A Channel Model for Wireless Infrared Communication Volker Pohl, Volker Jungnickel and Clemens von Helmolt Heinrich-Hertz-Institut für Nachrichtentechnik Berlin GmbH, Einsteinufer 37, 10587 Berlin, Germany, e-mail:
[email protected] ABSTRACT An simple analytic model for the light propagation in indoor environments is presented. Ray tracing simulations confirm that the model is applicable to the wireless infrared communication channel in rooms. INTRODUCTION Wireless infrared (IR) communication has attracted much attention during the last years since it is an alternative to radio transmission for high-speed indoor communication. IR systems occupy no radio frequency (RF) spectrum and they can be used where electromagnetic interference is critically (clinics, production halls, air planes). But of course, there are some peculiarities like the strong background light level, the transmitter power which is limited due to eye safety and the relatively poor sensitivity of the IR receivers in which direct detection of the intensity modulated light is performed [1, 2]. The basic knowledge on the indoor IR transmission channel is not yet complete. Unlike at RF transmission, the light is reflected diffusely at most walls in a room, and, already for the first reflection, there is a continuum of possible paths between the transmitter (Tx) and the receiver (Rx). The light may suffer many of these reflections, and the channel estimation is not trivial. Gfeller and Babst [1] estimated the delay spread for a single diffuse reflection at a wall in terms of a bandwidth times cell diameter product of 260 Mbit*m/s. Barry et al. [3] used a numerical approach to simulate the IR impulse response in rooms. A recursive method was introduced that was later on used in virtually all simulation work on the IR channel [4-8]. But because of numerical complexity, not more than 5 reflections [4] were considered. Hence, these models overestimate both the path loss and the channel bandwidth, systematically. Wireless channels do depend on the Tx and Rx configuration and on the actual situation in the environment. Consequently, the complexity of the simulation was increased from [3] to [8] to include, for instance, furniture or windows. On the other hand, an engineer trying to check the basic parameters of the IR channel rather needs some quick rule of thumb than specific simulations. But the former is not available at the moment. For this reason, an analytical model for the IR channel is needed, which is simple but includes all reflections. In this paper it is shown that the essential properties of the diffuse light propagation in rooms can be described, already, with a simple analytic formula which is derived
from the integrating sphere. It is also shown how the sphere formula can be scaled to a specific room. The model is confirmed by statistical ray tracing simulations which take all diffuse reflections into account. THE RAY TRACING TECHNIQUE The simulation method used here has already been employed by the authors to describe the diffuse light propagation in integrating spheres [9, 10]. The path of individual photons from the transmitter (Tx) through the room to the receiver (Rx) is tracked in detail (see Fig. 1).
diffuse reflections
Tx
LOS signal
Rx Fig. 1 Principle of ray tracing in a room. (LOS = line of sight)
At the Tx, a random start direction is created according to a generalised Lambertian law P (m + 1) I (θ ) = cos m θ (1) 2π (see Appendix A), where I is the radiant intensity, P is the total optical power, m is the Lambert exponent, and θ is the angle to the direction of maximum power. When the photon reaches the room surface (wall, ceiling, floor etc.), either it is absorbed or a random new direction is created according to a Lambertian distribution (set m = 1). A large number of photons N0 is tracked (up to 1010), and, occasionally, a photon reaches the Rx. The optical path loss a is calculated from the number of received photons NRx N a = −10 log Rx . (2) N0 It was checked that the sum of NRx and all absorbed photons is equal to N0. No photon is lost during the entire simulation and the number of reflections is not limited. During the flight of each photon, the total time of flight is accumulated. A histogram with a resolution of 167 ps is created for the overall time of flight distribution h(t) from the Tx to the Rx. The channel transfer function H(jf) is obtained from the histogram by
numerical Fourier transform. The cut-off frequency f½ is determined at |H(jf½)| = √0.5⋅|H(0)|1.
light from far source
∆t Θ d
photodiode
BASIC SIGNALS IN THE IR CHANNEL
The impulse response corresponding to the data in Fig. 3 is shown in Fig 4. It consists of a discrete, Dirac-like pulse due to the LOS signal followed by a continuous signal due to the diffuse reflections. Both components are well separated from each other so that it is useful to distinguish them in the following investigations. The channel model used from now on is thus decomposed into the two components hLOS(t) and hdiff(t) which are due to the LOS signal and the diffuse reflections, respectively (see insert in Fig. 4).
