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文章目录

Received Signal
Preface
Two-Ray Model
- Discussion for the relationship of Distances and Received Power
- Critical Distance for System Design
Dielectric Canyon (Ten-Ray Model)
- Assumptions
- Received Signal
- Received Power
- Discussion
General Ray Tracking

Received Signal

Consider a signal transmitted through free space to a receiver located at distance d from the transmitter. Assume there are no obstructions between the transmitter and receiver. The channel model associated with this transmission is called a Line-Of-Sight (LOS) channel, and the corresponding received signal is called the LOS signal or ray. Free-space path loss introduces a complex scale factor, resulting in the received signal:
r(t)=ℜ{λGle−j(2πd/λ)4πdu(t)ej2πfct},r(t)=\Re\left\{\frac{\lambda \sqrt{G_{l}} e^{-j(2 \pi d / \lambda)}}{4 \pi d} u(t) e^{j 2 \pi f_{c} t}\right\},r(t)=ℜ{4πdλGle−j(2πd/λ)u(t)ej2πfct}, where Gl\sqrt{G_l}Gl is the product of the transmit and receive antenna Field Radiation Patterns (FRP, 场辐射图) in the LOS direction, and the phase shift e−j(2πd/λ)e^{-j(2\pi d / \lambda)}e−j(2πd/λ) (e−j(2πfcτe^{-j(2 \pi f_c \tau}e−j(2πfcτ, where delay τ=d/c~\tau = d/c τ=d/c) is due to the distance ddd the wave travels.

Preface

In a typical urban or indoor environment, a radio signal transmitted from a fixed source will encounter multiple objects in the environment that produce reflected, diffracted, or scattered copies of the transmitted signal, as shown in the following figure.
These additional copies of the transmitted, called multiple signal components, can be attenuated in power, delayed in time, and shifted in phase and/or frequency from the LOS signal path at the receiver.
The 1) multipath and 2) transmitted signal are summed together at the receiver, which often produces distortion in the received signal.

Two-Ray Model

The two-ray model is used when a single ground reflection dominates the multipath effect, as illustrated in Figure 2.4.
The received signal consists of two components: 1) the LOS component or ray, which is just the transmitted signal propagating through free space, and 2) a reflected component or ray, which is the transmitted signal reflected off the ground.

The received LOS ray is given by the free-space propagation loss formulation r(t)={λGle−j(2πd/λ)4πdu(t)ej2πfct}r(t) = \{ \frac{\lambda \sqrt{G_l} e^{-j(2\pi d / \lambda)}}{4 \pi d} u(t) e^{j 2 \pi f_c t } \}r(t)={4πdλGle−j(2πd/λ)u(t)ej2πfct}.

The reflected ray is shown in Figure 2.42.42.4 by the segments xxx and x′x^\primex′. If we ignore the effect of surface wave attenuation then, by superposition, the received signal for the two-ray model is r2ray(t)=ℜ{λ4π[Glu(t)e−j(2πl/λ)l+RGru(t−τ)e−j2π(x+x′)/λx+x′]ej(2πfct+ϕ0)}.r_{2ray}(t) = \Re \left\{ \frac{\lambda}{4 \pi} \left[ \frac{\sqrt{G_l} u(t) e^{-j(2\pi l / \lambda)} }{l} + \frac{R\sqrt{G_r} u(t-\tau) e^{-j 2\pi (x+x^\prime)/\lambda} }{x + x^\prime} \right] e^{j(2\pi f_c t + \phi_0 )} \right\}.r2ray(t)=ℜ{4πλ[lGlu(t)e−j(2πl/λ)+x+x′RGru(t−τ)e−j2π(x+x′)/λ]ej(2πfct+ϕ0)}. where x+x′−lx+x^\prime - lx+x′−l is the time delay of the ground reflection relative to the LOS ray, Gl=GaGb\sqrt{G_l} = \sqrt{G_a G_b}Gl=GaGb is the product of transmit and receive antenna field radiation patterns in the LOS direction, RRR is the ground reflection coefficient, and Gr=GcGd\sqrt{G_r} = \sqrt{G_c G_d}Gr=GcGd is the product of transmit and receive antenna field radiation patterns corresponding to the rays of length xxx and x′x^\primex′, respectively. The delay spread of the two-ray model equals the delay between the LOS ray and reflected ray: (x+x′−l)/c(x+x^\prime - l)/c(x+x′−l)/c.

