patches;

– a vector representing the direction of the center of the emitting patch toward the center of the receiving patch;

– the average distance between two patches;

– the respective surface of the patches;

– the transmitted optical power; and

– the level of visibility between two patches (obstacle).

Some patches are defined as emission optical source. For each transmitter, we determine the receiving patches.

The radiosity process is recursive and illuminated patches at the current iteration will be emission optical sources for the next iteration. Specifically, the radiosity emitted by a patch i, (Bi), is equal to the self-emitted energy (Ei) added to the addition of all radiosity received from other patches j (Bj) weighted by a reflection factor depending on the material (Ri). The energy received by the plan patch i from patch j is equal to the radiosity emitted by j multiplied by a form factor (Fi-j), depending on the relative orientation of i and j, their distance, and the presence of other objects (obstacles) between the two patches.

It should be noted that the influence of the iterative component of order 4 and higher is negligible, and some authors consider that the component of order 3 or order 2 can also be considered negligible.

This computing model allows us the treatment of a 3D scene including objects and application examples provided in Chapters 6 and 11. The process is as follows:

– After creating a two-dimensional (2D) room, it is associated with a 3D scene generated by what is called a 3D engine (Ogre open-source module, for example);

– While processing, room data are transferred to the Nettle propagation module that get back the tree and the structure of 3D model. This operation is achieved by converting data into 2D triangular elementary objects and identified by their position in three dimensions.

– While calculating, for each emission source and each iteration, the elementary surfaces can be lit according to the room geometry and its constituent determined elements.

The result is done by:

– the transfer of the 2D propagation calculation process results to the 3D engine;

– the generation of a new 3D image file including the line of sight and diffuse optical propagation results.