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SECTION I

Accurate knowledge of the position of an object or person is important for a multitude of applications, while the global positioning system (GPS) works well in many cases, there are numerous situations, particularly indoors, where GPS is unavailable or the accuracy it provides is not high enough. Despite extensive research on RF-based solutions, the major challenge of strong multipath interference means that there is no accurate, low cost, low power system available [1], [2].

Because of their energy efficiency, white light emitting diodes are rapidly replacing conventional fluorescent and incandescent lights. Unlike conventional light sources, white LEDs designed for illumination can be modulated at frequencies up to 20 MHz [3], [4]. This offers an unprecedented opportunity to create an entirely new type of positioning system based on the light emitted by these LEDs, and has inspired rapidly growing interest in using these LEDs both to transmit high speed data [5]– [6] [7] [8] [9] [10][11], and more recently also for localization [12]– [13] [14] [15] [16] [17][18].

A positioning system based on white lighting LEDs has many advantages. Lights are typically placed at regular intervals on the ceiling, and have line of sight (LOS) to most positions within a room. Multipath transmission presents much less of a challenge in optical systems than RF, because if reflected components are present, they typically have much lower power than the LOS path [8]. As optical signals cannot pass through opaque obstacles, LED based positioning is free of interference from other systems working in different rooms. Installing a white LED based positioning system is potentially almost as “simple as changing a light bulb.” The advantages of positioning with LEDs also include low-cost frontends, simultaneous illumination and localization, and environmental friendliness. Because of the many advantages of LEDs, there are an increasing number of papers published describing a variety of schemes for indoor positioning using LEDs [12]– [13] [14] [15] [16] [17][18]. The ranging techniques used in these papers include received signal strength (RSS) [12]– [13][14], [17], [18], time difference of arrival (TDOA) [15], and phase difference of arrival (PDOA) [16]. Most of the papers published so far have presented either theoretical analyses or simulation results for very specific idealized scenarios in which parameters such as the optical power transmitted and the radiation pattern of each LED are precisely known. For the systems using TDOA or PDOA perfect synchronization has been assumed. For these idealized cases, very low estimation errors have been reported [12], [13], [15].

While the research so far on lighting LED-based systems has shown that accurate localization may be possible, little has been published on the theoretical limits. The determination of this limit will allow optimization of the parameters governing optical positioning systems.

A widely used approach to evaluate the theoretical accuracy in estimation theory is the Cramer–Rao bound (CRB), which is a lower bound on the mean square error of an unbiased estimate [19]– [20][21]. The CRB has been extensively studied for ranging using RF wireless signals. However, as discussed later, these results cannot be directly applied to optical wireless systems using LED transmitters. For RF systems, specific results have been calculated for the time-of-arrival (TOA) of low pass and bandpass signals [20]. These show that the CRB depends on the effective bandwidth of the signal, which in turn depends on the signal waveform. A number of papers have considered the problem of determining the optimum signal waveform for RF systems [22], [23]. In [22], the optimum waveform is found to have energy spectral density which is a Dirac delta function. Unfortunately, this energy density is shown not to correspond to any realizable waveform of finite energy [22]. In the work based on the GPS system [23], a systematic method for designing waveforms of the type used in GPS is presented. However, the resulting waveform is bipolar and therefore not applicable to intensity-modulated direct-detection (IM/DD), and the restriction to GPS type signals is not relevant to indoor localization using LEDs.

While the work on RF provides useful background, these results are not directly applicable to the white LED cases. Modulation and reception of the optical signals in these systems are fundamentally different from the RF case. With an LED transmitter, it is the intensity (not the field) of the electromagnetic light that is modulated. As a result, the transmitted signal must be nonnegative. Because of the properties of photodiodes, the current of the electrical signal after demodulation is proportional to the intensity of the received optical power. This is known as an IM/DD system. Because in the optical domain $x\left(t \right)$ is represented by intensity, but in the electrical domain it is represented by current, the average power of the optical signal is given by $E\left\{{x\left(t \right)} \right\}$ while the average power of the demodulated electrical signal is given by $E\left\{{x^2 \left(t \right)} \right\}$ [6]. In the literature, these two quantities are often referred to as optical power and electrical power, respectively.

