Deterministic radio propagation modeling and ray tracing 1) Introduction to deterministic propagation modelling 2) Geometrical Theory of Propagation I-The ray concept-Reflection and transmission 3) Geometrical Theory of Propagation II-Diffraction,multipath 4) Ray Tracing I 5) Ray Tracing II-Diffuse scattering modelling 6) Deterministic channel modelling I 7) Deterministic channel modelling II-Examples 8) Project -discussion
Deterministic radio propagation modeling and ray tracing 1) Introduction to deterministic propagation modelling 2) Geometrical Theory of Propagation I - The ray concept – Reflection and transmission 3) Geometrical Theory of Propagation II - Diffraction, multipath 4) Ray Tracing I 5) Ray Tracing II – Diffuse scattering modelling 6) Deterministic channel modelling I 7) Deterministic channel modelling II – Examples 8) Project - discussion
Envelope correlations (1/5) It is useful to define transfer function's envelope-correlations.Considering the module of the generic transfer function M(z)I in a e-kind domain z,the domain span△z and the average valueM,over△z we have:: "z-wise"correlation (envelope correlation) w(e-MlM(e+o-l] R(8)=4 R(0)=1,-1<R(⊙)s1 Jw(a-l了t {R(}=0 Especially frequency and space correlations are useful.The last one is fundamental for diversity techniques and MIMO
Envelope correlations (1/5) It is useful to define transfer function’s envelope-correlations. Considering the module of the generic transfer function |M(z)| in a e-kind domain z, the domain span Δz and the average value over Δz we have: “z-wise” correlation (envelope correlation) R z (δ ) = M (z) − M Δz ⎡ ⎣ ⎤ ⎦ M (z +δ ) − M Δz ⎡ ⎣ ⎤ ⎦ dz Δz ∫ M (z) − M Δz ⎡ ⎣ ⎤ ⎦ 2 dz Δz ∫ ; R z (0) = 1, −1< R z (δ ) ≤1 z M Δ Especially frequency and space correlations are useful. The last one is fundamental for diversity techniques and MIMO. lim δ→∞ R z { (δ )} = 0
Envelope correlations (2/5) Ex:frequency correlation Ja(r八-w]LHU+w-a风] R(w) JU-了 Space correlation (along the x direction) H(-l]He+-L] R(0=4 a(-l了
Envelope correlations (2/5) Space correlation (along the x direction) R x (l) = H (x) − H Δx ⎡ ⎣ ⎤ ⎦ H (x + l) − H Δx ⎡ ⎣ ⎤ ⎦ dx Δx ∫ H (x) − H Δx ⎡ ⎣ ⎤ ⎦ 2 dx Δx ∫ Ex: frequency correlation Rf (w) = H ( f ) − H Δf ⎡ ⎣ ⎤ ⎦ H ( f + w) − H Δf ⎡ ⎣ ⎤ ⎦ df Δf ∫ H ( f ) − H Δf ⎡ ⎣ ⎤ ⎦ 2 df Δf ∫
Envelope correlations (3/5) Ex.space correlation in a Rayleigh environment,i.e.with uniform 2D power- azimuth distribution with p)=1/2 is: 0元 R.(1)=J. 0.8 computation measurement -··Rayleigh(theoretic) 0.6 With Jo the zero-order Bessel's 0.4 function of the first kind.This means that the signal received from two Rx's 0.2 A2 apart is nearly uncorrelated (see figure),and this can be useful to decrease fast fading effects -0.2 0 0.5 11.5 22.5 33.5 Normalized distance d
Envelope correlations (3/5) Ex. space correlation in a Rayleigh environment, i.e. with uniform 2D powerazimuth distribution with pϕ(ϕ)=1/2π is: R x (l) = J 0 2πl λ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ With J0 the zero-order Bessel’ s function of the first kind. This means that the signal received from two Rx’s λ/2 apart is nearly uncorrelated (see figure), and this can be useful to decrease fast fading effects l/λ J0
Envelope correlations (4/5) Frequency correlation and time correlation allow a rigorous definition of coherence bandwidth and coherence time. Given a reference,residual frequency correlation "a",then coherence bandwidth is: B=币with R,(o)≤for w≥项 Similarly,given a reference,residual time correlation "a",then coherence time IS To=7 with R,()同sa fort≥i Coherence distance L.can also be defined in the following way: L9=I with R,(0)≤a forx≥l
Envelope correlations (4/5) Frequency correlation and time correlation allow a rigorous definition of coherence bandwidth and coherence time. Given a reference, residual frequency correlation “ a ”, then coherence bandwidth is: BC (a) = w with Rf (w) ≤ a for w ≥ w Similarly, given a reference, residual time correlation “ a ”, then coherence time is : TC (a) = t with R t(t ) ≤ a for t ≥ t Coherence distance Lc can also be defined in the following way: LC (a) = l with R x (l) ≤ a for x ≥ l