If the transmitted signal is narrowband relative to the delay spread (τ<<Bu−1)(\tau << B_u^{-1})(τ<<Bu−1), then u(t)≈u(t−τ)u(t) \approx u(t-\tau)u(t)≈u(t−τ). Thus, the received power of the two-ray model for narrowband transmission is Pr=Pt[λ4π]2∣Gll+RGre−jΔϕx+x′∣2,(2.12)P_{r}=P_{t}\left[\frac{\lambda}{4 \pi}\right]^{2}\left|\frac{\sqrt{G_{l}}}{l}+\frac{R \sqrt{G_{r}} e^{-j \Delta \phi}}{x+x^{\prime}}\right|^{2}, \qquad (2.12)Pr=Pt[4πλ]2∣∣∣∣lGl+x+x′RGre−jΔϕ∣∣∣∣2,(2.12) where △ϕ=2π(x+x′−l)/λ\triangle \phi= 2 \pi (x+x^\prime - l)/\lambda△ϕ=2π(x+x′−l)/λ is the phase difference between the two received signal components. The above equation has been shown to agree very closely with empirical data. If ddd donotes the horizontal separation of the antennas, hth_tht denotes the transmitter height, and hrh_rhr denotes the receiver height, then using geometry we can show that x+x′−l=(ht+hr)2+d2−(ht−hr)2+d2.x + x^\prime - l = \sqrt{(h_t + h_r)^2+d^2} - \sqrt{(h_t-h_r)^2+d^2}.x+x′−l=(ht+hr)2+d2−(ht−hr)2+d2.

When ddd is very large compared to ht+hrh_t + h_rht+hr, we use a Taylor series approximation in the above equation to get △ϕ=2π(x+x′−l)λ≈4πhthrλd\triangle \phi = \frac{2 \pi (x + x^\prime - l)}{\lambda} \approx \frac{4 \pi h_t h_r}{ \lambda d}△ϕ=λ2π(x+x′−l)≈λd4πhthr

The ground reflection coefficient is given by R=sin⁡θ−Zsin⁡θ+Z(2.15)R = \frac{\sin \theta - Z}{\sin \theta + Z } \qquad (2.15)R=sinθ+Zsinθ−Z(2.15) where Z={ϵr−cos⁡2θ/ϵrfor vertical polarizationϵr−cos⁡2θfor horizontal polarizationZ = \begin{cases} \sqrt{\epsilon_r - \cos^2 \theta} / \epsilon_r \text{~~~for vertical polarization} \\ \sqrt{\epsilon_r - \cos^2 \theta} \text{~~~for horizontal polarization} \end{cases}Z={ϵr−cos2θ/ϵr for vertical polarizationϵr−cos2θ for horizontal polarization and ϵr\epsilon_rϵr is the dielectric constant of the ground.

We see from Figure 2.4 and (2.15) that for asymptotically large ddd, x+x′≈l≈dx+x' \approx l \approx dx+x′≈l≈d, θ≈0\theta \approx 0θ≈0, Gl≈GrG_l \approx G_rGl≈Gr, and R≈−1R \approx -1R≈−1. Substituting these approximations into (2.12) yields that, in this asymptotic limit, the received signal power is approximately Pr≈[λGl4πd]2[4πhthrd2]2PtP_r \approx \left[ \frac{ \lambda \sqrt{G_l} }{4\pi d} \right]^2 \left[ \frac{ 4\pi h_t h_r }{ d^2 } \right]^2 P_tPr≈[4πdλGl]2[d24πhthr]2Pt, or equivalently, the dB\bf{dB}dB attenuation is given by
Pr(dBm)=Pt(dBm)+10log10(Gl)+20log10(hthr)−40log10(d).(2.18)P_r (\text{dBm}) = P_t (\text{dBm}) + 10log_{10} (G_l ) + 20log_{10} (h_t h_r ) − 40log_{10} (d). \qquad (2.18)Pr(dBm)=Pt(dBm)+10log10(Gl)+20log10(hthr)−40log10(d).(2.18) Thus, in the limit of asymptotically large ddd, the received power falls off inversely with the fourth power of ddd and is independent of the wavelangth λ\lambdaλ.

