Análisis para la ubicación

For a diffuse infrared channel employing IM/DD, a source _Si which emits an instantaneous optical power Xi(t), will produce an instantaneous photocurrent Yij(t) at receiver Rj with photodiode responsivityrj, in the presence of an additive, white Gaussian shot noiseNj(t), and can be modelled as the linear baseband system given by [129]

Yij(t) =rjXi(t)∗h(t;Si,Rj) +Nj(t) (3.1)

Whereh(t;_Si,Rj) is the impulse response given by (2.6), and is fixed for a given system config- uration of_Si and Rj.

For a multi spot diffuse configuration, if it is assumed that all I sources _Si emit an identical signal waveform, such that X1(t) =X2(t) =. . . =XI(t), but whose magnitude is individually scaled by a factorai, the instantaneous photocurrent generated at a given receiverYj(t) is simply the summation of (3.1) for all sources

Yj(t) = I

X

i=1

(rjaiXi(t)∗h(t;Si,Rj)) +Nj(t) (3.2)

Furthermore, as only single receiver design is concerned, the photodiode responsivityrj is con- stant for each receiver or receiver location, such that there may exist a set ofI scaling factorsai, that can be applied to theI identical signal waveformsXi(t), that will allow for theJ receivers, to attain the same or very similar instantaneous photocurrents

Y1(t)≈Y2(t)≈. . .≈YJ(t) (3.3)

Knowing that the IM/DD OW channel is linear, (3.2) can be rewritten as Yj(t) =

I

X

i=1

(rjXi(t)∗aih(t;Si,Rj)) +Nj(t) (3.4)

And as the method for determining the impulse response has already been described in chapter 2, and that it can be readily obtained, (3.3) can be solved by solving

I X i=1 aih(t;Si,R1)≈ I X i=1 aih(t;Si,R2)≈. . .≈ I X i=1 aih(t;Si,RJ) (3.5)

By inspection of equations (3.2) to (3.5), it can be seen that, in order to solve the problem, some scaling factors will be <1, and therefore will reduce the total amount of power received, compared to when all sources emit the same power. Ultimately, solving equation (3.3) for one environment will yield a different set of scaling factors than those needed to solve the problem for a different environment. Moreover, whilst all powers in both independent environments will be similar or the same, the relative magnitudes will be different.

An optical wireless system designer needs to achieve a high SNR at the receiver, which is pro- portional to the square of the received power [130], but also maintain system eye safety under IEC825 regulations [25]. These regulations mean that the maximum source power attainable will be determined by factors such as wavelength, exposure duration, pulse characteristics, distance from the eye and image size [131], which are all variable in terms of what the system designer is trying to achieve. In fact, as shown in section 2.3 determining the impulse response requires the use of a 1 W source by definition, and hence is almost certainly outside being classified as ‘eye safe’. Therefore, in any event, the impulse response of the system needs to be scaled using the source power to make it ‘eye safe’. The technique presented here will follow the same lines, such that once the scaling factors are determined for a given environment, they can be normalised and subsequently applied to the I sources of the same power magnitude, where the magnitude is set to be at the maximum acceptable exposure limit. The equality result of (3.3) is then independent of receiver power magnitude. In a second environment, where again the same set of steps is carried out to find the set of normalised scaling factors to solve (3.3), and it is found that the magnitude is different, adjustments can be made, for example, to the pulse characteristics in order to vary the magnitude of the power at the receiver such that both environments now have the same power distribution. This allows the use of the same receiver hardware, but where the system performance is different in the each environment. In many respects this concept draws many parallels to the IEEE 802.11a WiFi physical layer specification, that incorporates multi- rate transmission of up to 54Mbit/s, depending on channel characteristics [132, 133], and recent

work [40, 42, 134, 135, 136] in the IR domain has shown promising avenues for rate-adaptive transmission.

Considering the problem of solving (3.5) more closely: if an environment of dimensions with Xe = 5 m, Ye = 5 m, Ze = 3 m is assumed, similar to configuration A in [87], for a three reflection impulse response (k = 3), the longest time of flight for the radiation to travel is when it undergoes a path reflecting off the opposite corners of the room, where the time t = (4(52 + 52 + 32)0.5_{)/c ≈ 102.4 ns, and,using a sampling time of 0.1 ns would therefore} yield a 1024 sample impulse response train, for every combination ofI sources and J receivers, which will prove to be unwieldy, for two reasons.

Firstly, for all system designs and purposes, there will be an infinite number of environments, source and transmitter system configurations that can be modelled, and even if a solution could be formed for some of these environments, the computation time would be too large to allow the concept to be readily applicable except for system designers with state of the art computational facilities. Secondly, if no ideal solution could be found, it might be possible to produce several partial scaling factor solutions that each manage to equate some elements of (3.5) but not others. Therefore how does the system designer decide which of the solutions is optimal?

It is possible however, to simplify the task, replacing the need to evaluate each element of the impulse response train, with the need to find only the scaling factor solution for the time integral, or the DC value of the frequency response given by (2.15), H(0;_Si,Rj) =

R∞

−∞h(t;Si,Rj)dt,

such that the simplified task is to find a solution to I X i=1 aiH(0;Si,R1)≈ I X i=1 aiH(0;Si,R2)≈. . .≈ I X i=1 aiH(0;Si,RJ) (3.6) Whilst the solution of (3.6) may appear to be simpler than solving (3.5), the DC response only quantifies the amount of power received, not when the power was received. Therefore, as the technique for solving (3.6) is presented, the solution will be fed back into the original system

model to quantify how both system bandwidth and RMS delay spread have been affected. The worst case bandwidth and RMS delay spread are defined to be the smallest and largest values found at any receiver or location within the room, respectively.