In this paper, we analyze TOA-based ranging using LED transmitters. This is particularly novel because there is no published work on CRB for TOA positioning systems using IM/DD signals. We consider the particular case where $x\left(t \right)$ is a dc-biased windowed sinusoid. In contrast to the existing literature on CRB, due to the system we are analyzing, the dc bias is required to meet the nonnegativity constraint, while a window is required to create a signal from which TOA can be calculated, this latter condition also allows the use of the CRB [24]. Research in this new area is likely to evolve in the same way as CRB analysis of RF systems has done, where initially specific waveforms and ideal conditions were considered. We anticipate that, as with CRB calculations for RF, subsequent research will consider more general waveforms and the effect of nuisance parameters [25] such as the effect of errors in synchronization, and the stability of the LEDs. In this paper, we use a nonnegative “smooth” window which has a finite mean-square frequency deviation [21], i.e., $\int_{- \infty }^\infty {f^2 \left\vert {W\left(f \right)} \right\vert ^2 df} < \infty$, where $W\left(f \right)$ is the Fourier transform of the window function $w\left(t \right)$. By Parseval's law, this implies the derivative of the window function is square-integrable. The sinusoidal component is chosen to have a frequency $f_c$ which is much larger than the bandwidth of the window function. Signals of this form have been described in the radar literature, where it has been shown that they have an effective bandwidth that is approximately equal to the frequency of the sinusoid [21]. As a result TOA measurements based on these signals can have a very low CRB.

In Section II, expressions are derived for the CRB in terms of the properties of the signal and the noise at the receiver after optical to electrical conversion. In Section III, a typical indoor scenario using white lighting LEDs for localization is described. In Section IV, numerical values of the CRB are calculated for this indoor scenario using typical parameter values. It shows that CRBs in the order of centimeters or millimeters can be achieved for realistic parameter values. This demonstrates a significant potential for very accurate indoor localization based on white lighting LEDs.

SECTION II

We now derive expressions for the Cramer–Rao bound for ranging estimates based on intensity modulated signals transmitted by LEDs. We assume LOS with no multipath transmission, so the received optical signal is a delayed attenuated version of the transmitted ranging signal $x\left(t \right)$. Thus, the received signal after optical to electrical conversion is given by TeX Source$$r\left(t \right) = \alpha R_p x\left({t - \tau } \right) + n\left(t \right) \eqno{\hbox{(1)}}$$ where $\alpha$ is the attenuation of the optical channel, $\tau$ is the time taken for the light to travel from the transmitter to the receiver, $R_p$ is the responsivity of the photodiode, and $n\left(t \right)$ is the shot noise. We assume that $R_p$ and $\alpha$ are constant over the optical bandwidth of the ranging signal. Due to the properties of an IM/DD system, $\alpha$ is real and positive [6], [8]. We assume that the noise $n\left(t \right)$ is accurately modeled as additive white Gaussian noise (AWGN) with single-sided spectral density $N_0$ which is added in the electrical domain in the receiver. The distance between the transmitter and the receiver can be calculated from the delay using TeX Source$$d = c\tau \eqno{\hbox{(2)}}$$ where $c$ is the speed of light.

We consider the case where $x\left(t \right)$ is a dc-biased windowed sinusoid waveform with duration $T$ and is given by TeX Source$$\eqalignno{x(t) &= w(t)v(t) \cr &= w\left(t \right) + w\left(t \right)\cos \left({2\pi f_c t} \right) &\hbox{(3)}}$$ where TeX Source$$v\left(t \right) = 1 + \cos \left({2\pi f_c t} \right)\eqno{\hbox{(4)}}$$ and $f_c$ is the frequency of the sinusoid. Note that the first term of (3) is a baseband component and the second term is a bandpass component centered at $f_c$.