Discussion for the relationship of Distances and Received Power

A plot of (2.18) as a function of distance is illustrated in Figure 2.5 for f=900MHzf=900 \text{MHz}f=900MHz, R=−1R=-1R=−1, ht=50mh_t = 50 \text{m}ht=50m, hr=2mh_r = 2mhr=2m, Gl=1G_l = 1Gl=1, Gr=1G_r = 1Gr=1, and transmit power normalized so that the plot starts at 0dBm0 ~\text{dBm}0 dBm. This plot can be separated into three segments.

For small distances d<htd<h_td<ht, the path loss is roughly flat and proportional to 1(d2+ht2)\frac{1}{(d^2 + h_t ^2)}(d2+ht2)1, since, at these small distances, the distance between the transmitter and receiver is D=d2+(ht−hr)2D= \sqrt{d^2 + (h_t - h_r )^2}D=d2+(ht−hr)2 and thus 1D2≈1(d2+ht2)\frac{1}{D^2} \approx \frac{1}{(d^2 + h_t^2)}D21≈(d2+ht2)1 for ht>>hrh_t >> h_rht>>hr, which is typically the case.
For distances bigger than hth_tht and up to a certain critical distance dcd_cdc, the wave experiences constructive and destructive interference of the two rays, resulting in a wave pattern with a sequence of maxima and minima. These maxima and minima are also refered to small scale loss or multipath fading.
At the critical distance dcd_cdc, the final maxima is reached, after which the signal power falls off proportionally to d−4d^{-4}d−4.

The Properties of the Three Segments:

In this first segment, power falloff is constant and proportional to 1d2+ht2\frac{1}{d^2 + h_t^2}d2+ht21 ;
For distance between hth_tht and dcd_cdc, power falls off at −20-20−20 dB\text{dB}dB/decade ;
At the distances greater than dcd_cdc, power falls off at −40-40−40 dB\text{dB}dB/decade.

Critical Distance for System Design

The critical distance can be used for system design.

For example, if propagation in a cellular system obeys the two-ray model, then the critical distance would be a natural size for the cell radius, since the path los associated with the interference outside the cell would be much larger than path loss for desired signals inside the cell. However, setting the cell radius to dcd_cdc could result in very large cells.

Since smaller cells are more desirable, both to increase capacity and reduce transmit power, cell radii are typically much smaller than dcd_cdc.

Thus, with a two-ray propagation model, power falloff within these relatively small cells goes as distance squared. Moreover, propagation in cellular system rarely follows a two-ray model, since cancellation by reflected rays rarely occurs in all directions.

Example 2.2: Determine the critical distance for the two-ray model in an urban microcell (ht=10mh_t = 10 \text{m}ht=10m, hr=3mh_r = 3 \text{m}hr=3m) and an indoor microcell (ht=3m,hr=2mh_t = 3 \text{m}, h_r = 2 \text{m}ht=3m,hr=2m) for fc=2GHzf_c = 2 \text{GHz}fc=2GHz.

Solution:
urban microcell: dc=4hthrλ=800md_c = \frac{4h_t h_r}{\lambda} = 800~ \text{m}dc=λ4hthr=800 m
indoor microcell: dc=4hthrλ=160md_c = \frac{4h_t h_r}{\lambda} = 160~ \text{m}dc=λ4hthr=160 m

Discussions:
A cell radius of 800 m in an urban microcell system is a bit large, and usually the radius is 100 m to maintain large capacity. However, if we used a cell size of 800 m under these system parameters, signal power would fall off as d2d^2d2 inside the cell, and interference from neighboring cells would fall off as d4d^4d4, and thus would be greatly reduced. Since these are many wall in indoor environments, an indoor system would typically have a smaller cell radius, on the order of 10-20 m.