Fig. 2 The time resolution is limited by the detector size
magnitude response [dB]
-110 -112 -114 -116 -118
hLOS (t)
108 h diff (t) 7
10
diffuse reflections
6
10
105
0
20
40
1
10 100 frequency [MHz]
1000
Fig. 3 Comparison with previous results [3] (open circles).
|H(jf)| is related to the received optical power, and the electrical power after the photodiode scales with |H(jf)|2.
80
100
Fig. 4 Impulse response corresponding to the full line in Fig.3. Inset: The channel model used in this work.
THE SPHERE MODEL
The time response of the diffuse signal is very similar to that of an integrating sphere. Some more or less pronounced peaks with varying shape are observed which are related to the first reflections in the room. The time response then becomes smooth and a nearly perfect exponential decay is observed. The exponential decay results from the superposition of the higher order reflections. Due to the exponential decay, an IS in an optical transmission line is equivalent with a first order low-pass in the electrical domain [10]. Because the corresponding transfer function of the IS is a simple low pass, it shall now be adapted to the diffuse IR channel to obtain an approximate channel model. The transfer function reads 1 η H diff (jf) = with f 0 = (3) f 2πτ 1+ j f0 where the optical power efficiency
(4)
is related to the path loss by a = -10·log(η). Note that the leading term in (4) is the ratio of the detector area ARx and to the total area of the room surface Aroom. The average reflectivity ~ ρ is given by ρ~ =
1
60 time [ns]
A ρ~ η = Rx − 1 ρ~ Aroom
-120 -122 0,1
line-of-sight signal
109
number of photons
The simulation becomes faster when a large detector is used. On the other hand, the time resolution is limited by the detector diameter d (Fig. 2). When a light pulse from a far source is incident, the detector signal is broadened by ∆t. In a diffuse IR link, the light may arrive from all directions. When the figure ∆t is averaged with respect to the cosine-like detector characteristic, an overall time resolution =2d/3c is obtained where c is the speed of light. According to the sampling theorem, the transfer function is valid below fmax=3c/4d. Consequently, a cutoff near 200 MHz was observed in early simulation runs with d = 1 m (see dashed line in Fig. 3). In order to obtain reliable results below 1 GHz, a smaller detector (d = 10 cm) was generally used. The simulation results were carefully checked against previous work. A comparison with the data of Barry et al. [3] is shown in Fig. 3. The open circles mark their results. The same scenario (configuration A, Table 1 in Ref. [3]) was investigated with the ray tracing technique. For comparison, the magnitude data in Fig. 3 were normalised to the same Rx area. In general, the ray tracing technique creates similar results like Barrys approach. The results agree very well with each other, when the simulation is stopped after 3 reflections, like in [3] (dotted line). The complete ray tracing (full line) creates better results, especially at low frequencies, since the higher order reflections are properly taken into account.
1 Aroom
N
∑ Ai ρ i i =1
(5)
where N is the number of different areas Ai of the room surface with the individual reflectivities ρi. The cut-off frequency f0 in (3) is calculated using the decay time τ =− ~ (6) ln ρ where is the average time between two reflections in the actual room. A simple procedure to obtain for rectangular rooms is described in Appendix B. RESULTS ON THE DIFFUSE SIGNAL
In Fig. 5 it is shown how the properties of the diffuse IR ρ . For comparison, channel without LOS depend on ~ plots according to (3) are also shown (dashed lines). A rectangular room with uniform reflectivity ρ is investigated2 (see Fig. 6 for the scenario). Note the logarithmic axes in Fig. 5 which were scaled for better comparison with (3). The abscissa is multiplied with given in the figure while the ordinate is divided by ARx/Aroom.
bandwidth f0 in (3) g ⋅ f0 =
(6)
0
Fig. 5 do not really merge together at high frequencies. Eq. (4) predicts a constant power distribution in the room. This is a basic property of the IS, which is approximately applicable to the diffuse IR signal in rooms, too. But, of course, there are spatial variations when the Rx is moved (typically less than 5 dBopt).
ρ = 0.9
1
ρ = 0.75
85
ρ = 0.5
80
z
ρ = 0.3 0,1
= 8.5 ns 0,01
0,01
0,1
1
75
In general, the simulation results (full lines) can be well fitted using (3) when the same parameters are used like in the simulation. Obviously, formula (3) provides a good quantitative approximation both for the path loss and for the frequency response of the diffuse IR signal. As for the IS, significant deviations are observed only at high frequencies. z
5m
3m y
Tx x
1m
Rx
2.4 m
2m
70
y
35 m
x
Rx
Tx
z
z 45°
65
m=7 25° y
y x
sphere model
60 0
Fig. 5 Frequency response of the diffuse signal for different values of the wall reflectivity. The dashed lines are theoretical curves according to Eq. (3).