The accuracy of ranging using TOA can be lower bounded by the CRB, which for $r\left(t \right)$ as defined in (1) is given by [19]– [20][21] TeX Source$$\sqrt {{\rm var} ({\hat d})} \ge {c \over {2\sqrt 2 \pi \alpha R_p \sqrt {{{E_x } \mathord{\left/ {\vphantom {{E_x } {N_0}}} \right. \kern-\nulldelimiterspace} {N_0}}} \beta}} \eqno{\hbox{(5)}}$$ where $\hat d$ is the estimate of the distance (range) from LED to the receiver, $E_x$ is the electrical energy of the signal $x\left(t \right)$, which is defined as $E_x = \int_0^T {x^2 \left(t \right)} dt$, and $\beta$ is the effective bandwidth of the signal $x\left(t \right)$ defined as TeX Source$$\beta ^2 = {{\int_{- \infty }^\infty {f^2 \left\vert {X\left(f \right)} \right\vert ^2 df} \over \int_{- \infty }^\infty {\left\vert {X\left(f \right)} \right\vert ^2 df}}}. \eqno{\hbox{(6)}}$$ From (3) and using the properties of products of signals, this can be expressed as TeX Source$$\beta ^2 = {{\int_{- \infty }^\infty {f^2 \left\vert {V\left(f \right) \otimes W\left(f \right)} \right\vert ^2 df} \over \int_{- \infty }^\infty {\left\vert {V\left(f \right) \otimes W\left(f \right)} \right\vert ^2 df}}} \eqno{\hbox{(7)}}$$ where $X\left(f \right)$ and $V\left(f \right)$ are the Fourier transforms of $x\left(t \right)$ and $v\left(t \right)$, respectively, and “$\otimes$” denotes convolution. As $v\left(t \right)$ is a sinusoidal signal plus a unit dc bias, its Fourier transform is given by $V\left(f \right) = \delta \left(f \right) + {1 \over 2}\left[{\delta \left({f + f_c } \right) + \delta \left({f - f_c } \right)} \right]$, where $\delta \left(\cdot \right)$ is the Dirac delta function having the property: $W\left(f \right) \otimes \delta \left(f \right) = W\left(f \right)$ and $W\left(f \right) \otimes \delta \left({f \pm f_c } \right) = W\left({f \pm f_c } \right)$.