Dielectric Canyon (Ten-Ray Model)

Assumptions

This model assumes rectilinear streets with buildings along both sides of the street and transmitter and receiver antenna heights that are well below the tops of the buildings.
The building-lined streets acts as a dielectric canyon to the propagating signal.
Theoretically, an infinite number of rays can be reflected off the building fronts to arrive at the receiver. And, singal paths corresponding to more than three reflections can generally be ignored due their large energy dissipation.

As shown in the following figure, the ten rays incorporate all paths with one, two, or three reflections:

the LOS path;
the ground reflected (GR) path;
the single-wall (SW), the double-wall (DW), the triple-wall paths;
the wall-ground (WG), and ground-wall (GW) reflected paths.

Received Signal

For this model, the received signal is given by
r10ray(t)=ℜ{λ4π[Glu(t)e−j(2πl)/λl+∑i=19RiGxiu(t−τi)e−j(2πxi)/λxi]ej(2πfct+ϕ0),}r_{10 r a y}(t)=\Re\left\{\frac{\lambda}{4 \pi}\left[\frac{\sqrt{G_{l}} u(t) e^{-j(2 \pi l) / \lambda}}{l}+\sum_{i=1}^{9} \frac{R_{i} \sqrt{G_{x_{i}}} u\left(t-\tau_{i}\right) e^{-j\left(2 \pi x_{i}\right) / \lambda}}{x_{i}}\right] e^{j\left(2 \pi f_{c} t+\phi_{0}\right)},\right\}r10ray(t)=ℜ{4πλ[lGlu(t)e−j(2πl)/λ+i=1∑9xiRiGxiu(t−τi)e−j(2πxi)/λ]ej(2πfct+ϕ0),} where xix_ixi denotes the path length of the iiith reflected ray, τi=(xi−l)/c\tau_i = (x_i-l)/cτi=(xi−l)/c, and Gxi\sqrt{G_{x_i}}Gxi is the product of the transmit and receive antenna gains corresponding to the iiith ray. For each reflection path, the coefficient RiR_iRi is either a single reflection coefficient given by (2.15) or, if the path corresponding to mutiple reflections, the product of the reflection coefficients corrsponding to each reflection. The dielectric constants used in (2.15)(2.15)(2.15) are approximately the same as the ground dielectric, so ϵr=15\epsilon_r = 15ϵr=15 is used for all the calculations of RiR_iRi.

Received Power

If we again assume a narrow-band model such that u(t)≈u(t−τi)u(t) \approx u(t - \tau_i)u(t)≈u(t−τi) for all iii, then the received power corresponding to (2.19) is

Pr=Pt[λ4π]2∣Gll+∑i=19RiGxie−jΔϕixi∣2P_{r}=P_{t}\left[\frac{\lambda}{4 \pi}\right]^{2}\left|\frac{\sqrt{G_{l}}}{l}+\sum_{i=1}^{9} \frac{R_{i} \sqrt{G_{x_{i}}} e^{-j \Delta \phi_{i}}}{x_{i}}\right|^{2}Pr=Pt[4πλ]2∣∣∣∣∣lGl+i=1∑9xiRiGxie−jΔϕi∣∣∣∣∣2 where △ϕ=2π(xi−l)/λ\triangle \phi = 2 \pi (x_i - l)/\lambda△ϕ=2π(xi−l)/λ.

Discussion

Power falloff with distance in both the ten-ray model (2.20) and urban empirical data [13, 47, 48] for transmit antennas both above and below the building skyline is typically proportional to d−2d^{-2}d−2, even at relatively large distances.
Moreover, this falloff exponent is relatively insensitive to the transmitter height. This falloff with distance squared is due to the dominance of the multipath rays over the combination of LOS, which decay as d−2d^{-2}d−2, and ground-reflected rays (the two-ray model), which decay as d−4d^{-4}d−4.
Other empirical studies [15, 49, 50] have obtained power falloff with distance proportional to d−αd^{-\alpha}d−α , where α\alphaα lies anywhere between two and six.