Rx
Tx 1m
d=3 mm
x
Tx position
-1
frequency [ ]
5
10
15
20
25
Rx position [m]
Fig. 7 Path loss variation in a long corridor.
The variations become extraordinary large in a corridor scenario recently introduced by Gfeller et al. [11] which was again investigated with the ray tracing technique. As in [11], the LOS is blocked. The path loss as a function of the Rx position is plotted in Fig. 7. Along the corridor, a spatial variation of more than 20 dB is observed consistent with the measurements in [11]. Note that the sphere model still gives a reasonable estimate for the bandwidth in this environment. This is shown in Fig. 8 where |Hdiff| is plotted for two Rx positions in the room. While a cut-off frequency of f½=12.9 MHz is found at 10 m, it is reduced to 8.8 MHz at 15 m. The estimate according to (3) is 11.1 MHz. SUPERPOSITION WITH THE LOS
4m
4m
Fig. 6 The scenario investigated in Fig. 5.
2
ρ~ ln ρ~ ρ~ ≈ ~ 1− ρ < t > < t >
~ ) 〈〈 1. Note a nearly constant product is obtained for (1-ρ ρ so that the dashed curves in that g·f still depends on ~
path loss [dBopt]
|H(jf)| [ARx / Aroom]
10
When the reflectivity is increased, more optical power is received. At ρ = 0.9, the power is 10 times larger than for ρ = 0.5. This is explained by the increased light accumulation in the room3. On the other hand, the cut-off frequency f½ is 1.9 MHz for ρ = 0.9, but it is 13 MHz for ρ = 0.5. Apparently, the variations of the received power and of the available bandwidth are almost inverse to each other. This can also be shown analytically. When ~ /(1-ρ ~ ) in (4) is multiplied with the ρ the gain factor g=ρ ρ
No significant changes are observed, when the walls have ρ. individual reflectivities with the same ~
In practical IR systems, the diffuse signal and the LOS component may be present, simultaneously. The two signals add at the Rx, and noticeable superposition effects are observed. This is investigated in a scenario 3 This is a familiar phenomenon: Highly reflective walls considerably brighten a room.
θ = 90° ALOS = 0
|H(jf)| [ARx/Aroom]
10
|Hdiff(jf)| [ARx/Aroom]
Rx at 10 m
1
sphere model Rx at 15 m
1
0,1
= 9.6 ns
0,01
0,1
0
1
2
3
4
5
-1
frequency [ ] = 7.1 ns 0,01
0,1 -1 frequency [ ]
1
Fig. 8 Frequency response at two positions in the corridor. The dotted line indicates the theoretical curve.
depicted in Fig. 9. The distance between the Tx and the Rx is held fixed, and the Rx is tilted by an angle θ to modify the ratio between the two signal amplitudes Adiff and ALOS. A uniform reflectivity of ρ = 0.6 is assumed.
w = 5m Tx
θ
1m
Rx
m=3 1m
l=6m
5m
y
1
0,1
= 9.6 ns
0,01 0
1
2
3
4
5
-1
frequency [ ]
|H(jf)| [ARx/Aroom]
z 3m
θ = 89° ALOS/Adiff = 0.043
10
|H(jf)| [ARx/Aroom]
0,01
θ = 85° ALOS/Adiff = 0.22
1
0,1
= 9.6 ns
0,01 0
1
2
3
4
5
-1
frequency [ ]
Results for different angles θ are plotted in Fig. 10. When the Rx is directed to the ceiling (θ = 90°), |H(jf)| is due to the diffuse signal, only. At θ = 89°, a weak LOS signal is received having about 20 times less power than the diffuse signal. Although the direction of the Rx is nearly the same, the total frequency response is markedly changed. A distinct notch is observed near 210 MHz followed by some ripple at high frequencies. For θ = 85°, the LOS signal is lifted by a factor of 5, and the notch is now obtained near 52 MHz. The individual curves for |Hdiff(jf)| (dotted curve) and |HLOS(jf)| (dashed curve) are also plotted in Fig. 10. Note that the notches occur near the crossing point of these two signals. At θ = 45°, the frequency response is mainly determined by the LOS signal which has about 2 times more power than the diffuse light. A nearly flat response is then obtained. The notches are critical, when they should fall into the transmission band. In the following it is shown that there is a lower bound for the notch frequency fcrit. Adiff(f) is used to describe the diffuse signal amplitude and the LOS signal ALOS is added by taking a certain delay ∆T between both signals into account, which is clearly evident in Fig. 4. The total amplitude reads | H diff + H LOS | = 2 2 Adiff ( f ) + ALOS + 2 Adiff ( f ) ALOS cos[2πf∆T + Φ diff ]
. (7)
|H(jf)| [ARx/Aroom]
Fig. 9 The scenario investigated in Fig. 10.