Therefore, (7) can be expressed as TeX Source$$\eqalignno{\beta ^2 &= {{\int_{- \infty }^\infty {f^2 \left\vert {U\left(f \right)} \right\vert ^2 df} \over \int_{- \infty }^\infty {\left\vert {U\left(f \right)} \right\vert ^2 df}}} \cr &= {{\int_{- \infty }^\infty {f^2 U\left(f \right)U^* \left(f \right)df} \over \int_{- \infty }^\infty {U\left(f \right)U^* \left(f \right)df}}} &\hbox{(8)}}$$ where the superscript “∗” denotes complex conjugate and TeX Source$$U\left(f \right) = W\left(f \right) + {{W\left({f - f_c } \right)} \mathord{\left/ {\vphantom {{W\left({f - f_c } \right)} 2}} \right. \kern-\nulldelimiterspace} 2} + {{W\left({f + f_c } \right)} \mathord{\left/ {\vphantom {{W\left({f + f_c } \right)} 2}} \right. \kern-\nulldelimiterspace} 2}.\eqno{\hbox{(9)}}$$ Note that the window function $w\left(t \right)$ is a narrowband signal whose bandwidth is much lower than the frequency $f_c$, so $W\left(f \right)W\left({f \pm f_c } \right) \approx 0$ and $W\left({f + f_c } \right)W\left({f - f_c } \right) \approx 0$. Using this and with a simple change of variable, the numerator and the denominator of (8) can be simplified as TeX Source$$\eqalignno{&\int_{- \infty }^\infty {U\left(f \right)U^* \left(f \right)df} = \int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} \cr &\quad +{1 \over 4}\int_{- \infty }^\infty {\left\vert {W\left({f - f_c } \right)} \right\vert ^2 df} + {1 \over 4}\int_{- \infty }^\infty {\left\vert {W\left({f + f_c } \right)} \right\vert ^2 df} \cr &= \int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} \cr &\quad+{1 \over 4}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} + {1 \over 4}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} \cr &= {3 \over 2}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} &\hbox{(10)}}$$ and TeX Source$$\eqalignno{&\int_{- \infty }^\infty {f^2 U\left(f \right)U^* \left(f \right)df} \cr &= \int_{- \infty }^\infty {f^2 \left\vert {W\left(f \right)} \right\vert ^2 df} + {1 \over 4}\int_{- \infty }^\infty {f^2 \left\vert {W\left({f - f_c } \right)} \right\vert ^2 df} \cr &\quad + {1 \over 4}\int_{- \infty }^\infty {f^2 \left\vert {W\left({f + f_c } \right)} \right\vert ^2 df} \cr &= \int_{- \infty }^\infty {f^2 \left\vert {W\left(f \right)} \right\vert ^2 df} + {1 \over 4}\int_{- \infty }^\infty {\left({f + f_c } \right)^2 \left\vert {W\left(f \right)} \right\vert ^2 df} \cr &\quad + {1 \over 4}\int_{- \infty }^\infty {\left({f - f_c } \right)^2 \left\vert {W\left(f \right)} \right\vert ^2 df} \cr &= {3 \over 2}\int_{- \infty }^\infty {f^2 \left\vert {W\left(f \right)} \right\vert ^2 df} + {{f_c^2 } \over 2}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} &\hbox{(11)}}$$ respectively. Now, again using the fact that $f_c$ is much larger than the bandwidth of $w\left(t \right)$, so that TeX Source$${{f_c^2 } \over 2}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} >\!\! > {3 \over 2}\int_{- \infty }^\infty {f^2 \left\vert {W\left(f \right)} \right\vert ^2 df}.$$ Equation (8) can be further simplified to give TeX Source$$\beta ^2 \approx {{{{f_c^2 } \over 2}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df} \over {3 \over 2}\int_{- \infty }^\infty {\left\vert {W\left(f \right)} \right\vert ^2 df}}} = {1 \over 3}f_c^2. \eqno{\hbox{(12)}}$$ Substituting (12) into (5) gives the CRB for ranging using signals of this form as TeX Source$$\sqrt {{{\rm var}} ({\hat d})} \ge {{\sqrt 3 c \over 2\sqrt 2 \pi \alpha R_P \sqrt {{{E_x } \mathord{\left/ {\vphantom {{E_x } {N_0}}} \right. \kern-\nulldelimiterspace} {N_0}}} f_c}}. \eqno{\hbox{(13)}}$$ From (13), we can observe that the CRB is inversely proportional to the frequency $f_c$.

SECTION III

Fig. 1 shows the indoor scenario that will be analyzed. White LEDs are installed on the ceiling for illumination, and these also transmit the signals that will be used for ranging. The sensor of the device for which the position is to be estimated is assumed to be at a distance $h$ below the ceiling. It receives the signals from the LEDs and estimates its distance from each LED by measuring the TOA of the signal from that LED. The LEDs and the receiver are assumed to be perfectly synchronized to a common clock.

The block diagram of the receiver and estimator is shown in Fig. 2. The optical filter is used to limit the optical spectrum of the light reaching the photodiode. The choice of the bandwidth of the optical filter affects both the power of the wanted signal component and the level of the noise. In optical wireless systems, the main source of noise is shot noise, which is proportional to the total light power reaching the photodiode [5], [6], so in situations where there is a high level of ambient light, it is desirable to choose an optical filter that passes only the optical frequencies used for the ranging signal. The filtered signal is then sent to the photodiode, which generates the electrical signal $r\left(t \right)$ given by (1). $r\left(t \right)$ is then input to the estimator which consists of a matched filter, a square-law device, and a max search device [19]– [20][21]. The TOA estimate is given by the time instant corresponding to the maximum absolute value at the output of the matched filter over the observation interval. From (13), it can be seen that the CRB depends on $f_c, \alpha, E_x, R_p$, and $N_0$. We will now discuss how each of these parameters depends on the geometry of the system, the transmitted signal, and the characteristics of the LEDs and the photodiode.