General Ray Tracking

General Ray Tracking (GRT) can be used to predict field strength and delay spread for any building configuration and antenna placement.

For this model, the building database (height, location, and dielectric properties) and the transmitter and receiver locations relative to the buildings must be specified exactly. Since this model is site-specific, the GRT model is not used to obtain general theories about 1) system performance and 2) layout; rather, it explains the basic mechanism of urban propagation, and can be used to obtain 1) delay and 2) signal strength information for a particular transmitter and receiver configuration.

The GRT method uses geometrical optics to trace the propagation of the 1) LOS and 2) reflected signal components, as well as signal components from 1) building diffraction and 2) diffuse scattering. There is no limit to the number of multipath components at a given receiver location: the strength of each component is derived explicitly based on the 1) building locations and 2) dielectric properties.

In general, the LOS and reflected paths porvide the dominant components of the received signals, since 1) diffraction and 2) scattering losses are high. However, in regions close to scattering or diffracting surfaces, which are typically blocked from the LOS and reflecting rays, these other multipath components may dominate.

Diffraction occurs when the transmitted signal “bends around” an object in its path to the receiver, as shown in Figure 2.7. Diffraction results from many phenomena, including the curved surface of the earth, hilly or irregular terrain, building edges, or obstructions blocking the LOS path between the transmitter and receiver.

Diffraction is most commonly modeled by the Fresnel knife edge diffraction model due to its simplicity. The geometry of this model is shown in Figure 2.7, where the diffracting object is assumed to be asymptotically thin, which is not generally the case for hills, rough terrain, or wedge diffractors. In particular, this model does not consider diffractor parameters such as polarization, conductivity, and surfance roughness, which can lead to inaccuracies. The geometryof Figure 2.7 indicates that the diffracted signal travels distance d+d′d+d'd+d′ resulting in a phase shift of ϕ=2π(d+d′)/λ\phi = 2 \pi (d+d')/\lambdaϕ=2π(d+d′)/λ. The geometry indicates that for hhh small relative to ddd and d′d'd′, the singal must travel an additional distance relative to the LOS path of approximately △d=h22d+d′dd′,\triangle d = \frac{h^2}{2} \frac{d+d'}{dd'},△d=2h2dd′d+d′, and the corresponding phase shift relative to the LOS path is approximately ϕ=2π△dλ=π2v2,\phi = \frac{2 \pi \triangle d}{\lambda}=\frac{\pi}{2}v^2,ϕ=λ2π△d=2πv2, where v=h2(d+d′)λdd′v = h \sqrt{\frac{2(d+d')}{\lambda d d'}}v=hλdd′2(d+d′) is called the Fresnel-Kirchoff diffraction parameter. The path loss associated with knife-edge diffraction is generally a function of vvv.

Approximations for knife-edge diffraction path loss (in dB) relative to LOS path loss are given by Lee [14, Chapter 2] as

L(v)(dB)={20log⁡10[0.5−0.62v]−0.8≤v<020log⁡10[0.5e−.95v]0≤v<120log⁡10[0.4−.1184−(.38−.1v)2]1≤v<2.420log⁡10[.225/v]v>2.4L(v)(d B)= \begin{cases}20 \log _{10}[0.5-0.62 v] & -0.8 \leq v<0 \\ 20 \log _{10}\left[0.5 e^{-.95 v}\right] & 0 \leq v<1 \\ 20 \log _{10}\left[0.4-\sqrt{.1184-(.38-.1 v)^{2}}\right] & 1 \leq v<2.4 \\ 20 \log _{10}[.225 / v] & v>2.4\end{cases}L(v)(dB)=⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧20log10[0.5−0.62v]20log10[0.5e−.95v]20log10[0.4−.1184−(.38−.1v)2]20log10[.225/v]−0.8≤v<00≤v<11≤v<2.4v>2.4