θ = 45° ALOS/Adiff = 2.13
1
0,1
= 9.6 ns
0,01 0
1
2
3
4
5
-1
frequency [ ]
Fig. 10 Frequency response for different angles θ between the transmitter and the receiver. The dashed and dotted lines indicate the LOS signal and the diffuse component, respectively. The amplitude ratios of the two components are also given.
The second term Φdiff = -arctan(f/f0) in the cosine argument arises from the low pass behaviour of the diffuse signal. A zero sum signal requires both identical amplitudes and opposite phases of the two signals. It is relatively unlikely that both conditions are fulfilled, simultaneously. But also when the amplitudes are about the same, a deep notch is created when the phase condition 2πf crit ∆T = (2n − 1)π + arctan( f crit / f 0 ) (8) is fulfilled, where n is an integer. This is a transcendent equation depending both on ∆T and f0. In a worse case ∆T = can be assumed. The lowest possible notch frequency (n = 1) then varies from 0.5·-1 to 0.75·-1.
For scenario 2 similar response data were obtained so that only the path loss (2) and the delay spread σ
TRACKED DIRECTED LINKS It is well known, that the IR channel response can be markedly improved with tracked directed links. High speed transmissions at 140 Mbit/s [12] or even at 1 Gbit/s [13] have been reported. The transmitter power can be lowered by orders of magnitude, and a bandwidth limited by the Tx and Rx components is available. On the other hand, these links are based on the LOS and they are susceptible to shadowing and blocking. In the following, the ray tracing is used to study the improvements of the IR channel in a directed link. z
w=4m
Tx
3m
scenario 2 scenario 1
FOV
Rx 1m
M
M
σ =
∑ (ti − µ ) 2 h 2 (ti )
i =0
M
with µ =
∑ ti h 2 (ti )
i=0 M
(9)
∑ h 2 (ti )
∑ h (t i ) 2
i =0
i =0
were extracted from the data. M is the number of time slots in the histogram. The results are depicted in Fig. 13. Both the path loss and the delay spread are reduced with a narrow Tx beam. Assuming a 1 Gbit/s IR system, a FOV below 15° is required in scenario 1. Also a relatively precise tracking system is needed. In scenario 2, a Tx cone with m = 4 (half width ≈ 30°) and a FOV of 30° at the Rx will be sufficient to obtain a flat response. These demands increase when the diffuse signal is stronger.
0.5 m
5my
4m
|H(jf)| [ARx/Aroom]
m=1.5
35 m=7
30 m=45
25
20
40 60 FOV [°]
80
0
m=1 m=1.5
-1
m=7
10 10
m=45 -2
10
-3
10
-4
10
0
10
20
30
40 50 60 FOV [°]
70
80
90
Fig. 13 Path loss (top) and delay spread (bottom) in scenario 2 (Fig. 11) as a function of the FOV in the Rx. The Lambert exponent m of the Tx beam is indicated on the curves. The open symbols refer to scenario 1.
FOV = 90°
FOV = 40°
CONCLUSIONS
FOV = 20°
An analytic model for the diffuse IR indoor channel was proposed based on the integrating sphere. The model provides a quick rule of thumb for the path loss and for the bandwidth of the diffuse IR signal in rectangular rooms. The model was verified with a new ray tracing simulation technique which overcomes the limited number of reflections in previous works. A characteristic gain times bandwidth product of the diffuse IR indoor channel was revealed.