The CRB is inversely proportional to $f_c$. Thus, for accurate localization, it is desirable to use a frequency as high as possible, however the maximum value of $f_c$ depends in quite subtle ways on the electrical and optical characteristics of both the LED in the transmitter, and the photodiode in the receiver. There are two general types of white lighting LEDs. One uses three LEDs: red, green, and blue, which in combination are perceived as white by the human eye. The second type, which is currently most commonly used, consists of a single blue LED and a phosphor coating, which produces light with a broad spectrum in the green-yellow-red (GYR) visible spectrum. In the calculations, we assume the second form of white LED. The modulation bandwidth of an LED is the maximum frequency with which the intensity of the light can be varied. For phosphor coated LEDs, the blue component can typically be modulated at frequencies up to 20 MHz but the GYR component is limited to 2 MHz [4], [9]. In this paper, we consider the case where $f_c$ is limited to the modulation bandwidth of the GYR component, and the TOA estimate is derived from the entire spectrum of light transmitted by the white LED. (We note that using the approach in [9], it may be possible to improve the estimates by using only the blue component of light and using a higher value of $f_c$). We assume that the electrical bandwidth of the photodiode is greater than $f_c$ and so the transmitter, not the receiver limits the maximum value of $f_c$.

The parameter $\alpha$ is the signal attenuation of the optical channel between the LED and the photodiode in the scenario shown in Fig. 1. We assume that the receiver faces up and the white LEDs point down and are located within the field of view (FoV) of the receiver. Thus, the angle $\phi$ which the receiver makes with a particular LED equals the angle of incidence of the receiver $\varphi$, so $\cos \phi = \cos \varphi = {h \mathord{\left/ {\vphantom {h d}} \right. \kern-\nulldelimiterspace} d}$, where $d$ denotes the distance between this LED and the receiver. The area of the photodiode is $S$. We assume that the optical filter is an ideal bandpass filter which causes no attenuation in the passband and completely blocks any optical frequencies in the stop band. For LOS, given a generalized Lambertian LED with order $m$, the attenuation of the optical channel $\alpha$ can be expressed as [6], [8] TeX Source$$\alpha = {{\left({m + 1} \right)S \over 2\pi d^2}}\cos ^m \phi \cos \varphi = {{\left({m + 1} \right)S \over 2\pi}}\left({{{h^{m + 1} \over d^{m + 3}}}} \right).\eqno{\hbox{(14)}}$$ The parameter $E_x$ denotes the electrical energy of the ranging signal $x\left(t \right)$. It can be calculated as TeX Source$$\eqalignno{E_x &= \int_0^T {x^2 \left(t \right)dt} \cr &= \int_0^T {\left[{w\left(t \right) + w\left(t \right)\cos \left({2\pi f_c t} \right)} \right]^2 dt} \cr &= \int_0^T {w^2 \left(t \right)dt} + \int_0^T {2w^2 \left(t \right)\cos \left({2\pi f_c t} \right)dt} \cr &\quad + \int_0^T {w^2 \left(t \right)\cos ^2 \left({2\pi f_c t} \right)dt}. &\hbox{(15)}}$$ Since the squared waveform varies slowly relative to the cosine function, the net area of the second term is very small compared with the other two terms. Therefore, the second term can be neglected and the electrical energy can be expressed as TeX Source$$\eqalignno{&\int_0^T {w^2 \left(t \right)dt} + \int_0^T {w^2 \left(t \right)\cos ^2 \left({2\pi f_c t} \right)dt} \cr &= \int_0^T {w^2 \left(t \right)dt} + {1 \over 2}\int_0^T {w^2 \left(t \right)dt} \cr &= {3 \over 2}\int_0^T {w^2 \left(t \right)dt}. &\hbox{(16)}}$$ The ranging signal $x\left(t \right)$ is an intensity modulated optical signal whose brightness is determined by its average optical power. Unlike RF signals, the average power of an optical signal is given by its time average, i.e., $\bar P_x = {\lim }_{T \to \infty } {1 \over T}\int_0^T {x\left(t \right)dt} = {\lim }_{T \to \infty } {1 \over T}\int_0^T {w\left(t \right)dt}$. Hence, we can observe that both the electrical energy and the average optical power are affected only by the window function $w\left(t \right)$ as long as $f_c$ is large compared with the bandwidth of $w\left(t \right)$. Therefore, given an $f_c$, the optimum window is the “smooth” window which maximizes the electrical energy $E_x$ subject to a constraint on average optical power. In this paper, we choose a raised cosine signal given by TeX Source$${w\left(t \right) = A\left({1 + \cos \left({{{2\pi } \over T}t} \right)} \right),} \quad {- {T \over 2} \le t \le {T \over 2}}. \eqno{\hbox{(17)}}$$ We choose this window function because it is nonnegative in ${{- T} \mathord{\left/ {\vphantom {{- T} 2}} \right. \kern-\nulldelimiterspace} 2} \le t \le {T \mathord{\left/ {\vphantom {T 2}} \right. \kern-\nulldelimiterspace} 2}$. Therefore, it can be used for IM/DD. This window is also a “smooth” window with a continuous first order derivative in its time duration. Consequently, the CRB is applicable [24]. Thus, the average emitted optical power $\bar P_t$ is $A$ which determines the brightness of the LED and the electrical energy of the ranging signal is given by ${{\left({9A^2 T} \right)} \mathord{\left/ {\vphantom {{\left({9A^2 T} \right)} 4}} \right. \kern-\nulldelimiterspace} 4}$ by (16).