A similar approximation can be found in [40]. The knife-edge diffraction model yields the following formula for the received diffracted signal: r(t)=ℜ{L(v)Gdu(t−τ)e−j(2π(d+d′))λej(2πfct+ϕ0),}r(t)=\Re\left\{L(v) \sqrt{G_{d}} u(t-\tau) e^{\frac{-j\left(2 \pi\left(d+d^{\prime}\right)\right)}{\lambda} } e^{j\left(2 \pi f_{c} t+\phi_{0}\right)},\right\}r(t)=ℜ{L(v)Gdu(t−τ)eλ−j(2π(d+d′))ej(2πfct+ϕ0),} where Gd\sqrt{G_d}Gd is the antenna gain and τ=△d/c\tau = \triangle d/cτ=△d/c is the delay associated with the defracted ray relative to the LOS path.

In addition to the diffracted ray, there may also be multiple diffracted rays, or rays that are both reflected and diffracted. Models exist for including all possible permutations of reflection and diffraction [41]; however, the attenuation of the corresponding signal components is generally so large that these components are negligible relative to the noise.

A scattered ray, shown in Figure 2.8 by the segments s′s's′ and sss, has a path loss proportional to the product of sss and s′s's′. This multiplicative dependence is due to the additional spreading loss the ray experiences after scattering. The received signal due to a scattered ray is given by the bistatic radar equation [42]: r(t)=ℜ{u(t−τ)λGsσe−j(2π(s+s′)/λ)(4π)3/2ss′ej(2πfct+ϕ0)},r(t)=\Re\left\{u(t-\tau) \frac{\lambda \sqrt{G_{s} \sigma} e^{-j\left(2 \pi\left(s+s^{\prime}\right) / \lambda\right)}}{(4 \pi)^{3 / 2} s s^{\prime}} e^{j\left(2 \pi f_{c} t+\phi_{0}\right)}\right\}, r(t)=ℜ{u(t−τ)(4π)3/2ss′λGsσe−j(2π(s+s′)/λ)ej(2πfct+ϕ0)}, where τ=(s+s′−l)/cτ = (s+s' −l)/cτ=(s+s′−l)/c is the delay associated with the scattered ray, σ\sigmaσ (in m2m^2m2 ) is the radar cross section of the scattering object, which depends on the roughness, size, and shape of the scatterer, and Gs\sqrt{G_s}Gs is the antenna gain.

In addition to the diffracted ray, there may also be multiple diffracted rays, or rays that are both reflected and diffracted. Models exist for including all possible permutations of reflection and diffraction [41]; however, the attenuation of the corresponding signal components is generally so large that these components are negligible relative to the noise.

A scattered ray, shown in Figure 2.8 by the segments s′s's′ and sss, has a path loss proportional to the product of sss and s′s's′. This multiplicative dependence is due to the additional spreading loss the ray experiences after scattering. The received signal due to a scattered ray is given by the bistatic radar equation [42]: r(t)=ℜ{u(t−τ)λGsσe−j(2π(s+s′)/λ)(4π)3/2ss′ej(2πfct+ϕ0)}r(t)=\Re\left\{u(t-\tau) \frac{\lambda \sqrt{G_{s} \sigma} e^{-j\left(2 \pi\left(s+s^{\prime}\right) / \lambda\right)}}{(4 \pi)^{3 / 2} s s^{\prime}} e^{j\left(2 \pi f_{c} t+\phi_{0}\right)}\right\}r(t)=ℜ{u(t−τ)(4π)3/2ss′λGsσe−j(2π(s+s′)/λ)ej(2πfct+ϕ0)} where τ=(s+s′−l)/c\tau = (s+s' −l)/cτ=(s+s′−l)/c is the delay associated with the scattered ray, σ\sigmaσ (in m2m^2m2 ) is the radar cross section of the scattering object, which depends on the roughness, size, and shape of the scatterer, and GsG_sGs is the antenna gain.