FOV = 5°
1 = 8.5 ns 0,01
Fig. 12
40
0
delay spread [ns]
Two typical scenarios shown in Fig. 11 are investigated. An average reflectivity of 0.62 was assumed. In scenario 1, which is referred to as spot diffusing with directed links [12], a collimated beam is sent to the ceiling where it is diffusely reflected. The spot is observed by the Rx which is aligned with respect to the LOS. For the Rx, a “sharp-cut” field of view (FOV) was assumed. A photon is only accepted, when the angle between the final direction of flight and the direction of maximum Rx sensitivity is smaller than the FOV. In scenario 2, the spot is replaced by a base station which sends a generalised Lambertian Tx beam to the Rx. Scenario 2 can be compared with the adaptive array system in [14]. In Fig. 12 the results for scenario 1 are plotted for different FOV angles at the Rx. At a FOV of 90° the Rx signal is mainly due to the diffuse component. When the FOV is reduced, the diffuse signal is suppressed while the LOS component remains unchanged. At a FOV of 5°, the response becomes almost flat.
optical path loss [dB]
m=1 (scenario 1)
Fig. 11 The two scenarios investigated in Figs. 12 and 13 (floor: ρ = 0.4, ceiling: ρ = 0.9, wall: ρ = 0.6).
0,1 -1 frequency [ ]
1
10
Frequency response for different FOV angles at the Rx in scenario 1.
When the diffuse component is accompanied by a lineof-sight signal, notches may appear in the frequency response which are critical for data transmissions. A lower bound for the notch frequency was derived. Finally, remarkable improvements both of the path loss and of the delay spread can be obtained with tracked directed links. Quantitative data for these improvements in a specific indoor environment were reported.
distribution density
= 8.56 ns 1,0x10
8
5,0x10
7
l=5m ;w =4m ;h=3m 0,0
ACKNOWLEDGEMENTS
0
The authors wish to thank Dr. Gfeller (IBM Zurich) for a fruitful discussion on the different simulation techniques and Dr. Hermes (HHI Berlin) for his comments on the computer program. This work was supported in the ATMmobil project of the German ministry of research and education under contract No. 01 BK 611/3.
5
10 15 time [ns]
20
5,0 w/h=3
h
4,5 /h [ns/m]
4,0
w
w/h=2
APPENDIX A: RANDOM DIRECTION GENERATOR
l
w/h=4/3
3,5
w/h=1
3,0 w/h=2/3
2,5 2,0 0
The random generator for the transmitter and for the diffuse reflections is similar to Ref. [10]. When α, β and γ are three independent random numbers equally distributed over [0, 1], the new direction is
φ (α ) = 2πα and θ ( β ) = arccos(m +1 β ) (10) where m is the Lambert exponent (m = 1 for reflections). Number γ is only used to describe absorption at surfaces. When γ is larger than ρ, the random path is stopped. APPENDIX B: AVERAGE TIME BEWTEEN TWO REFLECTIONS IN ROOMS
The average time between two reflections is a basic parameter in the IS model. For the sphere, it is calculated in [10]. In an arbitrary room, however, the time of flight distribution depends on the position on the surface. Instead of analytical calculations, a numerical approach based on the ray tracing technique is used here. A single photon is sent to a random flight through an rectangular room in which the surface reflectivity is unity. During its flight, a histogram is created for the time of flight between two reflections. After about 108 reflections the normalised distribution converges. The various positions at the room surface have then been reached, frequently, and a numerical estimate for the time of flight distribution χ(t) is obtained. As an example, Fig. 14 (top) displays the distribution for a rectangular 3x4x5 m³ room. Note that the distribution for the sphere is a linear function which stops at D/c. The average time between two reflections is given by
2
3
4
5 l/h
6
7
8
9
10
3m ⋅2.9 ns/m = 8.7 ns. This value is inserted in (6) and (3) to estimate the cut-off frequency for the actual room. REFERENCES [1]
[2] [3]
[4]
[5]
[6]
[7]
(11)
i =1
where i is the index in the histogram. In order to allow a quick estimate of for a wide variety of rectangular rooms, the diagram in Fig. 14 (bottom) was prepared. Normalised values are used based on the room height h. For example, when h = 3 m, width w = 4 m and length l = 6 m, the curve for w/h = 4/3 is used and a value of 2.9 ns/m is obtained at l/h = 2. Hence reads
1
Fig. 14 Top: Distribution of the time of flight between two reflections in a rectangular room. Bottom: This diagram allows a quick estimate for in rectangular rooms. The dot refers to the example in the text.
M
< t >= ∑ ti χ (ti )
σ = 5.03 ns
[8]
[9]
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