The responsivity $R_p$ is the conversion factor from the optical to electrical domain. It is a function of the wavelength of the light received [6], [9]. For a silicon photodiode, the relationship between the wavelength and the responsivity is relatively linear in the range of visible light, peaking at around 900 nm and falling sharply to zero above the cutoff frequency [6], [9]. The typical value of the responsivity is in the range of 0.2 mA/mW at 400 nm up to 0.6 mA/mW at the peak [9]. Due to this linear relationship, the responsivity is usually taken at the central optical frequency of operation [5]. In this paper, we assume the responsivity of the photodiode is a constant taken at the central optical frequency of visible light.

Shot noise induced by ambient light is the dominant component of the noise at the receiver because most optical wireless systems operate in the presence of high ambient light levels. The single-sided spectral density $N_0$ of the shot noise is given by $N_0 = 2qR_p P_a$, where $q$ denotes the charge on an electron and $P_a$ is the power of ambient light incident on the photodiode [6]. For the case where the ambient light is isotropic, the power of ambient light is given by $P_a = p_n S\Delta \lambda$, where $p_n$ is the background spectral irradiance [6]. $\Delta \lambda$ is the bandwidth of the optical filter in front of the photodiode. Thus, the single-sided spectral density of the noise is given by TeX Source$$N_0 = 2qR_p p_n S\Delta \lambda.\eqno{\hbox{(18)}}$$

SECTION IV

We now present results for the CRB for the positioning system described in the previous section for a range of parameter values. We study a room of height 2.5 m with a Lambertian LED $(m = 1)$ installed on the ceiling. The receiver has an area $S$ with responsivity $R_p$ of 0.4 mA/mW [8]. We assume that the modulation bandwidth of the LED is 2 MHz. A value of $p_n = 5.8 \times 10^{- 6} \hbox{W}/{\rm cm}^2 \cdot {\rm nm}$ is used for background spectral irradiance as this is a value often used in the literature [6], [10]. The bandwidth of the optical filter $\Delta \lambda$ is chosen to be 360 nm (from 380 to 740 nm) as this covers the entire visible spectrum.