The model assumes that the signal propagates from the transmitter to the scatterer based on free space propagation, and is then reradiated by the scatterer with transmit power equal to σ times the received power at the scatterer. From (2.25) the path loss associated with scattering is Pr(dBm)=Pt(dBm)+10log⁡10(Gs)+10log⁡10(σ)−30log⁡(4π)−20log⁡10s−20log⁡10(s′).P_{r}(\mathrm{dBm})=P_{t}(\mathrm{dBm})+10 \log _{10}\left(G_{s}\right)+10 \log _{10}(\sigma)-30 \log (4 \pi)-20 \log _{10} s-20 \log _{10}\left(s^{\prime}\right).Pr(dBm)=Pt(dBm)+10log10(Gs)+10log10(σ)−30log(4π)−20log10s−20log10(s′). Empirical values of 10log⁡10σ10 \log 10 \sigma10log10σ were determined in [43] for different buildings in several cities. Results from this study indicate that σ=10log⁡10σ\sigma = 10 \log 10 \sigmaσ=10log10σ in dbm2\text{db}m^2dbm2 ranges from −4.5 dbm2\text{db}m^2dbm2 to 55.7 dbm2\text{db}m^2dbm2 , where dbm2\text{db}m^2dbm2 denotes the db\text{db}db value of the σ\sigmaσ measurement with respect to one square meter.

The received signal is determined from the superposition of all the components due to the multiple rays. Thus, if we have a LOS ray, NrN_rNr reflected rays, NdN_dNd diffracted rays, and NsN_sNs diffusely scattered rays, the total received signal is
rtotal (t)=ℜ{[λ4π][Glu(t)ej(2πl)/λl+∑i=1NrRxiGxiu(t−τi)e−j(2πri/λ)ri+∑j=1NdLj(v)Gdju(t−τj)e−j(2π(dj+dj′))/λej(2πfct+ϕ0),+∑k=1NsσkGsku(t−τk)ej(2π(sk+sk′))/λsksk′]ej(2πfct+ϕ0)},\begin{aligned} r_{\text {total }}(t) &=\Re\left\{[ \frac { \lambda } { 4 \pi } ] \left[\frac{\sqrt{G_{l}} u(t) e^{j(2 \pi l) / \lambda}}{l}+\sum_{i=1}^{N_{r}} \frac{R_{x_{i}} \sqrt{G_{x_{i}}} u\left(t-\tau_{i}\right) e^{-j\left(2 \pi r_{i} / \lambda\right)}}{r_{i}}\right.\right.\\ &+\sum_{j=1}^{N_{d}} L_{j}(v) \sqrt{G_{d_{j}}} u\left(t-\tau_{j}\right) e^{-j\left(2 \pi\left(d_{j}+d_{j}^{\prime}\right)\right) / \lambda} e^{j\left(2 \pi f_{c} t+\phi_{0}\right)}, \\ &\left.\left.+\sum_{k=1}^{N_{s}} \frac{\sigma_{k} \sqrt{G_{s_{k}}} u\left(t-\tau_{k}\right) e^{j\left(2 \pi\left(s_{k}+s_{k}^{\prime}\right)\right) / \lambda}}{s_{k} s_{k}^{\prime}}\right] e^{j\left(2 \pi f_{c} t+\phi_{0}\right)}\right\}, \end{aligned}rtotal (t)=ℜ{[4πλ][lGlu(t)ej(2πl)/λ+i=1∑NrriRxiGxiu(t−τi)e−j(2πri/λ)+j=1∑NdLj(v)Gdju(t−τj)e−j(2π(dj+dj′))/λej(2πfct+ϕ0),+k=1∑Nssksk′σkGsku(t−τk)ej(2π(sk+sk′))/λ]ej(2πfct+ϕ0)}, where τi\tau_iτi, τj\tau_jτj, τk\tau_kτk is, respectively, the time delay of the given reflected, diffracted, or scattered ray normalized to the delay of the LOS ray, as defined above.

Any of these multipath components may have an additional attenuation factor if its propagation path is blocked by buildings or other objects.

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Wireless Communications - 2.4 Ray Tracing

文章目录

Received Signal

Preface

Two-Ray Model

Discussion for the relationship of Distances and Received Power

Critical Distance for System Design

Dielectric Canyon (Ten-Ray Model)

Assumptions

Received Signal

Received Power

Discussion

General Ray Tracking

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