We first study the CRB for a range of values of $f_c$. Fig. 3 plots the CRB versus the emitted optical power for a receiver 5 m from the LED. The duration of the window $T$ is 0.01 s and the area $S$ of the receiver is 1 cm^{2} [4], [7]. We vary $f_c$ from 10 kHz up to 2 MHz. The CRB drops with increasing frequencies $f_c$ as indicated by (13). For an average transmitted optical power of only 1 W and $f_c = 2$ MHz the CR bound is less than 3 cm.

Next, we study the impact of the waveform duration. We use the same settings as in Fig. 3 but vary the window duration from 0.001 to 0.1 s. An $f_c$ of 2 MHz is used. In Fig. 4, the CRB is plotted as a function of the transmitted optical power. The CRB falls as the duration of the window $T$ increases. This is because, even though the average optical power is independent of $T$, the electrical energy $E_x$ is proportional to $T$. This graph again shows that low values of CRB are achievable with practical parameter values. For example, for a transmitted optical power of 1 W and a waveform lasting 10 ms, the CRB is less than 3 cm.

Now, we consider the effect of the distance from transmitter to receiver. The parameter settings are the same as in Fig. 4 except that $T$ is fixed at 0.01 s. In Fig. 5, the CRB is plotted versus optical power. As the attenuation $\alpha$ decreases with the increasing distance $d$ the receiver put at a larger distance receives lower optical power and therefore lower electrical energy, resulting in the increase of the CRB. However, even at 6 m, the CRB is around 5 cm for 1 W of transmitted optical power.

Finally, we investigate how the CRB behaves as we vary the area of the receiver photodetecting surface $S$. In Fig. 6, the CRB is plotted versus optical power for various receiver areas and for $T$ = 0.01 s, $f_c$ = 2 MHz, and $d$ = 5 m. It shows that for a given optical power, the CRB drops with increasing receiver area. At 1 W of optical power, the CRB for a receiver of 0.25 cm ^{2} is about 5 cm. This falls to just about 2 cm if the receiver area is increased to 1 cm^{2}.

The dependence on $S$ can be explained by analyzing the dependence of the channel and the noise on the receiver area. From (14) and (18), it can be seen that both $\alpha$ and $N_0$ depend linearly on the area $S$. Rearranging (13) and substituting (14) and (18) gives TeX Source$$\eqalignno{\sqrt {{{\rm var}} ({\hat d})} &\ge {c \over {2\sqrt 2 \pi \alpha R_p \sqrt {{{E_x } \mathord{\left/ {\vphantom {{E_x } {N_0}}} \right. \kern-\nulldelimiterspace} {N_0}}} \beta}} \cr &= {{c\sqrt {N_0 } \over 2\sqrt 2 \pi \alpha R_p \sqrt {E_x } \beta}} \cr &= {{c\sqrt {2qR_p p_n S\Delta \lambda } \over \sqrt 2 \left({m + 1} \right)S\left({{{h^{m + 1} \over d^{m + 3}}}} \right)R_p \sqrt {E_x } \beta}} \cr &= {{c\sqrt {2qR_p p_n \Delta \lambda } \over \sqrt 2 \left({m + 1} \right)\sqrt S \left({{{h^{m + 1} \over d^{m + 3}}}} \right)R_p \sqrt {E_x } \beta}} &\hbox{(19)}}$$ which shows that the CRB is inversely proportional to $\sqrt S$.

SECTION V

We have presented an analysis of the CRB of TOA-based ranging for an indoor positioning system based on IM/DD signals transmitted by white lighting LEDs. A dc-biased windowed sinusoid is used as the ranging function. It is shown that the CRB is inversely proportional to the frequency of the sinusoid. A detailed discussion on the parameters that determine the CRB and their dependence on the geometry of the system, the transmitted signal and the characteristics of the LEDs and the photodiode are also presented. Finally, we show that, in a realistic indoor application, assuming perfect synchronization between the transmitter and receiver, the bounds on position estimation accuracy are typically in the order of millimeters or centimeters depending on the geometry of the room, the frequency and power of the transmitted signal and the properties of the LED and the photoreceiver.

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